Quantum properties in the four-node network

By using quantum network one can exchange corresponding quantum information on different nodes through quantum channels, such as quantum entanglement 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 and quantum key distribution 15 ; 115 . Therefore, quantum network plays an important role in the study of quantum correlation 9 ; 10 ; 11 . The great progress has been made in the research and application of quantum network structure and quantum entangled states prepared in the quantum network in both theory and experiment 12 ; 13 ; 14 ; 16 ; 17 ; 18 . Of course, the practical application of quantum network technologies may also affect the real life in the future.

In the network system, quantum states are distributed to different nodes 12 ; 13 . It is assumed that sources are independent in the quantum network. It means different sources are separable. However, different particles in the same source can be entangled 14 ; 15 , so there are a lot of situations. For example, one can has the situation in which a source can only generate two particle entangled states. Of course, there also exists the situation in which three particle entangled states can be produced by the source.

Apparently, the quantum system consisting of different number particles have many quantum network structures and a lots of different preparable states 16 ; 17 . This paper only studies several different quantum network structures of four nodes. The structure of four-node quantum network has two different situations: one is the network structure of four nodes in a plane, the other is the network structure of four nodes in the space. We assume that the sources do not intersect. Under the assumption of no intersection of sources, the four nodes in a plane have only one quadrangular network structure, while the four nodes in the space have two kinds of network structures. Different quantum states can be prepared in different network structures. Based on the pioneer work in three nodes 18 , we will mainly discuss the network structures in the above three special situations, and deduce some properties and constraints of the quantum states that can be prepared.

The paper is organized as follows. In section II we define $n$-partite mutual information in quantum systems and obtain some properties of states prepared in a quadrangular network structure. The properties of quantum states prepared in two kinds of network structures with four nodes in the space are mainly investigated in section III. Among them, the two network structures of four nodes in the space with six sources, where each source generates the two-partite entangled state, and four sources, where every source produces three-partite entangled states are discussed respectively. A summary is given in section IV.

Let us discuss the situation in which four nodes are in a plane. This network is called independent quadrangular network (IQN). As shown in Fig.1(a), it has four nodes, namely A, B, C, and D, which are formed by pairing of two particles in each of the four sources $\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, and $\rho_{\delta}$ 19 . The states of two particles in each source are the two-partite entangled states 20 ; 21 ; 22 . $\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, and $\rho_{\delta}$ is shared respectively by two nodes of A, B, C, and D. It means that each of the four nodes A, B, C, and D receives two particles. Exactly, A receives two particles respectively from the source $\rho_{\alpha}$ and $\rho_{\beta}$; B receives the two particles respectively from the source $\rho_{\beta}$ and $\rho_{\gamma}$; C receives two particles respectively from the source $\rho_{\gamma}$ and $\rho_{\delta}$; D receives two particles respectively from the source $\rho_{\delta}$ and $\rho_{\alpha}$. The two particles at the each node can be applied by a local unitary matrix. The four local unitary matrices are denoted by $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ respectively.

We use $\Delta_{\mathrm{IQN}}$ to represent the set of quantum states that can be prepared in an independent quadrangular network structure shown in Fig.1(a). Obviously, the quantum state $\rho$ in the set $\Delta_{\mathrm{IQN}}$ can be written as

Here, $U_{A}$ only acts on the system consisting of a particle of $\rho_{\alpha}$ and a particle of $\rho_{\delta}$, it does not act on the other particles which are not at the node A at same time. Similarly, other unitary matrices cannot act on the states of $\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, and $\rho_{\delta}$ at same time.

Next we will discuss Von Neumann entropy of the quantum states in the set $\Delta_{\mathrm{IQN}}$. In this paper, entropy refers to Von Neumann entropy.

Because entropy $S(\rho)$ of quantum state $\rho$ is invariant under unitary transformation, and additive on tensor products, for the quantum states shown by Eq.(1), we have

The three-partite entropy is extended to

the two-partite entropy is extended to

the one-partite entropy is extended to

and similarly for others.

Based on entropy, one can study the mutual information of the quantum system. The two-partite quantum mutual information of a quantum system composed of X and Y is defined as

where $S(X|Y)$ stands for conditional entropy and $S(XY)$ is the joint entropy. Three-partite quantum mutual information measuring the common information of subsystems X, Y, and Z, was defined as 18

We define four-partite quantum mutual information of the quantum system composed of T, X, Y, and Z as

By substituting the formula of the three-partite quantum mutual information into the four-partite quantum mutual information, one can obtain

where

and

Obviously, the four-partite quantum mutual information is invariant under arbitrary permutation of the subsystems.

Furthermore, we define the $n$-partite quantum mutual information of the system consisting of subsystems $X_{1},X_{2},\cdots,X_{n}$. We use $\sigma_{X_{1}X_{2}\cdots X_{n}}$ to express a quantum state of the system. We define the $n$-partite quantum mutual information of the system as

where $\sigma_{X_{i}X_{j}\cdots X_{l}}=\mathrm{tr}_{X_{q}X_{r}\cdots X_{t}}\sigma_{X_{1}X_{2}\cdots X_{n}}$ is the reduced density matrix for the subsystems $X_{i},X_{j},\cdots,X_{l}$ and the set $\{q,r,\cdots,t\}=\{1,2,\cdots,n\}\setminus\{i,j,\cdots,l\}.$ Clearly, $I_{n}(X_{1}:X_{2}:\cdots:X_{n})$ is symmetry about the subsystems.

For the quantum states in the set $\triangle_{\mathrm{IQN}}$, it is easy to obtain the following result.

Conclusion 1-1. $I_{4}(A:B:C:D)=0$ for any $\rho\in\triangle_{\mathrm{IQN}}$.

Now we discuss the effect of the local channels on the four-partite quantum mutual information. Suppose that there are three local channels $\Lambda_{A}$, $\Lambda_{B}$, and $\Lambda_{C}$ acting on nodes A, B, and C respectively. The four-partite quantum mutual information of the quantum system is $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$. Here $I_{D}$ is the identity operator on the node $D$, and $\rho\in\triangle_{\mathrm{IQN}}$.

Clearly, in this case for two particles in D we have $S(D)=S(D_{\alpha})+S(D_{\delta})$. By considering the actions of the local quantum channels we can get

Therefore, we derive that

Hence, generally speaking, one can not determine whether $I_{4}$ is larger than zero or not for the quantum state $(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho$.

However, for two adjacent local channels 24 ; 25 ; 26 , the four-partite quantum mutual information of the quantum system becomes $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes I_{C}\otimes I_{D})\rho]$. For this case, one has

By using the strong subadditivity inequality

for three quantum systems $X,Y,Z,$ it is easy to obtain that the four-partite quantum mutual information of quantum system satisfies $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes I_{C}\otimes I_{D})\rho]\geq 0$.

For two non-adjacent local channels, the four-partite quantum mutual information of the quantum system turns out to be $I_{4}[(\Lambda_{A}\otimes I_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$. In this case one can deduce

By the subadditivity inequality

for two quantum systems $X,Y$, we arrive at that the four-partite quantum mutual information of the quantum system $I_{4}[(\Lambda_{A}\otimes I_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$ is still large than zero. Combining the above two cases we have the following result.

Conclusion 1-2. For any pair of local channels one finds that the four-partite quantum mutual information for the quantum states obtained by acting the pair of the local quantum channels on the quantum state in $\triangle_{\mathrm{IQN}}$ cannot be rendered negative.

Although in general we can not decide $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$ is large or not than zero, however, we have the following conclusion.

Conclusion 1-3. The four-partite quantum mutual information $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$ are bounded as

and

Here $\rho$ is a quantum state in the set $\Delta_{\mathrm{IQN}}$.

For detailed proof of above conclusion please refer to Appendix A.

As there exists quantum entanglement in Fig. 1, so the entanglement measure $\varepsilon[\sigma]$ should be introduced. The entanglement measure should satisfy three requirements: (1) if the quantum state is separable, the entanglement measure will disappear; (2) under local operation and classical communication, the average entanglement measure does not increase; (3) under local unitary transformation, the entanglement measure does not change. For present situation, we also require that the entanglement measure is additive on tensor products, that is

Furthermore, we require the entanglement measure satisfies the monogamy relation 23 ,

where $\sigma_{TX}=\mathrm{tr}_{YZ}\sigma_{TXYZ}$, $\sigma_{TY}=\mathrm{tr}_{XZ}\sigma_{TXYZ}$. Clearly, not arbitrary entanglement measure meets the above requirements, however the squashed entanglement does 35 ; 36 .

For the quantum state (1), as the local unitary matrices do not change the entanglement between the parties, we have

where $A_{\alpha}$ stands for a particle that $A$ receives from $\rho_{\alpha}$, similarly for others. Therefore for the entanglement measure $\varepsilon[\sigma]$ satisfying the above requirements, we have the following conclusion.

Conclusion 1-4. For any $\rho\in\triangle_{\mathrm{IQN}}$, $\varepsilon_{T|XYZ}[\rho]=\varepsilon_{T|X}[\mathrm{tr}_{YZ}\rho]+\varepsilon_{T|Z}[\mathrm{tr}_{XY}\rho]$ holds for all bipartitions, such as $A|BCD$, $B|ACD$, $C|ABD$, and $D|ABC$.

After that we discuss the ranks of the global state (1) and its marginals. Because the unitary matrices $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ don’t change the rank of the global state, for all $\rho\in\Delta_{\mathrm{IQN}}$, we have the rank of quantum state $\rho$

where $r_{\alpha}=\mathrm{rk}(\rho_{\alpha})$ is the rank of quantum state $\rho_{\alpha}$, similarly for others. The rank of the local reduced state is

the rank of the two-partite reduced state is

the rank of the three-partite reduced state is

Here $r_{\alpha}^{A}=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})$, similarly for others.

Then we arrive at

If the Hilbert space of every particle is d-dimensional, we have $r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$ $\in[1,d^{2}]$, $r_{\alpha}^{A}$, $r_{\alpha}^{D}$, $r_{\beta}^{A}$, $r_{\beta}^{B}$, $r_{\gamma}^{B}$, $r_{\gamma}^{C}$, $r_{\delta}^{C}$, $r_{\delta}^{D}$ $\in[1,d]$. Hence one has the conclusion as follows.

Conclusion 1-5. For $\rho\in\triangle_{\mathrm{IQN}}$ there exist integers $r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$$\in[1,d^{2}]$, $r_{\alpha}^{A}$, $r_{\alpha}^{D}$, $r_{\beta}^{A}$, $r_{\beta}^{B}$, $r_{\gamma}^{B}$, $r_{\gamma}^{C}$, $r_{\delta}^{C}$, $r_{\delta}^{D}$$\in[1,d]$, which satisfy rank relationship Eq. (16).

Next we discuss the scenario where the four nodes and the four sources are classically correlated 18 . As shown in Fig. 1(b), we assume that every source and every node are correlated by sharing the random variable $\lambda$ 18 . In this case, the quantum states are called classically correlated states, which read

Here $p(\lambda)$ is a probability distribution function of the random variable $\lambda$. We use CQN to denote this classically correlated quadrangular network, $\Delta_{\mathrm{CQN}}$ to represent the set of quantum states that can be prepared in CQN. In other words, a quantum state $\rho\in\Delta_{\mathrm{CQN}}$ can be written as $\rho=\sum_{\lambda}p_{\lambda}\rho_{\lambda}$, where $\rho_{\lambda}\in\Delta_{\mathrm{IQN}}$, $p_{\lambda}$ is a probability distribution.

Based on the properties of the ranks of quantum states, we will verify that there are no four-qubit genuine multipartite entangled states in the set $\Delta_{\mathrm{CQN}}$. Here four-qubit genuine multipartite entangled state refers to it can be embedded in larger dimensional systems. Evidently, if a four-qubit genuine multipartite entangled state is pure, then the rank of the global state $\rho$ is equal to 1 27 ; 28 ; 29 and it is entangled along every bipartition. Considering the Schmidt decomposition of the four-qubit genuine multipartite entangled state, one has that all single node reduced state has rank 2. However, as a matter of fact, when each source in the IQN is entangled among two partitions. According to the Schmidt decomposition, the rank of the reduced state of a particle in each source is 2. Therefore, in IQN as shown in Fig. 1(a), with sources which prepare 2-qubit entangled state, the local rank of the reduced state at each node is 4. So it is impossible to prepare four-qubit genuine multipartite entangled states in IQN. Similarly, we can show that the statement holds for other cases of resources. Furthermore, because four-qubit genuine entangled states need pure four-qubit genuine multipartite entangled states 30 , so it is also impossible to prepare four-qubit genuine multipartite entangled states in CQN. Hence we arrive at the following conclusion.

Conclusion 1-6. No four-qubit genuine multipartite entangled state can be prepared in IQN and CQN.

Fig. 1(c) shows the four-partite quantum states which can not be generated in IQN and CQN.

Clearly, four nodes in the space form a tetrahedron. By the assumption of no intersection of sources, in this case there are two different quantum network structures: one is the triangular cone network structure 1 with six two-partite sources as shown in Fig. 2(a); the other is the triangular cone network structure 2 with four three-partite sources as illustrated in Fig. 3(a).

We first study the triangular cone network structure 1. It has four nodes, namely A, B, C, and D. These four nodes are formed by pairs of two particles in each of the six sources $\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, $\rho_{\delta}$, $\rho_{\theta}$, and $\rho_{\tau}$. Two particles in each source are in two-partite entangled states. $\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, $\rho_{\delta}$, $\rho_{\theta}$ and $\rho_{\tau}$ are shared by $\{A,B\}$, $\{B,D\}$, $\{A,D\}$, $\{A,C\}$, $\{C,D\}$, $\{B,C\}$, respectively. Four nodes do not share common information. Each of four nodes A, B, C, and D receives three particles, for example, ${A}$ receives three particles from sources $\rho_{\alpha}$, $\rho_{\gamma}$, and $\rho_{\delta}$, similarly for others. One can apply a local unitary matrix to the three received particles. Four local unitary matrices are denoted by $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ respectively. The global quantum state in an independent triangular cone network structure 1 (ITCN1) is denoted by $\rho$. We use $\Delta_{\mathrm{ITCN1}}$ to represent the set of quantum states that can be prepared in ITCN1.

Apparently, the quantum states prepared in ITCN1 read

By using that entropy is invariant under unitary transformation, and for the tensor product state, entropy has additivity, we have the entropy of the quantum state (18)

The three-partite entropy, the two-partite entropy, the one-partite entropy are extended to

respectively, and similarly for the other cases. It is easy to obtain that $I_{4}(A:B:C:D)=0$. Hence we have the following result.

Conclusion 2-1-1. For any $\rho\in\triangle_{\mathrm{ITCN1}}$, the four-partite quantum mutual information $I_{4}(A:B:C:D)$ of quantum state $\rho$ is zero.

Now we discuss entanglement measure. We require the entanglement measure $\varepsilon$ satisfies the requirements stated in section II. It is easy to deduce

So the following result holds.

Conclusion 2-1-2. For any $\rho\in\triangle_{\mathrm{ITCN1}}$, we have that $\varepsilon_{T|XYZ}[\rho]=\varepsilon_{T|X}[\mathrm{tr}_{YZ}\rho]+\varepsilon_{T|Y}[\mathrm{tr}_{XZ}\rho]+\varepsilon_{T|Z}[\mathrm{tr}_{XY}\rho]$, for all the bipartitions, such as $A|BCD$, $B|ACD$, $C|ABD$, and $D|ABC$.

Later on we turn our attention to the rank of the quantum states. Because the unitary matrices $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ don’t change the rank of the whole quantum system, for all $\rho\in\Delta_{\mathrm{ITCN1}}$, we have the rank of the quantum state $\rho$,

the rank of the local reduced state $\mathrm{tr}_{BCD}\rho$ is

the rank of the two-partite reduced state $\mathrm{tr}_{CD}\rho$ is

the rank of the three-partite reduced state $\mathrm{tr}_{D}\rho$ is

Here $r_{\alpha}^{A}=\mathrm{rk}(\mathrm{tr}_{B}\rho_{\alpha})$ stands for the rank of the quantum state $\mathrm{tr}_{B}\rho_{\alpha}$, similarly for the others.

Therefore we have

Suppose that the Hilbert space of single particle is d-dimensional. So $r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$, $r_{\theta}$, $r_{\tau}$$\in[1,d^{2}]$, $r_{\alpha}^{A}$, $r_{\alpha}^{B}$, $r_{\beta}^{B}$, $r_{\beta}^{D}$, $r_{\gamma}^{A}$, $r_{\gamma}^{D}$, $r_{\delta}^{A}$, $r_{\delta}^{C}$, $r_{\theta}^{C}$, $r_{\theta}^{D}$, $r_{\tau}^{B}$, $r_{\tau}^{C}$$\in[1,d]$. Thus we arrive at the following conclusion.