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Centrally controlled quantum teleportation
Optics Communications 283 (2010) 4802–4809
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Centrally controlled quantum teleportation Xiu-Bo Chen a,b,⁎, Gang Xu c, Yi-Xian Yang a,b, Qiao-Yan Wen a a b c
State key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China Key Laboratory of network and information attack and defence technology of MOE, Beijing University of Posts and Telecommunications, Beijing 100876, China College of Mechanical Engineering, Taiyuan University of Technology, Taiyuan 030024, China
a r t i c l e
i n f o
Article history: Received 3 November 2009 Received in revised form 17 July 2010 Accepted 21 July 2010 Keywords: Controlled teleportation Quantum communication network Security Classical communication cost
a b s t r a c t Through designing a quantum communication network, we propose a protocol for the teleportation between multiple senders and multiple receivers via only one controller. In order to rationally employ the quantum entanglement resources, the controller shares the entangled state with every sender, while there is no directly shared entanglement link between sender and receiver. The security is analyzed in detail. Moreover, this protocol reduces the classical communication cost in the public channel by means of the coding. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction Quantum teleportation [1], which allows the disembodied transport of a quantum state between two parties, has been considered to be one of the singular fruits in the burgeoning field of quantum information theory. Since the seminal work of Bennett et al. [1], scientists have made dramatic progress with quantum teleportation in theory [2,3] and experiment [4,5]. In 1998, the idea of Bennett et al. [1] was generalized by Karlsson and Bourennane [6]. In their scheme, the unknown state of a qubit could be teleported to one receiver conditioned on the controller's measurement outcome. This means that different resources can be exchanged in the control manner. Yan et al. [7,8] investigated the three-party controlled teleportation via three-particle entangled state. Man et al. [9] constructed a (2 N + 1)qubit entangled state to perform controlled teleportation of an arbitrary N-qubit state under the control of an agent. Cao et al. [10] proposed a protocol in which an unknown 3-dimensional bipartite quantum state can be perfectly teleported from the sender to the receiver under the control of an agent. Since controlled teleportation is very useful in networked quantum processing and cryptographic conferences, it has become a particularly interesting topic. Recently, based on the communication network, many protocols for the controlled teleportation have been investigated. Zhang et al. [11] presented a general idea to construct methods for multi-qubit quantum teleportation between
⁎ Corresponding author. State key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China. Tel.: + 86 10 89189519. E-mail address: fl[email protected] (X.-B. Chen).
two remote parties with control of many agents in a network. Then, he [12] presented another scheme which allows an arbitrary 2-qubit quantum state teleportation between two remote parties with control of many agents in a network. Yang et al. [13] proposed a controlled teleportation that the receiver can successfully get access to the original state of each qubit, as long as all the agents collaborate through local operation and classical communication. Deng et al. [14] presented a way for symmetric controlled teleportation of an arbitrary two-particle entangled state based on Bell-basis measurements by using two Greenberger–Horne–Zeilinger (GHZ) states, i.e., a sender transmits an arbitrary two-particle entangled state to a distant receiver, via the control of the n agents in a network. Up to now, almost previous protocols for the controlled teleportation assume that in a network there are only one sender and only one receiver, while many controllers can exist. These protocols only operate over point-to-point communication between two users, and cannot be deployed over any arbitrary network topology. However, it is important for the multi-user to design a teleportation communication network. The quantum entanglement resources and the classical communication are necessary for realization of the teleportation. On one hand, the rational utilization of the entanglement resources should be considered. On the other hand, the classical communication cost can be used to better understand the fundamental laws of quantum information processing and also be regarded as the natural generalization of quantum communication complexity [15]. Recently, it has been paid much attention in some quantum systems, such as the remote state preparation (RSP) [16] in which the sender knows the state she wants receiver to prepare while in quantum teleportation sender need not know the state she wants to send. It is well known that the quantum teleportation process can transmit an unknown
0030-4018/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.07.058
X.-B. Chen et al. / Optics Communications 283 (2010) 4802–4809
quantum state from a sender to a spatially distant receiver with the help of some classical information. If the sender knows the state she wants receiver to obtain, the classical communication cost can be cut down. Lo [17] has shown that for some special ensembles of states, RSP requires less asymptotic classical communication than quantum teleportation. Pati [18] has found the RSP protocol more economical than teleportation and requires only one cbit from Alice to Bob for some special ensembles. Bennett et al. [19] has shown that in the high-entanglement limit the asymptotic classical communication cost of remotely preparing a general qubit is one cbit. As far as the controlled teleportation is concerned, it is necessary to research the classical communication cost even if the sender does not know the state transmitted to the receiver. As especially relevant to this paper, through designing a quantum communication network, we propose a protocol for the centrally controlled quantum teleportation. Different from some previous protocols [11–14], the goal of this paper is to propose a teleportation between multiple senders and multiple receivers via only one controller in a quantum communication network. Because some particles are required to be transmitted in the process of the communication, we should analyze the security of this protocol. Our protocol is secure. Eve not only is inevitably detected, but also obains no any useful information. To rationally employ the entanglement resources, the controller (TP) shares the entangled state with every sender, while there is no directly shared entanglement link between sender and receiver. This may be an appealing advantage in the implementation of a quantum communication network [20]. Maximizing information exchange over the classical communication networks has been a major subject among both the information theory and the networking societies. The classical network coding was first introduced by Ahlswede et al. [21]. As far as the quantum system is concerned, Refs. [22,23] investigated the transmission of a quantum state based on the quantum network. In this protocol, we will consider to save the classical communication cost in the public channel by means of the coding. The structure of this paper is as follows. In Section 2, through constructing a communication network, we propose a protocol for the centrally controlled teleportation. An arbitrary sender may teleport an unknown quantum state to an arbitrary receiver under the TP's assistance. In Section 3, a method for crossly teleporting two quantum states between two senders and two receivers is presented. Two receivers can simultaneously receive their own qubits. The security is analyzed in detail. A brief discussion and the concluding summary are given in Section 4.
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In general, there are many different communication networks and types thereof. Our multi-user quantum communication network for the centrally controlled teleportation is shown in Fig. 1. Entanglement is a valuable physical resource for accomplishing many useful quantum computing [24] and quantum information processing tasks [25,26]. Generally, in the protocol for the controlled teleportation including n senders and n receivers, all senders need share the entanglement with all receivers and the controller. However, the teleportation is implemented and the entanglement is used, if and only if there is a requirement for the communication. So, the huge waste for the valuable shared entanglement resources may occur in the realistic quantum communication network. The controller TP in our protocol shares the entangled state with every sender, while there is no directly shared entanglement link between sender Si and receiver Rj. The sender Si only connects corresponding receiver Ri with the Q-channel. It is an outstanding advantage in the implementation of a quantum communication network [20]. 2.1. Process of the protocol Now, let's describe how an unknown quantum state is teleported from an arbitrary sender to a distant arbitrary receiver via the control of the TP. As is shown in Fig. 2, suppose that an arbitrary sender Si wishes to teleport the following state to the distant arbitrary receiver Rj. jφ〉i = xi j0〉 + yi j1〉
ð4Þ
Where nothing is known about the coefficients xi and yi except that they satisfy the normalization condition |xi|2 + |yi|2 = 1. Like many probabilistic teleportation protocols, our protocol takes use of non-maximally entangled resources without first converting to a maximally entangled pair via local filtering [27] or entanglement concentration [28]. For example, Li et al. [29] investigated the probabilistic teleportation via the non-maximally entangled state. Zhou et al. [30] presented a scheme for controlled teleportation of an arbitrary m-qudit (d-dimensional quantum system) state by using non-maximally entangled states. In this protocol, we take the non-maximally entangled Einstein– Podolsky–Rosen (EPR) pairs as the information carriers. That is, TP shares the following states with senders Si and Sj. pffiffiffi pffiffiffiffiffiffiffiffiffiffi a j00〉 + 1−a j11〉 pffiffiffi Ti1 Si1 pffiffiffiffiffiffiffiffiffiffi = a j00〉 + 1−a j11〉
jΓ〉Ti1 Si1 = jΓ〉Tj1 Sj1
ð5Þ
Tj1 Sj1
2. Controlled teleportation between a sender and a receiver in which the first qubits Ti1 and Tj1 belong to TP, while the second qubits Si1 and Sj1 are in senders Si's and Sj's possession, respectively. Here, 0 b a b 1.
Four Bell-basis states, i.e. EPR pairs, are
jΦ 〉 = ð j00〉 j11〉Þ =
pffiffiffi 2;
jΨ 〉 = ð j01〉 j10〉Þ =
pffiffiffi 2
ð1Þ
The function U(T) is a unitary transformation defined by the Pauli operators. The T is independent variable which is expressed by two cbits, i.e. T ∈ {00, 01, 10, 11}. Uð00Þ=
1 0 1 0 0 1 0 1 ; Uð10Þ= ; Uð01Þ= ; Uð11Þ= 0 1 0 −1 1 0 −1 0
TP
S1
S2
…
Si
…
Sj
…
Sn
R1
R2
…
Ri
…
Rj
…
Rn
ð2Þ It is easy to prove that for two arbitrary independent variables T1 and T2, the function U satisfies the following relation U
−1
ðT1 ÞU
−1
ðT2 Þ = c⋅U
−1
ðT1 ⊕T2 Þðc = 1Þ
where, " ⊕ " denotes the addition module 2.
ð3Þ
Fig. 1. Quantum communication network for the centrally controlled teleportation. The line denotes the controller TP shares the entangled state with senders. The arrow line means that the sender Si connects the corresponding receiver Ri with the channel used to transmit the quantum state (Q-channel).
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TP Ti1
Suppose that the state of particle Sj1 conditioned on the measurement result T = 10 and X = 00 is
Tj1
1 ½ax j0〉−ð1−aÞyi j1〉Sj1 2 i
Si
ϕ
i
Sj1
Si1
Ri
Sj
Rj
Fig. 2. The centrally controlled teleportation between an arbitrary sender Si and receiver Rj. TP shares a pair of entangled state with Si and Sj, respectively. After TP makes the Bell-basis measurement, Sj sends his particle to the receiver Rj. The arrow line means that the particle Sj1 is transmitted to the receiver Rj through the Q-channel.
To begin the teleportation, TP performs Bell-basis measurements on particles Ti1Tj1 in her hand. TP's Bell-basis measurement outcome {|Φ+〉, |Φ−〉, |Ψ+〉, |Ψ−〉} is denoted as T = 00, 10, 01, 11, respectively. Then, one of the following entangled states can be obtained.
ð9Þ
TP broadcasts the classical messages m1m2m3 = 010. In the present case, the above state cannot be directly rotated back to the desired state owing to a lack of any knowledge of the state parameters xi and yi. So, receiver Rj introduces an auxiliary particle A with the state |0〉 and performs a collective unitary transformation U1 on particles Sj1 and A under the basis {|00〉, |10〉, |01〉, |11〉} Sj1A. The unitary transformation U1 takes the form of the following 4 × 4 matrices 0
1 0 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B a 2ffi a B 1− B0 B 1−a 1−a U1 = B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B a 2 a B B0 − 1− @ 1−a 1−a 0 0 0
0
1
C C 0C C C C C 0C A
ð10Þ
1
The state on particles Sj1 and A is changed into 1 j Θ〉Si1 Sj1 = pffiffiffi ½aj 00〉 ð1−aÞj11〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j Θ〉Si1 Sj1 = pffiffiffi að1−aÞ½ j01〉 j10〉Si1 Sj1 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2ffi a j0〉S j1〉A ½xi j0〉−yi j1〉Sj1 j0〉A −ð1−aÞ 1− j1 2 1−a
ðT = 00; 10Þ ðT = 01; 11Þ
ð6Þ
TP does not broadcast the measurement outcomes in this step. To reduce the classical communication cost, he shall take the coding technology in the following step. Supposed that T = 00 or 10. The combined state of total system composed of the transmitted state and the entangled pair can be represented as 1 jφ〉i jΘ〉Si1 Sj1 = pffiffiffi ½xi j0〉 + yi j1〉i ½a j00〉 ð1−aÞj11〉Si1 Sj1 2
ð7Þ
Then, receiver Rj measures the state of particle A in the basis {|0〉, |1〉}. From the above state, it can be seen that if the result is |1〉A, the teleportation fails. If the result is |0〉A, the particle Sj1 collapses into the state a[xi|0〉− yi|1〉]Sj1 /2 which is closely related to the transmitted state. Finally, receiver Rj makes the recovery operation U(m2m3) =U(10). He can obtain the state which is just a replica of sender Si's original state. Meanwhile, no trace of identity of the unknown state remains in sender Si's region, as is required in accordance with the no cloning theorem. Table 1 gives all cases. 2.2. Classical communication cost We first calculate the classical information entropy if TP encodes the measurements outcomes.
Sender Sj transmits her particle to receiver Rj through the Qchannel between them. Then, sender Si performs the Bell-basis measurements on the particle of state |φ〉i and the first particle Si1 of state |Θ〉Si1Sj1. The measurement outcome is denoted as X. The state on receiver Rj's particle is projected onto one of the following states. 1 ½ax j0〉 F ð1−aÞyi j1〉Sj1 2 i 1 ½ax j0〉 b ð1−aÞyi j1〉Sj1 2 i 1 ½F ð1−aÞxi j1〉 + ayi j0〉Sj1 2 1 ½F ð1−aÞxi j1〉−ayi j0〉Sj1 2
ð11Þ
111
H = ∑m1 m2 m3
= 000 −pm1 m2 m3
log pm1 m2 m3
2 2 h a + ð1−2aÞ 1−xi 2 i 2 log = − a + ð1−2aÞ 1− xi 2 2 2 h 2 i a + ð1−2aÞxi að1−aÞ 2 −2að1−aÞ log − a + ð1−2aÞxi log 2 4 ð12Þ
ð X = 00Þ
The figure of the function H in the above Eq. (12) is shown in the following Fig. 3.
ð X = 10Þ ð X = 01Þ ð X = 11Þ
ð8Þ
In order to make receiver Rj know the state he acquire with certainty, the measurement outcomes are required to be announced via the classical channel. Here, TP encodes the measurements outcomes into three cbits, i.e. m1m2m3. Here, the m1 is the second bit of T, and m2m3 = (X ⊕ T) When the receiver obtains the result m1 = 0, he first introduces an auxiliary particle and performs a unitary transformation, and then he performs the recovery operation. When the receiver receives the result m1 = 1, he performs one of four Pauli operators to obtain the qubit transmitted.
Table 1 Teleportation between an arbitrary sender and an arbitrary receiver. The classical communication cost is 3. After TP and sender obtain the Bell-basis measurement outcomes T and X, the state on particle Sj1 is |ϕ〉. The receiver performs the operation U(m2m3)to obtain the transmitted state. T
X
|ϕ〉
m1
R
00, 10
00 10 01 11 00 10 01 11
[axi|0〉 F (1 − a)yi|1〉]Sj1 /2 [axi|0〉 b (1 − a)yi|1〉]Sj1 /2 [F (1 − a)xi|1〉 + ayi|0〉]Sj1 /2 [Fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1 − a)x p ffi i|1〉 − ayi|0〉]Sj1 /2 að1−aÞffi½xi j 1〉 F yi j0〉Sj1 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−aÞffi½xi j 1〉 b yi j0〉Sj1 = 2 pa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−aÞffi½F xi j 0〉 + yi j 1〉Sj1 = 2 pa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1−aÞ½F xi j 0〉−yi j1〉Sj1 = 2
0
U(m2m3)
01, 11
1
X.-B. Chen et al. / Optics Communications 283 (2010) 4802–4809
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relevant to the message transmitted. In summary, the present protocol is secure. 3. Cross teleportation of two quantum states In this section, the teleportation for two unknown quantum states crossly transmitted between two senders and two receivers via only one controller is proposed. The feature is that two receivers can simultaneously receive their own qubits with the less classical communication cost. 3.1. Process of the protocol As is shown in Fig. 4, sender Si wants to transmit the quantum state |φ〉i to receiver Rj, while sender Sj wants to transmit the state |φ〉j to receiver Ri(i ≠ j).
Fig. 3. The figure of the classical information entropy H.
In general, the classical communication cost is 4 under the condition that TP does not encode the measurements outcomes. That is, the measurement outcomes X and T should be broadcasted, respectively. However, in our quantum communication network, TP not only controls the start of teleportation, but also is required to encode the measurements outcomes and broadcast it. The classical communication cost in our protocol is three cbits. Compared with 4 cbits, one cbit is saved. That is, the classical channel capacity can be decreased by means of the coding in our protocol. On the other hand, when the maximally entangled EPR pairs, i.e. a = 1/2, are used as the information carriers, the classical communication cost in Fig. 3 is 3. However, there is the redundancy information of one cbit. In fact, both of the outcome m1 = 0 and 1 are not required by the receiver. Under the circumstances, the controller TP should take the new strategy of coding. He encodes the measurement outcomes X ⊕ T, i.e. 2 cbits, and broadcasts it. It can be seen that 2 cbits can be saved when the maximally entangled states are utilized as the information carriers.
2.3. Security analysis The eavesdropping can happen, when sender Sj transmits her particle to receiver Rj through the Q-channel between them. For checking eavesdropping, sender Sj takes the following trick. She inserts some decoy photons, which are randomly in the four states {|0〉, |1〉, | + 〉, | − 〉}. Confirming that receiver Rj has received all the photons owned to themselves, sender Sj tells him the position and state of the decoy photons. Receiver Rj measures these decoy photons with the same basis as sender Sj for preparing them. If the error rate is low enough, they can go on. Otherwise, they discard this communication. In the present protocol, since the positions of the decoy photons is unknown to Eve and the four states of two conjugate bases {|0〉, |1〉} and {| + 〉, | − 〉} are indistinguishable, Eve's several kinds of attacks, such as the intercept-resend attack, the measurement-resend attack, entanglement-measure attack and the denial-of-service(DOS) attack will be detected with nonzero probability during the security checking process. It is obvious that our security checking method derives from the idea of the BB84 QKD [31]. The BB84 QKD has been proven to be unconditionally secure by several groups [32]. After sender Sj securely transmits the particle to the receiver Rj, outside eavesdropper has no chance to attack the secret message any more. The main reason is that there is only the communication of the classical information, and no longer the transmission of the qubits carried the message. Moreover, the classical information is not
jφ〉i = xi j0〉 + yi j1〉;
jφ〉j = xj j0〉 + yj j1〉
ð13Þ
Here, jxi j2 + jyi j2 = 1 and jxj j2 + jyj j2 = 1. TP shares the following state with sender Si pffiffiffi pffiffiffiffiffiffiffiffiffiffi a j00〉 + 1−a j11〉 ; pffiffiffi Ti1 Si1 pffiffiffiffiffiffiffiffiffiffi = a j00〉 + 1−a j11〉
jΓ〉Ti1 Si1 = jΓ〉Ti2 Si2
ð14Þ
Ti2 Si2
and sender Sj pffiffiffi pffiffiffiffiffiffiffiffiffiffi a j00〉 + 1−a j11〉 ; pffiffiffi Tj1 Sj1 pffiffiffiffiffiffiffiffiffiffi = a j00〉 + 1−a j11〉
jΓ〉Tj1 Sj1 = jΓ〉Tj2 Sj2
ð15Þ
Tj2 Sj2
Here, sender Si is in possession of particles Si1 and Si2, while particles Sj1 and Sj2 are held by sender Sj. The particles Ti1, Ti2, Tj1, Tj2 belong to TP. Here, a ≤ 1/2. The controller TP first performs two Bell-basis measurements on particles Ti1Tj1 and Ti2Tj2 in her hand to control the teleportation start. TP's Bell-basis measurement outcome on particles Ti1Tj1 and Ti2Tj2 is denoted as T1 and T2, respectively. Afterwards, two pairs of entangled states can be obtained. 1 jΘ〉Si1 Sj1 = pffiffiffi ½aj00〉 F ð1−aÞj11〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jΘ〉Si1 Sj1 = pffiffiffi að1−aÞ½ j01〉 F j10〉Si1 Sj1 2
ðT1 = 00; 10Þ ð16Þ ðT1 = 01; 11Þ
TP Ti2 Ti1
Si
Ri
ϕ
Si2 i
Si1
Tj2 Tj1
Sj1 Sj2 ϕ
i
Sj
Rj
Fig. 4. The controlled cross teleportation among two senders and two receivers. TP shares two pairs of entangled state with Si and Sj, respectively. TP makes two Bell-basis measurements.
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and 1 j Θ〉Si2 Sj2 = pffiffiffi ½aj 00〉 F ð1−aÞj11〉Si2 Sj2 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j Θ〉Si2 Sj2 = pffiffiffi að1−aÞ½ j01〉 F j10〉Si2 Sj2 2
ðT2 = 00; 10Þ ð17Þ ðT2 = 01; 11Þ
So, there are 16 combinations in all. It is found that they can be classified into three cases.
Depending on the classical messages broadcasted by TP, the receivers begin to recover the original state. Firstly, if m1 = 0 (or m4 = 0), receiver Ri (or Rj) introduces an auxiliary particle A with the state |0〉A and performs a collective unitary transformation U1 in Eq. (10) on particles Sj1 (or Si2) and A under the basis {|00〉, |10〉, |01〉, |11〉} Sj1A (or {|00〉, |10〉, |01〉, |11〉} Si2A). If m1 = 1 (or m4 = 1), receiver Ri (or Rj) introduces an auxiliary particle and performs the following unitary transformation U2. 0
Case 1. T1∈{00, 10} and T2∈{00, 10}. In fact, there are 4 kinds, i.e., (T1, T2)∈{(00, 00), (00, 10), (10, 00), (10, 10)}. Supposed that (T1, T2) = (00, 00), two entangled states shared by senders Si and Sj are as follows. 1 j Θ〉Si1 Sj1 = pffiffiffi ½aj 00〉 + ð1−aÞj11〉Si1 Sj1 2 1 j Θ〉Si2 Sj2 = pffiffiffi ½aj 00〉 + ð1−aÞj11〉Si2 Sj2 2
ð18Þ
As is shown in Fig. 5, sender Si performs Bell-basis measurements on the particle Si1 and the transmitted particle of state |φ〉i. Sender Sj performs Bell-basis measurements on the particle Sj2 and the particle of state |φ〉j. The measurement outcomes owned by senders Si and Sj are denoted as Xi and Xj, respectively. The state on the particle Sj1 held by the sender Sj becomes 1 ½ax j0〉 F ð1−aÞyi j1〉Sj1 2 i 1 ½ð1−aÞxi j1〉 F ayi j0〉Sj1 2
ðXi = 00; 10Þ
ð19Þ
Xj = 00; 10 Xj = 01; 11
ð20Þ
Senders Si and Sj perform the unitary operation U− 1(Xi) and the U (Xj) on the states of particles Si2 and Sj1 in their hand, respectively. Then, sender Si (and Sj) transmits the particle Si2 (and Sj1) to receiver Ri (and Rj) via the Q-channel. Here, in order to reduce the classical communication cost in the public channel, TP codes four Bell-basis measurement outcomes T1, T2, Xi, Xj into 6 cbits messages m1m2m3m4m5m6. Here, the cbit m1 is the second cbit of Xi, and m2m3 = Xi ⊕ Xj ⊕ T2. Similarly, The cbit m4 is the second bit of Xj, and m5m6 = Xj ⊕ Xi ⊕ T1. So, the classical communication cost in the public channel has decreased. −1
Ri
i
Si1 Si2
1 0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a 2ffi 1− 0C 1−a C C 0 0C C C a C 0C − A 1−a
0
0
ð21Þ
1
For example, suppose that T1 = T2 = 00, Xi = 01 and Xj = 10. When sender Si performs the unitary operation U− 1(Xi), the state in the remaining particle Si2 belonged to sender Si collapses into i 1h axj j1〉−ð1−aÞyj j0〉 : Si2 2
ð22Þ
After performing the unitary operation U− 1(Xj), sender Sj obtains the following state 1 ½−ð1−aÞxi j1〉 + ayi j0〉Sj1 : 2
ð23Þ
Then, the particles Si2 and Sj1 are transmitted to receivers Ri and Rj. TP broadcasts the classical messages m1m2m3m4m5m6 = 111011. According to m1 = 1, receiver Ri introduces an auxiliary particle and performs a collective unitary transformation U2. Due to m4 = 0, receiver Rj introduces an auxiliary particle and performs a collective unitary transformation U1. When the state of auxiliary particle is |0〉A, receivers Ri and Rj can obtain the following states on particles Si2 and Sj1. i ah xj j1〉−yj j0〉 ; Si2 2
a ½−xi j1〉 + yi j0〉Sj1 2
ð24Þ
Finally, receiver Ri performs the recovery operation U(Xi ⊕ Xj ⊕ T2) = U(m2m3) = U(11) to obtain the Sj's state |φ〉j. Receiver Rj performs the operation U(Xj ⊕ Xi ⊕ T1) = U(m5m6) = U(11) to recover the Si's state |φ〉i. It can be seen that two unknown quantum states can be crossly teleported between two senders and two receivers via only one controller. Table 2 gives all cases corresponding to the values of Xi and Xj under the condition of T1 = T2 = 00. Case 2. T1∈{00, 10} and T2∈{01, 11}. Or T1∈{01, 11} and T2∈{00, 10}. It can be seen that there are 8 kinds in all, i.e., (T1, T2)∈{(00, 01), (00, 11), (10, 01), (10, 11), (01, 00), (01, 10), (11, 00), (11, 10)}. In this case, it will be seen that the classical communication cost is 5 cbits.
TP
Si ϕ
0
ðXi = 01; 11Þ:
Similarly, sender Si can obtain one of the following states. i 1h axj j0〉 F ð1−aÞyj j1〉 Si2 2 i 1h ð1−aÞxj j1〉 F ayj j0〉 Si2 2
a B 1−a B B B 0 U2 = B B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2ffi B B 1− @ 1−a 0
S Sj2 j1
ϕ
j
Sj
Rj
Fig. 5. The controlled cross teleportation among two senders and two receivers.
The process of teleportation is also shown in Figs. 4 and 5. Supposed that (T1, T2) = (00, 01), two entangled states obtained are as follows. 1 jΘ〉Si1 Sj1 = pffiffiffi ½aj00〉 + ð1−aÞj11〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jΘ〉Si2 Sj2 = pffiffiffi að1−aÞ½ j01〉 + j10〉Si2 Sj2 2
ð25Þ
X.-B. Chen et al. / Optics Communications 283 (2010) 4802–4809
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Table 2 The cross teleportation under the condition of T1 = T2 = 00. The classical communication cost is 6 cbits. After Si (or Sj) performs the operation U− 1(Xi) (or U− 1(Xj)), he can obtain the state |ϕi〉 (or |ϕj〉) on the particle Si2 (or Sj1). Xi
Xj
|ϕi〉
m1
Ri
|ϕj〉
m4
Rj
00
00 10 01 11 00 10 01 11 00 10 01 11 00 10 01 11
[axj|0〉 ± (1 − a)yj|1〉]Si2 /2
0
U ðm2 m3 Þ
[axi|0〉 ± (1 − a)yi|1〉]Sj1 /2
0
U ðm5 m6 Þ
[± ayj|0〉 + (1 − a)xj|1〉]Si2 /2
[(1 − a)yi|0〉 ± axi|1〉]Sj1 /2
1
[axj|0〉 ∓ (1 − a)yj|1〉]Si2 /2
[axi|0〉 ∓ (1 − a)yi|1〉]Sj1 /2
0
[± ayj|0〉 − (1 − a)xj|1〉]Si2 /2
[− (1 − a)yi|0〉 ± axi|1〉]Sj1 /2
1
[ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2
0
[(1 − a)xj|0〉 ± ayj|1〉]Si2 /2
[(1 − a)xi|0〉 ± ayi|1〉]Sj1 /2
1
[± (1 − a)yj|0〉 − axj|1〉]Si2 /2
[− ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2
0
[(1 − a)xj|0〉 ∓ ayj|1〉]Si2 /2
[(1 − a)xi|0〉 ∓ ayi|1〉]Sj1 /2
1
10
01
11
[± (1 − a)yj|0〉 + axj|1〉]Si2 /2
1
After senders Si and Sj perform Bell-basis measurements and obtain the measurement outcomes Xi and Xj, the states on particles Sj1 and Si2collapse into 1 ½ax j0〉 F ð1−aÞyi j1〉Sj1 2 i 1 ½ð1−aÞxi j1〉 F ayi j0〉Sj1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i að1−aÞ xj j1〉 F yj j0〉 Si2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i að1−aÞ xj j0〉 F yj j1〉 Si2 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1−aÞ½x j1〉 F yi j0〉Sj1 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i að1−aÞ½xi j0〉 F yi j1〉Sj1 = 2 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih að1−aÞ xj j1〉 F yj j0〉 =2 iSi2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih =2 að1−aÞ xj j0〉 F yj j1〉
ðXi = 00; 10Þ ðXi = 01; 11Þ Xj = 00; 10 Xj = 01; 11 :
Si2
ð26Þ
Senders Si and Sj perform the operations U− 1(Xi) and U− 1(Xj) on the particles Si2 and Sj1, respectively. Then, senders transmit their particles to the corresponding receivers. Here, TP codes four Bell-basis measurement outcomes T1, T2, Xi, Xj into m1m2m3m4m5. Here, the cbit m3 is the second bit of Xj, and m1m2 = Xi ⊕ Xj ⊕ T2, and m4m5 = Xj ⊕ Xi ⊕ T1. Here, it is also shown that the classical communication cost is 5 if two receivers share a maximally entangled state and a non-maximally entangled state in the common teleportation. Depending on the classical messages broadcasted by TP, the receivers recover the original state. Firstly, if m3 = 0, receiver Rj introduces an auxiliary particle and performs a collective unitary transformation U1. If m3 = 1, receiver Rj introduces an auxiliary particle and performs a transformation U2. Finally, receivers Ri and Rj perform the operation U(Xi ⊕ Xj ⊕ T2) = U(m1m2) and U(Xj ⊕ Xi ⊕ T1) = U(m4m5) to recover the transmitted states. Table 3 gives all cases. Case 3. T1∈{01, 11} and T2∈{01, 11}. There are 4 kinds in all, i.e., (T1, T2)∈{(01, 01), (01, 11), (11, 01), (11, 11) }. In this case, it will be seen that the classical communication cost is 4 cbits. The process of teleportation is also shown in Figs. 4 and 5. Supposed that (T1, T2) = (01, 01), two entangled states obtained are as follows.
j Θ〉Si1 Sj1 j Θ〉Si2 Sj2
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = pffiffiffi að1−aÞ½ j01〉 + j10〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = pffiffiffi að1−aÞ½ j01〉 + j10〉Si2 Sj2 2
After sender Si and Sj perform Bell-basis measurements and obtain the measurement outcome Xi and Xj, the states on particles Sj1 and Si2 collapse into
ð27Þ
ðXi = 00; 10Þ ðX = 01; 11Þ i Xj = 00; 10 Xj = 01; 11 :
ð28Þ
Sender Si and Sj perform the operations U− 1(Xi) and U− 1(Xj) on the particles Si2 and Sj1, respectively. Then, senders transmit their particles to corresponding receivers. TP codes all the measurement outcomes into 4 cbits m1m2m3m4. Here, the cbits m1m2 = Xi ⊕ Xj ⊕ T2, m3m4 = Xj ⊕ Xi ⊕ T1. Sometimes, when T1 = T2, the classical communication cost are 2 cbits. Depending on the classical messages broadcasted by the TP, the receiver Ri makes the recovery operation U(Xi ⊕ Xj ⊕ T2) = U(m1m2), and receiver Rj performs the operation U(Xj ⊕ Xi ⊕ T1) = U(m3m4). Table 4 gives all cases. Similarly, it can be calculated that the classical information entropy is 8 if TP does not encode the measurements outcomes. However, in our quantum communication network, TP encodes the measurement outcomes and broadcasts it. The maximum classical communication cost is 6 cbits, i.e. m1m2m3m4m5m6. Compared with
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The cross teleportation under the condition of T1 = 00, T2 = 01. Here, að1−aÞ = 2 = b. Xi
Xj
00 00 10 01 11 10 00 10 01 11 01 00 10 01 11 11 00 10 01 11
|ϕi〉
Ri
|ϕj〉
m3 Rj
b[xj|1〉 F yj|0〉]Si2
U(m1m2)
[axi|0〉 ± (1 − a)yi|1〉]Sj1 /2
0
b[xj|0〉 F yj|1〉]Si2
[(1 − a)yi|0〉 ± axi|1〉]Sj1 /2
1
b[− xj|1〉 F yj|0〉]Si2
[axi|0〉 ∓ (1 − a)yi|1〉]Sj1 /2
0
b[xj|0〉 b yj|1〉]Si2
[− (1 − a)yi|0〉 ± axi|1〉]Sj1 /2 1
b[xj|0〉 F yj|1〉]Si2
[ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2
0
b[xj|1〉 F yj|0〉]Si2
[(1 − a)xi|0〉 ± ayi|1〉]Sj1 /2
1
b[xj|0〉 b yj|1〉]Si2
[− ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2 0
b[− xj|1〉 F yj|0〉]Si2
[(1 − a)xi|0〉 ∓ ayi|1〉]Sj1 /2
1
U(m4m5)
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Table 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The cross teleportation under the condition of T1 = 01, T2 = 01. Here, að1−aÞ = 2 = b. Xi
Xj
|ϕi〉
Ri
|ϕj〉
Rj
00
00 10 01 11 00 10 01 11 00 10 01 11 00 10 01 11
b[xj|1〉 ± yj|0〉]Si2
U(m1m2)
b[± xi|1〉 + yi|0〉]Sj1
U(m3m4)
10
01
11
b[xj|0〉 ± yj|1〉]Si2
b[xi|0〉 ± yi|1〉]Sj1
b[− xj|1〉 ± yj|0〉]Si2
b[± xi|1〉 − yi|0〉]Sj1
b[xj|0〉 ∓ yj|1〉]Si2
b[xi|0〉 ∓ yi|1〉]Sj1
b[xj|0〉 ± yj|1〉]Si2
b[xi|0〉 ± yi|1〉]Sj1
b[xj|1〉 ± yj|0〉]Si2
b[± xi|1〉 + yi|0〉]Sj1
b[xj|0〉 ∓ yj|1〉]Si2
b[xi|0〉 ∓ yi|1〉]Sj1
b[− xj|1〉 ± yj|0〉]Si2
b[± xi|1〉 − yi|0〉]Sj1
11
ρ = ∑m;n
Now, we analyze the security of this protocol. It can be seen that Eve has one chance to eavesdrop the messsge when sender Si (and Sj) transmits the particle Si2 (and Sj1) to receiver Ri (and Rj) via the Qchannel. That is, Eve can first intercept the particles transmitted, and then recover the message according to the classical information broadcasted by the TP. Unfortunately, the technology of checking eavesdropping in the Subsection 2.3 is still taken by communicators. That is, sender Si (and Sj) inserts some decoy photons randomly. If the error rate is enough low, the procedure of the protocol can go on. Otherwise, this communication is abandoned. Moreover, not only any eavesdropping will inevitably disturb the states of the decoy photons and be detected, but also Eve obtains no any useful information. It can be calculated that from Eve's point of view, the state carring the message is the mixed state. In fact, the particle transmitted from the sender to the receiver are in two kinds of states. One is that the particle is in one of the 16 states in the Table 2 or Table 3 with probablity a2/4. The other is that the particle is in one of the 16 states in the Table 3 or Table 4 with probablity a(1 − a)/4. For example, the partilce Si2 in the Table 3 is in one of the 16 states with probablity a(1 − a)/4. Thus, the density operator of the partilce Si2 system is i i 00 jϕmn 〉〈ϕmn j
=
ð29Þ
Notice that this state is a mixed state, since tr((2a(1 − a)I)2) = 8a (1 − a)2 b 1 (here, a b 1/2). This state has no dependence upon the state |φ〉j = xj|0〉 + yj|1〉 being transmitted, and thus any measure2
2
= 2a I:
ð30Þ
4. Conclusion
3.2. Security analysis
að1−aÞ h xj j1〉 + yj j0〉 xj 〈1 j + yj 〈0j 4 + xj j1〉−yj j0〉 xj 〈1j−yj 〈0 j i + ⋯⋯ + −xj j1〉−yj j0〉 −xj 〈1j−yj 〈0 j að1−aÞ h 2 2 2jxj j j1〉〈1j + 2 jyj j j0〉〈0j = 4 i 2 2 + 2jxj j j0〉〈0j + 2jyj j j1〉〈1j × 4 = 2að1−aÞI:
j j = 00 jϕmn 〉〈ϕmn j
So, it can prevent Eve from stealing any useful information.
8 cbits, 2 cbits are saved in this protocol. Moreover, it demonstrates that the classical communication cost is 4 when the maximally entangled EPR pairs are used as the information carriers, and more classical communication cost can be saved. In comparison with the protocol of Section 2, the feature of this section is that two receivers can simultaneously obtain their own qubits.
ρ = ∑11 m;n =
ments performed by Eve will contain no information about |φ〉j = xj| 0〉 + yj|1〉. Similarly, after implementing the unitary operation U, the particle Sj1 in the Table 3 is in one of the 16 states with probablity a2/4. Thus, the density operator of the partilce Sj1 system is
The central theme of this paper is that by constructing a quantum communication network, we propose the centrally controlled multiuser teleportation. We give a protocol for the controlled teleportation between an arbitrary sender and an arbitrary receiver, and a protocol for the controlled cross teleportation in which two receivers can simultaneously receive their own qubits. It is well known that the quantum entanglement resources and the classical communication are necessary for realization of the controlled teleportation. In order to save the entanglement resources, the controller shares the entangled state with every sender. All senders and receivers do not directly share the entangled state, while the sender only connects corresponding receiver with the Q-channel. This may be a merit in the implementation of a realistic quantum teleportation network. Moreover, the present protocol reduces the classical communication cost in the public channel by means of the coding. The security analysis is given because some particles are required to be transmitted in the process of the communication. It is shown that our protocol is secure. Not only is the eavesdropping inevitably detected, but also Eve obains no any useful information. Acknowledgements Project supported by the National Basic Research Program of China (973 Program) (No. 2007CB311203), the National Natural Science Foundation of China and the Research Grants Council of Hong Kong Joint Research Scheme (No. 60731160626), the National Natural Science Foundation of China (Nos. 61003287, 60821001, 60873191, and 60903152), the Beijing Natural Science Foundation (No. 4102055), and the Fundamental Research Funds for the Central Universities (No. BUPT2009RC 0220). References [1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [2] X.B. Chen, N. Zhang, S. Lin, Q.Y. Wen, F.C. Zhu, Opt. Commun. 281 (2008) 2331. [3] X.B. Chen, Q.Y. Wen, G. Xu, Y.X. Yang, F.C. Zhu, Phys. Rev. A 79 (2009) 036301. [4] D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Nature 390 (1997) 575. [5] D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, Phys. Rev. Lett. 80 (1998) 1121. [6] A. Karlsson, M. Bourennane, Phys. Rev. A 58 (1998) 4394. [7] F.L. Yan, D. Wang, Phys. Lett. A 316 (2003) 297. [8] J. Bae, J. Jin, J. Kim, C. Yoon, Y. Kwon, Chaos Solit. Fract. 24 (2005) 1047. [9] Z.X. Man, Y.J. Xia, N.B. An, Phys. Rev. A 75 (2007) 052306. [10] H.J. Cao, Z.H. Chen, H.S. Song, Phys. Scr. 78 (2008) 15002. [11] Z.J. Zhang, Z.X. Man, Phys. Lett. A 341 (2005) 55. [12] Z.J. Zhang, Phys. Lett. A 352 (2006) 55. [13] C.P. Yang, S.I. Chu, S.Y. Han, Phys. Rev. A 70 (2004) 022329. [14] F.G. Deng, C.Y. Li, Y.S. Li, H.Y. Zhou, Y. Wang, Phys. Rev. A 72 (2005) 022338. [15] H. Buhrman, R. Cleve, W. van Dam, Arxiv preprint quant-ph/9705033 (1997). [16] M.X. Luo, X.B. Chen, S.Y. Ma, Y.X. Yang, Z.M. Hu, J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 065501. [17] H.K. Lo, Phys. Rev. A 62 (2000) 012313. [18] A.K. Pati, Phys. Rev. A 63 (2000) 014302. [19] C.H. Bennett, D.P. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal, W.K. Wootters, Phys. Rev. Lett. 87 (2001) 077902. [20] C.A. Yen, S.J. Horng, H.S. Goan, T.W. Kao, Y.H. Chou, Arxiv preprint 0903.3444 (2009). [21] R. Ahlswede, N. Cai, S.Y.R. Li, R.W. Yeung, IEEE Trans. Inf. Theory 46 (2000) 1204.
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Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Centrally controlled quantum teleportation Xiu-Bo Chen a,b,⁎, Gang Xu c, Yi-Xian Yang a,b, Qiao-Yan Wen a a b c
State key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China Key Laboratory of network and information attack and defence technology of MOE, Beijing University of Posts and Telecommunications, Beijing 100876, China College of Mechanical Engineering, Taiyuan University of Technology, Taiyuan 030024, China
a r t i c l e
i n f o
Article history: Received 3 November 2009 Received in revised form 17 July 2010 Accepted 21 July 2010 Keywords: Controlled teleportation Quantum communication network Security Classical communication cost
a b s t r a c t Through designing a quantum communication network, we propose a protocol for the teleportation between multiple senders and multiple receivers via only one controller. In order to rationally employ the quantum entanglement resources, the controller shares the entangled state with every sender, while there is no directly shared entanglement link between sender and receiver. The security is analyzed in detail. Moreover, this protocol reduces the classical communication cost in the public channel by means of the coding. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction Quantum teleportation [1], which allows the disembodied transport of a quantum state between two parties, has been considered to be one of the singular fruits in the burgeoning field of quantum information theory. Since the seminal work of Bennett et al. [1], scientists have made dramatic progress with quantum teleportation in theory [2,3] and experiment [4,5]. In 1998, the idea of Bennett et al. [1] was generalized by Karlsson and Bourennane [6]. In their scheme, the unknown state of a qubit could be teleported to one receiver conditioned on the controller's measurement outcome. This means that different resources can be exchanged in the control manner. Yan et al. [7,8] investigated the three-party controlled teleportation via three-particle entangled state. Man et al. [9] constructed a (2 N + 1)qubit entangled state to perform controlled teleportation of an arbitrary N-qubit state under the control of an agent. Cao et al. [10] proposed a protocol in which an unknown 3-dimensional bipartite quantum state can be perfectly teleported from the sender to the receiver under the control of an agent. Since controlled teleportation is very useful in networked quantum processing and cryptographic conferences, it has become a particularly interesting topic. Recently, based on the communication network, many protocols for the controlled teleportation have been investigated. Zhang et al. [11] presented a general idea to construct methods for multi-qubit quantum teleportation between
⁎ Corresponding author. State key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China. Tel.: + 86 10 89189519. E-mail address: fl[email protected] (X.-B. Chen).
two remote parties with control of many agents in a network. Then, he [12] presented another scheme which allows an arbitrary 2-qubit quantum state teleportation between two remote parties with control of many agents in a network. Yang et al. [13] proposed a controlled teleportation that the receiver can successfully get access to the original state of each qubit, as long as all the agents collaborate through local operation and classical communication. Deng et al. [14] presented a way for symmetric controlled teleportation of an arbitrary two-particle entangled state based on Bell-basis measurements by using two Greenberger–Horne–Zeilinger (GHZ) states, i.e., a sender transmits an arbitrary two-particle entangled state to a distant receiver, via the control of the n agents in a network. Up to now, almost previous protocols for the controlled teleportation assume that in a network there are only one sender and only one receiver, while many controllers can exist. These protocols only operate over point-to-point communication between two users, and cannot be deployed over any arbitrary network topology. However, it is important for the multi-user to design a teleportation communication network. The quantum entanglement resources and the classical communication are necessary for realization of the teleportation. On one hand, the rational utilization of the entanglement resources should be considered. On the other hand, the classical communication cost can be used to better understand the fundamental laws of quantum information processing and also be regarded as the natural generalization of quantum communication complexity [15]. Recently, it has been paid much attention in some quantum systems, such as the remote state preparation (RSP) [16] in which the sender knows the state she wants receiver to prepare while in quantum teleportation sender need not know the state she wants to send. It is well known that the quantum teleportation process can transmit an unknown
0030-4018/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.07.058
X.-B. Chen et al. / Optics Communications 283 (2010) 4802–4809
quantum state from a sender to a spatially distant receiver with the help of some classical information. If the sender knows the state she wants receiver to obtain, the classical communication cost can be cut down. Lo [17] has shown that for some special ensembles of states, RSP requires less asymptotic classical communication than quantum teleportation. Pati [18] has found the RSP protocol more economical than teleportation and requires only one cbit from Alice to Bob for some special ensembles. Bennett et al. [19] has shown that in the high-entanglement limit the asymptotic classical communication cost of remotely preparing a general qubit is one cbit. As far as the controlled teleportation is concerned, it is necessary to research the classical communication cost even if the sender does not know the state transmitted to the receiver. As especially relevant to this paper, through designing a quantum communication network, we propose a protocol for the centrally controlled quantum teleportation. Different from some previous protocols [11–14], the goal of this paper is to propose a teleportation between multiple senders and multiple receivers via only one controller in a quantum communication network. Because some particles are required to be transmitted in the process of the communication, we should analyze the security of this protocol. Our protocol is secure. Eve not only is inevitably detected, but also obains no any useful information. To rationally employ the entanglement resources, the controller (TP) shares the entangled state with every sender, while there is no directly shared entanglement link between sender and receiver. This may be an appealing advantage in the implementation of a quantum communication network [20]. Maximizing information exchange over the classical communication networks has been a major subject among both the information theory and the networking societies. The classical network coding was first introduced by Ahlswede et al. [21]. As far as the quantum system is concerned, Refs. [22,23] investigated the transmission of a quantum state based on the quantum network. In this protocol, we will consider to save the classical communication cost in the public channel by means of the coding. The structure of this paper is as follows. In Section 2, through constructing a communication network, we propose a protocol for the centrally controlled teleportation. An arbitrary sender may teleport an unknown quantum state to an arbitrary receiver under the TP's assistance. In Section 3, a method for crossly teleporting two quantum states between two senders and two receivers is presented. Two receivers can simultaneously receive their own qubits. The security is analyzed in detail. A brief discussion and the concluding summary are given in Section 4.
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In general, there are many different communication networks and types thereof. Our multi-user quantum communication network for the centrally controlled teleportation is shown in Fig. 1. Entanglement is a valuable physical resource for accomplishing many useful quantum computing [24] and quantum information processing tasks [25,26]. Generally, in the protocol for the controlled teleportation including n senders and n receivers, all senders need share the entanglement with all receivers and the controller. However, the teleportation is implemented and the entanglement is used, if and only if there is a requirement for the communication. So, the huge waste for the valuable shared entanglement resources may occur in the realistic quantum communication network. The controller TP in our protocol shares the entangled state with every sender, while there is no directly shared entanglement link between sender Si and receiver Rj. The sender Si only connects corresponding receiver Ri with the Q-channel. It is an outstanding advantage in the implementation of a quantum communication network [20]. 2.1. Process of the protocol Now, let's describe how an unknown quantum state is teleported from an arbitrary sender to a distant arbitrary receiver via the control of the TP. As is shown in Fig. 2, suppose that an arbitrary sender Si wishes to teleport the following state to the distant arbitrary receiver Rj. jφ〉i = xi j0〉 + yi j1〉
ð4Þ
Where nothing is known about the coefficients xi and yi except that they satisfy the normalization condition |xi|2 + |yi|2 = 1. Like many probabilistic teleportation protocols, our protocol takes use of non-maximally entangled resources without first converting to a maximally entangled pair via local filtering [27] or entanglement concentration [28]. For example, Li et al. [29] investigated the probabilistic teleportation via the non-maximally entangled state. Zhou et al. [30] presented a scheme for controlled teleportation of an arbitrary m-qudit (d-dimensional quantum system) state by using non-maximally entangled states. In this protocol, we take the non-maximally entangled Einstein– Podolsky–Rosen (EPR) pairs as the information carriers. That is, TP shares the following states with senders Si and Sj. pffiffiffi pffiffiffiffiffiffiffiffiffiffi a j00〉 + 1−a j11〉 pffiffiffi Ti1 Si1 pffiffiffiffiffiffiffiffiffiffi = a j00〉 + 1−a j11〉
jΓ〉Ti1 Si1 = jΓ〉Tj1 Sj1
ð5Þ
Tj1 Sj1
2. Controlled teleportation between a sender and a receiver in which the first qubits Ti1 and Tj1 belong to TP, while the second qubits Si1 and Sj1 are in senders Si's and Sj's possession, respectively. Here, 0 b a b 1.
Four Bell-basis states, i.e. EPR pairs, are
jΦ 〉 = ð j00〉 j11〉Þ =
pffiffiffi 2;
jΨ 〉 = ð j01〉 j10〉Þ =
pffiffiffi 2
ð1Þ
The function U(T) is a unitary transformation defined by the Pauli operators. The T is independent variable which is expressed by two cbits, i.e. T ∈ {00, 01, 10, 11}. Uð00Þ=
1 0 1 0 0 1 0 1 ; Uð10Þ= ; Uð01Þ= ; Uð11Þ= 0 1 0 −1 1 0 −1 0
TP
S1
S2
…
Si
…
Sj
…
Sn
R1
R2
…
Ri
…
Rj
…
Rn
ð2Þ It is easy to prove that for two arbitrary independent variables T1 and T2, the function U satisfies the following relation U
−1
ðT1 ÞU
−1
ðT2 Þ = c⋅U
−1
ðT1 ⊕T2 Þðc = 1Þ
where, " ⊕ " denotes the addition module 2.
ð3Þ
Fig. 1. Quantum communication network for the centrally controlled teleportation. The line denotes the controller TP shares the entangled state with senders. The arrow line means that the sender Si connects the corresponding receiver Ri with the channel used to transmit the quantum state (Q-channel).
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TP Ti1
Suppose that the state of particle Sj1 conditioned on the measurement result T = 10 and X = 00 is
Tj1
1 ½ax j0〉−ð1−aÞyi j1〉Sj1 2 i
Si
ϕ
i
Sj1
Si1
Ri
Sj
Rj
Fig. 2. The centrally controlled teleportation between an arbitrary sender Si and receiver Rj. TP shares a pair of entangled state with Si and Sj, respectively. After TP makes the Bell-basis measurement, Sj sends his particle to the receiver Rj. The arrow line means that the particle Sj1 is transmitted to the receiver Rj through the Q-channel.
To begin the teleportation, TP performs Bell-basis measurements on particles Ti1Tj1 in her hand. TP's Bell-basis measurement outcome {|Φ+〉, |Φ−〉, |Ψ+〉, |Ψ−〉} is denoted as T = 00, 10, 01, 11, respectively. Then, one of the following entangled states can be obtained.
ð9Þ
TP broadcasts the classical messages m1m2m3 = 010. In the present case, the above state cannot be directly rotated back to the desired state owing to a lack of any knowledge of the state parameters xi and yi. So, receiver Rj introduces an auxiliary particle A with the state |0〉 and performs a collective unitary transformation U1 on particles Sj1 and A under the basis {|00〉, |10〉, |01〉, |11〉} Sj1A. The unitary transformation U1 takes the form of the following 4 × 4 matrices 0
1 0 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B a 2ffi a B 1− B0 B 1−a 1−a U1 = B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B a 2 a B B0 − 1− @ 1−a 1−a 0 0 0
0
1
C C 0C C C C C 0C A
ð10Þ
1
The state on particles Sj1 and A is changed into 1 j Θ〉Si1 Sj1 = pffiffiffi ½aj 00〉 ð1−aÞj11〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j Θ〉Si1 Sj1 = pffiffiffi að1−aÞ½ j01〉 j10〉Si1 Sj1 2
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2ffi a j0〉S j1〉A ½xi j0〉−yi j1〉Sj1 j0〉A −ð1−aÞ 1− j1 2 1−a
ðT = 00; 10Þ ðT = 01; 11Þ
ð6Þ
TP does not broadcast the measurement outcomes in this step. To reduce the classical communication cost, he shall take the coding technology in the following step. Supposed that T = 00 or 10. The combined state of total system composed of the transmitted state and the entangled pair can be represented as 1 jφ〉i jΘ〉Si1 Sj1 = pffiffiffi ½xi j0〉 + yi j1〉i ½a j00〉 ð1−aÞj11〉Si1 Sj1 2
ð7Þ
Then, receiver Rj measures the state of particle A in the basis {|0〉, |1〉}. From the above state, it can be seen that if the result is |1〉A, the teleportation fails. If the result is |0〉A, the particle Sj1 collapses into the state a[xi|0〉− yi|1〉]Sj1 /2 which is closely related to the transmitted state. Finally, receiver Rj makes the recovery operation U(m2m3) =U(10). He can obtain the state which is just a replica of sender Si's original state. Meanwhile, no trace of identity of the unknown state remains in sender Si's region, as is required in accordance with the no cloning theorem. Table 1 gives all cases. 2.2. Classical communication cost We first calculate the classical information entropy if TP encodes the measurements outcomes.
Sender Sj transmits her particle to receiver Rj through the Qchannel between them. Then, sender Si performs the Bell-basis measurements on the particle of state |φ〉i and the first particle Si1 of state |Θ〉Si1Sj1. The measurement outcome is denoted as X. The state on receiver Rj's particle is projected onto one of the following states. 1 ½ax j0〉 F ð1−aÞyi j1〉Sj1 2 i 1 ½ax j0〉 b ð1−aÞyi j1〉Sj1 2 i 1 ½F ð1−aÞxi j1〉 + ayi j0〉Sj1 2 1 ½F ð1−aÞxi j1〉−ayi j0〉Sj1 2
ð11Þ
111
H = ∑m1 m2 m3
= 000 −pm1 m2 m3
log pm1 m2 m3
2 2 h a + ð1−2aÞ 1−xi 2 i 2 log = − a + ð1−2aÞ 1− xi 2 2 2 h 2 i a + ð1−2aÞxi að1−aÞ 2 −2að1−aÞ log − a + ð1−2aÞxi log 2 4 ð12Þ
ð X = 00Þ
The figure of the function H in the above Eq. (12) is shown in the following Fig. 3.
ð X = 10Þ ð X = 01Þ ð X = 11Þ
ð8Þ
In order to make receiver Rj know the state he acquire with certainty, the measurement outcomes are required to be announced via the classical channel. Here, TP encodes the measurements outcomes into three cbits, i.e. m1m2m3. Here, the m1 is the second bit of T, and m2m3 = (X ⊕ T) When the receiver obtains the result m1 = 0, he first introduces an auxiliary particle and performs a unitary transformation, and then he performs the recovery operation. When the receiver receives the result m1 = 1, he performs one of four Pauli operators to obtain the qubit transmitted.
Table 1 Teleportation between an arbitrary sender and an arbitrary receiver. The classical communication cost is 3. After TP and sender obtain the Bell-basis measurement outcomes T and X, the state on particle Sj1 is |ϕ〉. The receiver performs the operation U(m2m3)to obtain the transmitted state. T
X
|ϕ〉
m1
R
00, 10
00 10 01 11 00 10 01 11
[axi|0〉 F (1 − a)yi|1〉]Sj1 /2 [axi|0〉 b (1 − a)yi|1〉]Sj1 /2 [F (1 − a)xi|1〉 + ayi|0〉]Sj1 /2 [Fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (1 − a)x p ffi i|1〉 − ayi|0〉]Sj1 /2 að1−aÞffi½xi j 1〉 F yi j0〉Sj1 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−aÞffi½xi j 1〉 b yi j0〉Sj1 = 2 pa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−aÞffi½F xi j 0〉 + yi j 1〉Sj1 = 2 pa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1−aÞ½F xi j 0〉−yi j1〉Sj1 = 2
0
U(m2m3)
01, 11
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relevant to the message transmitted. In summary, the present protocol is secure. 3. Cross teleportation of two quantum states In this section, the teleportation for two unknown quantum states crossly transmitted between two senders and two receivers via only one controller is proposed. The feature is that two receivers can simultaneously receive their own qubits with the less classical communication cost. 3.1. Process of the protocol As is shown in Fig. 4, sender Si wants to transmit the quantum state |φ〉i to receiver Rj, while sender Sj wants to transmit the state |φ〉j to receiver Ri(i ≠ j).
Fig. 3. The figure of the classical information entropy H.
In general, the classical communication cost is 4 under the condition that TP does not encode the measurements outcomes. That is, the measurement outcomes X and T should be broadcasted, respectively. However, in our quantum communication network, TP not only controls the start of teleportation, but also is required to encode the measurements outcomes and broadcast it. The classical communication cost in our protocol is three cbits. Compared with 4 cbits, one cbit is saved. That is, the classical channel capacity can be decreased by means of the coding in our protocol. On the other hand, when the maximally entangled EPR pairs, i.e. a = 1/2, are used as the information carriers, the classical communication cost in Fig. 3 is 3. However, there is the redundancy information of one cbit. In fact, both of the outcome m1 = 0 and 1 are not required by the receiver. Under the circumstances, the controller TP should take the new strategy of coding. He encodes the measurement outcomes X ⊕ T, i.e. 2 cbits, and broadcasts it. It can be seen that 2 cbits can be saved when the maximally entangled states are utilized as the information carriers.
2.3. Security analysis The eavesdropping can happen, when sender Sj transmits her particle to receiver Rj through the Q-channel between them. For checking eavesdropping, sender Sj takes the following trick. She inserts some decoy photons, which are randomly in the four states {|0〉, |1〉, | + 〉, | − 〉}. Confirming that receiver Rj has received all the photons owned to themselves, sender Sj tells him the position and state of the decoy photons. Receiver Rj measures these decoy photons with the same basis as sender Sj for preparing them. If the error rate is low enough, they can go on. Otherwise, they discard this communication. In the present protocol, since the positions of the decoy photons is unknown to Eve and the four states of two conjugate bases {|0〉, |1〉} and {| + 〉, | − 〉} are indistinguishable, Eve's several kinds of attacks, such as the intercept-resend attack, the measurement-resend attack, entanglement-measure attack and the denial-of-service(DOS) attack will be detected with nonzero probability during the security checking process. It is obvious that our security checking method derives from the idea of the BB84 QKD [31]. The BB84 QKD has been proven to be unconditionally secure by several groups [32]. After sender Sj securely transmits the particle to the receiver Rj, outside eavesdropper has no chance to attack the secret message any more. The main reason is that there is only the communication of the classical information, and no longer the transmission of the qubits carried the message. Moreover, the classical information is not
jφ〉i = xi j0〉 + yi j1〉;
jφ〉j = xj j0〉 + yj j1〉
ð13Þ
Here, jxi j2 + jyi j2 = 1 and jxj j2 + jyj j2 = 1. TP shares the following state with sender Si pffiffiffi pffiffiffiffiffiffiffiffiffiffi a j00〉 + 1−a j11〉 ; pffiffiffi Ti1 Si1 pffiffiffiffiffiffiffiffiffiffi = a j00〉 + 1−a j11〉
jΓ〉Ti1 Si1 = jΓ〉Ti2 Si2
ð14Þ
Ti2 Si2
and sender Sj pffiffiffi pffiffiffiffiffiffiffiffiffiffi a j00〉 + 1−a j11〉 ; pffiffiffi Tj1 Sj1 pffiffiffiffiffiffiffiffiffiffi = a j00〉 + 1−a j11〉
jΓ〉Tj1 Sj1 = jΓ〉Tj2 Sj2
ð15Þ
Tj2 Sj2
Here, sender Si is in possession of particles Si1 and Si2, while particles Sj1 and Sj2 are held by sender Sj. The particles Ti1, Ti2, Tj1, Tj2 belong to TP. Here, a ≤ 1/2. The controller TP first performs two Bell-basis measurements on particles Ti1Tj1 and Ti2Tj2 in her hand to control the teleportation start. TP's Bell-basis measurement outcome on particles Ti1Tj1 and Ti2Tj2 is denoted as T1 and T2, respectively. Afterwards, two pairs of entangled states can be obtained. 1 jΘ〉Si1 Sj1 = pffiffiffi ½aj00〉 F ð1−aÞj11〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jΘ〉Si1 Sj1 = pffiffiffi að1−aÞ½ j01〉 F j10〉Si1 Sj1 2
ðT1 = 00; 10Þ ð16Þ ðT1 = 01; 11Þ
TP Ti2 Ti1
Si
Ri
ϕ
Si2 i
Si1
Tj2 Tj1
Sj1 Sj2 ϕ
i
Sj
Rj
Fig. 4. The controlled cross teleportation among two senders and two receivers. TP shares two pairs of entangled state with Si and Sj, respectively. TP makes two Bell-basis measurements.
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and 1 j Θ〉Si2 Sj2 = pffiffiffi ½aj 00〉 F ð1−aÞj11〉Si2 Sj2 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j Θ〉Si2 Sj2 = pffiffiffi að1−aÞ½ j01〉 F j10〉Si2 Sj2 2
ðT2 = 00; 10Þ ð17Þ ðT2 = 01; 11Þ
So, there are 16 combinations in all. It is found that they can be classified into three cases.
Depending on the classical messages broadcasted by TP, the receivers begin to recover the original state. Firstly, if m1 = 0 (or m4 = 0), receiver Ri (or Rj) introduces an auxiliary particle A with the state |0〉A and performs a collective unitary transformation U1 in Eq. (10) on particles Sj1 (or Si2) and A under the basis {|00〉, |10〉, |01〉, |11〉} Sj1A (or {|00〉, |10〉, |01〉, |11〉} Si2A). If m1 = 1 (or m4 = 1), receiver Ri (or Rj) introduces an auxiliary particle and performs the following unitary transformation U2. 0
Case 1. T1∈{00, 10} and T2∈{00, 10}. In fact, there are 4 kinds, i.e., (T1, T2)∈{(00, 00), (00, 10), (10, 00), (10, 10)}. Supposed that (T1, T2) = (00, 00), two entangled states shared by senders Si and Sj are as follows. 1 j Θ〉Si1 Sj1 = pffiffiffi ½aj 00〉 + ð1−aÞj11〉Si1 Sj1 2 1 j Θ〉Si2 Sj2 = pffiffiffi ½aj 00〉 + ð1−aÞj11〉Si2 Sj2 2
ð18Þ
As is shown in Fig. 5, sender Si performs Bell-basis measurements on the particle Si1 and the transmitted particle of state |φ〉i. Sender Sj performs Bell-basis measurements on the particle Sj2 and the particle of state |φ〉j. The measurement outcomes owned by senders Si and Sj are denoted as Xi and Xj, respectively. The state on the particle Sj1 held by the sender Sj becomes 1 ½ax j0〉 F ð1−aÞyi j1〉Sj1 2 i 1 ½ð1−aÞxi j1〉 F ayi j0〉Sj1 2
ðXi = 00; 10Þ
ð19Þ
Xj = 00; 10 Xj = 01; 11
ð20Þ
Senders Si and Sj perform the unitary operation U− 1(Xi) and the U (Xj) on the states of particles Si2 and Sj1 in their hand, respectively. Then, sender Si (and Sj) transmits the particle Si2 (and Sj1) to receiver Ri (and Rj) via the Q-channel. Here, in order to reduce the classical communication cost in the public channel, TP codes four Bell-basis measurement outcomes T1, T2, Xi, Xj into 6 cbits messages m1m2m3m4m5m6. Here, the cbit m1 is the second cbit of Xi, and m2m3 = Xi ⊕ Xj ⊕ T2. Similarly, The cbit m4 is the second bit of Xj, and m5m6 = Xj ⊕ Xi ⊕ T1. So, the classical communication cost in the public channel has decreased. −1
Ri
i
Si1 Si2
1 0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a 2ffi 1− 0C 1−a C C 0 0C C C a C 0C − A 1−a
0
0
ð21Þ
1
For example, suppose that T1 = T2 = 00, Xi = 01 and Xj = 10. When sender Si performs the unitary operation U− 1(Xi), the state in the remaining particle Si2 belonged to sender Si collapses into i 1h axj j1〉−ð1−aÞyj j0〉 : Si2 2
ð22Þ
After performing the unitary operation U− 1(Xj), sender Sj obtains the following state 1 ½−ð1−aÞxi j1〉 + ayi j0〉Sj1 : 2
ð23Þ
Then, the particles Si2 and Sj1 are transmitted to receivers Ri and Rj. TP broadcasts the classical messages m1m2m3m4m5m6 = 111011. According to m1 = 1, receiver Ri introduces an auxiliary particle and performs a collective unitary transformation U2. Due to m4 = 0, receiver Rj introduces an auxiliary particle and performs a collective unitary transformation U1. When the state of auxiliary particle is |0〉A, receivers Ri and Rj can obtain the following states on particles Si2 and Sj1. i ah xj j1〉−yj j0〉 ; Si2 2
a ½−xi j1〉 + yi j0〉Sj1 2
ð24Þ
Finally, receiver Ri performs the recovery operation U(Xi ⊕ Xj ⊕ T2) = U(m2m3) = U(11) to obtain the Sj's state |φ〉j. Receiver Rj performs the operation U(Xj ⊕ Xi ⊕ T1) = U(m5m6) = U(11) to recover the Si's state |φ〉i. It can be seen that two unknown quantum states can be crossly teleported between two senders and two receivers via only one controller. Table 2 gives all cases corresponding to the values of Xi and Xj under the condition of T1 = T2 = 00. Case 2. T1∈{00, 10} and T2∈{01, 11}. Or T1∈{01, 11} and T2∈{00, 10}. It can be seen that there are 8 kinds in all, i.e., (T1, T2)∈{(00, 01), (00, 11), (10, 01), (10, 11), (01, 00), (01, 10), (11, 00), (11, 10)}. In this case, it will be seen that the classical communication cost is 5 cbits.
TP
Si ϕ
0
ðXi = 01; 11Þ:
Similarly, sender Si can obtain one of the following states. i 1h axj j0〉 F ð1−aÞyj j1〉 Si2 2 i 1h ð1−aÞxj j1〉 F ayj j0〉 Si2 2
a B 1−a B B B 0 U2 = B B rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2ffi B B 1− @ 1−a 0
S Sj2 j1
ϕ
j
Sj
Rj
Fig. 5. The controlled cross teleportation among two senders and two receivers.
The process of teleportation is also shown in Figs. 4 and 5. Supposed that (T1, T2) = (00, 01), two entangled states obtained are as follows. 1 jΘ〉Si1 Sj1 = pffiffiffi ½aj00〉 + ð1−aÞj11〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jΘ〉Si2 Sj2 = pffiffiffi að1−aÞ½ j01〉 + j10〉Si2 Sj2 2
ð25Þ
X.-B. Chen et al. / Optics Communications 283 (2010) 4802–4809
4807
Table 2 The cross teleportation under the condition of T1 = T2 = 00. The classical communication cost is 6 cbits. After Si (or Sj) performs the operation U− 1(Xi) (or U− 1(Xj)), he can obtain the state |ϕi〉 (or |ϕj〉) on the particle Si2 (or Sj1). Xi
Xj
|ϕi〉
m1
Ri
|ϕj〉
m4
Rj
00
00 10 01 11 00 10 01 11 00 10 01 11 00 10 01 11
[axj|0〉 ± (1 − a)yj|1〉]Si2 /2
0
U ðm2 m3 Þ
[axi|0〉 ± (1 − a)yi|1〉]Sj1 /2
0
U ðm5 m6 Þ
[± ayj|0〉 + (1 − a)xj|1〉]Si2 /2
[(1 − a)yi|0〉 ± axi|1〉]Sj1 /2
1
[axj|0〉 ∓ (1 − a)yj|1〉]Si2 /2
[axi|0〉 ∓ (1 − a)yi|1〉]Sj1 /2
0
[± ayj|0〉 − (1 − a)xj|1〉]Si2 /2
[− (1 − a)yi|0〉 ± axi|1〉]Sj1 /2
1
[ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2
0
[(1 − a)xj|0〉 ± ayj|1〉]Si2 /2
[(1 − a)xi|0〉 ± ayi|1〉]Sj1 /2
1
[± (1 − a)yj|0〉 − axj|1〉]Si2 /2
[− ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2
0
[(1 − a)xj|0〉 ∓ ayj|1〉]Si2 /2
[(1 − a)xi|0〉 ∓ ayi|1〉]Sj1 /2
1
10
01
11
[± (1 − a)yj|0〉 + axj|1〉]Si2 /2
1
After senders Si and Sj perform Bell-basis measurements and obtain the measurement outcomes Xi and Xj, the states on particles Sj1 and Si2collapse into 1 ½ax j0〉 F ð1−aÞyi j1〉Sj1 2 i 1 ½ð1−aÞxi j1〉 F ayi j0〉Sj1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i að1−aÞ xj j1〉 F yj j0〉 Si2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i að1−aÞ xj j0〉 F yj j1〉 Si2 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1−aÞ½x j1〉 F yi j0〉Sj1 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i að1−aÞ½xi j0〉 F yi j1〉Sj1 = 2 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih að1−aÞ xj j1〉 F yj j0〉 =2 iSi2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih =2 að1−aÞ xj j0〉 F yj j1〉
ðXi = 00; 10Þ ðXi = 01; 11Þ Xj = 00; 10 Xj = 01; 11 :
Si2
ð26Þ
Senders Si and Sj perform the operations U− 1(Xi) and U− 1(Xj) on the particles Si2 and Sj1, respectively. Then, senders transmit their particles to the corresponding receivers. Here, TP codes four Bell-basis measurement outcomes T1, T2, Xi, Xj into m1m2m3m4m5. Here, the cbit m3 is the second bit of Xj, and m1m2 = Xi ⊕ Xj ⊕ T2, and m4m5 = Xj ⊕ Xi ⊕ T1. Here, it is also shown that the classical communication cost is 5 if two receivers share a maximally entangled state and a non-maximally entangled state in the common teleportation. Depending on the classical messages broadcasted by TP, the receivers recover the original state. Firstly, if m3 = 0, receiver Rj introduces an auxiliary particle and performs a collective unitary transformation U1. If m3 = 1, receiver Rj introduces an auxiliary particle and performs a transformation U2. Finally, receivers Ri and Rj perform the operation U(Xi ⊕ Xj ⊕ T2) = U(m1m2) and U(Xj ⊕ Xi ⊕ T1) = U(m4m5) to recover the transmitted states. Table 3 gives all cases. Case 3. T1∈{01, 11} and T2∈{01, 11}. There are 4 kinds in all, i.e., (T1, T2)∈{(01, 01), (01, 11), (11, 01), (11, 11) }. In this case, it will be seen that the classical communication cost is 4 cbits. The process of teleportation is also shown in Figs. 4 and 5. Supposed that (T1, T2) = (01, 01), two entangled states obtained are as follows.
j Θ〉Si1 Sj1 j Θ〉Si2 Sj2
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = pffiffiffi að1−aÞ½ j01〉 + j10〉Si1 Sj1 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = pffiffiffi að1−aÞ½ j01〉 + j10〉Si2 Sj2 2
After sender Si and Sj perform Bell-basis measurements and obtain the measurement outcome Xi and Xj, the states on particles Sj1 and Si2 collapse into
ð27Þ
ðXi = 00; 10Þ ðX = 01; 11Þ i Xj = 00; 10 Xj = 01; 11 :
ð28Þ
Sender Si and Sj perform the operations U− 1(Xi) and U− 1(Xj) on the particles Si2 and Sj1, respectively. Then, senders transmit their particles to corresponding receivers. TP codes all the measurement outcomes into 4 cbits m1m2m3m4. Here, the cbits m1m2 = Xi ⊕ Xj ⊕ T2, m3m4 = Xj ⊕ Xi ⊕ T1. Sometimes, when T1 = T2, the classical communication cost are 2 cbits. Depending on the classical messages broadcasted by the TP, the receiver Ri makes the recovery operation U(Xi ⊕ Xj ⊕ T2) = U(m1m2), and receiver Rj performs the operation U(Xj ⊕ Xi ⊕ T1) = U(m3m4). Table 4 gives all cases. Similarly, it can be calculated that the classical information entropy is 8 if TP does not encode the measurements outcomes. However, in our quantum communication network, TP encodes the measurement outcomes and broadcasts it. The maximum classical communication cost is 6 cbits, i.e. m1m2m3m4m5m6. Compared with
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The cross teleportation under the condition of T1 = 00, T2 = 01. Here, að1−aÞ = 2 = b. Xi
Xj
00 00 10 01 11 10 00 10 01 11 01 00 10 01 11 11 00 10 01 11
|ϕi〉
Ri
|ϕj〉
m3 Rj
b[xj|1〉 F yj|0〉]Si2
U(m1m2)
[axi|0〉 ± (1 − a)yi|1〉]Sj1 /2
0
b[xj|0〉 F yj|1〉]Si2
[(1 − a)yi|0〉 ± axi|1〉]Sj1 /2
1
b[− xj|1〉 F yj|0〉]Si2
[axi|0〉 ∓ (1 − a)yi|1〉]Sj1 /2
0
b[xj|0〉 b yj|1〉]Si2
[− (1 − a)yi|0〉 ± axi|1〉]Sj1 /2 1
b[xj|0〉 F yj|1〉]Si2
[ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2
0
b[xj|1〉 F yj|0〉]Si2
[(1 − a)xi|0〉 ± ayi|1〉]Sj1 /2
1
b[xj|0〉 b yj|1〉]Si2
[− ayi|0〉 ± (1 − a)xi|1〉]Sj1 /2 0
b[− xj|1〉 F yj|0〉]Si2
[(1 − a)xi|0〉 ∓ ayi|1〉]Sj1 /2
1
U(m4m5)
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Table 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The cross teleportation under the condition of T1 = 01, T2 = 01. Here, að1−aÞ = 2 = b. Xi
Xj
|ϕi〉
Ri
|ϕj〉
Rj
00
00 10 01 11 00 10 01 11 00 10 01 11 00 10 01 11
b[xj|1〉 ± yj|0〉]Si2
U(m1m2)
b[± xi|1〉 + yi|0〉]Sj1
U(m3m4)
10
01
11
b[xj|0〉 ± yj|1〉]Si2
b[xi|0〉 ± yi|1〉]Sj1
b[− xj|1〉 ± yj|0〉]Si2
b[± xi|1〉 − yi|0〉]Sj1
b[xj|0〉 ∓ yj|1〉]Si2
b[xi|0〉 ∓ yi|1〉]Sj1
b[xj|0〉 ± yj|1〉]Si2
b[xi|0〉 ± yi|1〉]Sj1
b[xj|1〉 ± yj|0〉]Si2
b[± xi|1〉 + yi|0〉]Sj1
b[xj|0〉 ∓ yj|1〉]Si2
b[xi|0〉 ∓ yi|1〉]Sj1
b[− xj|1〉 ± yj|0〉]Si2
b[± xi|1〉 − yi|0〉]Sj1
11
ρ = ∑m;n
Now, we analyze the security of this protocol. It can be seen that Eve has one chance to eavesdrop the messsge when sender Si (and Sj) transmits the particle Si2 (and Sj1) to receiver Ri (and Rj) via the Qchannel. That is, Eve can first intercept the particles transmitted, and then recover the message according to the classical information broadcasted by the TP. Unfortunately, the technology of checking eavesdropping in the Subsection 2.3 is still taken by communicators. That is, sender Si (and Sj) inserts some decoy photons randomly. If the error rate is enough low, the procedure of the protocol can go on. Otherwise, this communication is abandoned. Moreover, not only any eavesdropping will inevitably disturb the states of the decoy photons and be detected, but also Eve obtains no any useful information. It can be calculated that from Eve's point of view, the state carring the message is the mixed state. In fact, the particle transmitted from the sender to the receiver are in two kinds of states. One is that the particle is in one of the 16 states in the Table 2 or Table 3 with probablity a2/4. The other is that the particle is in one of the 16 states in the Table 3 or Table 4 with probablity a(1 − a)/4. For example, the partilce Si2 in the Table 3 is in one of the 16 states with probablity a(1 − a)/4. Thus, the density operator of the partilce Si2 system is i i 00 jϕmn 〉〈ϕmn j
=
ð29Þ
Notice that this state is a mixed state, since tr((2a(1 − a)I)2) = 8a (1 − a)2 b 1 (here, a b 1/2). This state has no dependence upon the state |φ〉j = xj|0〉 + yj|1〉 being transmitted, and thus any measure2
2
= 2a I:
ð30Þ
4. Conclusion
3.2. Security analysis
að1−aÞ h xj j1〉 + yj j0〉 xj 〈1 j + yj 〈0j 4 + xj j1〉−yj j0〉 xj 〈1j−yj 〈0 j i + ⋯⋯ + −xj j1〉−yj j0〉 −xj 〈1j−yj 〈0 j að1−aÞ h 2 2 2jxj j j1〉〈1j + 2 jyj j j0〉〈0j = 4 i 2 2 + 2jxj j j0〉〈0j + 2jyj j j1〉〈1j × 4 = 2að1−aÞI:
j j = 00 jϕmn 〉〈ϕmn j
So, it can prevent Eve from stealing any useful information.
8 cbits, 2 cbits are saved in this protocol. Moreover, it demonstrates that the classical communication cost is 4 when the maximally entangled EPR pairs are used as the information carriers, and more classical communication cost can be saved. In comparison with the protocol of Section 2, the feature of this section is that two receivers can simultaneously obtain their own qubits.
ρ = ∑11 m;n =
ments performed by Eve will contain no information about |φ〉j = xj| 0〉 + yj|1〉. Similarly, after implementing the unitary operation U, the particle Sj1 in the Table 3 is in one of the 16 states with probablity a2/4. Thus, the density operator of the partilce Sj1 system is
The central theme of this paper is that by constructing a quantum communication network, we propose the centrally controlled multiuser teleportation. We give a protocol for the controlled teleportation between an arbitrary sender and an arbitrary receiver, and a protocol for the controlled cross teleportation in which two receivers can simultaneously receive their own qubits. It is well known that the quantum entanglement resources and the classical communication are necessary for realization of the controlled teleportation. In order to save the entanglement resources, the controller shares the entangled state with every sender. All senders and receivers do not directly share the entangled state, while the sender only connects corresponding receiver with the Q-channel. This may be a merit in the implementation of a realistic quantum teleportation network. Moreover, the present protocol reduces the classical communication cost in the public channel by means of the coding. The security analysis is given because some particles are required to be transmitted in the process of the communication. It is shown that our protocol is secure. Not only is the eavesdropping inevitably detected, but also Eve obains no any useful information. Acknowledgements Project supported by the National Basic Research Program of China (973 Program) (No. 2007CB311203), the National Natural Science Foundation of China and the Research Grants Council of Hong Kong Joint Research Scheme (No. 60731160626), the National Natural Science Foundation of China (Nos. 61003287, 60821001, 60873191, and 60903152), the Beijing Natural Science Foundation (No. 4102055), and the Fundamental Research Funds for the Central Universities (No. BUPT2009RC 0220). References [1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [2] X.B. Chen, N. Zhang, S. Lin, Q.Y. Wen, F.C. Zhu, Opt. Commun. 281 (2008) 2331. [3] X.B. Chen, Q.Y. Wen, G. Xu, Y.X. Yang, F.C. Zhu, Phys. Rev. A 79 (2009) 036301. [4] D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Nature 390 (1997) 575. [5] D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, Phys. Rev. Lett. 80 (1998) 1121. [6] A. Karlsson, M. Bourennane, Phys. Rev. A 58 (1998) 4394. [7] F.L. Yan, D. Wang, Phys. Lett. A 316 (2003) 297. [8] J. Bae, J. Jin, J. Kim, C. Yoon, Y. Kwon, Chaos Solit. Fract. 24 (2005) 1047. [9] Z.X. Man, Y.J. Xia, N.B. An, Phys. Rev. A 75 (2007) 052306. [10] H.J. Cao, Z.H. Chen, H.S. Song, Phys. Scr. 78 (2008) 15002. [11] Z.J. Zhang, Z.X. Man, Phys. Lett. A 341 (2005) 55. [12] Z.J. Zhang, Phys. Lett. A 352 (2006) 55. [13] C.P. Yang, S.I. Chu, S.Y. Han, Phys. Rev. A 70 (2004) 022329. [14] F.G. Deng, C.Y. Li, Y.S. Li, H.Y. Zhou, Y. Wang, Phys. Rev. A 72 (2005) 022338. [15] H. Buhrman, R. Cleve, W. van Dam, Arxiv preprint quant-ph/9705033 (1997). [16] M.X. Luo, X.B. Chen, S.Y. Ma, Y.X. Yang, Z.M. Hu, J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 065501. [17] H.K. Lo, Phys. Rev. A 62 (2000) 012313. [18] A.K. Pati, Phys. Rev. A 63 (2000) 014302. [19] C.H. Bennett, D.P. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal, W.K. Wootters, Phys. Rev. Lett. 87 (2001) 077902. [20] C.A. Yen, S.J. Horng, H.S. Goan, T.W. Kao, Y.H. Chou, Arxiv preprint 0903.3444 (2009). [21] R. Ahlswede, N. Cai, S.Y.R. Li, R.W. Yeung, IEEE Trans. Inf. Theory 46 (2000) 1204.
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