simultaneous intraportation of many quantum states within the quantum computing network

16 April 2001

Physics Letters A 282 (2001) 138–144www.elsevier.nl/locate/pla

Simultaneous intraportation of many quantum states within thequantum computing network

Mang Fenga,b

a Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Street 38, D-01187 Dresden, Germanyb Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,

Academia Sinica, Wuhan 430071, PR China

Received 12 December 2000; received in revised form 9 February 2001; accepted 6 March 2001Communicated by A.R. Bishop

Abstract

A scheme is proposed for simultaneous intraportation of many unknown quantum states within a quantum computingnetwork. It is shown that our scheme, much different from the teleportation in the strict sense, can be very similar to the originalteleportation proposal (Phys. Rev. Lett. 70 (1993) 1895) and the efficiency of the scheme for quantum state transmission is veryhigh. The possible applications of our scheme are also discussed. 2001 Elsevier Science B.V. All rights reserved.

PACS: 03.65.Bz; 42.50.Dv; 89.70.+c

Quantum computing [1] is an interesting and hottopic in the quantum theory, which can treat effi-ciently some nondeterministic polynomial-time prob-lems inaccessible for the existing computer, such asfactorization of large numbers [2], or solve someproblems more rapidly, e.g., searching a certain itemfrom a large disordered system [3], etc. It has beenproven that any operation in the quantum computingcan be decomposed into a series of two basic opera-tions [4]. One is controlled-NOT (CN) gate, defined as|ε1〉|ε2〉 → |ε1〉|ε1 ⊕ ε2〉 with ε1,2 = 0,1, and the otheris Hadamard gate

1√2

(1 11 −1

),

which transforms|0〉 and |1〉 to (1/√

2)(|0〉 + |1〉)and (1/

√2)(|0〉 − |1〉), respectively. As Hadamard

E-mail address: [email protected] (M. Feng).

gate is a rotation on a single qubit, which is easilyrealized, the physical realization of the CN operationis the key to the achievement of an actual quantumcomputing. The experimental demonstration of the CNoperation on a trapped ultracold Be+ showed that thequantum computing can be actually carried out aftersome technical difficulties, such as the decoherenceand ultracold cooling for large quantities of trappedions, have been overcome [5,6].

The teleportation of an unknown quantum state overarbitrary distance is another interesting and hot topicin the quantum theory. It is also a striking demonstra-tion of the nonlocal character of quantum states [7],which starts at a joint measurement on a particle (la-belled as particlea) in an unknown state and a par-ticle b being one-half of a maximally entangled pairof particles (b and c), and ends at a suitable unitaryrotation of particlec to restore the state of particleawith the help of two bits of classical message. It hasbeen proven [8,9] that the teleportation is a special re-

0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(01)00170-0

M. Feng / Physics Letters A 282 (2001) 138–144 139

versible quantum operation, which hides the quantuminformation within the correlation between the sys-tem and the environment. Recently, the scheme hasbeen extended to the teleportation of continued vari-ables [10], and teleportation with GHZ state [11]. Mil-burn et al. [12] demonstrated theoretically the telepor-tation via a two-mode squeezed vacuum state in lightof the proposal in Ref. [13], and the teleportation wasalso proposed to construct a variety of quantum gates,associated with other operations [14]. As far as weknow, the teleportation has been demonstrated exper-imentally by using parametric down-conversion in in-terferometric Bell state analyzers and ink-vector en-tanglement [15,16], NMR method [17] and continuousvariables of electromagnetic field [18].

Since the essence of the teleportation is based onthe quantum entanglement which is also the basis ofthe quantum parallelism in the quantum computing, ascheme [19] for achieving teleportation via quantumcomputing was proposed recently. In that scheme,three quantum states are put into a circuit consistingof Hadamard operations and CN ones, in which one(input from channela) is the unknown quantum stateneeded to be teleported and the other two are auxiliaryones (input from channelsb and c, respectively) tobe entangled. After the entanglement of the states inchannelsa and b (one-half of the entanglement ofb and c), one carries out a series of CN and Hada-mard operations on the state in channelc with thehelp of the quantum information from channelsa

and b. Finally, the quantum state put in the channela reappears at the output of the channelc. Obviously,this scheme is somewhat different from the originalscheme proposed by Bennett et al. [7]. First, there isno need of joint measurement with Bell basis states.Although the authors made some discussions for themeasurement on the channelsa and b at the borderline of the sender Alice and the receiver Bob, itis easily found that the final result is irrelative tothese measurements. Secondly, Bob needs not onlya unitary rotation to the quantum state in channelc,but also CN operations to the quantum state with thehelp of the information from channelsa andb beingcontrol bits. Particularly, the CN operation betweenthe channelsa–c and b–c implies thata and b arenot arbitrarily far away fromc. So this scheme isnot actually referred to the teleportation, but, in somesense, theintraportation which transmits a state from

a channel at the sender to another at the receiver withina network system.

Nevertheless, this scheme is an interesting exten-sion for the original teleportation proposal, which en-lightens us to make a more efficient transmission of theunknown states via quantum computing network. Inthis Letter, we will extend this idea to try to transmit si-multaneously many unknown quantum states with thecircuit similar to Ref. [19]. It can be found that thepure state, instead of the entangled state required inRef. [7], can be used as the auxiliary state for the statetransmission, and fewer classical messages are neededin our scheme than in Ref. [7]. However, the entangle-ment is still essential to our scheme. A comparison ofour scheme with Ref. [7] will be made to show the ad-vantages of our scheme. The possible applications ofour scheme will be discussed in the secure transmis-sion of quantum states as well as the preparation ofnonclassical states.

Suppose first that two quantum statesa|1〉 + b|0〉and e|1〉 + f |0〉 put, respectively, in channels 1 and3 will be intraported, and the other statec|1〉 + d|0〉in the channel 2 is auxiliary. (In this Letter, we setchannels 1, 2 and 3 corresponding to the the channelsfrom the top to the bottom.) Alice need not makeany measurement at the position of the vertical dottedline denoting the border line of Alice and Bob, butinforms Bob by the broadcast or telephone with onebits of message about the values ofc andd . If c = d =1/

√2, Bob will carry out a series of corresponding

operations, as shown in Fig. 1 where and in thefollowing figures the channels for transmitting theclassical message are omitted. Then he will obtaina|1〉 + b|0〉 and e|1〉 + f |0〉 at the outputs of thechannels 2 and 1, respectively. Ifc = 0 andd = 1,the operation by Bob is shown in Fig. 2, where heobtainse|1〉 + f |0〉 anda|1〉 + b|0〉 at the outputs ofthe channels 1 and 3, respectively. Similarly, ifc = 1andd = 0, the operation by Bob is demonstrated inFig. 3. The later two procedures in fact result in theswapping of quantum states, or identity interchange ofquantum states [20].

The scheme can be generalized to the intraportationof the unknown quantum states input, respectively,from channels 1 and 2, or channels 2 and 3, wherethe values ofe andf , or values ofa andb should beinformed to Bob. The circuit shown in Fig. 4 is an ex-ample in these respects. From Figs. 1–3 we know that

140 M. Feng / Physics Letters A 282 (2001) 138–144

Fig. 1. Intraportation of two quantum states, where the auxiliary state is|0〉 + |1〉 and input from the middle channel.

Fig. 2. The same as Fig. 1, but the auxiliary state is|0〉.


Fig. 4. The same as Fig. 1, but the auxiliary state|0〉 is input from the lowest channel.

there are three kinds of possible outputs for a certainkind of input under the present operations. For exam-ple, for the input case in Fig. 1, the three kinds of pos-sible outputs are(e|1〉 + f |0〉)1(a|1〉 + b|0〉)2(|1〉 +|0〉)3, (|1〉 + |0〉)1(e|1〉 + f |0〉)2(a|1〉 + b|0〉)3, and(e|1〉+f |0〉)1(|1〉+ |0〉)2(a|1〉+b|0〉)3. So if the aux-iliary state input by Alice is restricted to three cases,that is,c = d = 1/

√2, c = 0 andd = 1, andc = 1 and

d = 0, then there are totally nine different cases foreach intraportation, which in fact constitute the pro-tocol between Bob and Alice for different operationsperformed by Bob corresponding to different inputsset by Alice. Obviously, in the protocol, it is important

to make clear from which channel the auxiliary stateis input, since different situation for input channels ofauxiliary states correspond to different operations Bobshould perform.

As we know, three CN operation sequences willresult in the swapping of two quantum states, i.e.,CN12CN21CN12 |Ψ 〉1|Φ〉2 = |Φ〉1|Ψ 〉2. So with dif-ferent group of these operations, three input quan-tum states can also reappear at the outputs of differ-ent channels via quantum computing, as demonstratedin Fig. 5, whereΨ,Φ, θ are arbitrary quantum states.But this swapping operation is different from our in-traportation scheme. The major difference is that, in


Fig. 5. Quantum swapping of three quantum states.

our scheme of intraportation, the quantum informa-tion is divided into two parts: one is related to the en-tangled state, and the other is the purely classical in-formation. Moreover, the three input quantum statesare fully entangled before they are sent to the receiverin our scheme, instead of the swapping operation de-pending on the interaction between arbitrary two statesof the three input states. However, by means of theswapping operations, we can simplify above protocolfor intraportation, that is, Bob only needs to considerone kind of output case for each input case. For exam-ple, in Fig. 1, Bob chooses the operations to produce(e|1〉+f |0〉)1(a|1〉+ b|0〉)2(|1〉+ |0〉)3, and the othertwo kinds of outputs can be obtained via a group ofthe swapping operations. Therefore, the protocol forthe present intraportation scheme can be reduced to in-cluding only three different cases, and the other differ-ent output cases are resorted to the post-intraportationtreatment.

The teleportation of the entangled state is alsoan interesting topic for the quantum communication.With the original teleportation proposal [7], it iseasily proven that an unknown entangled state can beteleported via a known entangled state as well as twobits of classical message. Our scheme simultaneouslytransmitting two pure quantum states, in some sense,also means that it can intraport entangled states. Asshown in Fig. 6, two quantum statesc|1〉 + d|0〉 ande|1〉 + f |0〉 are entangled, after the Hadamard gateand CN gate, to beΨ = (1/

√2)(c + d)(e|01〉 +

f |00〉) + (1/√

2)(d − c)(e|10〉 + f |11〉). With thehelp of the auxiliary state|0〉 and the correspondingoperations by Bob,Ψ reappears at the outputs ofchannels 1 and 2. In this procedure, the intraportationonly needs a pure state to be auxiliary, and one bitsof classical message about the auxiliary state fromAlice to Bob, which is more efficient than that with theoriginal teleportation proposal. Obviously, before theintraportation, the protocol should be made betweenAlice and Bob for Bob’s operation corresponding tothe different auxiliary state input by Alice.

The present scheme can be readily extended tointraporting simultaneously many unknown quantumstates and the intraportation of many-particle entan-gled states, with the increase of the channels trans-mitting the quantum states, and a little bit modifi-cation of the circuit. For example, for the case offour channels, we can intraport simultaneously at mostthree unknown quantum states via one bit of classi-cal message after the protocol has been made by Al-ice and Bob, as shown in Figs. 7, 8 and 9, where theauxiliary states were input from the lowest channel.Comparing these figures with Figs. 1–4, and by di-rect deductions for other many-channel situations, wefind that, for the case ofN channels, no matter fromwhich channel the auxiliary state is input, the opera-tions performed by Alice can be HN−1CNN−1NHN−2CNN−2N−1HN−3 . . .CN23CN12H1, and the first fiveoperations by Bob are CNN−1NHNCN1NH1HN . Theother operations by Bob should be performed in termsof the specific input situation. These characteristicsare useful for the production of the protocol for in-traportation. According to above discussion, we knowthat, even for the many-channel situation, there are stillthree different cases in the protocol if we use the swap-ping operations for the post-intraportation treatment.

As the quantum states are fully entangled beforethey are reconstructed at the output locations, one pos-sible application of our scheme is the secure transmis-sion of theeveryday quantum messages within a quan-tum computing network. Suppose an eavesdropperwants to get messages from the channels between thesender Alice and the distant receiver Bob. The eaves-dropper will not succeed even if he eavesdrops thequantum information simultaneously from all chan-nels, and meanwhile receives the classical messagesend by Alice, as long as he do not know the proto-col made by Alice and Bob. As the eavesdropper isnot clear from which channel the auxiliary state is in-put, the probability for his successful eavesdroppingdecreases as the increase of the number of the chan-nels, and the action of the eavesdropper can be de-tected by Bob from the comparison of the output re-sults with the expected results given by the protocol.Although it also needs the judgement for whether thereexists the eavesdropping, however, different from thestandard cryptographic scheme [21], our scheme doesnot involve the quantum key distribution process be-cause we have supposed that Alice and Bob are not


Fig. 6. Intraportation of an entangled state.

Fig. 7. Intraportation of three quantum states, where the auxiliary state is|0〉 + |1〉 and input from the lowest channel.



much far away from each other. The protocol in ourscheme can be made by Alice and Bob meeting ata common place. One may ask: why does not Alicesend her quantum message to Bob directly when theymeet each other? The key point in our scheme is that,once the protocol has been made, Alice can send herquantum messages to Bob efficiently and safely at anytime when required. They may meet each other oncea month or longer to change their protocol for keep-

ing secure transmission of the quantum information inthe next several days. Strictly speaking, our scheme isless practical for two much distant users than the stan-dard quantum cryptography, whereas it might be moreconvenient and practical for the everyday transmissionof the quantum information within a limited quantumcomputing network in future. Moreover, a by-productof this application is the preparation of different quan-tum states at the output location. By suitably choos-


ing the operations as well as the coefficients of the in-put states, some useful quantum states, such as Bellstates [7], Greenberger–Horne–Zeilinger states [22]and so on, can be obtained from the output states. Forexample, in Fig. 1, if Bob does not perform the lastCN23, but make a measurement at the channel 3, thenhe will obtain ea|11〉 + f b|00〉 or eb|10〉 + f a|01〉from channels 1 and 2 by projecting the output state ofthe channel 3 on|0〉 or |1〉.

A shortcoming of our scheme is the increase of thequantum gates with the increase of the number of thechannels. In practical application, we may restrict thenumber of the channels in terms of the quantity ofthe quantum message to be transmitted. We can alsotry to integrate some quantum gates, according to theregularity of operations referred to above, to reducethe number of the operation.

In summary, we pointed out that a former telepor-tation scheme via quantum computing network is ac-tually an intraportation within a quantum computingnetwork, and along this idea, we studied how to in-traport simultaneously many unknown quantum states.As the classical message from Alice to Bob is neces-sary in the scheme, our scheme is more in tune with theoriginal teleportation proposal than Ref. [19]. More-over, our scheme is also more efficient than the trans-mission of unknown quantum states with the originalteleportation proposal. Different from Ref. [7], how-ever, the present scheme do not obviously depend onthe entanglement characteristic of the auxiliary stateitself since the pure state can also act as the auxiliarystate, whereas the entanglement is still the heart of ourscheme. So from this viewpoint, the present schemedoes not follow closely the original teleportation pro-posal. On the other hand, for more coincidence withthe original teleportation proposal, we can use twoauxiliary states for the many-channel cases. For ex-ample, in Fig. 7, we can sete = 0 andf = 1, thus theHadamard gate and the following CN operation makethe two auxiliary states entangled. Then one part ofthe entanglement is correlated with the two unknownquantum states, and the other one is sent directly toBob. As a result, Bob can reproduce in his side thetwo unknown quantum states by two bits of classicalmessages sent from Alice and some corresponding op-erations performed according to the protocol he madewith Alice. Obviously, with this change, our scheme iswell consistent with the proposal in Ref. [7], whereas

our scheme is still more efficient as many unknownquantum states (e.g., two unknown quantum states inFig. 7) are transmitted simultaneously by means oftwo bits of classical message. From above analysis, weknow that, the realization of the present scheme at leastrequires two prerequisites. One is the quantum chan-nels between Alice and Bob, whereas with what mater-ial to construct these quantum channels is still an openquestion. The other is the experimental achievementof the quantum computing with many-qubit. As theseven-qubit quantum computing has been carried outvia NMR [23], and many-qubit quantum computingvia trapped ions will be achieved in the near future [24,25], we believe that our scheme will be helpful for theexploration of efficient transmission of quantum infor-mation in future actual quantum computing network.

Note added

After finishing this work, the author was told that awork [26] extending Ref. [19] along another directionhad been carried out in the field of condensed matterphysics.

Acknowledgements

The valuable discussion with Prof. Xiwen Zhu andKelin Gao is highly acknowledged. The author alsothanks Dr. Yurong Jiang for critically checking thededuction in this work.

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simultaneous intraportation of many quantum states within the quantum computing network

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