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Entropic measure and hypergraph states

We investigate some properties of the entanglement of hypergraph states in purely hypergraph theoretical terms. We first introduce an approach for computing local entropic measure on qubit t of a hypergraph state by using the Hamming weight of the so-called t-adjacent subhypergraph. Then we quantify and characterize the entanglement of hypergraph states in terms of local entropic measures obtained by using the above approach. Our results show that a class of n-qubit hypergraph states can not be converted into any graph state under local unitary transformations.


I. INTRODUCTION
The understanding of the subtle properties of multipartite entangled states [1] is at the very heart of quantum information theory [2]. But the ultimate goal to cope with the properties of arbitrary multipartite states is far from being reached. Therefore, several special classes of entangled states have been introduced and identified to be useful for certain tasks. It is well known that graph states [3,4] are an example of these classes. Any graph state can be constructed on the basis of a (simple and undirected) graph. Although graph states can describe a large family of entangled states including cluster states [5], GHZ states, stabilizer states [6], etc., it is clear that they cannot represent all entangled states. To go beyond graph states and still keep the appealing connection to graphs, Ref. [7] introduces an axiomatic framework for mapping graphs to quantum states of a suitable physical system, and extends this framework to directed graphs and weighted graphs. Several classes of multipartite entangled states, such as qudit graph states [8], Gaussian cluster states [9], projected entangled pair states [10], and quantum random networks [11], emerge from the axiomatic framework. In [12], we generalize the above axiomatic framework to encoding hypergraphs into so-called quantum hypergraph states.
It has been known that hypergraph states include graph states [12]. One may ask whether hypergraph states are equivalent to graph states under local unitary transformations, that is, whether hypergraph states can describe more quantum states than graph states under local unitary transformations. Ref. [13] has shown that one class of three-qubit hypergraph states can not be converted into any graph state under local unitaries. The main aim of this work is to answer the above question for n qubits. For this, we will address the issue of using the entropic measure [14] to quantify and characterize the entanglement of hypergraph states in purely hypergraph theoretical terms. Several literatures have shown that there are several approaches for studying the properties of the entanglement of hypergraph states in the hypergraph theoretical terms. For graph states, Ref. [3] presents various upper and lower bounds to the Schmidt measure [15] in graph theoretical terms. For hypergraph states, similar work is done in [12]. Moreover, Ref. [12] qualitatively studies the entanglement structure of hypergraph states in purely hypergraph theoretical terms. In this paper, we will present an approach for computing local entropic measure on qubit t of a hypergraph state by using the Hamming weight of the so-called t-adjacent subhypergraph. Then we will investigate some properties of the entanglement by using local entropic measures. Furthermore, we will show a class of n-qubit hypergraph states is not equivalent to any graph state under local unitary transformations. This paper is organized as follows. In Sec. II, we recall notations of hypergraphs, hypergraph states, etc. In Sec. III, we define the Hamming weight of a hypergraph. Then we show how to calculate it by using the hypergraph theoretical terms. In Sec. IV, we present an approach for computing local entropic measures of a hypergraph state by means of the Hamming weights of some special subhypergraphs. Furthermore, we investigate some properties of the entanglement of hypergraph states by means of local entropic measures. We also indicate that a family of hypergraph states can not be converted into any graph state under local unitary transformations. Section V contains our conclusions.
It is known that real equally weighted states [16] are equivalent to hypergraph states [12]. In and define a mapping c on ( Then we can construct a 1-1 mapping u between hypergraphs and Boolean functions which where ⊕ denotes the addition operator over 2 » . Thus we have is just the real equally weighted state associate with the Boolean function ( ) u g .
Let g and g' be two hypergraphs with n vertices. We say that they are LU equivalent if there exist local unitraries { } 1,2,..., i.e., g and ' g are equivalent under local unitary operations.

III. HAMMING WEIGHT
In this section we discuss how to obtain the Hamming weight of a Boolean function by means of hypergraph theory. It is known that the Hamming weight of a Boolean function f is defined as where S denotes the cardinality of the set S. By (4), we also can define the Hamming weight of a hypergraph g with n vertices as We give the following proposition to calculate the Hamming weight of g.
be a hypergraph with n vertices. Then  (4), we can obtain that ( )( ) 1 2 , ,..., 0 n u g x x x = for any 1 2 , ,..., n x x x . Thus (7). (ii) When 1 m = , it is easily seen that (8) is true by (4) and (7). Assume that (8) is true when m t = . Now we will prove that (8) is also true when n e e n e e e i i j t Since g' has t hyperedges, we can obtain By (9) and (10), we can obtain the equation (8) for . ■ According to (8), it is easy to obtain the Hamming weight of a hypergraph. For instance, ( ) 4 hw g of the hypergraph 4 g in Fig. 1(d) is equal to 6 by (8). According to the above proposition, the Hamming weight of a hypergraph has some properties as follows. We will first discuss how to identify whether the Hamming weight of a hypergraph is odd or not.
Clearly, ( ) hw g is even. By (8), it is known that By simple calculation, we can obtain ( ) ( ) Thus ( ) hw g is odd since ( ) ' hw g is even and ∆ is odd. (ii) "only if". Assume that [ ] n E ∉ . Clearly, we can obtain that ( ) hw g would be even by (8).

■
According to the proof of the above corollary, we can easily obtain the following corollary. We say that a hypergraph is odd (even) if its Hamming weight is odd (even). By the corollary 2, it is known that the sufficiency and necessary condition of an n-vertex odd hypergraph g is ( ) ran g n = .
In the following, we will investigate the properties of the hamming weight of the hypergraph g for ( ) ran g n < . Proof. It is clear that (i) is true. Now we prove (ii). Let By (17), we can obtain ( ) ■ Note that the converse proposition of (ii) in the above proposition is not true. In fact, it is clear that the Hamming weight of a 3-vertex complete graph is equal to 4 while its rank is 2.

IV. LOCAL ENTROPIC MEASURES
In this paper, the local entropic measure From (20) and (21), we can obtain ( ) According to the definition of the local entropic measure, we can get Thus it is import for calculating ( )  ■ Note that the converse proposition of (ii) in the above proposition is not true. For g' shown in Fig. 1(b), it is clear that 4 ' g is a 3-vertex complete graph, which is shown in Sec. III. By (22) and (23)   g is just of graph state and it is locally maximally entangled by the above proposition. It is easily seen that all graph states are locally maximally entangled. One may ask whether every locally maximally entangled hypergraph belongs to graph states. Our answer is "no". In fact, the hypergraph { } ( ) ' t hw g are respectively odd and even for any vertex t. By (22) and (23), it is clear that By the above proposition and the properties of local entropic measure, we can obtain main conclusion of this work as follows. V. CONCLUSIONS This work uses the local entropic measures to quantify and characterize the entanglement of hypergraph states of n qubits. For this, we introduce an approach for computing the local entropic measures in purely hypergraph theoretical terms. At first, we define the Hamming weight of a hypergraph. Then we give a method to compute the Hamming weight of the hypergraph by using hypergraph theory. And we prove that one can use the Hamming weight of the t-adjacent subhypergraph to calculate the local entropic measure on qubit t of a hypergraph state. Our research shows that the n-vertex hypergraphs including the hyperedge [ ] n are not LU equivalent to any other hypergraph. Since (simple and undirected) graphs with n ( )