Chinese Science Bulletin

© 2007 Science in China Press

Springer-Verlag

www.scichina.com www.springerlink.com Chinese Science Bulletin | May 2007 | vol. 52 | no. 9 | 1162-1165

Complete multiple round quantum dense coding with quantum logical network

LI ChunYan1,2, LI XiHan1,2, DENG FuGuo1,2,3†, ZHOU Ping1,2 & ZHOU HongYu1,2,3 1 Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875,

China; 2 Institute of Low Energy Nuclear Physics, and Department of Material Science and Engineering, Beijing Normal University, Beijing

100875, China; 3 Beijing Radiation Center, Beijing 100875, China

We present a complete multiple round quantum dense coding scheme for improving the source ca-pacity of that introduced recently by Zhang et al. The receiver resorts to two qubits for storing the four local unitary operations in each round.

quantum communication, quantum dense coding, quantum logical network, complete

Quantum entanglement provides some novel ways to transmit and process quantum information, such as quantum computation and quantum communication[1―11]. It presents the non-locality of a quantum system. Two subsystems consisting of the quantum system cannot be considered to be independent even if they are far apart. Any measurement on the two subsystems separately cannot give all the information about the quantum sys-tem. Quantum entanglement has been widely used in quantum computation and quantum communication, such as quantum dense coding[12―14], quantum key dis- tribution[15 ― 17], quantum secure direct communica- tion[18―22], and quantum secret sharing[23―31].

Quantum dense coding (QDC) was proposed by Bennett and Wiesner in 1992[12]. Two qubits of informa-tion are transmitted by transporting only one particle. This scheme has the largest capacity. One of the two parties of communication, say Alice, prepares two parti-cles A and B in an Einstein-Podolsky-Rosen (EPR) state described by one of the four Bell states,

AB A B A B1| (| 0 | 0 |1 |1 ),2

φ± ⟩ = ⟩ ⟩ ± ⟩ ⟩ (1)

AB A B A B1| (| 0 |1 |1 | 0 ),2

ψ ± ⟩ = ⟩ ⟩ ± ⟩ ⟩ (2)

where |0⟩ and |1⟩ are the two eigenstates of the Pauli operator σz. She sends the particle A to the other party, say Bob, and keeps the particle B. Bob chooses one of the four local unitary operations Ui (i = 0, 1, 2, 3) to en-code his information on the particle A and then send it back to Alice, where U0 ≡ I, U1 ≡σz, U2 ≡σx, U3 ≡ iσy. Now the two particles are changed into a new one of the four Bell states. When Alice makes a joint Bell-state measurement on them, she will know which operation Bob has done. In this way, Bob can transmit more than one bit of information by manipulating only one particle in a maximally entangled state. Compared with classical communication, the QDC doubles the source capacity as one particle in an entangled state can carry two bits of classical information from one party to the other. Using d-dimensional quantum systems, the capacity of infor-mation of the QDC can be further enhanced[13,14,32]. By far, the QDC has been implemented in various experi-ments[33―39]. Received October 25, 2006; accepted November 11, 2006 doi: 10.1007/s11434-007-0148-6 †Corresponding author (email: fgdeng@bnu.edu.cn) Supported by the National Natural Science Foundation of China (Grant Nos 10604008, 10435020, 10254002, A0325401 and 10374010) and Beijing Education Committee (Grant No. XK100270454)

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Recently, an alternative QDC scheme was proposed by Zhang et al.[40]. It is a multiple-round quantum dense coding scheme (let us name it ZXWDLL QDC scheme). With m + 1 particles in a product state, Bob can transmit 2m+1 messages to Alice by manipulating only one qubit for m times. These operations are represented as

( )b z kO R ϕ , where bO I= , xσ , and 2( ) k ziz kR e ϕ σϕ =

with 2 2mk kϕ π= − (k = 0, 1, 2, …, 2m−1). Obviously

Ob denotes the bit operations, and ( )z kR ϕ denotes the phase operations. Alice does the inverse of quantum Fourier transform (QFT) on the m particles and meas-ures all the m + 1 quantum systems. Then she can get the 2m+1 information. In their work[40], they also imple-mented their scheme in a three-qubit nuclear magnetic resonance (NMR) system, and gave out a good experi-ment result. In this paper, we will improve the source capacity of the ZXWDLL QDC scheme, i.e., 22m en-coded operations can be achieved by transmitting the travelling particle for m rounds. That is, the travelling qubit carries two bits of information in each round. In this way, the present scheme is a model for the complete multiple-round quantum dense coding.

The principle of our QDC scheme is shown in Figure 1 particularly. Qubit 0 is the travelling qubit, and the other qubits are the stationary qubits, where m particles are used to store the phase operations, i.e., pj (j = 1, 2, …, m), and the other m−1 particles (bj) are used to store the bit operations (in the last round, the travelling particle 0 is also used to store the bit operation). The transmission

of one of 22m messages is described in detail with the following steps.

Step 0: Alice prepares 2m+1 qubits which are all in the same original state |0⟩, and setting the mark j = 1. j records the number of round the qubit 0 travelled.

Step 1: Alice starts to store the phase operation into qubit pj by manipulating qubit pj and qubit 0. Alice ap-plies a Hadamard (H) transform to qubit pj, and then a controlled-NOT (CNOT) gate (Cpj0, where the pj is the control bit and the 0 is the target bit) to qubits pj and 0 to transform the two qubits into an entangled state. Then she sends qubit 0 to Bob, who applies (2 )m j

j z kO R ϕ−

to qubit 0, noting 2(2 ) ( )m jm j

z k z kR Rϕ ϕ−− = , i. e., re-

peating ( )z kR ϕ for 2m−j times. Then, he returns qubit 0 to Alice. One should note that the two qubits are still entangled. Alice applies Cpj0 to qubits pj and 0 for the second time to disentangle qubit 0 from qubit pj. Then pj has noted the phase change of the entangled state.

Step 2: Alice starts to store the bit operation into qubit bj by applying two CNOT gates Cbj0C0bj. If Oj is I, the bj will not be changed and still in the initial state |0⟩. Oth-erwise, bj will transform into |1⟩ state after the two CNOT operations. Moreover, the two CNOT gates Cbj0C0bj will make qubit 0 return to |0⟩ for the next round.

Step 3: Check whether j < m. If it is, put change j to j +1 and go to step 1. If not, i. e., j = m, go to step 4.

Step 4: Make an inverse Fourier transform to qubits

Figure 1 The principle of the complete multiple-round quantum dense coding. Every horizontal line denotes a qubit. All the qubits are in the same origi-nal state |0⟩. Qubit 0 is the travelling qubit, and the other qubits are the stationary qubits, where m qubits are used to store the phase operations, and the other m qubits (including qubit 0 in the last round) are used to store the bit operations. They are denoted by the subscript p and b, respectively. Oj (j = 1,

2, …, m) = I, σx, and 2( ) i k zz kR e ϕ σϕ = with 2 2m

k kϕ π= − (k = 0, 1, 2, …, 2m－1). H denotes the Hadamard transform and F－1 denotes the inverse of

quantum Fourier transform.

1164 LI ChunYan et al. Chinese Science Bulletin | May 2007 | vol. 52 | no. 9 | 1162-1165

p1, p2, …, pm, and then make a joint measurement on all the stationary qubits. From this result, Alice will read out the information encoded by Bob.

The detail of the scheme can be represented as the compound operation for encoding in round j 0 0 0 0(2 )

j j j j j

m jj b b p j z k p pT C C C O R C Hϕ−= . (3)

Through calculations, one obtains

00 0 0j j

j p bT

20

1 ( 0 e 1 ) 0 02

m jk

j j j

ijp p bOϕ−−= + . (4)

The m round encoding is expressed as

1 1m mT T T T−= ⋅ ⋅ ⋅ . After the completion of the m rounds, one obtains

1 1 1 1 00 0 0 0 0 0 0m m m mp p p b b bT

− −L L

201

1 ( 0 e 1 ) 0 02

m jk

j

mi

jj j bm jOϕ−−

== ⊗ + . (5)

The decoding process requires an inverse of quantum Fourier transform, which transforms eq. (5) into

1 2 1 21 2 1 2m mm mp p b bp bx x x y y yL L . Through

measurement of the 2m-1 stationary qubits and the trav-eling qubit, Alice knows which manipulation Bob has made, and the 22m information exchanging process was completed.

In experiment, this complete multiple round quantum dense coding scheme can also be implemented in princi-ple with NMR, similar to the ZXWDLL QDC scheme[40]. It can be proved that the composite operation

( ) , ,0 20 0 e jz p z

j j

ip j z p jC O R C O

θσ σθ = . (6)

In NMR, , ,0 2z p zjieθϕ σ

can be realized by J-coupling between qubits pj and 0 through the standard spin-refo- cus technique[41]. Eq. (6) provides the experimental convenience to implement our scheme. The CNOT gates Cbj0 and C0bj can be realized through radio-frequency pulses and J-couplings[42]. The other operations, such as the Hadamard transform, Oj, and the inverse of quantum Fourier transform, can be implemented in the same way in the ZXWDLL QDC scheme[40]. Demonstrating the non-trivial results of our scheme requires at least five qubit systems. Up to now, seven qubit NMR quantum computer has been realized[43], and the control of twelve qubits for quantum information processor has be dem-onstrated[44]. Hence it is feasible to implement our scheme within the current NMR techniques.

In summary, a complete multiple round quantum dense coding scheme has been proposed. In this scheme, there are 2m quantum systems, i.e., 2m−1 stationary qubits and one travelling qubit. The stationary qubits were divided into two groups. One is used to record the phase operations and the other for the bit operations. Alice can get 2m qubits of information by manipulating only one qubit travelling between Alice and Bob for m rounds. Compared with the ZXWDLL QDC scheme[40], our scheme has the high capacity because the travelling qubit can carry two bits of information in each round.

We thank Dr. Jingfu Zhang for his kind help.

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