Continuous-Variable Quantum Network Coding Based on Quantum Discord

: Establishing entanglement is an essential task of quantum communication technology. Beyond entanglement, quantum discord, as a measure of quantum correlation, is a necessary prerequisite to the success of entanglement distribution. To realize efficient quantum communication based on quantum discord, in this paper, we consider the practical advantages of continuous variables and propose a feasible continuous-variable quantum network coding scheme based on quantum discord. By means of entanglement distribution by separable states, it can achieve quantum entanglement distribution from sources to targets in a butterfly network. Compared with the representative discrete-variable quantum network coding schemes, the proposed continuous-variable quantum network coding scheme has a higher probability of entanglement distribution and defends against eavesdropping and forgery attacks. Particularly, the deduced relationship indicates that the increase in entanglement is less than or equal to quantum discord.

decode operation, while Gaussian cloning is used to simulate the copy operation of QNC. Also, considering practical advantage, we utilize continuous variables for QNC to increase the success probability of transmitting the ancillary mode to the corresponding target. The main contributions of our work are: (1) A CVQNC scheme based on quantum discord is proposed. The basic operation of ADD/SUB operators and Gaussian cloning are provided. Based on the butterfly network, two separable auxiliary modes are transmitted between source nodes and target nodes, achieving continuous-variable entanglement distribution. As a result, the fidelity of the link for transmitting the ancillary modes from sources to targets is 2 / 1 . Each target node receives N 2 log 4 bits of classical information via one network transmission. Also, our scheme can defend against eavesdropping and forgery attacks.
(2) Theoretical results based on quantum discord are deduced. The relationship between entanglement distribution and quantum discord is quantified and the theorem that the entanglement gain between source nodes and target nodes is less than or equal to quantum discord is deduced. This paper is organized as follows. In Section 2, we introduce related works, including continuous-variable EDSS, ADD/SUB operators and Gaussian cloning. Section 3 gives a CVQNC scheme based on quantum discord. Section 4 focuses on the scheme analysis in term of performance and security. Section 5 is our conclusion.

Related works 2.1 Continuous-variable entanglement distribution by separable states (CV-EDSS)
By sending a separable ancillary mode c from Alice to Bob, the CV-EDSS protocol [Mišta Jr. and Korolkova (2009)] aims to entangle mode a at Alice with separable mode b at Bob (see Fig. 1).
Step 1 where the squeezing parameter is 0 ≥ t , I is a two-dimensional identity matrix. Then according to the Gaussian distribution with correlation matrix ) (x Q , Alice and Bob respectively displace their own modes by random correlated displacements where z σ denotes the z Pauli matrix and . As a result, a three-mode fully separable Gaussian state with a CM Eq. 3 will be prepared. (3) Figure 1: This is the process of continuous variables entanglement distribution by separable Gaussian states. Empty ellipses represent that modes a and c which are in the momentum and position-squeezed vacuum states, and empty circle represents that mode b is in a vacuum state Step 2. The modes a and c are superimposed on a balanced beam splitter . The state is separable with respect to ac b − splitting and for 2 , also with respect to ab c − splitting.
Step 3. The mode c is mixed with mode b on a balanced beam splitter We denote the reduced state of the mode a and the mode b is the matrix By calculating the logarithmic negativity given by the formula υ , therefore the modes a and b are entangled for an arbitrarily small nonzero squeezing parameter. In this protocol, two modes of the state are entangled by mixing them sequentially with the third separable mode on two beam splitters. Beyond point-to-point communication, it can help construct the entanglement between sources and targets via intermediate nodes, especially in a butterfly network. By encoding the quantum states, the information of the third mode in EDSS will be hidden and guarantee that the modes a and b will be entangled ultimately between sources and targets.

ADD/SUB operators
After mixing two single-mode states  .We give a diagram of ADD/SUB operators (see Fig. 2). The 50 : 50 beam splitter BS mixes the input states. One of the beams of BS is selected as the input of the noiseless linear amplifier (NLA). After the amplification process with a factor 2 = g , we get the desired state 2 1 α α α

Figure 2: Diagram of ADD/SUB operators
Hence the ADD operator and the SUB operator are defined [Shang, Li and Liu (2017)] as follows: We apply the ADD operator and the SUB operator to encode and decode coherent states (see Fig. 3).

Figure 3: Encode and decode coherent states by applying the ADD operator and SUB operator
At the source node, we obtain the encoded state by applying the ADD operator to two coherent states At the target node, we obtain the decoded state by applying the SUB operator to e α , 2 α .
So the state 1 α or 2 α which is input to the ADD operator can be decoded by the SUB operator.

Gaussian cloning (GC)
Cloning is an important step in the implementation of QNC. For discrete variables, quantum cloning techniques can be classified into two types. Definitive cloning performs unitary transformations during the entire cloning process. Probabilistic cloning performs unitary transformation and quantum measurement during cloning. Quantum no-cloning theorem governs both types of quantum cloning. Duan et al. Guo (1998a, 1998b)] proposed the probabilistic cloning technique which introduces quantum measurements to accurately clone a set of linearly independent quantum states with a certain probability. For continuous variables, approximate cloning schemes are used to simulate the copying operation. Cerf et al. [Cerf, Ipe and Rottenberg (2000)] proposed that the set of input states to be copied was restricted to Gaussian states and derived the optimal cloning fidelity. Here, we introduce a Gaussian cloning machine for continuous variables. Gaussian cloning machine can be used to simulate the copying operation for coherent states. 0 α denotes the input coherent state and the output of the Gaussian cloning machine is consists of the position error x and the phase error p . x and p obey the bivariate Gaussian distribution with zero mean and a variance of 4 / 1 , i.e., So the distribution function of The fidelity of the Gaussian cloning machine is calculated as follows: In summary, the set of input states to be copied is restricted to Gaussian states and the optimal cloning fidelity is 3 / 2 . So Gaussian cloning can be an effective operation used to simulate the copy operation of QNC.

CVQNC scheme based on quantum discord
The CV-EDSS protocol constructs entanglement between two distant locations. Gaussian cloning clones quantum states with a certain probability. These two basic operations provide basic conditions to design a CVQNC scheme. Fig. 4 shows the setting of the proposed CVQNC scheme.
Our CVQNC scheme is described as follows: Step 1 are the vacuum state. The CMs are Then Alice and Bob displace locally their modes by random correlated displacements distributed according to the Gaussian distribution with correlation matrix As a result, they prepare by LOCC a three-mode fully separable Gaussian state with CM as follows: , and the beam splitter is described by the matrix Step 3. Step 4. (Encoding at node 0 t ) The new mode 3 c is introduced at 0 s . By applying the ADD operator to 2 1 ,c c , we obtain the encoded state ) , ( Step 5. 0 s sends 1 c to 2 t via 1 s and 0 s sends 2 c to 1 t via 1 s . 3 c is sent to the node 0 t .
Step 6 Step 8. At the node 1 t , the mode 1 c which from Step 7 mixes the mode 1 b on another balanced beam splitter

Scheme analysis 4.1 Performance analysis
In this section, we will provide performance analysis of our CVQNC scheme from the perspectives of fidelity and network throughput.

Fidelity
We consider the fidelity of the link which indicates that this operation will not amplify quantum fluctuation. By applying the ideal ADD operation to 2 1 ,c c at the node 0 s , we obtain the quantum state where the displacement error q obeys a Gaussian distribution of After the GC operation at the node 0 t , the replicas of 3 c are r q c c r q c c q qG d r rG where r follows Gaussian distribution For the reason of symmetry [Shang, Li and Liu (2017)], the fidelity of the link 2 2 t s → for transmitting the mode 2 c is also 2 / 1 .

Network throughput
Network throughput is one of criteria for evaluating the performance of network coding schemes. The extension from discrete variables to continuous variables means to vary from finite to infinite spaces. After one network transmission, the target node 1 t receives 31 2 ,c c and the target node 2 t receives 32 1 ,c c .
Theorem 2: Each target node receives N 2 log 4 bits of classical information via one network transmission.
Proof: Suppose that a coherent state ip x + is modulated with classical characters, each classical character x or p has N elements which are 1 , , That is, the amount of information of a coherent state is N 2 log 2 . So each target node receives N 2 log 4 bits via one network transmission.
Similarly, we suppose qubits 0 and 1 are used to carry classical bits 0 and 1. In this case, the information of one qubit is the same as that of one bit. Also, the network throughput of DVQNC schemes can be measured in terms of classical bits. In XQQ, each target node receives 2 bits of classical information. In the QNC scheme with prior entanglement between senders [Hayashi (2007)], each target node receives one bit of classical information. Compared with the DVQNC schemes, the CVQNC schemes contain more information. As a result, our CVQNC scheme has a larger network throughput than the DVQNC schemes.

Network throughput
Quantum discord is in a primitive place than entanglement, so we can give insight into the role of discord in entanglement distribution. We calculate the relative entropy of entanglement [Piani, Gharibian, Adesso et al. (2011)] and the relative entropy of discord [Nielsen and Chuang (2007)] to search for a relationship between the increase in entanglement and quantum discord. The von Neumann entropy of quantum state ρ is The quantum relative entropy between two states ρ andσ is defined as The relative entropy of entanglement in the bipartition x -versusy is defined as (39) The relative entropy of discord is defined as which is the minimum relative entropy between ρ and the set of quantum-classical states where j is an orthonormal basis for y [Abeyesinghe, Devetak, Hayden et al. (2009)].
The key step in our scheme is the transmission of ancillary modes i c from source nodes i s to target nodes i t . The bipartitions b ac : and cb a : corresponds to the situations before and after the transmission of the ancillary modes. The difference between the relative entropy of entanglement in the bipartition b ac : and the relative entropy of entanglement in the bipartition cb a : can be limited [Madhok and Datta (2013)].
Lemma 1: For any tripartite state abc ρ , the difference of entanglement is bounded by the relative entropy of discord, i.e., Reference [Bennett and Shor (1998)] indicates that under any completely positive tracepreserving map M the relative entropy is monotonic, i.e., )) ( By combining it with lemma 1, we can obtain the following theorem. It can be deduced that ) The local encoding operation ac M in our scheme corresponds to the mixing operation of modes i a and i c on a balanced beam splitter i i c a BS .
This theorem indicates that the increase in entanglement is less than or equal to quantum discord measured in the communication system. So quantum discord is a necessary prerequisite to the entanglement distribution.

Security analysis
The purpose of our CVQNC scheme based on quantum discord is to entangle two modes existing in source node and target node, respectively. In the process of entanglement distribution, the quantum channels are tentatively used. After the evolution of the system, channels are unnecessary, so attackers cannot eavesdrop the links to obtain information. We analyze whether the ancillary mode i c can be calculated and forged by eavesdropping the modes transmitted in quantum channels. For the reason of symmetry, we analyze the link For 0 > t and 2 , we obtain 1 < υ [Mišta Jr. and Korolkova (2009)], therefore the modes a and b are entangled for an arbitrarily small nonzero squeezing.

Conclusion
In this paper, from the perspective of quantum discord, we proposed a feasible CVQNC scheme in which a tripartite state is established between sources nodes and target nodes.
By virtue of the CV-EDSS protocol, the scheme achieves quantum entanglement distribution from sources to targets on a butterfly network. The fidelity of the link for transmitting the ancillary modes from sources to targets is 2 / 1 . Our CVQNC scheme has a larger network throughput than the DVQNC schemes. Security analysis proves that our scheme defends against eavesdropping and forgery which means it can be applied to the case of high security. The proposed CVQNC scheme provides a model for constructing entanglement in quantum network and a guidance for future work.

Conflicts of Interest:
There is no conflict of interests or disclose all the conflicts of interest regarding the manuscript submitted.