Planar two-region multi-partite maximally entangled states

Multipartite quantum entanglement Einstein ; Werner ; Horodecki is a distinctive feature of quantum mechanics and responsible for numerous practical tasks Nielsen . So quantum entanglement is an essential constituent part of quantum information and at the same time the subject of philosophical debates Horodecki . By far bipartite entangled states have been studied sufficiently, however the knowledge about multipartite states is far from perfect. From both a theoretical and practical perspective, multipartite entangled states are a very important resource in quantum information science.

It is well-known that the maximally entangled states (MES) play a very important role in practical quantum information processing. However, in fact there are different ways to consider one state to be more entangled than another. A method for defining ”maximally” entangled states in multipartite systems is based on an axiomatic approach. By using this method Gisin and Bechmann-Pasquinucci identify $n$-qubit Greenberger-Horne-Zeilinger (GHZ) states as maximally entangled in $n$-qubit systems Gisin . The reason for regarding these states as maximally entangled may be that GHZ states of $n$ qubits maximally violate the Bell-Klyshko inequalities Klyshko . Moreover, Chen proved that GHZ states in $n$-qubit systems and the states obtained from them by local unitary transformations are the unique family of states which show such a maximal violation Chen . Recently, the absolutely maximally entangled (AME) states have been defined and discussed Helwig ; Raissi ; limaosheng . The characterization of AME states of an $n$-particle quantum system is that any collection of $\lfloor\frac{n}{2}\rfloor$ particles are in a maximally mixed state, where, $\lfloor\cdot\rfloor$ is the floor function. Well-known examples are the Bell and GHZ states on two and three parties respectively. Obviously, the 4-qubit GHZ state is not a absolutely maximally entangled state.

According to different criteria 1 ; 2 ; 3 ; 4 ; 5 ; 6 , the maximally entangled states can be divided into multiple types 7 ; 8 ; 9 ; 10 ; 11 , such as the AME state and planar maximally entangled (PME) state. PME states have the property that any collection of $\lfloor\frac{n}{2}\rfloor$ adjacent particles in a planar $n$-particle quantum system are in a completely mixed state 26 . Various maximally entangled states can be applied to quantum parallel teleportation Helwig ; 12 ; 13 ; 14 ; 15 , quantum secret sharing 16 ; 17 ; 18 and so on.

AME states do not exist in all possible Hilbert spaces . It has been demonstrated that for any $n$-particle system, when the dimension $d$ of the Hilbert space of the subsystem is chosen large enough, one can find the AME states; however, for low local dimensions there are severe constraints. The existence of AME states composed of two-level systems was recently solved. For qubit systems ($d$=2), AME states do only exist for $n=2,3,5$, and 6 parties, whereas it has been shown that no AME states exist for 4, 7, 8, and more than 8 qubits Helwig ; 20 ; 21 ; 22 ; Higuchi ; Scott ; FHuber .

Compared with AME state, the PME state is one with a lower number of constraints. Hence there are many more PME states than AME states, and the PME states are a wider class of entangled states than the AME states. The AME states are a subclass of PME states, that is, any AME state is a PME state, but the converse is not true 26 . Is there other class of maximally entangled states except the AME states and PME states? In this paper we solve this problem by proposing a new class of maximally entangled states, called the planar two-region multipartite maximally entangled states. In Sec. II, we investigate a special case, the planar two-region four-partite maximally entangled states. Then in Sec. III, we study the exactly definition of this kind of maximally entangled states and prove that there exists this kind of maximally entangled state. A summary with some remarks is provided in Sec. IV.

It was shown that there exist various PME states for any even number of qudits, and there are two distinct multiparameter classes of four-qudit PMEs 26 . One may ask if it is possible to search for a wider class of entangled states except AME states and PME states.

In this section we will answer the above problem by defining the planar two-region four-partite maximally entangled state first. Then we will show that there is this kind of maximally entangled state.

Assume that the structure of the planar quantum system consisting of $n$ particles can be divided into two regions $A$, $B$, and each region has two disconnected parts. In the region $A$, one part has $k$ adjacent particles, and the other part has $\lfloor n/2\rfloor-k$ adjacent particles. At the same time we also require that one part of region $B$ has $k$ adjacent particles and the other part of the region $B$ has $n-\lfloor n/2\rfloor-k$ adjacent particles. As the two parts of region $A$ are not connected, so the only neighboring parts of each part of region $A$ are that of region $B$. Evidently, for a given planar quantum system there can exist many structures. If a quantum state of the system satisfies that every subset of region $A$ in each possible quantum structure, is in a completely mixed state, then this state is called a planar two-region four-partite maximally entangled state, simply written as PKME$(n,k,4,d)$ state. Here $d$ is the dimension of the Hilbert space of single particle. Apparently, the requirement condition of this kind of maximally entangled state is weak than that of AME state and different from that of the planar maximally entangled state.

Next we will prove that there are PKME$(n,k,4,d)$ states for even $n$ and odd $n$, respectively.

Suppose that a partition of $n$ particles of a quantum system is parted into parts $A$ and $B$ with $|A|\leq|B|$, then a pure state $|\Psi\rangle$ of the quantum system can be written as 26

So the density matrix of part $A$ is

For $\rho_{A}$ to be the completely mixed state $I_{A}$, it is necessary to demand that the states $\{|\phi_{\mathbf{k}}\rangle\}$ are orthogonal and with equal norm, and the states $\{|\mathbf{k}\rangle\}$ form a basis for the Hilbert space of the part $A$ 26 .

In this section, we will generalize the definition of two-region four-partite maximally entangled state to the more generalized case. That is we will define two-region multi-partite maximally entangled state and prove that this kind of maximally entangled states do exist.

Consider a quantum system composed of $n$ particles in a planar circle, where the Hilbert space of each particle is a $d$-dimensional complex space $C^{d}$. Suppose that one can divide the $n$ particles into two-region $\{A,B\}$, where $A$ ($B$) consists of parts $\{A_{1},A_{2},\cdots,A_{m}\}$ ($\{B_{1},B_{2},\cdots,B_{m}\}$), and the particles in each part are adjacent, however $A_{1},A_{2},\cdots,A_{m}$ are not neighbors, so do $B_{1},B_{2},\cdots,B_{m}$. Each $\{A_{1},A_{2},\cdots,A_{m}\,B_{1},B_{2},\cdots,B_{m}\}$ is called a quantum structure. We also demand that region $A$ contains $\lfloor n/2\rfloor$ particles, region $B$ consists of $n-\lfloor n/2\rfloor$ particles. We use $k_{A_{1}},k_{A_{2}},\cdots,k_{A_{m}},k_{B_{1}},k_{B_{2}},\cdots,k_{B_{m}}$ to denote the corresponding particle numbers of parts $A_{1},A_{2},\cdots,A_{m},B_{1},B_{2},\cdots,B_{m}$ respectively. Here $2m\leq n,4\leq n$, $|A|\leq|B|$. Obviously, for an $n$ particle system and fixed $k_{A_{1}},k_{A_{2}},\cdots,k_{A_{m}},k_{B_{1}},k_{B_{2}},\cdots,k_{B_{m}}$ there are many quantum structures.

If a quantum state in the Hilbert space $(C^{d})^{\bigotimes n}$ of the $n-$particle system satisfies that every subset of region $A$ in each possible quantum structure, is in a completely mixed state, then this state is called a planar two-region multi-partite maximally entangled state, simply written as PKME$(n,\{z\},2m,d)$ state. Here $d$ is the dimension of the Hilbert space of single particle and $\{z\}=\{k_{A_{1}},k_{A_{2}},\cdots,k_{A_{m}},k_{B_{1}},k_{B_{2}},\cdots,k_{B_{m}}\}$.

Clearly, the requirement condition of PKME states is weak than that of the absolutely maximally entangled state and different from that of the planar maximally entangled state.

Next we will give two examples of the planar two-region multi-partite maximally entangled states.

Example 1. Consider a $2mk$-qubit system. It is easy to demonstrate that the quantum state in Hilbert space $(C^{2})^{\bigotimes 2mk}$

is a PKME$(2mk,\{z\},m,2)$ state, with $\{z\}=\{k_{A_{1}},k_{A_{2}},\cdots,k_{A_{m}},k_{B_{1}},k_{B_{2}},\cdots,k_{B_{m}}|k_{A_{1}}=k_{A_{2}}=\cdots=k_{A_{m}}=k_{B_{1}}=k_{B_{2}}=k_{B_{m}}=k\}$.

Therefore, there exists the planar two-region multi-partite maximally entangled state of the planar $2mk$-qubit system, where both $k$ and $m$ are positive integers larger than or equal to 1.

Example 2. Assume $m$, $k$ are positive integers larger than or equal to 1. Let us consider a $2mk+1$ qubit system. One can verify that a quantum state in the Hilbert space $(C^{2})^{\bigotimes 2mk+1}$

is just a PKME$(2mk+1,\{z\},m,2)$ state, with $\{z\}=\{k_{A_{1}},k_{A_{2}},\cdots,k_{A_{m}},k_{B_{1}},k_{B_{2}},\cdots,k_{B_{m}}|k_{A_{1}}=k_{A_{2}}=\cdots=k_{A_{m}}=k_{B_{1}}=k_{B_{2}}=k_{B_{m-1}}=k,k_{B_{m}}=k+1\}$.