Effect of correlation on Perfect K pair Coding

M. Mahdian 1 , R. Bayramzadeh 1

1Faculty of Physics, Theoretical and Astrophysics Department ,University of Tabriz, 51665-163, Tabriz, Iran.

We find a protocol transmitting K quantum states crossly in the quantum networks with K pair sender-receiver and one bottleneck channels only with sharing quantum correlations between senders. In these networks senders want to multicast perfect quantum states to receivers. Perfect send on the quantum networks is impossible just using some additional resources like quantum correlation. This protocol has been used maximum entangled states and Quantum Discord which is best indicator of the quantum nature of the correlation to sharing between senders. At first, to send perfect K states in quantum network applied some maximum entanglement states to teleport quantum states, Enistein – Podolsky – Rosen (EPR) states as simple and two states and Grrenberger – Horne – Zeilinger (GHZ) states and W states as kinds of three qubit states. Then reached that there is no difference between sharing these maximum entanglement states and must send same amount of classical bits in bottleneck channel. In the second part of this paper of sharing Quantum Correlation (quantum discord) perfect send of two quantum states is possible in the Butterfly network with fidelity equal one.

Ι. Introduction

The excellent idea of network coding proposed by Ahlswede, et al [1] in 2000, opened up a new communication method of transmition information through networks. The network itself is given as a weighted, directed acyclic graph with the weights denoting the capacities of the edges with noiseless channels.Maximizing information exchange over classical communication networks has been a bigger subject among both the and the Figure 1 :The Classical 2 pair Butterfly network that total networking societies. Especially, a large scale flow going through each edge in the network does not network often has a bottleneck point that causes a exceed its capacity. transmission rate relatively small for the size of its communication resource, Bottleneck restricts the Network communication was traditionally done by total performance of communication. The important routing also known as the store and forward method idea is to allow coding and replication of information in which received data is simply copied and locally at any intermediate node of the network for forwarded without data processing, network coding instance, this allows to send two bits simultaneously in general offers throughput benefits as compared to between two source-target pairs over several the routing.Also network coding makes the upper networks for which the same task cannot be solved bound on the achievable rate given by the min-cut by routing in the Butterfly network depicted in Fig. (1),then must used swap operation where a coding max-flow condition is in fact always achievable.In scheme is a choice of operations for internal nodes in other words, network coding allows one to send graph. mmassages if and only if the value of any cut between the source node and each target node is at

E-mail: [email protected] least mthen network coding are good for high much as possible, the work of Leung, et al [6], they communication rate transmission. We refer to Refs investigated various settings of quantum network [2-5] for extensive treatment of the classical network coding assisted with supplied resources such as free coding.The multicast problem is a task that can be classical communication or entanglement. They elegantly solved by network coding; in this K pair considered the case where classical communication problem all messages at one source node must be sent can be sent only between each pair of nodes to each of several target nodes;For each iϵ {1…, K}a connected by a quantum channel and only in the massage xiis given at the source Si, and has to be sent direction of that quantum channel. They found that quantum two pair problem on the butterfly network to the target ti through the network under the condition that each edge has unit capacity, The cannot be solved even under this model.In 2009, Butterfly network is an instance of the two pair under the assumption that there was no prior problem. Note that the classical K pair problem, that entanglement shared among any of the parties, but the channels and messages are classical, is one of the that classical communication was free, Kobayashi et most important network coding problems [2]. al, gave a perfect quantum network communication protocol based on classical network coding and Since the faults of its must because of classical communication does not increase replaced with general but the amount of entanglement of the system [7-9].This making the copy of digital information is convenient protocol requires only one qubit transmission or two but it is not possible exactly for . bits transmission in each channel in the network.The The important question is whether or not quantum single-qubit quantum and 2 bit classical channels are network coding is possible, we consider a problem of mutually in equivalent resources. Note that super quantum communication between parties that are dense coding implies that a single qubit quantum connected through a network of quantum channels. In channel and shared 1 ebit entanglement together have spite of no-cloning theorem and has not quantum the capacity of 2 bit classical channels, and copy operation, we can send quantum information in teleportation shows 2 bit classical channel and shared quantum networks that have quantum channels and 1 ebit entanglement together have the capacity of a transmit in long distance using single qubit quantum channel. Other additional network coding to reduce the amount of quantum resources are entanglement and Prior entanglement communication. At 2007 Hayashi , et al [11] provides some miracle performances in quantum performed network coding discuss to quantum information. Moreover, prior entanglement enables systems and their resulted are that perfect quantum the transmission of quantum state only by sending network coding is impossible and we can't send classical information. Hayashi finded a protocol to quantum states in quantum networks with fidelities sending perfect states in the butterfly network with equal one. They concluded that we can achieve prior entanglement among senders [13]. fidelities greater than 1/2 and lower that 1. For more studying on quantum network coding [6, 11-13] In real situations, however, it is most of the time not discusses about quantum transmission on quantum possible to have a maximum entangled state at one’s networks with quantum channels. In the quantum disposal. Because of the interaction with the case, the methods used in the classical case cannot be environment, the quantum state of any system will applied directly and must used additional resources became the mixed state after a certain period then for example classical communications and always maximum entanglement states is not entanglement [6,13]. The K pair problem is one of accessible, entanglement is susceptible to interaction the most important network coding problem, and in with the environment which results in quantum discussion quantum K pair problem is not disentanglement. So our group wants to study perfectly solvable, we motivated with this papers to quantum network coding with quantum correlation investigate K pair quantum network coding. Perfect that are more interesting and powerful than network coding is impossible for all [18]. Entanglement may disappear completely after a finite time, an effect known as protocols, the fidelities at nodes t 1 and t 2 are <1; without additional resources whereas imperfect entanglement sudden death. There exist, however, network coding is possible: there exists a quantum quantum correlations more general and more protocol whose fidelities at targets are >1/2. To fundamental than entanglement.Resistance of reduce the amount of classical communication as quantum correlation against decoherency is more

E-mail: [email protected] than entangled states and we have high speed , , and transition with quantum correlation. There are many quantum computation and so on. Entangled states, application of quantum correlation in quantum which are called quantum channels in teleportation information theory,one of the usage is quantum processing, make it possible to send an unknown teleportation [14, 15] such also used in this paper. quantum state in long distance that maximum One of the other quantum correlation types which is entangled states, i.e. Bell states, are used as quantum more general than entanglement, is quantum discord. channels in perfect quantum teleportation. As So quantum discord , a more robust and easy to gain additional resources to the quantum network coding access to phenomenon than entanglement, can also we allow sharing maximum entanglement states deliver a quantum advantage introduced by Olivier between senders to send perfect quantum states.If and Zurek [16] for a rather limited set of bipartite two bit transmission is allowed instead of one qubit systems that has nonzero value for some mixed transmission, only using quantum teleportation, and as an informational-theoretical perfect transmission of quantum state is available. measure of quantumness of correlations. Quantum Therefore perfect state transfer is possible, we also discord is defined as the difference between general want to achieve it with as few uses of the network as mutual information and classical information [16-18]. possible then must used Network coding.We have generalized quantum 2 pair perfect quantum network This paper investigates the K pair quantum network coding with sharing two maximum entanglement coding and sending perfect quantum states on the EPRstates [13, 15] in Eq. (1) to Kpair quantum networks. This paper used K maximum entanglement network coding and calculated the capacity of states to sharing between senders as additional bottleneck channel to send classical communications. resources to remove limitation on quantum information processing. Using results of teleportation 1 φ EPR =()00 + 11 (1) K quantum states and amount of classical 2 communication, we concluded for quantum network coding must send same amount of classical bits. As the results, find that same amount of classical For teleportation two quantum states [14] in the communication needed for these maximum quantum butterfly network for sending perfect states entanglement states; EPR states, GHZ states and W showed in Fig. (1), we need two classical bits and states; that calculated 2(K-1) bits on bottleneck must send two bits in bottleneck channel to decoding channel for crossly sending of course this protocol arbitrary states in targets according to [13,15]. alloweither or bits classical communication in networks. Main discussion of the paper is thatin the quantum network if senders want to send some quantum states with special angles in the Bloch sphere to receivers, fidelity one can approached with quantum discord.

This paper has organized as follows: In Sec.2 we described K pair perfect quantum network coding with sharing maximum entanglement EPR states, In Sec.3 we review K pair Quantum network coding with sharing GHZ and W states, in Sec. 4 we introduces the Quantum correlation and described how Quantum teleportation and Quantum network coding can be perfect with using quantum discord, Conclusions are then presented in Sec.5. ΙΙ . Kpair Quantum network coding with sharing Maximum entanglement EPR states

Entanglement is considered to be a fundamental resources of quantum information processing such as

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Figure 2: K pair quantum network that it’s possible perfect sources to their targets.For simplicity, first we treat 3 send with sharing K maximum entangled states. pair quantum network and presented its in Fig. (3) To sending perfect three states with K pair problem which have K sources and K targets sharing three EPR entangled states. in Fig. (2) Must send K quantum states of specific

ψ1

EPR φ1

ψ 3

ψ 2

EPR φ2

ψ 1

ψ 3

EPR φ3 ψ 2

Figure 3: The Quantum circuit for perfect 3 pair quantum network with sharing three maximum entangled EPR states.

Now, we have initial state according to Eq. (2) which The replaced form of quantum states multiply in Eq. every represent the information of sender, and (4) is the same with Eq. (3)for example the goal| is send this state perfectly. described three multiple states belong to sources|000 that all of them were zero. initial state :

ψ⊗ ψ ⊗ ψ ααα123000+ ααβ 123 001 1 2 3 (2) ( ()SSS123,,() SSS 123 ,, +αβα123010 + αββ 123 011 ()SSS123,,() SSS 123 ,, +βαα123100 + ββα 123 110 Then shared three maximum entangled states ()SSS123,,() SSS 123 ,, between senders and send last qubit of entanglement +βαβ123101 + βββ 123 111 ()SSS123,,() SSS 123 ,, ) states or state of Bob in Teleportation processing with sender which we want to send crossly arbitrary ⊗ 00 00 00 quantum states to proper receivers, from this point ( hh1,1, 2,1 hh 1,2 , 2,2 hh 1,3 , 2,3 + 00 00 11 Eq. (3) duplicated the quantum state. hh1,1, 2,1 hh 1,2 , 2,2 hh 1,3 , 2,3 + 00 11 00 hh1,1, 2,1 hh 1,2 , 2,2 hh 1,3 , 2,3 ψ1⊗ ψ 2 ⊗ ψ 3 EPR EPR EPR + 11 00 00 hh1,1, 2,1 hh 1,2 , 2,2 hh 1,3 , 2,3 ⊗φ1 ⊗ φ 2 ⊗ φ 3 + 00 11 11 ≡Q ⊗ Q ⊗ Q hh, hh , hh , S1 S 2 S 3 1,1 2,1 1,2 2,2 1,3 2,3

⊗(,hh1,1 2,1 + hh 1,1 ,) 2,1 + 11 11 00 hh1,1, 2,1 hh 1,2 , 2,2 h 1,3 , h2,3 ⊗(,hh + hh ,) 1,2 2,2 1,2 2,2 + 11 00 11 hh1,1, 2,1 hh 1,2 , 2,2 hh 1,3 , 2,3 ⊗(,hh1,3 2,3 + hh 1,3 ,) 2,3

=()α10 + β 1 1 + 11 11 11 hh1,1, 2,1 hh 1,2 , 2,2 hh 1,3 , 2,3 ) (4) ⊗()α20 + β 2 1

⊗()α30 + β 3 1 ⊗()00 + 11 For perform protocol to perfect send in 3 pair ⊗()00 + 11 quantum network, apply the Hadamard gate on ⊗()00 + 11 (3)

E-mail: [email protected] quantum states of three sources states and CNOT gate measurement of the first, second and third quantum to proper states Eq. (5) which these gates act states. For example in Fig. (3), the state h2, 2 should be according to Eq. (6), send second state to its receiver as same as teleportation must apply XlZkgates on this state to HQ[ , Q , Q ] S1 S 2 S 3 receive.But this state belongs to third of the Alice n m CNOT[,, Qh Q ,,,] h Q h stateand in this circuit X Z transformation effected. ()()S11,1 S 2 1,2() S 3 2,3 (5) Now, using of coding and summing received bits in 1 xz H=∑ () − 1 z x module two on the internal nodes, perfect sending 2 x, z∈ F 2 can be approached with sending two bits n+l and m+k on the bottleneck channel. CNOT(1,2 ) = xxy + xy ∑ 1 212 x, y∈ F 2

σ x =∑ x + 1 x x∈ F 2

z σ z =∑ () − 1 z z z∈ F 2 (6)

For EPR states sharing on the teleportation processing there are four measurement basis that is same for network coding to access classical bits for sending to targets, basis of measurement : 00 , 01 , 10 , 11 (7) Figure 4: Capacity of channels in 3 pair quantum network for perfect sending three quantum states.

According to Eq. (6) after earn two classical bits of Thereforein 3 pair network toperfect send three each sources and applying X and Z Pauli quantum states to its receivers there should be sent 4 transformation to remove additional phases on Bob bits (i+k, j+l, k+m, l+n) on the bottleneck channel,this qubit,those bits must send to internal nodes in the is represented in Fig. (4).It’s possible to generalize 3 network and because of limitation on the capacity of pair quantum network to K pair quantum network bottleneck channel,encoding of classical bits is coding.This paper introduce a new generalization for necessary. K pair network that must send 2(K-1) bits on bottleneck channel whereas K pair problem does not After applying CNOT and Hadamard transformation solved without network coding.If the ordering of theAlice states must be measured on suitable basis. receiver places be clockwise or non-clockwise and if The attained classical bits that are two quantity 0 and there is a definite permutation in ordering of receiver 1. For recovering perfect states on the last qubit of places, for perfect transverse of K quantum states in entanglement states must a proper X and Zgates quantum networks number of bits is needed are 2(K- applied.At the end of using classical coding the 1). We can define K as the place of K receivers targets can access final state such as Eq. (8),thatis which replaced in the network and any receivers do equal to initial state which every represent the not be in front of its senders for perfect send thenit information of targets. | needs2(K-1) bits on the bottleneck channel. If all receivers placed in front of its senders and one final state : quantum channel supposed as pass channel for

ψ⊗ ψ ⊗ ψ perfect send we don’t need any classical bits 1 2 3 whereasin these setting perfect quantum network =Q ⊗ Q ⊗ Q T1 T 2 T 3 (8) needs K maximum entanglement EPR states.

ΙΙΙ . Quantum network coding with sharing Interesting point is that for transverse information of GHZ states the senders to the targets must Coding used to reduce classical mass of the communications. Assuming (i, Entanglement in three qubits is more complicated j),(k, l),(m, n) are classical bits that obtained after than that in two qubits. Recently, the entanglement of

E-mail: [email protected] three qubits was classified by separable, bi-separable, otherqubits in the middleof the stateno longer is GHZ and W states. The GHZ class state cannot be needed. transformed to the W class by the local operation and classical communication. Although many suggestion has utilized the GHZ state in quantum teleportation and some proposals have suggested an 1 φ GHZ =()000 + 111 (9) implementation of the W state. The GHZ state and the 2 W state can be used as the quantum channel for the For sending K quantum states in the perfect teleportation of an unknown qubitstate. In quantumnetworkwith sharing GHZ statespresented in these type of teleportation and sending quantum Fig. (5). Therefore, there is no difference between states in quantum channels, Alice channel has three EPR and GHZ states for sharing and in both of them qubit channels but in EPR states sharing the Alice it sends same amount of bits on bottleneck channel has two qubit channels. At the first look, channels.If share K states of EPR or GHZ states in K decoding of any states in any receivers it’s obvious pair network must send 2(K-1) bits to perfect send. that it need three bits indeed on the teleportation with sharing GHZ state that agreeswithEq. (9) Two bits are necessary for decoding and perfect sending and some

ψ1

GHZ φ1

ψ K

ψ 2

GHZ φ2

ψ1

ψ K

GHZ φK ψ K −1

Figure 5: The Quantum circuit for Perfect K pair quantum network sharing Ktimes of GHZ states.

A. Quantum network coding with sharing W states The method that how we share these states dependent The fact that W statein Eq. (10) having different type to how we want to multicast quantum states to of GHZ states entanglement is an interesting topic for receivers.For instance, with supposition exist one studying. The basis of measurement in teleportation quantum channel of one sender to another receiver with sharing W state are same as basis of GHZ state directly with onequbit capacity quantum channel.If sharing then the classical bits needed for decoding we want to send quantum states of one sender to its and applying Pauli transformation for both of them receiver thatexist in front of another sender, we share are same. maximum entanglement states between these two senders. We attend to share three qubit states so W 1 every three qubit quantum states for teleportation φ =()100 + 010 + 001 (10) 3 have two parts that called the Alice and Bob channels.The two qubitof three qubit states belong to

E-mail: [email protected] the Alice that it want to send her quantum state and . Since perfectly to Bob that have just one qubitchannel. To quantifies the classical| = correlation, / the difference in have perfect sending we must share Bob's qubit with Eq. (14), definite sender. The quantum gates that needs for sending Kquantum states with sharing W states as (14) same as the quantum circuit presented in Fig. (5) Just = − these difference that must share Ktimes of W states Defines the quantum discord that quantifies the instead GHZ states and the bits needed in the quantum correlation. Also the minimum in Eq. (14) bottleneck channel were equal. can be taken over all von Neuman measurement and IV. Quantum Network Coding using we denote the corresponding classical correlation as Quantum Discord and quantum discord as , respectively Ref (20). A. Introduction of Quantum Correlation

Entanglement measure of concurrence and quantum discord are two different types of the quantum B. Application of Quantum Correlation correlations. Several measures of quantum on the Teleportation process correlation have been investigated in the literature To perform teleportation procedure, we must have a and among them the quantum discord has recently quantum state to be shared between Alice and Bob as received a great deal of attention. The quantum quantum channel. Correlations in this quantum state, discord measures quantum correlations of a more has an important role in teleportation is that the general type than entanglement; such as there process won’t be done without these correlations. We existsseparable mixed states having nonzero discord. consider single qubit state which want to teleport as Interestingly, proved both theoretically and experimentally that such states provide computational speedup compared to classical states in some θiφ θ ψ in =cos 0 + e sin 1 (15) quantum computation models Ref. [19]. For a given 2 2 quantum state ρ of a composite system AB the total | in Eq. (15), amount of correlations, is quantified by the quantum mutual information defined as Eq. (11). Where ϴ and ϕare real number define a point on the

Itot(ρ) = S( ρ A) + S( ρ B ) − S ( ρ ) (11) unit three dimensional Bloch sphere, it provides auseful means of visualizing the state of single qubit.For simplicity, we take as initial states of the Where S( ρ)denotes the von Neumann composite system of channels a class of states with entropyaccording to Eq. (12),

S(ρ) = − Tr ( ρlog ρ ) (12) 1 3  ρc=I ⊗+ I∑ c iii σ ⊗ σ  (16) 4 i=1  maximally mixed marginal, as ρcfrom Eq. (16), And ρA(B) and ρ are the reduced of subsystem A(B) and the density matrix of the total system, respectively where ρA= Tr B(ρ) , ρB= Tr A(ρ). Where ci is a real number such that for An alternative version of the mutual information can every i and I the identity operator0 ≤ of| the| ≤ 1 total be defined as Eq. (13), systemand σi are and Pauli operators, this class of states includes the Bell states if . (13) .Quantum discord computed for| two-qubit| = || =X | states| = = − | 1Ref. [18] with this, we obtain quantum discord for density matrix that is defined in Eq. (16).Thenwe Where the minimum is taken over all possible want to see the effect of quantum discord on quantum POVM s on subsystem A with correlation, so investigate the teleportation with =

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1 − iφ2 i φ 2 θ Fee=( cos (2() − t 1 cosθ cos φ 2 2 −isinφ() 1 +−+ c 3 () 1 2cos θ ) θ −sin2 (() − 2 − t cosθ cos φ 2 1 +isinφ() −+ 1 c 3 () 1 + 2cos θ )) (22 ) channel that its entanglement is zero but discord is non-zero.

For this purpose, use the separable condition cc12−=−1 ccc 312 , +=+ 1 c 3 (17)

(Positive - Partial - Transpose) Ref. [21], of quantum 1 2θ 2  θ  ρout =(sint1 + t 2 cos   00 channel and obtain the following equation Eq. (17) 2 2  2 

 −iφθθ i φ  θθ  +te1cos sin + te 2 cos  sin 01 1+c3 0 01 − c 3   22  22    1 01−c3 1 + c 3 0  ρ =  −iφθθ i φ  θθ  c te te 4 01+c 1 − c 0  + 2cos sin + 1 cos  sin 10 3 3   22  22  1−c3 0 01 + c 3  2θ  2  θ   (18) +t2sin  + t 1 cos    1 1 ) 2   2   (19) on c ; i

Which for simplicity have Eq. (20); By substituting Eq. (17) in Eq.(16), we get the channel density matrix equal to Eq. (18). To measure the overlap between input and output states there are quantity that is called Fidelity and can If use the Eq. (18) as channel, the entanglement is be given by Eq. (21), zero, Ofcourse; we suppose that Alice wants to teleport state expressed as Eq. (15) to Bob while the channel described as Eq. (16). According to Eq. (15) And then the fidelity calculated according to Eq. and Eq. (18), we have output according to Eq. (19), (22),

By considering ϕ = 0 , we plotted the fidelity of quantum teleportation versus ϴ and ϕ in Fig. (6).

Noting that in this figure it is assumed that c3 = 0 . Fig. (7) plotted the fidelity as function of ϴ such that

c3 = ϕ = 0 .This figure shows that the fidelity only at point ϴ = π / 2 is equal to one and elsewhere is t=−1 ct , =+ 1 c (20 ) 1 32 3 always smaller than one. So we can transfer one quantum state perfectly under certain condition by using the quantum discord.

F = ψin ρ out ψ in (21)

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1 3  ρc=I ⊗+ I∑ c iii σ ⊗ σ  4   i=1 1 3  ρc′ =I ⊗+ I∑ c iii′ σ ⊗ σ  4 i=1  (24)

Using of results teleportation process with quantum discord, we can generalized to the butterfly network of two pair source- targets and send two quantum states perfectly of sources to targets. Then if quantum Figure 6 : The Fidelity of quantum Teleportation as states belong to sources were special states with function of ϴ and ϕ for c =0. 3 particular angles on Bloch sphere namely Eq. (25) perfect send accessible.

1.0 0.9 = = , 2 (25) 0.8 = = 0 , = = 0 . 0.7

Fidelity This condition on transmission quantum states can 0.6 applied to K pair quantum networks that the goal is sending perfect K quantum states of sources to 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 receivers. q

Figure 7 : The Fidelity of quantum Teleportation as function of ϴ for ϕ=0 and c 3=0. V. Conclusion

At end we concluded that, both of quantum discord It’s possible to send K quantum states of K senders to and entanglement played an important role in K receivers with fidelity equal one in quantum quantum teleportation and in absence of networks. In this paper have been used of quantum entanglement, discord causes non-zero fidelity. correlation to send perfect quantum states. Using of maximum entangled states; EPR, GHZ and W states we must used classical communication in bottleneck C. Application of Quantum Correlation channels. In Kpair networks must send 2(K-1) times of bits on bottleneck channel for all three entangled in the Quant.um Network Coding states sharing if targets placed on the permutation In the famous Butterfly network, we consider to places.Using of quantum correlation can send sending two quantum states with sharing two quantum states of senders to receivers that have not quantum states that have not entanglement and using any prior entanglement. quantum discord as channels. In these setup, we have two quantum statessame as Eq. (23) that must send to receivers with different special angles, VI. References

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