Journal of the Korean Physical Society, Vol. 61, No. 1, July 2012, pp. 1∼5
Multi-user Quantum Network System and Quantum Communication Using χ-type Entangled States Chang Ho Hong, Jin O Heo and Jong In Lim Graduate School of Information Management and Security, Korea University, Seoul 136-701, Korea and Center for Information Security Technologies (CIST), Korea University, Seoul 136-701, Korea
Hyung Jin Yang∗ Graduate School of Information Management and Security, Korea University, Seoul 136-701, Korea, Center for Information Security Technologies (CIST), Korea University, Seoul 136-701, Korea, Department of Physics, Korea University, Yeongi 339-700, Korea and Graduate School of Information Security (GSIS), Korea University, Seoul 136-701, Korea (Received 16 March 2012, in final form 2 May 2012) A multi-user quantum direct communication network system for N users utilizing χ-type entangled states is proposed. The network system is composed of a communication center, N users, and N quantum lines linking the center and the N users. There is no quantum line among users; therefore, only N quantum lines are necessary for the communication between users. Using one χ-type entangled state, in this protocol, we are able to send two bits of information through direct communication and, at the same time, share two bits of quantum keys. The security of the protocol will be analyzed. PACS numbers: 03.67.Hk, 03.67.Dd, 03.65.Ud Keywords: Quantum communication network, Quantum direct communication, Quantum key distribution, χ-type entangled state DOI: 10.3938/jkps.61.1
I. INTRODUCTION
Many quantum communication protocols use entangled states as a resource. Entangled states have played an important role as a resource of quantum communication. In entangled states, the qubits show perfect correlations that cannot be explained with classical physics if they are spatially separated. Two-qubit entangled states, the Bell states, are well studied [34] while multi-qubit entangled states are still under investigation. Multipartite entangled states have been shown to have more useful properties in quantum communication protocols than two-qubit entanglements have. Hillery et al. [14] used not Bell states but Greenberger-Horne-Zeilinger (GHZ) states as a resource for their QSS protocol. Recently, Lin et al. [28] proposed a QSDC scheme using χ-type entangled states:
The first quantum communication protocol for quantum key distribution (QKD) was proposed by Bennett and Brassard in 1984 [1] and proved to be unconditionally secure. After the first QKD, many quantum communication protocols were suggested, including other QKDs [2–13], quantum secret sharing (QSS) [14–21], quantum secure direct communication (QSDC) [11,22-33], and so on. The QKD is built to share a sequence of secret keys between two users. The security of the QKD is guaranteed by the fact that the keys are distributed between users only when there is no eavesdropper in the quantum channel. In QSS, the message sender Alice splits secret information into two parts and distributes each of them to two legitimate users Bob and Charlie separately, and Bob and Charlie can reconstruct the message only by collaboration. QSDC is intended to transmit secret messages directly without using keys to encrypt them. Beige et al. proposed the first QSDC protocol in 2002 [30]. In the same year, Bostrom and Felbinger [11] suggested a “ping-pong” protocol that allowed information to be transmitted in a deterministic manner. ∗ E-mail:
|χ =
1 √ (|0000 − |0011 − |0101 + |0110 2 2 +|1001 + |1010 + |1100 + |1111)3214 . (1)
Entering the 2000s, research on the χ-type entangled states composed of four qubits has been actively pursued. Wu et al. [35] studied quantum nonlocality of the χ-type entangled state to show that the state violated a new Bell inequality. The χ-type entangled state is a genuine four-qubit entangled state in the sense that it
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Journal of the Korean Physical Society, Vol. 61, No. 1, July 2012
is not reduced to EPR pairs [36]. Wang and Zhang [37] argued in 2009 that χ-type entangled state could be effectively generated. In the same year, Wang [38] showed that χ-type entangled state could be produced through a large-detuning interaction of atoms. Liu and Kuang [39] suggested a method to generate the χ-type entangled state by using a separated optical cavity in which four atoms were entangled to form the state. Hong-Wu et al. [36] showed that linear optics elements and four one-sided cavities could be used to create the χ-type entangled state. These studies would increase the possibility of implementation of a protocol utilizing the χ-type entangled state. Furthermore, many quantum communication protocols using the χ-type entangled state have been proposed. Employing the χ-type entangled state, Wang et al. [20] proposed a hierarchical QSS protocol, and Yeo and Chua presented a quantum teleportation and dense coding scheme [40]. Utilizing the χ-type entangled state, Lin et al. [28] proposed a highly efficient QSDC protocol in 2008 and recently Yang et al. suggested a three-partite QSS protocol of secure direct communication [41] based on the two-step protocol proposed by Deng et al. [7,25]. This paper is organized as follows: The structure of our quantum network system employing χ-type entangled states is explained in the following section. Here, using the physical properties of χ-type entangled states, we show that the network system of N users needs only N quantum channels linking communication center Trent and each of the N users. After introducing the network system, we present the proposed quantum direct communication protocol in detail. We analyze the security of our protocol against the intercept-and-resend attack in Sec. III. In Sec. IV, we compare our protocol discussed in Sec. II with Song’s protocol (SL protocol) [28]. Finally, we present concluding remarks in Sec. V.
II. QUANTUM SECURE DIRECT COMMUNICATION AND QUANTUM KEY DISTRIBUTION SCHEME IN MULTI-USER QUANTUM COMMUNICATION NETWORK SYSTEM Our quantum secure direct communication and quantum key distribution protocol (QSDC-QKD) in a quantum communication network is designed for any two users among N users of the quantum network system to securely communicate with each other with no quantum line directly connecting them. Using this protocol, any two users are able both to share a sequence of secret keys and to send secret messages directly over a long distance. Figure 1 shows a schematic representation of the multi-user quantum communication network system. In this system, the communication center, Trent, who corresponds to the present telephone company, is connected to N legitimate users with N quantum lines. This re-
Fig. 1. Multi-user quantum communication network system.
quires only N quantum lines joining Trent to N users because our protocol enables any two users to communicate with each other without a quantum line directly linking them. Our communication scheme is composed of two stages: The first stage is for the network establishment, represented by (N-) in the following, and the second is for the QSDC-QKD protocol, represented by (C-). (N-1) Bob informs Trent that he wants to communicate with Alice. (N-2) After hearing Bob’s request, Trent prepares a string of M ordered χ-type entangled states. The state is shown in Eq. (1), and it is rearranged as Eq. (2) by simple grouping: 1 |χ = √ (|0000 − |0011 − |0101 + |0110 2 2 +|1001 + |1010 + |1100 + |1111)Ai ai Bi bi (2) 1 = √ |00Ai ai |Φ− Bi bi − |01Ai ai |Ψ− Bi bi 2 2 +|10Ai ai |Ψ+ Bi bi + |11Ai ai |Φ+ Bi bi , (3) where |Φ± = √12 (|00 ± |11), |Ψ± = √12 (|01 ± |10), and i = 1, 2, 3,. . ., M . The subscripts (Ai , ai , Bi , bi ) represent each particle in a χ-type entangled state |χ. Trent takes these particles to form four distinct sequences of particles as follows: A-sequence: [A1 , A2 , . . . , AN ], asequence: [a1 , a2 , . . . , aN ], B-sequence: [B1 , B2 , . . . , BN ], and b-sequence: [b1 , b2 , . . . , bN ]. He sends A-sequences to Alice and B-sequences to Bob. (N-3) Both Alice and Bob announce that they received the sequences. Alice randomly chooses the checking particles of the ordered A-sequence and measures them with a σz -basis. She publicly announces the position of checking particles to Bob and Trent. (N-4) Bob and Trent measure the corresponding particles by using the same σz -basis and tell the measurement outcomes to Alice. Now, she can check the security of the channel by using Eq. (1). If the error rate in the security check is lower than the allowed limit, Alice and Bob confirm that there is no eavesdropper on the channel and
Multi-user Quantum Network System and Quantum Communication · · · – Chang Ho Hong et al.
continue to the next step. Otherwise, they abandon these steps and restart setting up the communication network. (N-5) After Alice confirms that the quantum channels are secure from step N-4, Trent transmits the a-sequence and the b-sequence to Alice and Bob, respectively. Alice and Bob confirm that they received the a- and the b-sequences, then, Alice chooses checking particles of the ordered A- and a-sequences and measures them with the σz -basis. Alice publicly announces to Bob the position and the measurement outcomes of the checking particles. Bob performs Bell-state measurements on the corresponding checking particles. The form of Bob’s checking particle groups are [(Bh , bh ), (Br , br ), . . . , (Bu , bu )] in which each group forms a Bell state. Bob estimates the security of the quantum channel by using Eq. (2). If the error rate exceeds the threshold, the process is aborted. Otherwise, they start on the following communication steps. (C-1: Measurement) Let us suppose a case where Bob sends messages to Alice and she receives his messages. Alice performs σz –basis measurements on the remaining A-sequence and a-sequence, and Bob makes a Bell-state measurement (BSM) on the pair of particles (Bi , bi ) of the B-sequence and the b-sequence. (C-2: Encoding Estimate) Obtaining the measurement outcome on the pair (Bi , bi ), Bob estimates the states to which the pair (Bi , bi ) would be transformed if he applied one of the Pauli operators (U00 , U01 , U10 , U11 ). The Pauli operator is the message he wants to send Alice. Here, the Pauli operators and mean messages of two bits, U00 , U01 , U10 , and U11 , respectively, and they are represented as follows: U00 U01 U10 U11
= = = =
I = |00| + |11|, σx = |10| + |01|, σy = |01| − |10|, σz = |00| − |11|.
(4) (5) (6) (7)
(C-3: Message Sending) Bob publically announces the estimated outcome to Alice, not the measured outcome. (C-4: Message Decoding and Quantum Key Distribution) Alice already knows Bob’s BSM outcomes because of her measurement outcomes in (C-1) and Eq. (3). From this, she can infer Bob’s secret messages based on Bob’s announcement in (C-3). Furthermore, Bob can figure out Alice’s measurement outcomes by using his BSM outcomes and Eq. (3). Alice’s measurement outcomes can be used as quantum keys; thus, two bits of quantum keys are distributed. For example, according to Eq. (3), Bob’s Bell state is |Ψ− when Alice’s measurement outcomes are 01. After the step (C-1), both Alice and Bob know the state of the counterpart of the communication. If Bob announces the estimated state is |Φ− in (C-3), Alice recognizes that his encoding operator is U10 ; thus, Bob’s messages are “10”. In addition to this direct communication, Bob notices that Alice’s measurement outcome is “01”. The information “01” is able to be shared as a key between
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Fig. 2. (Color online) QSDC-QSS scheme.
Alice and Bob. In this scheme, hence, we both send two-bits of messages directly and share two-bits key by using a χ-type entangled state. QSDC-QKD scheme enables us to share four-bits of classical information between Alice and Bob. A schematic representation of QSDC-QSS between Alice and Bob is shown in Fig. 2.
III. SECURITY OF THE PROTOCOL Since in the communication steps from (C-1) to (C-5), information is not transferred through the quantum lines of the quantum network, the security of the QSDC-QKD protocol is guaranteed if the quantum bits of chi-type entangled states are securely distributed. The security of the network establishment is, in our protocol, equivalent to the secure distribution of chi-type entangled states in our protocol, which is confirmed in the steps (N-3), (N-4), and (N-5). During the process of network establishment from (N-1) to (N-5), the distributed quantum particles do not contain any information; thus, the attacker, Eve, cannot get any messages or keys. We consider the intercept-and-resend attack in order to certify that our protocol is secure against the attack. A schematic representation of the intercept-and-resend attack is shown in Fig. 3, where Eve prepares a sequence of ordered χ-type entangled states which form A , B , A , and B -sequences. When Trent sends Alice and Bob the A-sequence and the B-sequence, respectively, in the step (N-3), Eve steals both sequences, restores them, and sends Alice and Bob the A -sequence and the B -sequence, respectively. As such, her attack is useless because not only does it break the correlation of entangled states among legitimate users but also she gets no information. Furthermore, Eve’s existence is revealed in the checking procedures of (N-4) and (N-5). Since Bob encodes his secret messages and announces the measurement results only after the confirmation of the security of the quantum channels, Eve cannot get any information. Before the communication, her malevolent
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Journal of the Korean Physical Society, Vol. 61, No. 1, July 2012
attack is checked by Alice and Bob. The probability that Eve’s attack is revealed is 1/2 per checking particle. If n is the total number of χ-type states, n = N + Nd , where N is the total number of χ-type entangled states used for the message delivery and key distribution, and Nd is the number of pairs of checking particles for checking security. Then, the total probability of detecting Eve is Nd 1 D =1− . 2
(8)
We can infer that for sufficiently large value Nd , the detection probability approaches unity. Therefore, the
proposed protocol is secure against the intercept-andresend attack because the total detection probability D tends to unity in the long-message limit.
IV. COMPARISON WITH THE SL PROTOCOL In Ref. 28, another highly efficient protocol (SL protocol) for QSDC using the χ-type entangled states was proposed. The process of the protocol is as follows: Eq. (2) can be rearranged as
1 √ |Φ− Ai ai |0+Bi bi + |Φ+ Ai ai |0−Bi bi + |Ψ− Ai ai |1+Bi bi + |Ψ+ Ai ai |1−Bi bi . 2 2
Alice prepares the χ-type entangled states and sends the quantum particles, the B- and the b-sequences, to Bob. Bob chooses randomly a sufficiently large subset from the B- and the b-sequences and measures these particles with the bases BMB 1 = {|0+, |0−, |1+, |1−} or BMB 2 = {| + 0, | − 0, | + 1, | − 1}. He announces to Alice the positions of the checking particles and his measurement basis. Alice performs a measurement with the proper basis and judges the security of the channel. If the error rate exceeds the limit, Alice and Bob abort the protocol. Otherwise, they continue the protocol. According to her secret message, Alice applies the Pauli operator on the remaining particles in her lab. Alice transmits these encoding particles to Bob. After receiving these encoding particles, Bob measures the particles in the ba sis FMB = |χklmn = Ukl Umn |χ|k, l, m, n = 0, 1, 2, 3 and obtains Alice’s secret message. According to the SL protocol’s authors, it has high efficiency and source capacity because it transmits four-bit classical information with a χ-type entangled state. However, the SL protocol must transmit encoding particles that can be good targets for Eve to attack. The transmission of encoding particles increases the number of quantum channels. Bob must have an ability that measures the four-particles in the basis of four-qubit entangled states, FMB. The measurement of four-qubit entangled states is still quite inconvenient in practice. For this reason, the SL protocol is still likely far from practical applications. Compared to the SL protocol, our protocol enhances the security in the sense that encoding particles containing information are not transferred through lines; thus, information cannot be exposed to an eavesdropper. Furthermore, our protocol improves the ease of implementation because it requires only a single particle measurement and BSM. Our QSDC-QKD protocol is also highly efficient like the SL protocol: four-bits of classical information are shared using one χ-type entangled state.
(9)
V. CONCLUSION We have shown that the χ-type entangled states can be used for the multi-user QSDC-QKD protocol. The fact that communicators in our protocol publically announce the estimated states instead of sending encoded quantum states enhances the security of the communication. The implementation of our system is relatively easy because it needs only a single-particle measurement and BSM. The great feature of our communication network system is that any two users among N subscribers can communicate with each other without a quantum channel directly linking them. Our network system does not require N (N −)/2 quantum channels linking N users, but uses only N channels connecting Trent and N users. Our protocol has higher security and easier implementation than the SL protocol. Compared with the SL protocol based on superdense coding with 4-dimension systems, our protocol is more feasible because the measurement of a high-dimensional quantum system is more difficult than that of a two-level system. The protocol is able to send two bits of message and two bits of random key at the same time. In practice, there are noise and information loss in the quantum channel. A quantum error correction and privacy amplification may be required to distribute the key and transmit messages. We may adopt [11,42] the Hamming code to correct the error introduced by the channel noise. For implementation of our protocol, quantum memory is required to store the quantum states. Certainly, the technology of quantum memory is not perfectly developed at the present day. However, meaningful results of research work have been published recently. A notable example is an ultra-long quantum memory technique by Ham. He suggested “Coherent control of the collective atom phase for ultra-long, inversion-free photon echoes”, which quantum state can be stored for hours [43]. It is believed that the quantum
Multi-user Quantum Network System and Quantum Communication · · · – Chang Ho Hong et al.
memory technique will be available in the future. The security of the protocol was discussed by examining and proving its security against the intercept-and-resend attack. We have shown that our protocol is more practical than the SL protocol [28].
ACKNOWLEDGMENTS This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program (NIPA-2012-H0301-12-3007) supervised by the NIPA (National IT Industry Promotion Agency).
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