Research ArticleA General Scheme for Information Interception in thePing-Pong Protocol

Piotr Zawadzki1 and JarosBaw Adam Miszczak2

1 Institute of Electronics Silesian University of Technology Akademicka 16 44-100 Gliwice Poland2Institute of Theoretical and Applied Informatics Polish Academy of Sciences Bałtycka 5 44-100 Gliwice Poland

Correspondence should be addressed to Piotr Zawadzki ZawadzkiPiotrusgmailcom

Received 22 March 2016 Revised 25 May 2016 Accepted 6 June 2016

Academic Editor Kamil Bradler

Copyright copy 2016 P Zawadzki and J A Miszczak This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The existence of undetectable eavesdropping of dense coded information has been already demonstrated by Pavicic for the quantumdirect communication based on the ping-pong paradigm However (a) the explicit scheme of the circuit is only given and no designrules are provided (b) the existence of losses is implicitly assumed (c) the attack has been formulated against qubit based protocolonly and it is not clear whether it can be adapted to higher dimensional systems These deficiencies are removed in the presentedcontribution A new generic eavesdropping scheme built on a firm theoretical background is proposed In contrast to the previousapproach it does not refer to the properties of the vacuum state so it is fully consistent with the absence of losses assumptionMoreover the scheme applies to the communication paradigm based on signal particles of any dimensionality It is also shown thatsome well known attacks are special cases of the proposed scheme

1 Introduction

Quantum direct communication (QDC) aims at provision ofconfidentiality without resorting to classic encryptionThis isin contrast to quantum key distribution (QKD) technique asno shared key is established and quantum resources take overits role In QDC similar to QKD it is assumed that legitimateparties can communicate over open and authenticated classicchannel

The roots of QDC can be traced out to the QKD protocolof Long and Liu [1] that after slight modification proposed asthe two-step protocol [2] can be considered the first protocolof this kind The ping-pong protocol [3] is another QDCscheme which is easier to implement at the price of lessersecurity margin and capacity These initial works exploitedthe entanglement of EPR pairs to protect transmission ofsensitive information Ideas of these proposals have beenfurther adapted to higher dimensional systems [4ndash7] andormodified to enhance capacity via dense coding [8 9] Theentanglement is a very fragile quantum resource and itshandling is technically challenging This motivated the work

towards exploiting quantum uncertainty a resource used bymost QKD protocols The first single-photon QDC protocolproposed by Deng and Long [10] has been recently demon-strated experimentally [11] The LM05 protocol [12] is theother proposal of this kind that is worth notingThehistory ofthe development and the review of the early QDC proposalscan be found in [13]

QDC protocols offer different level of security whichusually results from the tradeoff between practical feasibilityand type of quantum resource available to communicatingparties QDC protocols which process particles in blocks[2 4] can be parametrized in such a way that probability ofrevealing sensitive information is arbitrarily small Howeverthey assume that legitimate parties have long-term quantummemory Protocols that process particles individually arequasi-secure [13ndash15] Quasi-securitymeans that before eaves-dropping detection which is inevitable for long sequencespart of the sensitive informationmay be revealed to the eaves-dropper QDC is a more versatile cryptographic primitivethan QKD In fact QDC protocols can be used as enginesfor key agreement Any key agreement protocol executed

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 3162012 7 pageshttpdxdoiorg10115520163162012

2 Advances in Mathematical Physics

in a private channel provided by a QDC protocol offeringunconditional security has security comparable with QKDAlso quasi-secureQDCprotocols can realize unconditionallysecure QKD However in this case QDC phase deliversshared sequence that is partially known to the eavesdropperBy the appropriate postprocessing that is privacy amplifica-tion the eavesdropperrsquos knowledge on the resulting sequencecan be reduced to arbitrarily small value provided that hisinformation on the initial sequence is less than mutualinformation of the legitimate parties The realization of theQKD via QDC can be potentially more efficient as the basisreconciliation step which severely plagues efficiency of manyQKD protocols can be avoided [16ndash18] Protocols of this typeare referred to as deterministic QKD and some of them havebeen recently experimentally demonstrated [19 20]

This paper is devoted to the analysis of the (in)securityof the ping-pong protocol an entanglement based QDCscheme [3] Quasi-security is provided only for perfect quan-tum channels [14] and the scheme becomes insecure whenlosses [21] andor communication errors and imperfection ofdevices are taken into account [22] Protocol offers capacityof single bit per protocol cycle because the authenticity ofthe shared EPR pair is verified only by a measurement ina single basis This limits the available encoding to phaseflips Possible capacity enhancement via dense coding leads toundetectable information leakage as demonstrated in [2] andusage of mutually unbiased bases in control measurementsis required to preserve quasi-security of the communica-tion [8] In our previous work we have proved that thisobservation also holds for the qudit based protocol andthat detection probability depends on the number of basesused in the control mode [7 23] Anyway no explicit attacktransformation has been given in the aforementioned papersThe present contribution is motivated by the appearance ofthe circuit [24] (further it will be referred to as P-circuit)capable of undetectably intercepting information transmittedin the qubit based ping-pong protocol with the followingconfiguration quantum channel is perfect legitimate partiesuse single basis for control measurements and information isdense coded In other words the instantiation of the attack isforecasted in [2] Although P-circuit is applicable to perfectchannels it assumes the appearance of the vacuum states inthe eavesdropperrsquos ancilla In consequence it does not wellfit the existing analyses Shortly after its appearance a controlmode that addresses detection of this specific circuit has beenproposed [25]

We propose a generic scheme for construction of attacksthat permit undetectable eavesdropping under the sameassumptions quantum channel is perfect control measure-ments are executed in a single basis and sensitive informationis dense coded Thus our contribution can be considered asthe generalization of the result given in [24] The presentedmethod is applicable to systems of any dimension so itcan be used to construct a plethora of new transformsUsing introduced generalization we also demonstrate theequivalence of the attack from [24] and CNOT operation Inconsequence we claim that there is no need for constructionof specific control modes as in [25] because any controlmode able to detect CNOT operation is also able to detect

circuit proposed in [24] We do not propose the attack that isundetectable by control measurements in unbiased bases Infact we think that the opposite is true control measurementsinmutually unbiased bases are sufficient to statistically detectcoherence break of the shared entangled state and in thatwayreveal the presence of the eavesdropper [23]

The paper is organized as follows In Section 2 we providenotation and concepts used in the text Section 3 presents themain contribution In particular we provide a general bit-flip detection scheme demonstrate its equivalence with theexisting approaches and introduce an attack on the quditbased protocol In Section 4 we summarize the presentedwork

2 Preliminaries

21 Ping-Pong Protocol The communication protocoldescribed below is a ping-pong paradigm variant analysed in[24] Compared to the seminal version [3] it differs only inthe encoding operation the sender uses dense coding insteadof phase flipsThe remaining elements of the communicationscenario are left intact

Bob starts the communication process by creation of EPRpair (the assumed initial state is the same as in [3 24] tomaintain compatibility of mathematical expressions for thequdit version of the protocol considered in Section 31 it isassumed that Bob starts from the generalization of |Φ+⟩ =

(|0ℎ⟩|0119905⟩ + |1ℎ⟩|1119905⟩)radic2)

1003816100381610038161003816Ψinit⟩ =1003816100381610038161003816Ψminus

⟩ =(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ minus10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)

radic2 (1)

Then he sends one of the qubits further referred to as thesignaltravel qubit to Alice Alice can in principle encode twoclassic bits 120583 and ] applying unitary transformation A

120583] =

X120583Z] where X = |1⟩⟨0| + |0⟩⟨1| and Z = |0⟩⟨0| minus

|1⟩⟨1| are bit-flip and phase-flip operations respectively Thesignal particle is sent back to Bob who detects appliedtransformation by a collective measurement of both qubits(Figure 1)

Passive eavesdropping is impossible Eve has access onlyto the travel qubit which before and after encoding lookslike maximally mixed state Unfortunately the describedcommunication scenario is vulnerable to the intercept-resendattack and Alice has to check whether the received qubit isgenuine As a countermeasure Alice measures the receivedqubit in computational basis (|0⟩ |1⟩) in randomly selectedprotocol cycles and asks Bob over authenticated classicchannel to do the same with his qubit (Figure 2) Hermeasurement causes the collapse of the shared state (1) Theperfect (anti)correlation of the outcomes is preserved only ifthe qubit measured by Alice is the same one that was sent byBob If Eve inserts fake qubit then the measured qubits areno longer correlated and some discrepancies which are thesign of the eavesdropping do occur In that way Alice andBob can convince themselves with confidence approachingcertainty that the quantum channel is not spoofed providedthat they have executed a sufficient number of control cycles

Advances in Mathematical Physics 3

Bell

BobBob Alice

h

t120583

|120595init⟩

119987119989120583

120583

Figure 1 The schematic diagram of a message mode in the ping-pong protocol

BobBob Alice

h

t

|120595init⟩120572

120573

Figure 2 The schematic diagram of a control mode in the ping-pong protocol

However the intercept-resend attack is not the onlypossible way of active sensitive information interceptionThesignal particle that travels back and forth between legitimateparties can be the subject of any quantum action introducedby Eve (Figure 3) Introduced coupling causes the encodingoperation to also modify Eversquos ancilla state and Eve hopesto detect and decipher Alicersquos actions by its inspectionActions of Eve not necessarily unitary in the affected qubitrsquosspace can be described as unitary operation Q acting in thespace extended with two additional qubits as follows fromStinespringrsquos dilation theorem The control state shared bylegitimate parties then takes the form

1003816100381610038161003816120595ℎ119905119864⟩ = (Iℎotimes Q) (

1003816100381610038161003816Ψinit⟩ otimes1003816100381610038161003816120594119864⟩) (2)

where |120594119864⟩ is some initial state of Eversquos ancilla Eve presence

is detected with probability

119901det (Q) = Tr (Pℎ119905Tr119864(1003816100381610038161003816120595ℎ119905119864⟩ ⟨120595ℎ119905119864

1003816100381610038161003816)) (3)

where projectionPℎ119905depends on initial state and the consid-

ered case is defined as

Pℎ119905= Iℎ119905minus10038161003816100381610038160ℎ⟩ ⟨0ℎ

1003816100381610038161003816 otimes10038161003816100381610038161119905⟩ ⟨1119905

1003816100381610038161003816 minus10038161003816100381610038161ℎ⟩ ⟨1ℎ

1003816100381610038161003816

otimes10038161003816100381610038160119905⟩ ⟨0119905

1003816100381610038161003816 (4)

22 Pavicic Attack Pavicicrsquos attack demonstrates the viola-tion of ping-pong protocol security when dense coding isusedThe attack does not introduce errors or losses in controlandmessagemode and it permits eavesdropping informationencoded as bit-flip operation

The P-circuit presented by Pavicic (Figure 4) is a result ofa cut-and-try procedure [24 section IV] applied to Wojcikrsquoscircuit [21] It is composed of two Hadamard gates followedby the controlled polarization beam splitter (CPBS) whichis a generalization of the polarization beam splitter (PBS)concept The PBS is a two-port gate that swaps horizontally

Bell

EveEve

BobBob Alice

120583

h

t

|120595init⟩

119987119989120583

⟩|120594E

119980minus1119980

120583

Figure 3 A schematic diagram of an individual attack

t

x H

Hy

PBS

CPBS

Figure 4 P-circuit Q119905119909119910

[24 eq (2)]

polarized photons |0119909⟩ (|0119910⟩) entering its input to the other

port |0119910⟩ (|0119909⟩) on output while vertically polarized ones |1

119909⟩

(|1119910⟩) remain in their port |1

119909⟩ (|1119910⟩) that is

PBS 1003816100381610038161003816V119909⟩100381610038161003816100381610038160119910⟩ =

10038161003816100381610038160119909⟩10038161003816100381610038161003816V119910⟩

PBS 1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩ =

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩

(5a)

PBS 10038161003816100381610038160119909⟩10038161003816100381610038161003816V119910⟩ =

1003816100381610038161003816V119909⟩100381610038161003816100381610038160119910⟩

PBS 10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩

(5b)

where |V⟩ denotes the vacuum state The CPBS behavesas normal PBS if control qubit is set to |0

119905⟩ The roles of

horizontal and vertical polarization are exchanged for controlqubit set to |1

119905⟩

CPBS 10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ =

10038161003816100381610038160119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

CPBS 10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ =

10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

(6a)

CPBS 10038161003816100381610038160119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

CPBS 10038161003816100381610038161119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

(6b)

CPBS 10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩ =

10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩

CPBS 10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩ =

10038161003816100381610038161119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩

(6c)

CPBS 10038161003816100381610038160119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038160119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩

CPBS 10038161003816100381610038161119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩

(6d)

4 Advances in Mathematical Physics

Initially Eversquos ancilla is initialized to the state |1205940⟩ = |V

119909⟩|0119910⟩

The action of the P-circuit from Figure 4 is then described bythe following formulas

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038160119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

(7a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038161119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(7b)

For the purpose of future analysis let us also identify actionsof the circuit under consideration onto the state |120594

1⟩ =

|0119909⟩|V119910⟩

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038160119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

(8a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038161119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(8b)

The control state (2) after entangling with Eversquos ancilla reads

1003816100381610038161003816120595ℎ119905119864⟩ =(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

radic2 (9)

This state is further used by Alice and Bob for eavesdroppingcheck It is clear from (3) that the attack does not introduceerrors or losses in control mode and the expected correlationof outcomes is preserved in the computational basis

Phase Flip The phase-flip encoding applied to the coupledstate leads to

10038161003816100381610038161003816120595phase⟩ = (I

ℎotimesZ119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119886119864⟩ minus

10038161003816100381610038160ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩) (10)

The signal qubit is then sent back to Bob who afterdisentangling on a basis of (7a) and (7b) observes

10038161003816100381610038161003816120601phase⟩ = (Q

119905119909119910)minus1 10038161003816100381610038161003816

120595phase⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩ minus10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩)1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

= [(IℎotimesZ119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205940⟩

(11)

Bit Flip The bit-flip operation transforms Alicersquos state to1003816100381610038161003816120595bit⟩ = (I

ℎotimesX119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119886119864⟩ +

10038161003816100381610038160ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩) (12)

The system state after disentangling can be deduced from(8a) and (8b)

1003816100381610038161003816120601bit⟩ = (Q119905119909119910)minus1 1003816100381610038161003816120595bit⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

= [(IℎotimesX119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205941⟩

(13)

In both cases that is phase-flip and bit-flip encoding thesignalling subsystem behaves as if there was no couplingwith the ancilla However Alicersquos bit-flip encoding modifiesEversquos register (|120594

0⟩ rarr |120594

1⟩) The states |120594

0⟩ and |120594

1⟩ are

orthogonal and perfectly distinguishable In consequenceEve can eavesdrop on bit-flip operations without introducingerrors and losses in message mode as well

3 Results

This section is devoted to the analysis of the general form ofthe incoherent attack shown diagrammatically in Figure 3Each cycle of the protocol is considered to be independentof the other ones Consequently the effectiveness of theattack is expressed in a fraction of bits eavesdropped onper communication cycle Throughout the analysis it isalso assumed that legitimate parties rely on control modeused in the seminal version of the protocol They locallymeasure possessed particles in the computational basis andverify expected correlation via the public discussion overauthenticated classic channel

31 Generic Bit-Flip Detection Scheme for Qubit Based Proto-col As the controlmode explores outcomes of localmeasure-ments in computational basis for intrusion detection themapQ has to be of trivial form

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(14)

to not induce errors andor losses in control cycles It followsthat under attack Alice operates on the state

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩) (15)

Let the entangling transformation Q additionally satisfy

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(16)

for some state |120601119864⟩ = |120594

119864⟩ The process of information

encoding and disentangling from the ancilla is then describedby the expressions

Qminus1

(IℎotimesI119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

Advances in Mathematical Physics 5

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

=1003816100381610038161003816Ψminus

⟩1003816100381610038161003816120594119864⟩

(17a)

Qminus1

(IℎotimesX119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

=1003816100381610038161003816Φminus

⟩1003816100381610038161003816120601119864⟩

(17b)

Qminus1

(IℎotimesZ119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

= minus1003816100381610038161003816Ψ+

⟩1003816100381610038161003816120594119864⟩

(17c)

Qminus1

(IℎotimesX119905Z119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

= minus1003816100381610038161003816Φ+

⟩1003816100381610038161003816120601119864⟩

(17d)

As a result the registers used for signalling are left untouchedand decoupled but Eversquos register is flipped from |120594

119864⟩ to |120601

119864⟩

whenAlice applies bit-flip operation In consequence Eve cansuccessfully decode half of the message content provided thatthe detection states |120594

119864⟩ and |120601

119864⟩ are perfectly distinguish-

able It follows that any unitary coupling transformation Qthat satisfies (14) and (16) can be used for bit-flip detection

32 Equivalence of P-Circuit and CNOT Circuit The prop-erties of the above generic scheme and the P-circuit [24]perfectly coincide As follows from (7a) (7b) (8a) and (8b)the states |120594

0⟩ = |V

119909⟩|0119910⟩ and |120594

1⟩ = |0

119909⟩|V119910⟩ play the role

of detection states |120594119864⟩ and |120601

119864⟩ respectively It is also clear

that transformation Q119905119909119910

has properties claimed in (14) and(16) Thus the P-circuit can be considered as an instance ofthe generic scheme described in Section 31

However the operator Q satisfying (14) and (16) can berealized in many ways It seems that CNOT operation actingon a single qubit of Eversquos ancilla Q = CNOT

119905119909 |120594119864⟩ = |0

119909⟩

|120601119864⟩ = |1

119909⟩ |119886119864⟩ = |0

119909⟩ and |119889

119864⟩ = |1

119909⟩ is the simplest

realization of the logic behind the attack Such version is alsopractically feasible as the attacks involving probes entangledvia the CNOT operation have been already proposed in theQKD context [26 27] As a result both the CNOT circuitand P-circuit are equivalent in terms of provided information

gain detectability and practical feasibility Consequentlythere is no need for the design of control modes that addressP-circuit in a special manner [25]

33 An Attack on Qudit Based Protocol The P-circuit has nostraightforward generalization to qudit based version of theprotocol In contrast the presented approach can be adaptedwith ease Let Bob start communication process with creationof EPR pair

10038161003816100381610038161003816120573(00)

ℎ119905⟩ =

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩ (18)

where119863 is the qudit dimension The travel qudit is then sentto Alice for encoding or control measurement In controlmode the home and travel qubits are measured in thecomputational basis so the projection P

ℎ119905used in control

equation (3) takes the form

Pℎ119905= Iℎ119905minus

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩ ⟨119896ℎ1003816100381610038161003816 otimes

1003816100381610038161003816119896119905⟩ ⟨1198961199051003816100381610038161003816 (19)

Let by an analogy to the qubit case |120572(119896)119864⟩ and |119886(119896)

119864⟩ be the

sets of 119863 orthonormal states of the ancilla system Thesestates will be further referred to as detection and probe statesrespectively The map used by Eve must be of the form

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(0)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119896)

119864⟩ 119896 = 0 119863 minus 1 (20)

to not introduce errors in control measurements Let usadditionally postulate that Q satisfies

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119898+119896 mod 119863)119864

⟩ (21)

that is Q advances index 119896 positions in a set of Eversquos probestates Similarly Qminus1 decrements the index 119896 positions

Qminus1 1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816120572(119898minus119896 mod 119863)119864

⟩ (22)

Let us recall that for qudits Alice uses

Z =

119863minus1

sum119896=0

120596119896

|119896⟩ ⟨119896|

X =

119863minus1

sum119896=0

|119896 + 1 mod 119863⟩ ⟨119896|

120596 = 1198901198952120587119863

(23)

to encode classic 120583 ] ldquocditsrdquo in the following way

100381610038161003816100381610038161003816120573(120583])ℎ119905

⟩ = X120583

119905Z

]119905

10038161003816100381610038161003816120573(00)

ℎ119905⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

(24)

6 Advances in Mathematical Physics

Under attack Alice applies encoding (24) to the state coupledaccording to rule (20)

1003816100381610038161003816120595enc⟩ = X120583

119905Z

]119905Q10038161003816100381610038161003816120573(00)

ℎ119905⟩10038161003816100381610038161003816120572(0)

119864⟩

= X120583

119905Z

]119905

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119896)

119864⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩

(25)

The travel qubit is affected by Qminus1 in its way back to Bob

1003816100381610038161003816120601dec⟩ = Qminus1 1003816100381610038161003816120595enc⟩ =

1

radic119863

sdot

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩ (Q

minus1 1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩)

= 1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

sdot100381610038161003816100381610038161003816120572(minus120583 mod 119863)119864

⟩

(26)

The expression in curly braces is exactly the state thatBob expects to receive when there is no Eve (see (24))so eavesdropping also does not affect the message At thesame time the initial state of the ancilla is moved by 120583

positions within the set of detection states As a result Evecan unambiguously identify the value of cdit 120583 as long as thedetection states are mutually orthogonal

The C119883

(controlled X) gate seems to be the simplestinstance of the attack paradigm Let the detection and probesets of states be the elements of the computational basis(|120572(119898)119864

⟩ = |119898119864⟩ |119886(119898)119864

⟩ = |119898119864⟩) and the ancilla is composed of

the single qudit register The attack operation Q can be thenimplemented as

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ = C

119883

1003816100381610038161003816119896119905⟩1003816100381610038161003816119898119864⟩ =

1003816100381610038161003816119896119905⟩X119896

119864

1003816100381610038161003816119898119864⟩

=1003816100381610038161003816119896119905⟩

1003816100381610038161003816(119898 + 119896 mod 119863)119864⟩

(27)

In an obvious way requirements (21) regarding properties ofQ are then fulfilled

The existence of attacks able to undetectably eavesdropon half of the dense coded information has been alreadyforecasted in relation to qubit [2] qutrit [6] and qudit[23] based protocol However no explicit form of the attacktransformation has been given The presented result fills inthis gap and provides some general guidelines on how toconstruct coupling transformation with desired properties

34 Control Mode Able to Detect Bit-Flip Eavesdropping Theinsecurity of the considered protocol results from inabilityto detect coupling Q

ℎ119905with the control measurements in a

single basis Let us consider a qubit based protocol fromSection 21 with control mode enhanced to measurementsin two bases namely computational basis and its dual

basis that is eigenvectors of X gate In the new controlmode Alice randomly selects measurement basis performsmeasurement and asks Bob to make local measurement inthe same basisThe control state (9) in the absence of couplingtakes the form

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038161205940⟩

=1

radic2(1003816100381610038161003816+ℎ⟩

1003816100381610038161003816+119905⟩ minus1003816100381610038161003816minusℎ⟩

1003816100381610038161003816minus119905⟩)10038161003816100381610038161205940⟩

(28)

where |plusmn⟩ = (|0⟩ plusmn |1⟩)radic2 are eigenvectors of X It followsthat legitimate parties expect anticorrelation (correlation) ofoutcomes in the computational (dual) basis Under attackundetectable in the computational basis (14) the controlequation (15) takes the following form in the dual basis

1003816100381610038161003816120595ℎ119905119864⟩

=1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ minus1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ +

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816+119905⟩

minus1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ +1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ minus

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816minus119905⟩

(29)

Alice measurement causes the collapse to one of the statesin the curly braces It follows that Bob can obtain plusmn1

outcome with equal probability which in turn renders Evedetectability If control bases are selected with equal proba-bility then bit-flip attack is detected with 119901det = 14 Theabove qualitative discussion addresses bit-flip attack Themore advanced discussion on the properties of controlmodesbased on mutually unbiased bases and in relation to attacksof any form can be found in [23]

4 Conclusion

A generic scheme that provides undetectable eavesdroppingof bit-flip operations in the seminal version of the ping-pongprotocol is introduced It can be considered as a general-ization of the P-circuit [24] but in contrast it is deducedfrom the very basic properties of the coupling transformationMoreover the proposed scheme can be realized withoutreferring to the vacuum states so it is fully consistent withthe absence of losses assumption The CNOT gate and P-circuit are special cases of the introduced scheme so bothapproaches are equivalent It follows that any control modeable to detect CNOT coupling is also able to detect thepresence of the P-circuit The control mode based on localmeasurements in randomly selected unbiased bases is anexample of such procedure Consequently there is no needfor special addressing of P-circuit in the security analysesAlso the introduced scheme can be adapted to higherdimensional systems It can be considered as the constructiveproof of the existence of attacks forecasted in [2 6 23]

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Mathematical Physics

in a private channel provided by a QDC protocol offeringunconditional security has security comparable with QKDAlso quasi-secureQDCprotocols can realize unconditionallysecure QKD However in this case QDC phase deliversshared sequence that is partially known to the eavesdropperBy the appropriate postprocessing that is privacy amplifica-tion the eavesdropperrsquos knowledge on the resulting sequencecan be reduced to arbitrarily small value provided that hisinformation on the initial sequence is less than mutualinformation of the legitimate parties The realization of theQKD via QDC can be potentially more efficient as the basisreconciliation step which severely plagues efficiency of manyQKD protocols can be avoided [16ndash18] Protocols of this typeare referred to as deterministic QKD and some of them havebeen recently experimentally demonstrated [19 20]

This paper is devoted to the analysis of the (in)securityof the ping-pong protocol an entanglement based QDCscheme [3] Quasi-security is provided only for perfect quan-tum channels [14] and the scheme becomes insecure whenlosses [21] andor communication errors and imperfection ofdevices are taken into account [22] Protocol offers capacityof single bit per protocol cycle because the authenticity ofthe shared EPR pair is verified only by a measurement ina single basis This limits the available encoding to phaseflips Possible capacity enhancement via dense coding leads toundetectable information leakage as demonstrated in [2] andusage of mutually unbiased bases in control measurementsis required to preserve quasi-security of the communica-tion [8] In our previous work we have proved that thisobservation also holds for the qudit based protocol andthat detection probability depends on the number of basesused in the control mode [7 23] Anyway no explicit attacktransformation has been given in the aforementioned papersThe present contribution is motivated by the appearance ofthe circuit [24] (further it will be referred to as P-circuit)capable of undetectably intercepting information transmittedin the qubit based ping-pong protocol with the followingconfiguration quantum channel is perfect legitimate partiesuse single basis for control measurements and information isdense coded In other words the instantiation of the attack isforecasted in [2] Although P-circuit is applicable to perfectchannels it assumes the appearance of the vacuum states inthe eavesdropperrsquos ancilla In consequence it does not wellfit the existing analyses Shortly after its appearance a controlmode that addresses detection of this specific circuit has beenproposed [25]

We propose a generic scheme for construction of attacksthat permit undetectable eavesdropping under the sameassumptions quantum channel is perfect control measure-ments are executed in a single basis and sensitive informationis dense coded Thus our contribution can be considered asthe generalization of the result given in [24] The presentedmethod is applicable to systems of any dimension so itcan be used to construct a plethora of new transformsUsing introduced generalization we also demonstrate theequivalence of the attack from [24] and CNOT operation Inconsequence we claim that there is no need for constructionof specific control modes as in [25] because any controlmode able to detect CNOT operation is also able to detect

circuit proposed in [24] We do not propose the attack that isundetectable by control measurements in unbiased bases Infact we think that the opposite is true control measurementsinmutually unbiased bases are sufficient to statistically detectcoherence break of the shared entangled state and in thatwayreveal the presence of the eavesdropper [23]

The paper is organized as follows In Section 2 we providenotation and concepts used in the text Section 3 presents themain contribution In particular we provide a general bit-flip detection scheme demonstrate its equivalence with theexisting approaches and introduce an attack on the quditbased protocol In Section 4 we summarize the presentedwork

2 Preliminaries

21 Ping-Pong Protocol The communication protocoldescribed below is a ping-pong paradigm variant analysed in[24] Compared to the seminal version [3] it differs only inthe encoding operation the sender uses dense coding insteadof phase flipsThe remaining elements of the communicationscenario are left intact

Bob starts the communication process by creation of EPRpair (the assumed initial state is the same as in [3 24] tomaintain compatibility of mathematical expressions for thequdit version of the protocol considered in Section 31 it isassumed that Bob starts from the generalization of |Φ+⟩ =

(|0ℎ⟩|0119905⟩ + |1ℎ⟩|1119905⟩)radic2)

1003816100381610038161003816Ψinit⟩ =1003816100381610038161003816Ψminus

⟩ =(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ minus10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)

radic2 (1)

Then he sends one of the qubits further referred to as thesignaltravel qubit to Alice Alice can in principle encode twoclassic bits 120583 and ] applying unitary transformation A

120583] =

X120583Z] where X = |1⟩⟨0| + |0⟩⟨1| and Z = |0⟩⟨0| minus

|1⟩⟨1| are bit-flip and phase-flip operations respectively Thesignal particle is sent back to Bob who detects appliedtransformation by a collective measurement of both qubits(Figure 1)

Passive eavesdropping is impossible Eve has access onlyto the travel qubit which before and after encoding lookslike maximally mixed state Unfortunately the describedcommunication scenario is vulnerable to the intercept-resendattack and Alice has to check whether the received qubit isgenuine As a countermeasure Alice measures the receivedqubit in computational basis (|0⟩ |1⟩) in randomly selectedprotocol cycles and asks Bob over authenticated classicchannel to do the same with his qubit (Figure 2) Hermeasurement causes the collapse of the shared state (1) Theperfect (anti)correlation of the outcomes is preserved only ifthe qubit measured by Alice is the same one that was sent byBob If Eve inserts fake qubit then the measured qubits areno longer correlated and some discrepancies which are thesign of the eavesdropping do occur In that way Alice andBob can convince themselves with confidence approachingcertainty that the quantum channel is not spoofed providedthat they have executed a sufficient number of control cycles

Advances in Mathematical Physics 3

Bell

BobBob Alice

h

t120583

|120595init⟩

119987119989120583

120583

Figure 1 The schematic diagram of a message mode in the ping-pong protocol

BobBob Alice

h

t

|120595init⟩120572

120573

Figure 2 The schematic diagram of a control mode in the ping-pong protocol

However the intercept-resend attack is not the onlypossible way of active sensitive information interceptionThesignal particle that travels back and forth between legitimateparties can be the subject of any quantum action introducedby Eve (Figure 3) Introduced coupling causes the encodingoperation to also modify Eversquos ancilla state and Eve hopesto detect and decipher Alicersquos actions by its inspectionActions of Eve not necessarily unitary in the affected qubitrsquosspace can be described as unitary operation Q acting in thespace extended with two additional qubits as follows fromStinespringrsquos dilation theorem The control state shared bylegitimate parties then takes the form

1003816100381610038161003816120595ℎ119905119864⟩ = (Iℎotimes Q) (

1003816100381610038161003816Ψinit⟩ otimes1003816100381610038161003816120594119864⟩) (2)

where |120594119864⟩ is some initial state of Eversquos ancilla Eve presence

is detected with probability

119901det (Q) = Tr (Pℎ119905Tr119864(1003816100381610038161003816120595ℎ119905119864⟩ ⟨120595ℎ119905119864

1003816100381610038161003816)) (3)

where projectionPℎ119905depends on initial state and the consid-

ered case is defined as

Pℎ119905= Iℎ119905minus10038161003816100381610038160ℎ⟩ ⟨0ℎ

1003816100381610038161003816 otimes10038161003816100381610038161119905⟩ ⟨1119905

1003816100381610038161003816 minus10038161003816100381610038161ℎ⟩ ⟨1ℎ

1003816100381610038161003816

otimes10038161003816100381610038160119905⟩ ⟨0119905

1003816100381610038161003816 (4)

22 Pavicic Attack Pavicicrsquos attack demonstrates the viola-tion of ping-pong protocol security when dense coding isusedThe attack does not introduce errors or losses in controlandmessagemode and it permits eavesdropping informationencoded as bit-flip operation

The P-circuit presented by Pavicic (Figure 4) is a result ofa cut-and-try procedure [24 section IV] applied to Wojcikrsquoscircuit [21] It is composed of two Hadamard gates followedby the controlled polarization beam splitter (CPBS) whichis a generalization of the polarization beam splitter (PBS)concept The PBS is a two-port gate that swaps horizontally

Bell

EveEve

BobBob Alice

120583

h

t

|120595init⟩

119987119989120583

⟩|120594E

119980minus1119980

120583

Figure 3 A schematic diagram of an individual attack

t

x H

Hy

PBS

CPBS

Figure 4 P-circuit Q119905119909119910

[24 eq (2)]

polarized photons |0119909⟩ (|0119910⟩) entering its input to the other

port |0119910⟩ (|0119909⟩) on output while vertically polarized ones |1

119909⟩

(|1119910⟩) remain in their port |1

119909⟩ (|1119910⟩) that is

PBS 1003816100381610038161003816V119909⟩100381610038161003816100381610038160119910⟩ =

10038161003816100381610038160119909⟩10038161003816100381610038161003816V119910⟩

PBS 1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩ =

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩

(5a)

PBS 10038161003816100381610038160119909⟩10038161003816100381610038161003816V119910⟩ =

1003816100381610038161003816V119909⟩100381610038161003816100381610038160119910⟩

PBS 10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩

(5b)

where |V⟩ denotes the vacuum state The CPBS behavesas normal PBS if control qubit is set to |0

119905⟩ The roles of

horizontal and vertical polarization are exchanged for controlqubit set to |1

119905⟩

CPBS 10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ =

10038161003816100381610038160119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

CPBS 10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ =

10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

(6a)

CPBS 10038161003816100381610038160119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

CPBS 10038161003816100381610038161119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

(6b)

CPBS 10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩ =

10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩

CPBS 10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩ =

10038161003816100381610038161119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩

(6c)

CPBS 10038161003816100381610038160119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038160119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩

CPBS 10038161003816100381610038161119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩

(6d)

4 Advances in Mathematical Physics

Initially Eversquos ancilla is initialized to the state |1205940⟩ = |V

119909⟩|0119910⟩

The action of the P-circuit from Figure 4 is then described bythe following formulas

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038160119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

(7a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038161119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(7b)

For the purpose of future analysis let us also identify actionsof the circuit under consideration onto the state |120594

1⟩ =

|0119909⟩|V119910⟩

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038160119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

(8a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038161119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(8b)

The control state (2) after entangling with Eversquos ancilla reads

1003816100381610038161003816120595ℎ119905119864⟩ =(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

radic2 (9)

This state is further used by Alice and Bob for eavesdroppingcheck It is clear from (3) that the attack does not introduceerrors or losses in control mode and the expected correlationof outcomes is preserved in the computational basis

Phase Flip The phase-flip encoding applied to the coupledstate leads to

10038161003816100381610038161003816120595phase⟩ = (I

ℎotimesZ119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119886119864⟩ minus

10038161003816100381610038160ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩) (10)

The signal qubit is then sent back to Bob who afterdisentangling on a basis of (7a) and (7b) observes

10038161003816100381610038161003816120601phase⟩ = (Q

119905119909119910)minus1 10038161003816100381610038161003816

120595phase⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩ minus10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩)1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

= [(IℎotimesZ119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205940⟩

(11)

Bit Flip The bit-flip operation transforms Alicersquos state to1003816100381610038161003816120595bit⟩ = (I

ℎotimesX119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119886119864⟩ +

10038161003816100381610038160ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩) (12)

The system state after disentangling can be deduced from(8a) and (8b)

1003816100381610038161003816120601bit⟩ = (Q119905119909119910)minus1 1003816100381610038161003816120595bit⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

= [(IℎotimesX119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205941⟩

(13)

In both cases that is phase-flip and bit-flip encoding thesignalling subsystem behaves as if there was no couplingwith the ancilla However Alicersquos bit-flip encoding modifiesEversquos register (|120594

0⟩ rarr |120594

1⟩) The states |120594

0⟩ and |120594

1⟩ are

orthogonal and perfectly distinguishable In consequenceEve can eavesdrop on bit-flip operations without introducingerrors and losses in message mode as well

3 Results

This section is devoted to the analysis of the general form ofthe incoherent attack shown diagrammatically in Figure 3Each cycle of the protocol is considered to be independentof the other ones Consequently the effectiveness of theattack is expressed in a fraction of bits eavesdropped onper communication cycle Throughout the analysis it isalso assumed that legitimate parties rely on control modeused in the seminal version of the protocol They locallymeasure possessed particles in the computational basis andverify expected correlation via the public discussion overauthenticated classic channel

31 Generic Bit-Flip Detection Scheme for Qubit Based Proto-col As the controlmode explores outcomes of localmeasure-ments in computational basis for intrusion detection themapQ has to be of trivial form

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(14)

to not induce errors andor losses in control cycles It followsthat under attack Alice operates on the state

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩) (15)

Let the entangling transformation Q additionally satisfy

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(16)

for some state |120601119864⟩ = |120594

119864⟩ The process of information

encoding and disentangling from the ancilla is then describedby the expressions

Qminus1

(IℎotimesI119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

Advances in Mathematical Physics 5

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

=1003816100381610038161003816Ψminus

⟩1003816100381610038161003816120594119864⟩

(17a)

Qminus1

(IℎotimesX119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

=1003816100381610038161003816Φminus

⟩1003816100381610038161003816120601119864⟩

(17b)

Qminus1

(IℎotimesZ119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

= minus1003816100381610038161003816Ψ+

⟩1003816100381610038161003816120594119864⟩

(17c)

Qminus1

(IℎotimesX119905Z119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

= minus1003816100381610038161003816Φ+

⟩1003816100381610038161003816120601119864⟩

(17d)

As a result the registers used for signalling are left untouchedand decoupled but Eversquos register is flipped from |120594

119864⟩ to |120601

119864⟩

whenAlice applies bit-flip operation In consequence Eve cansuccessfully decode half of the message content provided thatthe detection states |120594

119864⟩ and |120601

119864⟩ are perfectly distinguish-

able It follows that any unitary coupling transformation Qthat satisfies (14) and (16) can be used for bit-flip detection

32 Equivalence of P-Circuit and CNOT Circuit The prop-erties of the above generic scheme and the P-circuit [24]perfectly coincide As follows from (7a) (7b) (8a) and (8b)the states |120594

0⟩ = |V

119909⟩|0119910⟩ and |120594

1⟩ = |0

119909⟩|V119910⟩ play the role

of detection states |120594119864⟩ and |120601

119864⟩ respectively It is also clear

that transformation Q119905119909119910

has properties claimed in (14) and(16) Thus the P-circuit can be considered as an instance ofthe generic scheme described in Section 31

However the operator Q satisfying (14) and (16) can berealized in many ways It seems that CNOT operation actingon a single qubit of Eversquos ancilla Q = CNOT

119905119909 |120594119864⟩ = |0

119909⟩

|120601119864⟩ = |1

119909⟩ |119886119864⟩ = |0

119909⟩ and |119889

119864⟩ = |1

119909⟩ is the simplest

realization of the logic behind the attack Such version is alsopractically feasible as the attacks involving probes entangledvia the CNOT operation have been already proposed in theQKD context [26 27] As a result both the CNOT circuitand P-circuit are equivalent in terms of provided information

gain detectability and practical feasibility Consequentlythere is no need for the design of control modes that addressP-circuit in a special manner [25]

33 An Attack on Qudit Based Protocol The P-circuit has nostraightforward generalization to qudit based version of theprotocol In contrast the presented approach can be adaptedwith ease Let Bob start communication process with creationof EPR pair

10038161003816100381610038161003816120573(00)

ℎ119905⟩ =

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩ (18)

where119863 is the qudit dimension The travel qudit is then sentto Alice for encoding or control measurement In controlmode the home and travel qubits are measured in thecomputational basis so the projection P

ℎ119905used in control

equation (3) takes the form

Pℎ119905= Iℎ119905minus

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩ ⟨119896ℎ1003816100381610038161003816 otimes

1003816100381610038161003816119896119905⟩ ⟨1198961199051003816100381610038161003816 (19)

Let by an analogy to the qubit case |120572(119896)119864⟩ and |119886(119896)

119864⟩ be the

sets of 119863 orthonormal states of the ancilla system Thesestates will be further referred to as detection and probe statesrespectively The map used by Eve must be of the form

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(0)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119896)

119864⟩ 119896 = 0 119863 minus 1 (20)

to not introduce errors in control measurements Let usadditionally postulate that Q satisfies

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119898+119896 mod 119863)119864

⟩ (21)

that is Q advances index 119896 positions in a set of Eversquos probestates Similarly Qminus1 decrements the index 119896 positions

Qminus1 1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816120572(119898minus119896 mod 119863)119864

⟩ (22)

Let us recall that for qudits Alice uses

Z =

119863minus1

sum119896=0

120596119896

|119896⟩ ⟨119896|

X =

119863minus1

sum119896=0

|119896 + 1 mod 119863⟩ ⟨119896|

120596 = 1198901198952120587119863

(23)

to encode classic 120583 ] ldquocditsrdquo in the following way

100381610038161003816100381610038161003816120573(120583])ℎ119905

⟩ = X120583

119905Z

]119905

10038161003816100381610038161003816120573(00)

ℎ119905⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

(24)

6 Advances in Mathematical Physics

Under attack Alice applies encoding (24) to the state coupledaccording to rule (20)

1003816100381610038161003816120595enc⟩ = X120583

119905Z

]119905Q10038161003816100381610038161003816120573(00)

ℎ119905⟩10038161003816100381610038161003816120572(0)

119864⟩

= X120583

119905Z

]119905

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119896)

119864⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩

(25)

The travel qubit is affected by Qminus1 in its way back to Bob

1003816100381610038161003816120601dec⟩ = Qminus1 1003816100381610038161003816120595enc⟩ =

1

radic119863

sdot

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩ (Q

minus1 1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩)

= 1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

sdot100381610038161003816100381610038161003816120572(minus120583 mod 119863)119864

⟩

(26)

The expression in curly braces is exactly the state thatBob expects to receive when there is no Eve (see (24))so eavesdropping also does not affect the message At thesame time the initial state of the ancilla is moved by 120583

positions within the set of detection states As a result Evecan unambiguously identify the value of cdit 120583 as long as thedetection states are mutually orthogonal

The C119883

(controlled X) gate seems to be the simplestinstance of the attack paradigm Let the detection and probesets of states be the elements of the computational basis(|120572(119898)119864

⟩ = |119898119864⟩ |119886(119898)119864

⟩ = |119898119864⟩) and the ancilla is composed of

the single qudit register The attack operation Q can be thenimplemented as

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ = C

119883

1003816100381610038161003816119896119905⟩1003816100381610038161003816119898119864⟩ =

1003816100381610038161003816119896119905⟩X119896

119864

1003816100381610038161003816119898119864⟩

=1003816100381610038161003816119896119905⟩

1003816100381610038161003816(119898 + 119896 mod 119863)119864⟩

(27)

In an obvious way requirements (21) regarding properties ofQ are then fulfilled

The existence of attacks able to undetectably eavesdropon half of the dense coded information has been alreadyforecasted in relation to qubit [2] qutrit [6] and qudit[23] based protocol However no explicit form of the attacktransformation has been given The presented result fills inthis gap and provides some general guidelines on how toconstruct coupling transformation with desired properties

34 Control Mode Able to Detect Bit-Flip Eavesdropping Theinsecurity of the considered protocol results from inabilityto detect coupling Q

ℎ119905with the control measurements in a

single basis Let us consider a qubit based protocol fromSection 21 with control mode enhanced to measurementsin two bases namely computational basis and its dual

basis that is eigenvectors of X gate In the new controlmode Alice randomly selects measurement basis performsmeasurement and asks Bob to make local measurement inthe same basisThe control state (9) in the absence of couplingtakes the form

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038161205940⟩

=1

radic2(1003816100381610038161003816+ℎ⟩

1003816100381610038161003816+119905⟩ minus1003816100381610038161003816minusℎ⟩

1003816100381610038161003816minus119905⟩)10038161003816100381610038161205940⟩

(28)

where |plusmn⟩ = (|0⟩ plusmn |1⟩)radic2 are eigenvectors of X It followsthat legitimate parties expect anticorrelation (correlation) ofoutcomes in the computational (dual) basis Under attackundetectable in the computational basis (14) the controlequation (15) takes the following form in the dual basis

1003816100381610038161003816120595ℎ119905119864⟩

=1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ minus1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ +

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816+119905⟩

minus1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ +1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ minus

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816minus119905⟩

(29)

Alice measurement causes the collapse to one of the statesin the curly braces It follows that Bob can obtain plusmn1

outcome with equal probability which in turn renders Evedetectability If control bases are selected with equal proba-bility then bit-flip attack is detected with 119901det = 14 Theabove qualitative discussion addresses bit-flip attack Themore advanced discussion on the properties of controlmodesbased on mutually unbiased bases and in relation to attacksof any form can be found in [23]

4 Conclusion

A generic scheme that provides undetectable eavesdroppingof bit-flip operations in the seminal version of the ping-pongprotocol is introduced It can be considered as a general-ization of the P-circuit [24] but in contrast it is deducedfrom the very basic properties of the coupling transformationMoreover the proposed scheme can be realized withoutreferring to the vacuum states so it is fully consistent withthe absence of losses assumption The CNOT gate and P-circuit are special cases of the introduced scheme so bothapproaches are equivalent It follows that any control modeable to detect CNOT coupling is also able to detect thepresence of the P-circuit The control mode based on localmeasurements in randomly selected unbiased bases is anexample of such procedure Consequently there is no needfor special addressing of P-circuit in the security analysesAlso the introduced scheme can be adapted to higherdimensional systems It can be considered as the constructiveproof of the existence of attacks forecasted in [2 6 23]

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 3

Bell

BobBob Alice

h

t120583

|120595init⟩

119987119989120583

120583

Figure 1 The schematic diagram of a message mode in the ping-pong protocol

BobBob Alice

h

t

|120595init⟩120572

120573

Figure 2 The schematic diagram of a control mode in the ping-pong protocol

However the intercept-resend attack is not the onlypossible way of active sensitive information interceptionThesignal particle that travels back and forth between legitimateparties can be the subject of any quantum action introducedby Eve (Figure 3) Introduced coupling causes the encodingoperation to also modify Eversquos ancilla state and Eve hopesto detect and decipher Alicersquos actions by its inspectionActions of Eve not necessarily unitary in the affected qubitrsquosspace can be described as unitary operation Q acting in thespace extended with two additional qubits as follows fromStinespringrsquos dilation theorem The control state shared bylegitimate parties then takes the form

1003816100381610038161003816120595ℎ119905119864⟩ = (Iℎotimes Q) (

1003816100381610038161003816Ψinit⟩ otimes1003816100381610038161003816120594119864⟩) (2)

where |120594119864⟩ is some initial state of Eversquos ancilla Eve presence

is detected with probability

119901det (Q) = Tr (Pℎ119905Tr119864(1003816100381610038161003816120595ℎ119905119864⟩ ⟨120595ℎ119905119864

1003816100381610038161003816)) (3)

where projectionPℎ119905depends on initial state and the consid-

ered case is defined as

Pℎ119905= Iℎ119905minus10038161003816100381610038160ℎ⟩ ⟨0ℎ

1003816100381610038161003816 otimes10038161003816100381610038161119905⟩ ⟨1119905

1003816100381610038161003816 minus10038161003816100381610038161ℎ⟩ ⟨1ℎ

1003816100381610038161003816

otimes10038161003816100381610038160119905⟩ ⟨0119905

1003816100381610038161003816 (4)

22 Pavicic Attack Pavicicrsquos attack demonstrates the viola-tion of ping-pong protocol security when dense coding isusedThe attack does not introduce errors or losses in controlandmessagemode and it permits eavesdropping informationencoded as bit-flip operation

The P-circuit presented by Pavicic (Figure 4) is a result ofa cut-and-try procedure [24 section IV] applied to Wojcikrsquoscircuit [21] It is composed of two Hadamard gates followedby the controlled polarization beam splitter (CPBS) whichis a generalization of the polarization beam splitter (PBS)concept The PBS is a two-port gate that swaps horizontally

Bell

EveEve

BobBob Alice

120583

h

t

|120595init⟩

119987119989120583

⟩|120594E

119980minus1119980

120583

Figure 3 A schematic diagram of an individual attack

t

x H

Hy

PBS

CPBS

Figure 4 P-circuit Q119905119909119910

[24 eq (2)]

polarized photons |0119909⟩ (|0119910⟩) entering its input to the other

port |0119910⟩ (|0119909⟩) on output while vertically polarized ones |1

119909⟩

(|1119910⟩) remain in their port |1

119909⟩ (|1119910⟩) that is

PBS 1003816100381610038161003816V119909⟩100381610038161003816100381610038160119910⟩ =

10038161003816100381610038160119909⟩10038161003816100381610038161003816V119910⟩

PBS 1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩ =

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩

(5a)

PBS 10038161003816100381610038160119909⟩10038161003816100381610038161003816V119910⟩ =

1003816100381610038161003816V119909⟩100381610038161003816100381610038160119910⟩

PBS 10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩

(5b)

where |V⟩ denotes the vacuum state The CPBS behavesas normal PBS if control qubit is set to |0

119905⟩ The roles of

horizontal and vertical polarization are exchanged for controlqubit set to |1

119905⟩

CPBS 10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ =

10038161003816100381610038160119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

CPBS 10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ =

10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

(6a)

CPBS 10038161003816100381610038160119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

CPBS 10038161003816100381610038161119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119905⟩10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

(6b)

CPBS 10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩ =

10038161003816100381610038160119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩

CPBS 10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩ =

10038161003816100381610038161119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩

(6c)

CPBS 10038161003816100381610038160119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038160119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩

CPBS 10038161003816100381610038161119905⟩10038161003816100381610038161119909⟩

10038161003816100381610038161003816V119910⟩ =

10038161003816100381610038161119905⟩1003816100381610038161003816V119909⟩

100381610038161003816100381610038161119910⟩

(6d)

4 Advances in Mathematical Physics

Initially Eversquos ancilla is initialized to the state |1205940⟩ = |V

119909⟩|0119910⟩

The action of the P-circuit from Figure 4 is then described bythe following formulas

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038160119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

(7a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038161119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(7b)

For the purpose of future analysis let us also identify actionsof the circuit under consideration onto the state |120594

1⟩ =

|0119909⟩|V119910⟩

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038160119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

(8a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038161119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(8b)

The control state (2) after entangling with Eversquos ancilla reads

1003816100381610038161003816120595ℎ119905119864⟩ =(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

radic2 (9)

This state is further used by Alice and Bob for eavesdroppingcheck It is clear from (3) that the attack does not introduceerrors or losses in control mode and the expected correlationof outcomes is preserved in the computational basis

Phase Flip The phase-flip encoding applied to the coupledstate leads to

10038161003816100381610038161003816120595phase⟩ = (I

ℎotimesZ119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119886119864⟩ minus

10038161003816100381610038160ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩) (10)

The signal qubit is then sent back to Bob who afterdisentangling on a basis of (7a) and (7b) observes

10038161003816100381610038161003816120601phase⟩ = (Q

119905119909119910)minus1 10038161003816100381610038161003816

120595phase⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩ minus10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩)1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

= [(IℎotimesZ119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205940⟩

(11)

Bit Flip The bit-flip operation transforms Alicersquos state to1003816100381610038161003816120595bit⟩ = (I

ℎotimesX119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119886119864⟩ +

10038161003816100381610038160ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩) (12)

The system state after disentangling can be deduced from(8a) and (8b)

1003816100381610038161003816120601bit⟩ = (Q119905119909119910)minus1 1003816100381610038161003816120595bit⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

= [(IℎotimesX119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205941⟩

(13)

In both cases that is phase-flip and bit-flip encoding thesignalling subsystem behaves as if there was no couplingwith the ancilla However Alicersquos bit-flip encoding modifiesEversquos register (|120594

0⟩ rarr |120594

1⟩) The states |120594

0⟩ and |120594

1⟩ are

orthogonal and perfectly distinguishable In consequenceEve can eavesdrop on bit-flip operations without introducingerrors and losses in message mode as well

3 Results

This section is devoted to the analysis of the general form ofthe incoherent attack shown diagrammatically in Figure 3Each cycle of the protocol is considered to be independentof the other ones Consequently the effectiveness of theattack is expressed in a fraction of bits eavesdropped onper communication cycle Throughout the analysis it isalso assumed that legitimate parties rely on control modeused in the seminal version of the protocol They locallymeasure possessed particles in the computational basis andverify expected correlation via the public discussion overauthenticated classic channel

31 Generic Bit-Flip Detection Scheme for Qubit Based Proto-col As the controlmode explores outcomes of localmeasure-ments in computational basis for intrusion detection themapQ has to be of trivial form

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(14)

to not induce errors andor losses in control cycles It followsthat under attack Alice operates on the state

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩) (15)

Let the entangling transformation Q additionally satisfy

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(16)

for some state |120601119864⟩ = |120594

119864⟩ The process of information

encoding and disentangling from the ancilla is then describedby the expressions

Qminus1

(IℎotimesI119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

Advances in Mathematical Physics 5

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

=1003816100381610038161003816Ψminus

⟩1003816100381610038161003816120594119864⟩

(17a)

Qminus1

(IℎotimesX119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

=1003816100381610038161003816Φminus

⟩1003816100381610038161003816120601119864⟩

(17b)

Qminus1

(IℎotimesZ119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

= minus1003816100381610038161003816Ψ+

⟩1003816100381610038161003816120594119864⟩

(17c)

Qminus1

(IℎotimesX119905Z119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

= minus1003816100381610038161003816Φ+

⟩1003816100381610038161003816120601119864⟩

(17d)

As a result the registers used for signalling are left untouchedand decoupled but Eversquos register is flipped from |120594

119864⟩ to |120601

119864⟩

whenAlice applies bit-flip operation In consequence Eve cansuccessfully decode half of the message content provided thatthe detection states |120594

119864⟩ and |120601

119864⟩ are perfectly distinguish-

able It follows that any unitary coupling transformation Qthat satisfies (14) and (16) can be used for bit-flip detection

32 Equivalence of P-Circuit and CNOT Circuit The prop-erties of the above generic scheme and the P-circuit [24]perfectly coincide As follows from (7a) (7b) (8a) and (8b)the states |120594

0⟩ = |V

119909⟩|0119910⟩ and |120594

1⟩ = |0

119909⟩|V119910⟩ play the role

of detection states |120594119864⟩ and |120601

119864⟩ respectively It is also clear

that transformation Q119905119909119910

has properties claimed in (14) and(16) Thus the P-circuit can be considered as an instance ofthe generic scheme described in Section 31

However the operator Q satisfying (14) and (16) can berealized in many ways It seems that CNOT operation actingon a single qubit of Eversquos ancilla Q = CNOT

119905119909 |120594119864⟩ = |0

119909⟩

|120601119864⟩ = |1

119909⟩ |119886119864⟩ = |0

119909⟩ and |119889

119864⟩ = |1

119909⟩ is the simplest

realization of the logic behind the attack Such version is alsopractically feasible as the attacks involving probes entangledvia the CNOT operation have been already proposed in theQKD context [26 27] As a result both the CNOT circuitand P-circuit are equivalent in terms of provided information

gain detectability and practical feasibility Consequentlythere is no need for the design of control modes that addressP-circuit in a special manner [25]

33 An Attack on Qudit Based Protocol The P-circuit has nostraightforward generalization to qudit based version of theprotocol In contrast the presented approach can be adaptedwith ease Let Bob start communication process with creationof EPR pair

10038161003816100381610038161003816120573(00)

ℎ119905⟩ =

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩ (18)

where119863 is the qudit dimension The travel qudit is then sentto Alice for encoding or control measurement In controlmode the home and travel qubits are measured in thecomputational basis so the projection P

ℎ119905used in control

equation (3) takes the form

Pℎ119905= Iℎ119905minus

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩ ⟨119896ℎ1003816100381610038161003816 otimes

1003816100381610038161003816119896119905⟩ ⟨1198961199051003816100381610038161003816 (19)

Let by an analogy to the qubit case |120572(119896)119864⟩ and |119886(119896)

119864⟩ be the

sets of 119863 orthonormal states of the ancilla system Thesestates will be further referred to as detection and probe statesrespectively The map used by Eve must be of the form

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(0)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119896)

119864⟩ 119896 = 0 119863 minus 1 (20)

to not introduce errors in control measurements Let usadditionally postulate that Q satisfies

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119898+119896 mod 119863)119864

⟩ (21)

that is Q advances index 119896 positions in a set of Eversquos probestates Similarly Qminus1 decrements the index 119896 positions

Qminus1 1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816120572(119898minus119896 mod 119863)119864

⟩ (22)

Let us recall that for qudits Alice uses

Z =

119863minus1

sum119896=0

120596119896

|119896⟩ ⟨119896|

X =

119863minus1

sum119896=0

|119896 + 1 mod 119863⟩ ⟨119896|

120596 = 1198901198952120587119863

(23)

to encode classic 120583 ] ldquocditsrdquo in the following way

100381610038161003816100381610038161003816120573(120583])ℎ119905

⟩ = X120583

119905Z

]119905

10038161003816100381610038161003816120573(00)

ℎ119905⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

(24)

6 Advances in Mathematical Physics

Under attack Alice applies encoding (24) to the state coupledaccording to rule (20)

1003816100381610038161003816120595enc⟩ = X120583

119905Z

]119905Q10038161003816100381610038161003816120573(00)

ℎ119905⟩10038161003816100381610038161003816120572(0)

119864⟩

= X120583

119905Z

]119905

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119896)

119864⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩

(25)

The travel qubit is affected by Qminus1 in its way back to Bob

1003816100381610038161003816120601dec⟩ = Qminus1 1003816100381610038161003816120595enc⟩ =

1

radic119863

sdot

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩ (Q

minus1 1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩)

= 1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

sdot100381610038161003816100381610038161003816120572(minus120583 mod 119863)119864

⟩

(26)

The expression in curly braces is exactly the state thatBob expects to receive when there is no Eve (see (24))so eavesdropping also does not affect the message At thesame time the initial state of the ancilla is moved by 120583

positions within the set of detection states As a result Evecan unambiguously identify the value of cdit 120583 as long as thedetection states are mutually orthogonal

The C119883

(controlled X) gate seems to be the simplestinstance of the attack paradigm Let the detection and probesets of states be the elements of the computational basis(|120572(119898)119864

⟩ = |119898119864⟩ |119886(119898)119864

⟩ = |119898119864⟩) and the ancilla is composed of

the single qudit register The attack operation Q can be thenimplemented as

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ = C

119883

1003816100381610038161003816119896119905⟩1003816100381610038161003816119898119864⟩ =

1003816100381610038161003816119896119905⟩X119896

119864

1003816100381610038161003816119898119864⟩

=1003816100381610038161003816119896119905⟩

1003816100381610038161003816(119898 + 119896 mod 119863)119864⟩

(27)

In an obvious way requirements (21) regarding properties ofQ are then fulfilled

The existence of attacks able to undetectably eavesdropon half of the dense coded information has been alreadyforecasted in relation to qubit [2] qutrit [6] and qudit[23] based protocol However no explicit form of the attacktransformation has been given The presented result fills inthis gap and provides some general guidelines on how toconstruct coupling transformation with desired properties

34 Control Mode Able to Detect Bit-Flip Eavesdropping Theinsecurity of the considered protocol results from inabilityto detect coupling Q

ℎ119905with the control measurements in a

single basis Let us consider a qubit based protocol fromSection 21 with control mode enhanced to measurementsin two bases namely computational basis and its dual

basis that is eigenvectors of X gate In the new controlmode Alice randomly selects measurement basis performsmeasurement and asks Bob to make local measurement inthe same basisThe control state (9) in the absence of couplingtakes the form

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038161205940⟩

=1

radic2(1003816100381610038161003816+ℎ⟩

1003816100381610038161003816+119905⟩ minus1003816100381610038161003816minusℎ⟩

1003816100381610038161003816minus119905⟩)10038161003816100381610038161205940⟩

(28)

where |plusmn⟩ = (|0⟩ plusmn |1⟩)radic2 are eigenvectors of X It followsthat legitimate parties expect anticorrelation (correlation) ofoutcomes in the computational (dual) basis Under attackundetectable in the computational basis (14) the controlequation (15) takes the following form in the dual basis

1003816100381610038161003816120595ℎ119905119864⟩

=1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ minus1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ +

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816+119905⟩

minus1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ +1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ minus

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816minus119905⟩

(29)

Alice measurement causes the collapse to one of the statesin the curly braces It follows that Bob can obtain plusmn1

outcome with equal probability which in turn renders Evedetectability If control bases are selected with equal proba-bility then bit-flip attack is detected with 119901det = 14 Theabove qualitative discussion addresses bit-flip attack Themore advanced discussion on the properties of controlmodesbased on mutually unbiased bases and in relation to attacksof any form can be found in [23]

4 Conclusion

A generic scheme that provides undetectable eavesdroppingof bit-flip operations in the seminal version of the ping-pongprotocol is introduced It can be considered as a general-ization of the P-circuit [24] but in contrast it is deducedfrom the very basic properties of the coupling transformationMoreover the proposed scheme can be realized withoutreferring to the vacuum states so it is fully consistent withthe absence of losses assumption The CNOT gate and P-circuit are special cases of the introduced scheme so bothapproaches are equivalent It follows that any control modeable to detect CNOT coupling is also able to detect thepresence of the P-circuit The control mode based on localmeasurements in randomly selected unbiased bases is anexample of such procedure Consequently there is no needfor special addressing of P-circuit in the security analysesAlso the introduced scheme can be adapted to higherdimensional systems It can be considered as the constructiveproof of the existence of attacks forecasted in [2 6 23]

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Mathematical Physics

Initially Eversquos ancilla is initialized to the state |1205940⟩ = |V

119909⟩|0119910⟩

The action of the P-circuit from Figure 4 is then described bythe following formulas

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038160119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

(7a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205940⟩ =

1

radic2

10038161003816100381610038161119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(7b)

For the purpose of future analysis let us also identify actionsof the circuit under consideration onto the state |120594

1⟩ =

|0119909⟩|V119910⟩

Q119905119909119910

10038161003816100381610038160119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038160119905⟩ (1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩ +

10038161003816100381610038161119909⟩10038161003816100381610038161003816V119910⟩)

=10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

(8a)

Q119905119909119910

10038161003816100381610038161119905⟩10038161003816100381610038161205941⟩ =

1

radic2

10038161003816100381610038161119905⟩ (10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩ +

1003816100381610038161003816V119909⟩100381610038161003816100381610038161119910⟩)

=10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(8b)

The control state (2) after entangling with Eversquos ancilla reads

1003816100381610038161003816120595ℎ119905119864⟩ =(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

radic2 (9)

This state is further used by Alice and Bob for eavesdroppingcheck It is clear from (3) that the attack does not introduceerrors or losses in control mode and the expected correlationof outcomes is preserved in the computational basis

Phase Flip The phase-flip encoding applied to the coupledstate leads to

10038161003816100381610038161003816120595phase⟩ = (I

ℎotimesZ119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119886119864⟩ minus

10038161003816100381610038160ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩) (10)

The signal qubit is then sent back to Bob who afterdisentangling on a basis of (7a) and (7b) observes

10038161003816100381610038161003816120601phase⟩ = (Q

119905119909119910)minus1 10038161003816100381610038161003816

120595phase⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩ minus10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩)1003816100381610038161003816V119909⟩

100381610038161003816100381610038160119910⟩

= [(IℎotimesZ119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205940⟩

(11)

Bit Flip The bit-flip operation transforms Alicersquos state to1003816100381610038161003816120595bit⟩ = (I

ℎotimesX119905)1003816100381610038161003816120595ℎ119905119864⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119886119864⟩ +

10038161003816100381610038160ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩) (12)

The system state after disentangling can be deduced from(8a) and (8b)

1003816100381610038161003816120601bit⟩ = (Q119905119909119910)minus1 1003816100381610038161003816120595bit⟩

=1

radic2(10038161003816100381610038161ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038160119909⟩

10038161003816100381610038161003816V119910⟩

= [(IℎotimesX119905)1003816100381610038161003816120595init⟩]

10038161003816100381610038161205941⟩

(13)

In both cases that is phase-flip and bit-flip encoding thesignalling subsystem behaves as if there was no couplingwith the ancilla However Alicersquos bit-flip encoding modifiesEversquos register (|120594

0⟩ rarr |120594

1⟩) The states |120594

0⟩ and |120594

1⟩ are

orthogonal and perfectly distinguishable In consequenceEve can eavesdrop on bit-flip operations without introducingerrors and losses in message mode as well

3 Results

This section is devoted to the analysis of the general form ofthe incoherent attack shown diagrammatically in Figure 3Each cycle of the protocol is considered to be independentof the other ones Consequently the effectiveness of theattack is expressed in a fraction of bits eavesdropped onper communication cycle Throughout the analysis it isalso assumed that legitimate parties rely on control modeused in the seminal version of the protocol They locallymeasure possessed particles in the computational basis andverify expected correlation via the public discussion overauthenticated classic channel

31 Generic Bit-Flip Detection Scheme for Qubit Based Proto-col As the controlmode explores outcomes of localmeasure-ments in computational basis for intrusion detection themapQ has to be of trivial form

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120594119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119889119864⟩

(14)

to not induce errors andor losses in control cycles It followsthat under attack Alice operates on the state

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩) (15)

Let the entangling transformation Q additionally satisfy

Q10038161003816100381610038160119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038160119905⟩

1003816100381610038161003816119889119864⟩

Q10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩ 997888rarr10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩

(16)

for some state |120601119864⟩ = |120594

119864⟩ The process of information

encoding and disentangling from the ancilla is then describedby the expressions

Qminus1

(IℎotimesI119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

Advances in Mathematical Physics 5

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

=1003816100381610038161003816Ψminus

⟩1003816100381610038161003816120594119864⟩

(17a)

Qminus1

(IℎotimesX119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

=1003816100381610038161003816Φminus

⟩1003816100381610038161003816120601119864⟩

(17b)

Qminus1

(IℎotimesZ119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

= minus1003816100381610038161003816Ψ+

⟩1003816100381610038161003816120594119864⟩

(17c)

Qminus1

(IℎotimesX119905Z119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

= minus1003816100381610038161003816Φ+

⟩1003816100381610038161003816120601119864⟩

(17d)

As a result the registers used for signalling are left untouchedand decoupled but Eversquos register is flipped from |120594

119864⟩ to |120601

119864⟩

whenAlice applies bit-flip operation In consequence Eve cansuccessfully decode half of the message content provided thatthe detection states |120594

119864⟩ and |120601

119864⟩ are perfectly distinguish-

able It follows that any unitary coupling transformation Qthat satisfies (14) and (16) can be used for bit-flip detection

32 Equivalence of P-Circuit and CNOT Circuit The prop-erties of the above generic scheme and the P-circuit [24]perfectly coincide As follows from (7a) (7b) (8a) and (8b)the states |120594

0⟩ = |V

119909⟩|0119910⟩ and |120594

1⟩ = |0

119909⟩|V119910⟩ play the role

of detection states |120594119864⟩ and |120601

119864⟩ respectively It is also clear

that transformation Q119905119909119910

has properties claimed in (14) and(16) Thus the P-circuit can be considered as an instance ofthe generic scheme described in Section 31

However the operator Q satisfying (14) and (16) can berealized in many ways It seems that CNOT operation actingon a single qubit of Eversquos ancilla Q = CNOT

119905119909 |120594119864⟩ = |0

119909⟩

|120601119864⟩ = |1

119909⟩ |119886119864⟩ = |0

119909⟩ and |119889

119864⟩ = |1

119909⟩ is the simplest

realization of the logic behind the attack Such version is alsopractically feasible as the attacks involving probes entangledvia the CNOT operation have been already proposed in theQKD context [26 27] As a result both the CNOT circuitand P-circuit are equivalent in terms of provided information

gain detectability and practical feasibility Consequentlythere is no need for the design of control modes that addressP-circuit in a special manner [25]

33 An Attack on Qudit Based Protocol The P-circuit has nostraightforward generalization to qudit based version of theprotocol In contrast the presented approach can be adaptedwith ease Let Bob start communication process with creationof EPR pair

10038161003816100381610038161003816120573(00)

ℎ119905⟩ =

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩ (18)

where119863 is the qudit dimension The travel qudit is then sentto Alice for encoding or control measurement In controlmode the home and travel qubits are measured in thecomputational basis so the projection P

ℎ119905used in control

equation (3) takes the form

Pℎ119905= Iℎ119905minus

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩ ⟨119896ℎ1003816100381610038161003816 otimes

1003816100381610038161003816119896119905⟩ ⟨1198961199051003816100381610038161003816 (19)

Let by an analogy to the qubit case |120572(119896)119864⟩ and |119886(119896)

119864⟩ be the

sets of 119863 orthonormal states of the ancilla system Thesestates will be further referred to as detection and probe statesrespectively The map used by Eve must be of the form

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(0)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119896)

119864⟩ 119896 = 0 119863 minus 1 (20)

to not introduce errors in control measurements Let usadditionally postulate that Q satisfies

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119898+119896 mod 119863)119864

⟩ (21)

that is Q advances index 119896 positions in a set of Eversquos probestates Similarly Qminus1 decrements the index 119896 positions

Qminus1 1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816120572(119898minus119896 mod 119863)119864

⟩ (22)

Let us recall that for qudits Alice uses

Z =

119863minus1

sum119896=0

120596119896

|119896⟩ ⟨119896|

X =

119863minus1

sum119896=0

|119896 + 1 mod 119863⟩ ⟨119896|

120596 = 1198901198952120587119863

(23)

to encode classic 120583 ] ldquocditsrdquo in the following way

100381610038161003816100381610038161003816120573(120583])ℎ119905

⟩ = X120583

119905Z

]119905

10038161003816100381610038161003816120573(00)

ℎ119905⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

(24)

6 Advances in Mathematical Physics

Under attack Alice applies encoding (24) to the state coupledaccording to rule (20)

1003816100381610038161003816120595enc⟩ = X120583

119905Z

]119905Q10038161003816100381610038161003816120573(00)

ℎ119905⟩10038161003816100381610038161003816120572(0)

119864⟩

= X120583

119905Z

]119905

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119896)

119864⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩

(25)

The travel qubit is affected by Qminus1 in its way back to Bob

1003816100381610038161003816120601dec⟩ = Qminus1 1003816100381610038161003816120595enc⟩ =

1

radic119863

sdot

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩ (Q

minus1 1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩)

= 1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

sdot100381610038161003816100381610038161003816120572(minus120583 mod 119863)119864

⟩

(26)

The expression in curly braces is exactly the state thatBob expects to receive when there is no Eve (see (24))so eavesdropping also does not affect the message At thesame time the initial state of the ancilla is moved by 120583

positions within the set of detection states As a result Evecan unambiguously identify the value of cdit 120583 as long as thedetection states are mutually orthogonal

The C119883

(controlled X) gate seems to be the simplestinstance of the attack paradigm Let the detection and probesets of states be the elements of the computational basis(|120572(119898)119864

⟩ = |119898119864⟩ |119886(119898)119864

⟩ = |119898119864⟩) and the ancilla is composed of

the single qudit register The attack operation Q can be thenimplemented as

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ = C

119883

1003816100381610038161003816119896119905⟩1003816100381610038161003816119898119864⟩ =

1003816100381610038161003816119896119905⟩X119896

119864

1003816100381610038161003816119898119864⟩

=1003816100381610038161003816119896119905⟩

1003816100381610038161003816(119898 + 119896 mod 119863)119864⟩

(27)

In an obvious way requirements (21) regarding properties ofQ are then fulfilled

The existence of attacks able to undetectably eavesdropon half of the dense coded information has been alreadyforecasted in relation to qubit [2] qutrit [6] and qudit[23] based protocol However no explicit form of the attacktransformation has been given The presented result fills inthis gap and provides some general guidelines on how toconstruct coupling transformation with desired properties

34 Control Mode Able to Detect Bit-Flip Eavesdropping Theinsecurity of the considered protocol results from inabilityto detect coupling Q

ℎ119905with the control measurements in a

single basis Let us consider a qubit based protocol fromSection 21 with control mode enhanced to measurementsin two bases namely computational basis and its dual

basis that is eigenvectors of X gate In the new controlmode Alice randomly selects measurement basis performsmeasurement and asks Bob to make local measurement inthe same basisThe control state (9) in the absence of couplingtakes the form

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038161205940⟩

=1

radic2(1003816100381610038161003816+ℎ⟩

1003816100381610038161003816+119905⟩ minus1003816100381610038161003816minusℎ⟩

1003816100381610038161003816minus119905⟩)10038161003816100381610038161205940⟩

(28)

where |plusmn⟩ = (|0⟩ plusmn |1⟩)radic2 are eigenvectors of X It followsthat legitimate parties expect anticorrelation (correlation) ofoutcomes in the computational (dual) basis Under attackundetectable in the computational basis (14) the controlequation (15) takes the following form in the dual basis

1003816100381610038161003816120595ℎ119905119864⟩

=1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ minus1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ +

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816+119905⟩

minus1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ +1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ minus

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816minus119905⟩

(29)

Alice measurement causes the collapse to one of the statesin the curly braces It follows that Bob can obtain plusmn1

outcome with equal probability which in turn renders Evedetectability If control bases are selected with equal proba-bility then bit-flip attack is detected with 119901det = 14 Theabove qualitative discussion addresses bit-flip attack Themore advanced discussion on the properties of controlmodesbased on mutually unbiased bases and in relation to attacksof any form can be found in [23]

4 Conclusion

A generic scheme that provides undetectable eavesdroppingof bit-flip operations in the seminal version of the ping-pongprotocol is introduced It can be considered as a general-ization of the P-circuit [24] but in contrast it is deducedfrom the very basic properties of the coupling transformationMoreover the proposed scheme can be realized withoutreferring to the vacuum states so it is fully consistent withthe absence of losses assumption The CNOT gate and P-circuit are special cases of the introduced scheme so bothapproaches are equivalent It follows that any control modeable to detect CNOT coupling is also able to detect thepresence of the P-circuit The control mode based on localmeasurements in randomly selected unbiased bases is anexample of such procedure Consequently there is no needfor special addressing of P-circuit in the security analysesAlso the introduced scheme can be adapted to higherdimensional systems It can be considered as the constructiveproof of the existence of attacks forecasted in [2 6 23]

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 5

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

=1003816100381610038161003816Ψminus

⟩1003816100381610038161003816120594119864⟩

(17a)

Qminus1

(IℎotimesX119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ minus

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

=1003816100381610038161003816Φminus

⟩1003816100381610038161003816120601119864⟩

(17b)

Qminus1

(IℎotimesZ119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩1003816100381610038161003816120594119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038160119905⟩

1003816100381610038161003816120594119864⟩)

= minus1003816100381610038161003816Ψ+

⟩1003816100381610038161003816120594119864⟩

(17c)

Qminus1

(IℎotimesX119905Z119905otimesI119864)1003816100381610038161003816120595ℎ119905119864⟩

= Qminus1 minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816119889119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816119886119864⟩)

=minus1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038160119905⟩1003816100381610038161003816120601119864⟩ +

10038161003816100381610038161ℎ⟩10038161003816100381610038161119905⟩

1003816100381610038161003816120601119864⟩)

= minus1003816100381610038161003816Φ+

⟩1003816100381610038161003816120601119864⟩

(17d)

As a result the registers used for signalling are left untouchedand decoupled but Eversquos register is flipped from |120594

119864⟩ to |120601

119864⟩

whenAlice applies bit-flip operation In consequence Eve cansuccessfully decode half of the message content provided thatthe detection states |120594

119864⟩ and |120601

119864⟩ are perfectly distinguish-

able It follows that any unitary coupling transformation Qthat satisfies (14) and (16) can be used for bit-flip detection

32 Equivalence of P-Circuit and CNOT Circuit The prop-erties of the above generic scheme and the P-circuit [24]perfectly coincide As follows from (7a) (7b) (8a) and (8b)the states |120594

0⟩ = |V

119909⟩|0119910⟩ and |120594

1⟩ = |0

119909⟩|V119910⟩ play the role

of detection states |120594119864⟩ and |120601

119864⟩ respectively It is also clear

that transformation Q119905119909119910

has properties claimed in (14) and(16) Thus the P-circuit can be considered as an instance ofthe generic scheme described in Section 31

However the operator Q satisfying (14) and (16) can berealized in many ways It seems that CNOT operation actingon a single qubit of Eversquos ancilla Q = CNOT

119905119909 |120594119864⟩ = |0

119909⟩

|120601119864⟩ = |1

119909⟩ |119886119864⟩ = |0

119909⟩ and |119889

119864⟩ = |1

119909⟩ is the simplest

realization of the logic behind the attack Such version is alsopractically feasible as the attacks involving probes entangledvia the CNOT operation have been already proposed in theQKD context [26 27] As a result both the CNOT circuitand P-circuit are equivalent in terms of provided information

gain detectability and practical feasibility Consequentlythere is no need for the design of control modes that addressP-circuit in a special manner [25]

33 An Attack on Qudit Based Protocol The P-circuit has nostraightforward generalization to qudit based version of theprotocol In contrast the presented approach can be adaptedwith ease Let Bob start communication process with creationof EPR pair

10038161003816100381610038161003816120573(00)

ℎ119905⟩ =

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩ (18)

where119863 is the qudit dimension The travel qudit is then sentto Alice for encoding or control measurement In controlmode the home and travel qubits are measured in thecomputational basis so the projection P

ℎ119905used in control

equation (3) takes the form

Pℎ119905= Iℎ119905minus

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩ ⟨119896ℎ1003816100381610038161003816 otimes

1003816100381610038161003816119896119905⟩ ⟨1198961199051003816100381610038161003816 (19)

Let by an analogy to the qubit case |120572(119896)119864⟩ and |119886(119896)

119864⟩ be the

sets of 119863 orthonormal states of the ancilla system Thesestates will be further referred to as detection and probe statesrespectively The map used by Eve must be of the form

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(0)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119896)

119864⟩ 119896 = 0 119863 minus 1 (20)

to not introduce errors in control measurements Let usadditionally postulate that Q satisfies

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816119886(119898+119896 mod 119863)119864

⟩ (21)

that is Q advances index 119896 positions in a set of Eversquos probestates Similarly Qminus1 decrements the index 119896 positions

Qminus1 1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119898)

119864⟩ 997888rarr

1003816100381610038161003816119896119905⟩10038161003816100381610038161003816120572(119898minus119896 mod 119863)119864

⟩ (22)

Let us recall that for qudits Alice uses

Z =

119863minus1

sum119896=0

120596119896

|119896⟩ ⟨119896|

X =

119863minus1

sum119896=0

|119896 + 1 mod 119863⟩ ⟨119896|

120596 = 1198901198952120587119863

(23)

to encode classic 120583 ] ldquocditsrdquo in the following way

100381610038161003816100381610038161003816120573(120583])ℎ119905

⟩ = X120583

119905Z

]119905

10038161003816100381610038161003816120573(00)

ℎ119905⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

(24)

6 Advances in Mathematical Physics

Under attack Alice applies encoding (24) to the state coupledaccording to rule (20)

1003816100381610038161003816120595enc⟩ = X120583

119905Z

]119905Q10038161003816100381610038161003816120573(00)

ℎ119905⟩10038161003816100381610038161003816120572(0)

119864⟩

= X120583

119905Z

]119905

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119896)

119864⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩

(25)

The travel qubit is affected by Qminus1 in its way back to Bob

1003816100381610038161003816120601dec⟩ = Qminus1 1003816100381610038161003816120595enc⟩ =

1

radic119863

sdot

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩ (Q

minus1 1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩)

= 1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

sdot100381610038161003816100381610038161003816120572(minus120583 mod 119863)119864

⟩

(26)

The expression in curly braces is exactly the state thatBob expects to receive when there is no Eve (see (24))so eavesdropping also does not affect the message At thesame time the initial state of the ancilla is moved by 120583

positions within the set of detection states As a result Evecan unambiguously identify the value of cdit 120583 as long as thedetection states are mutually orthogonal

The C119883

(controlled X) gate seems to be the simplestinstance of the attack paradigm Let the detection and probesets of states be the elements of the computational basis(|120572(119898)119864

⟩ = |119898119864⟩ |119886(119898)119864

⟩ = |119898119864⟩) and the ancilla is composed of

the single qudit register The attack operation Q can be thenimplemented as

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ = C

119883

1003816100381610038161003816119896119905⟩1003816100381610038161003816119898119864⟩ =

1003816100381610038161003816119896119905⟩X119896

119864

1003816100381610038161003816119898119864⟩

=1003816100381610038161003816119896119905⟩

1003816100381610038161003816(119898 + 119896 mod 119863)119864⟩

(27)

In an obvious way requirements (21) regarding properties ofQ are then fulfilled

The existence of attacks able to undetectably eavesdropon half of the dense coded information has been alreadyforecasted in relation to qubit [2] qutrit [6] and qudit[23] based protocol However no explicit form of the attacktransformation has been given The presented result fills inthis gap and provides some general guidelines on how toconstruct coupling transformation with desired properties

34 Control Mode Able to Detect Bit-Flip Eavesdropping Theinsecurity of the considered protocol results from inabilityto detect coupling Q

ℎ119905with the control measurements in a

single basis Let us consider a qubit based protocol fromSection 21 with control mode enhanced to measurementsin two bases namely computational basis and its dual

basis that is eigenvectors of X gate In the new controlmode Alice randomly selects measurement basis performsmeasurement and asks Bob to make local measurement inthe same basisThe control state (9) in the absence of couplingtakes the form

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038161205940⟩

=1

radic2(1003816100381610038161003816+ℎ⟩

1003816100381610038161003816+119905⟩ minus1003816100381610038161003816minusℎ⟩

1003816100381610038161003816minus119905⟩)10038161003816100381610038161205940⟩

(28)

where |plusmn⟩ = (|0⟩ plusmn |1⟩)radic2 are eigenvectors of X It followsthat legitimate parties expect anticorrelation (correlation) ofoutcomes in the computational (dual) basis Under attackundetectable in the computational basis (14) the controlequation (15) takes the following form in the dual basis

1003816100381610038161003816120595ℎ119905119864⟩

=1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ minus1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ +

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816+119905⟩

minus1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ +1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ minus

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816minus119905⟩

(29)

Alice measurement causes the collapse to one of the statesin the curly braces It follows that Bob can obtain plusmn1

outcome with equal probability which in turn renders Evedetectability If control bases are selected with equal proba-bility then bit-flip attack is detected with 119901det = 14 Theabove qualitative discussion addresses bit-flip attack Themore advanced discussion on the properties of controlmodesbased on mutually unbiased bases and in relation to attacksof any form can be found in [23]

4 Conclusion

A generic scheme that provides undetectable eavesdroppingof bit-flip operations in the seminal version of the ping-pongprotocol is introduced It can be considered as a general-ization of the P-circuit [24] but in contrast it is deducedfrom the very basic properties of the coupling transformationMoreover the proposed scheme can be realized withoutreferring to the vacuum states so it is fully consistent withthe absence of losses assumption The CNOT gate and P-circuit are special cases of the introduced scheme so bothapproaches are equivalent It follows that any control modeable to detect CNOT coupling is also able to detect thepresence of the P-circuit The control mode based on localmeasurements in randomly selected unbiased bases is anexample of such procedure Consequently there is no needfor special addressing of P-circuit in the security analysesAlso the introduced scheme can be adapted to higherdimensional systems It can be considered as the constructiveproof of the existence of attacks forecasted in [2 6 23]

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Mathematical Physics

Under attack Alice applies encoding (24) to the state coupledaccording to rule (20)

1003816100381610038161003816120595enc⟩ = X120583

119905Z

]119905Q10038161003816100381610038161003816120573(00)

ℎ119905⟩10038161003816100381610038161003816120572(0)

119864⟩

= X120583

119905Z

]119905

1

radic119863

119863minus1

sum119896=0

1003816100381610038161003816119896ℎ⟩1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816119886(119896)

119864⟩

=1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩

(25)

The travel qubit is affected by Qminus1 in its way back to Bob

1003816100381610038161003816120601dec⟩ = Qminus1 1003816100381610038161003816120595enc⟩ =

1

radic119863

sdot

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩ (Q

minus1 1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩10038161003816100381610038161003816119886(119896)

119864⟩)

= 1

radic119863

119863minus1

sum119896=0

120596119896] 1003816100381610038161003816119896ℎ⟩

1003816100381610038161003816(119896 + 120583 mod 119863)119905⟩

sdot100381610038161003816100381610038161003816120572(minus120583 mod 119863)119864

⟩

(26)

The expression in curly braces is exactly the state thatBob expects to receive when there is no Eve (see (24))so eavesdropping also does not affect the message At thesame time the initial state of the ancilla is moved by 120583

positions within the set of detection states As a result Evecan unambiguously identify the value of cdit 120583 as long as thedetection states are mutually orthogonal

The C119883

(controlled X) gate seems to be the simplestinstance of the attack paradigm Let the detection and probesets of states be the elements of the computational basis(|120572(119898)119864

⟩ = |119898119864⟩ |119886(119898)119864

⟩ = |119898119864⟩) and the ancilla is composed of

the single qudit register The attack operation Q can be thenimplemented as

Q1003816100381610038161003816119896119905⟩

10038161003816100381610038161003816120572(119898)

119864⟩ = C

119883

1003816100381610038161003816119896119905⟩1003816100381610038161003816119898119864⟩ =

1003816100381610038161003816119896119905⟩X119896

119864

1003816100381610038161003816119898119864⟩

=1003816100381610038161003816119896119905⟩

1003816100381610038161003816(119898 + 119896 mod 119863)119864⟩

(27)

In an obvious way requirements (21) regarding properties ofQ are then fulfilled

The existence of attacks able to undetectably eavesdropon half of the dense coded information has been alreadyforecasted in relation to qubit [2] qutrit [6] and qudit[23] based protocol However no explicit form of the attacktransformation has been given The presented result fills inthis gap and provides some general guidelines on how toconstruct coupling transformation with desired properties

34 Control Mode Able to Detect Bit-Flip Eavesdropping Theinsecurity of the considered protocol results from inabilityto detect coupling Q

ℎ119905with the control measurements in a

single basis Let us consider a qubit based protocol fromSection 21 with control mode enhanced to measurementsin two bases namely computational basis and its dual

basis that is eigenvectors of X gate In the new controlmode Alice randomly selects measurement basis performsmeasurement and asks Bob to make local measurement inthe same basisThe control state (9) in the absence of couplingtakes the form

1003816100381610038161003816120595ℎ119905119864⟩ =1

radic2(10038161003816100381610038160ℎ⟩

10038161003816100381610038161119905⟩ +10038161003816100381610038161ℎ⟩

10038161003816100381610038160119905⟩)10038161003816100381610038161205940⟩

=1

radic2(1003816100381610038161003816+ℎ⟩

1003816100381610038161003816+119905⟩ minus1003816100381610038161003816minusℎ⟩

1003816100381610038161003816minus119905⟩)10038161003816100381610038161205940⟩

(28)

where |plusmn⟩ = (|0⟩ plusmn |1⟩)radic2 are eigenvectors of X It followsthat legitimate parties expect anticorrelation (correlation) ofoutcomes in the computational (dual) basis Under attackundetectable in the computational basis (14) the controlequation (15) takes the following form in the dual basis

1003816100381610038161003816120595ℎ119905119864⟩

=1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ minus1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ +

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816+119905⟩

minus1

2radic21003816100381610038161003816+ℎ⟩ (

1003816100381610038161003816119889119864⟩ +1003816100381610038161003816119886119864⟩) +

1003816100381610038161003816minusℎ⟩ (1003816100381610038161003816119889119864⟩ minus

1003816100381610038161003816119886119864⟩)

sdot1003816100381610038161003816minus119905⟩

(29)

Alice measurement causes the collapse to one of the statesin the curly braces It follows that Bob can obtain plusmn1

outcome with equal probability which in turn renders Evedetectability If control bases are selected with equal proba-bility then bit-flip attack is detected with 119901det = 14 Theabove qualitative discussion addresses bit-flip attack Themore advanced discussion on the properties of controlmodesbased on mutually unbiased bases and in relation to attacksof any form can be found in [23]

4 Conclusion

A generic scheme that provides undetectable eavesdroppingof bit-flip operations in the seminal version of the ping-pongprotocol is introduced It can be considered as a general-ization of the P-circuit [24] but in contrast it is deducedfrom the very basic properties of the coupling transformationMoreover the proposed scheme can be realized withoutreferring to the vacuum states so it is fully consistent withthe absence of losses assumption The CNOT gate and P-circuit are special cases of the introduced scheme so bothapproaches are equivalent It follows that any control modeable to detect CNOT coupling is also able to detect thepresence of the P-circuit The control mode based on localmeasurements in randomly selected unbiased bases is anexample of such procedure Consequently there is no needfor special addressing of P-circuit in the security analysesAlso the introduced scheme can be adapted to higherdimensional systems It can be considered as the constructiveproof of the existence of attacks forecasted in [2 6 23]

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Mathematical Physics 7

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

Piotr Zawadzki acknowledges the support from the statutorysources and Jarosław Adam Miszczak was supported bythe Polish National Science Center (NCN) under Grant201103DST600413

References

[1] G L Long and X S Liu ldquoTheoretically efficient high-capacityquantum-key-distribution schemerdquo Physical Review A vol 65no 3 Article ID 032302 2002

[2] F-GDengG L Long andX-S Liu ldquoTwo-step quantumdirectcommunication protocol using the Einstein-Podolsky-Rosenpair blockrdquo Physical Review A vol 68 no 4 Article ID 0423176 pages 2003

[3] K Bostrom and T Felbinger ldquoDeterministic secure directcommunication using entanglementrdquo Physical Review Lettersvol 89 no 18 pp 1879021ndash1879024 2002

[4] C Wang F G Deng and G L Long ldquoMulti-step quan-tum secure direct communication using multi-particle Green-HorneZeilinger staterdquo Optics Communications vol 253 no 13pp 15ndash20 2005

[5] C Wang F G Deng and G L Long ldquoErratum to lsquoMulti-stepquantum secure direct communication using multi-particleGreen-Horne-Zeilinger statersquo [Opt Commun 253 (2005) 15ndash20]rdquo Optics Communications vol 262 no 1 p 134 2006

[6] E V Vasiliu ldquoNon-coherent attack on the ping-pong protocolwith completely entangled pairs of qutritsrdquo Quantum Informa-tion Processing vol 10 no 2 pp 189ndash202 2011

[7] P Zawadzki ldquoSecurity of ping-pong protocol based on pairs ofcompletely entangled quditsrdquoQuantum Information Processingvol 11 no 6 pp 1419ndash1430 2012

[8] Q-Y Cai and B-W Li ldquoImproving the capacity of the Bostrom-Felbinger protocolrdquo Physical Review AmdashAtomic Molecular andOptical Physics vol 69 no 5 Article ID 054301 2004

[9] C Wang F-G Deng Y-S Li X-S Liu and G L LongldquoQuantum secure direct communication with high-dimensionquantum superdense codingrdquo Physical Review A vol 71 no 4Article ID 044305 2005

[10] F-G Deng and G L Long ldquoSecure direct communication witha quantum one-time padrdquo Physical Review A vol 69 no 5Article ID 052319 2004

[11] J Hu B Yu M Jing et al Experimental quantum secure directcommunication with single photons LSA 2016

[12] M Lucamarini and S Mancini ldquoSecure deterministic commu-nicationwithout entanglementrdquo Physical Review Letters vol 94no 14 Article ID 140501 2005

[13] G-L Long F-G Deng C Wang X-H Li K Wen andW-Y Wang ldquoQuantum secure direct communication anddeterministic secure quantum communicationrdquo Frontiers ofPhysics in China vol 2 no 3 pp 251ndash272 2007

[14] K Bostrom and T Felbinger ldquoOn the security of the ping-pongprotocolrdquoPhysics Letters A vol 372 no 22 pp 3953ndash3956 2008

[15] P Zawadzki ldquoImproving security of the ping-pong protocolrdquoQuantum Information Processing vol 12 no 1 pp 149ndash155 2013

[16] H Lu C-H F Fung X Ma and Q-Y Cai ldquoUnconditionalsecurity proof of a deterministic quantum key distributionwith a two-way quantum channelrdquo Physical Review AmdashAtomicMolecular and Optical Physics vol 84 no 4 Article ID 0423442011

[17] N J BeaudryM Lucamarini SMancini and R Renner ldquoSecu-rity of two-way quantum key distributionrdquo Physical Review Avol 88 no 6 Article ID 062302 2013

[18] Y-G Han Z-Q Yin H-W Li et al ldquoSecurity of modified Ping-Pong protocol in noisy and lossy channelrdquo Scientific Reports vol4 article 4936 2014

[19] A Cere M Lucamarini G Di Giuseppe and P TombesildquoExperimental test of two-way quantum key distribution in thepresence of controlled noiserdquo Physical Review Letters vol 96no 20 Article ID 200501 2006

[20] H Chen Z-Y Zhou A J J Zangana et al ldquoExperimentaldemonstration on the deterministic quantum key distributionbased on entangled photonsrdquo Scientific Reports vol 6 ArticleID 20962 2016

[21] A Wojcik ldquoEavesdropping on the lsquoping-pongrsquo quantum com-munication protocolrdquo Physical Review Letters vol 90 no 15Article ID 157901 2003

[22] F-G Deng X-H Li C-Y Li P Zhou and H-Y ZhouldquoEavesdropping on the lsquoping-pongrsquo quantum communicationprotocol freely in a noise channelrdquo Chinese Physics vol 16 no2 pp 277ndash281 2007

[23] P Zawadzki Z Pucha la and J A Miszczak ldquoIncreasing thesecurity of the ping-pong protocol by using many mutuallyunbiased basesrdquo Quantum Information Processing vol 12 no1 pp 569ndash576 2013

[24] M Pavicic ldquoIn quantumdirect communication an undetectableeavesdropper can always tellΨ fromΦBell states in themessagemoderdquo Physical Review A vol 87 no 4 Article ID 042326 2013

[25] B Zhang W-X Shi J Wang and C-J Tang ldquoQuantumdirect communication protocol strengthening against Pavicicrsquosattackrdquo International Journal of Quantum Information vol 13no 7 Article ID 1550052 2015

[26] H E Brandt ldquoEntangled eavesdropping in quantum key distri-butionrdquo Journal of Modern Optics vol 53 no 16-17 pp 2251ndash2257 2006

[27] J H Shapiro ldquoPerformance analysis for Brandtrsquos conclusiveentangling proberdquo Quantum Information Processing vol 5 no1 pp 11ndash24 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of