Qudit hypergraph states and their properties
Fei-Lei Xiong Yi-Zheng Zhen Wen-Fei Cao Kai Chen Zeng-Bing Chen








zbchen@ustc.edu.cn


Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

CAS Center for Excellence and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China



Hypergraph states, a generalization of graph states, constitute a large class of quantum states with intriguing non-local properties and have promising applications in quantum information science and technology. In this paper, we generalize hypergraph states to qudit hypergraph states, i.e., each vertex in the generalized hypergraph (multi-hypergraph) represents a d-level quantum system instead of a qubit. It is shown that multi-hypergraphs and d-level hypergraph states have a one-to-one correspondence. We prove that if one part of a multi-hypergraph is connected with the other part, the corresponding subsystems are entangled. More generally, the structure of a multi-hypergraph reveals the entanglement property of the corresponding quantum state. Furthermore, we discuss their relationship with some well-known state classes, e.g., real equally weighted states and stabilizer states. These states' responses to the generalized Z (X) operations and Z (X) measurements are studied. The Bell non-locality, an important resource in fulfilling many quantum information tasks, is also investigated.

03.67.-a, 03.67.Ac, 03.65.Ud















Qudit hypergraph states and their properties
    Zeng-Bing Chen
    
============================================



§ INTRODUCTION


In quantum information science and technology, graph states constitute  an almost unique family of states for their appealing properties and applications <cit.>. They can be used to implement one-way quantum computation <cit.> and construct quantum codes <cit.>. Moreover, they can be used to characterize many kinds of widely used entangled states, such as cluster states <cit.>,  the Greenberger-Horne-Zeilinger (GHZ) states <cit.> and more generally, stabilizer states <cit.>. To make quantum states of suitable physical systems describable in the framework as that of graph states, Ref. <cit.> introduced an axiomatic method. Later, Ref. <cit.> generalized this approach and introduced a new class of quantum states named hypergraph states.

Like graph states, given a hypergraph, one can define  an associated qubit hypergraph state, i.e., hypergraphs can be encoded into quantum states <cit.>. Besides this feature, every qubit hypergraph state corresponds to a stabilizer group <cit.>. However, generally speaking, the stabilizers are no longer products of local operators <cit.>. As a new class of quantum states, they possess lots of new properties, e.g., local unitary symmetries <cit.>, entanglement properties <cit.> and non-local properties <cit.>. Besides these fundamental properties, these states also have many applications. Qubit hypergraph states are real equally weighted states <cit.>,
which have important applications in Grover <cit.> and Deutsch-Jozsa <cit.> algorithms. Recently, Ref. <cit.> has shown that, if one has a black box that can tell whether an input qubit hypergraph state is a product state, he/she can solve the NP-complete SAT problem efficiently <cit.>. Fully connected k-uniform qubit hypergraph states, a generalization of GHZ states, are applicable in Heisenberg-limited quantum metrology with more robustness to noise and particle losses <cit.>. Superior to one-way quantum computation based on graph states, measurement-based quantum computation with qubit hypergraph states is non-adaptive, making the measurement scheme simpler <cit.>.

In this paper, we implement the concept of multi-hypergraphs <cit.> and encode these multi-hypergraphs into multi-qudit quantum states. We investigate the relationship between the multi-hypergraphs and their corresponding qudit hypergraph states, mainly about the map from the set of multi-hypergraphs to the set of qudit hypergraph states, and the relationship between the connectivity and entanglement. We investigate the local unitary transformations and local measurements on qudit hypergraph states, for  pushing their potential applications in error-correcting quantum codes and quantum computation. The Bell non-locality, a useful resource in quantum computation and quantum high precision measurement, is also studied. Furthermore, a systematic approach for the experimental detection is provided.

The paper is organized as follows: In Sec. <ref>, we give some preliminary knowledge of hypergraph and qubit hypergraph states, and explain related terminologies. We then generalize these concepts to represent a larger class of quantum states, which we call qudit hypergraph states, using a similar formalism. We show how the notion hypergraph should be modified when each vertex represents a qudit instead of a qubit. In Sec. <ref>, we discuss  the connection between the generalized hypergraphs and the qudit hypergraph states, including the correspondence and relationship between the connectivity and the entangled properties.  In Sec. <ref>, we discuss the relationship among qudit hypergraph states and some well-known state classes, like real equally weighted states, qudit graph states and stabilizer states. In Sec. <ref>, qudit hypergraph states' responses to local Z and X operations (measurements) are investigated. In Sec. <ref>, we investigate the Bell non-locality <cit.> of N-uniform qudit hypergraph states and expound a general detection scheme for illustrating the Bell non-locality of a general qudit hypergraph state. Conclusions are drawn in Sec. <ref>.



§ MULTI-HYPERGRAPHS AND QUDIT HYPERGRAPH STATES


In this section, we will introduce some preliminary knowledge of hypergraph and qubit hypergraph state and propose our main generalization of these concepts. Some important properties of qubit hypergraph states and qudit hypergraph states will be discussed.



 §.§ Preliminary: hypergraphs and qubit hypergraph states


A hypergraph H is composed of a set of vertices V and a set of hyperedges E <cit.>, i.e., H=(V,E) (For simplification, in this subsection, H represents such a hypergraph.). Suppose that the vertices are labeled as 1,2,⋯,N, then V={1, 2, ⋯, N}. Unlike the edges defined in standard graphs, hyperedges in hypergraphs may connect more (or less) than 2 vertices, i.e., elements in E has a form e={k_1, k_2, ⋯, k_|e|}, where k_1, k_2, ⋯, k_|e| are the vertices connected by e, and |e|, the cardinality of e, ranges from 0 to N. If all the hyperedges
in H are of the same cardinality k, then H is called k-uniform <cit.>. Standard graphs are in fact 2-uniform hypergraphs. Some examples of hypergraphs are shown in Fig. <ref>.

Hypergraphs can be encoded into a class of quantum states named qubit hypergraph states, in which every vertex represents a two-level quantum system whose computational basis is {|0⟩, |1⟩}. The operator corresponding to the hyperedge e={k_1, ⋯, k_|e|} is defined as

    C_e=[                                                      -1                                                  |e|=0,;                                                       Z                                                  |e|=1,; i_k_1,⋯,i_k_|e|=01∑(-1)^i_k_1⋯ i_k_|e|Π̂_i_k_1⋯ i_k_|e|                                                 |e|≥ 2, ]

where Π̂_i_k_1⋯ i_k_|e|=|i_k_1⋯ i_k_|e|⟩⟨ i_k_1⋯ i_k_|e|| and i_k_1, ⋯, i_k_|e| denotes the value of the vertices k_1, ⋯, k_|e|, respectively. The qubit hypergraph state corresponding to H is

    |H⟩=e∈ E∏C_e|+⟩^⊗ N,

where |+⟩=(|0⟩+|1⟩)/√(2). The state |H⟩ can be interpreted as applying a series of C_e operations to |+⟩ ^⊗ N. As all the C_es are commutative with respect to each other, the order of the operations makes no difference, and a hypergraph corresponds to a definite qubit hypergraph state (see Fig. <ref> for the examples).

The same as the graph states, qubit hypergraph states can also be characterized within the framework of stabilizers. Define a set of operators

    g_k=(∏ _e∈ E C_e) X_k (∏ _e'∈ E C_e')^†=X_k∏ _{e|k∈ e, e∈ E} C_e\{k},

where X_k is the Pauli-X operator of the kth vertex, then (see Fig. <ref> for the examples)

    g_k|H⟩=|H⟩.

Because

    [g_k,g_k']=(∏ _e∈ E C_e) [X_k,X_k'] (∏ _e'∈ E C_e')^†=0,

the set {g_k |k ∈ V} can generate an Abelian cyclic group called the stabilizer group of |H⟩. Either {g_k|k∈ V} or the stabilizer group can determine a qubit hypergraph state up to a phase factor <cit.>.




Qubit hypergraph states have interesting properties and important applications. The formalism offers a systematically pictorial representation of the real equally weighted states, which is a vivid way of demonstrating entanglement <cit.>. The entanglement and Bell non-locality make this class of quantum states have a broad range of applications in quantum computation and quantum metrology <cit.>.



 §.§ Multi-hypergraphs and qudit hypergraph states



Multi-hypergraph, whose hyperedge can have a multiplicity larger than 1, is a generalization of hypergraph (see Fig. <ref> for the examples). A multi-hypergraph whose vertices represent d-level quantum systems can be denoted as H_d=(V, E), where V={1, 2, ⋯, N} is the set of vertices, and E is a multiset of the hyperedges. The times an element e occurs in E is called multiplicity of e and is denoted as m_e (m_e ∈{1,2,⋯,d-1}) <cit.>. For the e that satisfies e∈ 2^V (2^V denotes the power set of V, which constitutes of all the subsets of V) and e∉ E, its multiplicity m_e is defined to be 0. With this generalization, every H_d is associated with a definite multiplicity function e → m_e, here e∈ 2^V and m_e∈{0,1,⋯,d-1}. In the following, if not particularly specified, H_d refers to such a multi-hypergraph, and the multiplicity of e is denoted as m_e.



Now we define qudit hypergraph states corresponding to H_d. Suppose the computational basis of each vertex is {|0⟩, |1⟩, ⋯, |d-1⟩}, then in this basis the generalized Pauli-X and Pauli-Z operators are <cit.>

    X=[
    [ 0 1 0 ⋯ 0; 0 0 1 ⋯ 0; ⋮ ⋮ ⋮ ⋱ ⋮; 0 0 0 ⋯ 1; 1 0 0 ⋯ 0 ]],
    
    Z=[
    [        1        0        0        ⋯        0;        0      ω_d        0        ⋯        0;        0        0   ω ^2_d        ⋯        0;        ⋮        ⋮        ⋮        ⋱        ⋮;        0        0        0        ⋯ ω ^d-1_d ]],

in which ω_d=e^i  2 π/d and X Z=ω_d Z X (Manin's quantum plane algebra <cit.>). The operator corresponding to the hyperedge e={k_1, k_2, ⋯, k_|e|} is defined as

    C_e=[                                                         ω_d                                                      |e|=0,;                                                           Z                                                      |e|=1,; i_k_1,⋯, i_k_|e|=0d-1∑ω ^i_k_1⋯ i_k_|e|_d Π̂_i_k_1⋯ i_k_|e|                                                     |e|≥ 2, ]

where Π̂_i_k_1⋯ i_k_|e|=|i_k_1⋯ i_k_|e|⟩⟨ i_k_1⋯ i_k_|e|| and i_k_1, ⋯, i_k_|e| denote the possible values of the vertices k_1, ⋯, k_|e| (in the computational basis), respectively. The unitary operators X, Z, and C_e satisfy

    X^k   =𝕀 k=0  d,
    
    Z^k   =𝕀 k=0  d,
    
    C_e^k   =𝕀 k=0  d.

Denote that |+⟩_d=∑_k=0^d-1|k⟩/√(d),
then the d-level hypergraph state corresponding to H_d can be defined as

    |H_d⟩ =e∈2^V∏C_e^m_e|+⟩^⊗ N_d.

Here the condition “e∈2^V” is equivalent to “e∈ E”, because C_e^0=𝕀 (∀ e∈2^V). For simplicity, in the following we will not express it explicitly.

A qudit hypergraph state is also associated with a stabilizer group through which it can be determined up to a phase factor. For H_d=(V,E), define

    g_k=ł(∏ C_e^m_e)̊ X_k ł(∏ C_e'^m_e')̊^†=X_ke:k∈ e∏(C_e\{k}^†)^m_e,

then

    g_k .|H_d.⟩=.|H_d.⟩,

and

    [g_k,g_k']=ł(∏ C_e^m_e)̊ [X_k,X_k'] ł(∏ C_e'^m_e')̊^†=0.

 
Note that the form of g_k in Eq. (<ref>) is different from that in Eq. (<ref>). The reason is that when d=2, ∀ e, 
C_e is Hermitian, while for general d, this property cannot always hold. 
The set {g_k|k∈ V} generates a cyclic Abelian group named the stabilizer group of .|H_d.⟩. Generally speaking, like those of qubit hypergraph states <cit.>, the stabilizers of qudit hypergraph states are also non-local operators.



§ RELATION BETWEEN MULTI-HYPERGRAPHS AND QUDIT HYPERGRAPH STATES: CORRESPONDENCE AND ENTANGLEMENT PROPERTY


In this section, we will discuss the relation between multi-hypergraphs and qudit hypergraph states. Theorem <ref> shows that the map from {H_d|H_d=(V,E)} to {|H_d⟩|H_d=(V,E)}, where H_d is mapped to |H_d⟩, is a bijection. Theorem <ref> demonstrates that the connectivity of a multi-hypergraph is closely related to the entanglement property of the corresponding quantum state. To prove these two theorems, we shall prove several lemmas first.

lemmaLemma

Divide the hyperedge e={1, 2,⋯, n} into the control part e_C={1, 2, ⋯, m} and the target part e_T={m+1, m+2, ⋯, n}, then

    C_e=i_1,⋯ ,i_m=0d-1∑|i_1 ⋯ i_m⟩⟨ i_1 ⋯ i_m|C_e_T^i_1 ⋯ i_m.




From the definition in Eq. (<ref>),

    C_e   =i_1,⋯ ,i_n=0d-1∑ω _d ^i_1⋯ i_nΠ̂_i_1 ⋯ i_n,
    
    C_e_C   =i_1,⋯ ,i_m=0d-1∑ω _d ^i_1⋯ i_mΠ̂_i_1 ⋯ i_m,
    
    C_e_T   =i_m+1,⋯ ,i_n=0d-1∑ω _d^i_m+1⋯ i_nΠ̂_i_m+1⋯ i_n.

Because

    i_m+1,⋯ ,i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_n
    
    =   i_m+1,⋯ ,i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_mΠ̂_i_m+1⋯ i_n
    
    =   Π̂_i_1 ⋯ i_mi_m+1,⋯ ,i_n=0d-1∑(ω_d^i_m+1⋯ i_n)^i_1 ⋯ i_mΠ̂_i_m+1⋯ i_n
    
    =   Π̂_i_1 ⋯ i_m C_e_T^i_1 ⋯  i_m,


    C_e   =i_1, ⋯, i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_n
       =i_1, ⋯, i_m=0d-1∑i_m+1, ⋯, i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_n
       =i_1, ⋯, i_m=0d-1∑Π̂_i_1 ⋯ i_m C_e_T^i_1 ⋯ i_m,

which is exactly the conclusion in Lemma <ref>.


Lemma <ref> demonstrates that a hyperedge operation can be interpreted as a controlled operation: the products of the vertices in C determine the operations imposed on the target part T. In fact, one can choose an arbitrary subset of e as the control part, and the remaining part as the target, which originates from the symmetry of C_e. 


Consider a system composed of A and B, whose associated Hilbert spaces are ℋ_A and ℋ_B, respectively. Suppose {|1⟩, |2⟩, ⋯, |n⟩} is an orthonormal basis of ℋ_A and |ψ _1⟩, |ψ _2⟩, ⋯, |ψ _n⟩ are normalized vectors in ℋ_B. The vector

    .......|1⟩|ψ_1.⟩ +|2⟩ |ψ _2⟩ +⋯ +|n⟩ |ψ _n⟩

is a product state if and only if all the |ψ_j⟩s (1≤ j≤ n) are parallel.



 (i)“if”. If all the |ψ _j⟩s are parallel, then each |ψ _j⟩ has a form .e^i ϕ_j|ψ _0⟩. So

    |1⟩|ψ _1⟩ +|2⟩ |ψ_2⟩ +⋯ +|n⟩ |ψ _n⟩ =(∑ _j=1^n e^i ϕ_j|j⟩) |ψ _0⟩,

which is a product state.

(ii)“only if". Suppose the total system is in a product state, B remains the same physical state no matter what measurement is made to A and whatever the result is. By implementing the von Neumann measurement {M_j=|j ⟩⟨ j||j∈{1, 2, ⋯, n}} to A, the part B will collapse to one of the states in ł{|ψ_1⟩, |ψ_2⟩, ⋯, |ψ_n⟩}̊. So all the |ψ_j⟩s are physically equivalent, i.e., they are parallel.



Qudit hypergraph state |H_d⟩ equals |+⟩ ^⊗ N_d if and only if E=∅.


(i)“if”. If E=∅, by definition for all e∈2^V, m_e=0, so |H_d⟩=|+⟩ ^⊗ N_d.

(ii)“only if". The stabilizer group of .|H_d.⟩ is generated by {X_ke:k∈ e∏(C_e\{k}^†)^m_e| k∈ V} while that of |+⟩^⊗ N_d is generated by {X_k |k∈ V}.

If

    e∏C_e^m_e|+⟩^⊗ N_d=|+⟩ ^⊗ N_d,

the two qudit hypergraph states will have the same stabilizer group, leading to

    X_k e:k∈ e∏(C_e\{k}^†)^m_e=X_k j ≠ k∏ X^p_j_j,

where k ∈ V and p_j ∈{0,1,⋯,d-1}. The factor e:k∈ e∏(C_e\{k}^†)^m_e
is always diagonal in the computational basis, while j ≠ k∏ X^p_j_j is diagonal only if p_j=0 (∀ j ≠ k). That is to say, to make Eq. (<ref>) hold,

    e:k∈ e∏(C_e\{k}^†)^m_e=j ≠ k∏ X^0_j=𝕀,

thus (notice that C_e\{k} is unitary)

    e:k∈ e∏C_e\{k}^m_e|+⟩^⊗ N-1_d=|+⟩^⊗ N-1_d.

Implement the above procedure several times, generally, one arrives

    e:k_1,⋯,k_n∈ e∏C_e\{k_1,⋯,k_n}^m_e|+⟩^⊗ N-n_d=|+⟩^⊗ N-n_d.


When n=N-1, Eq. (<ref>) becomes

    C_∅^m_{k_1,k_2,⋯,k_N-1}C_{k_N}^m_{k_1,k_2,⋯,k_N}|+⟩_d=|+⟩_d,

indicating that

    m_{k_1,k_2,⋯,k_N-1}=m_{k_1,k_2,⋯,k_N}=0.

Because all the k_i (i ∈{1,2,⋯,N}) are arbitrarily arranged in order, for all the e that satisfy |e|=N or N-1, m_e=0.

When n=N-2, Eq. (<ref>) becomes

    e:k_1,⋯,k_N-2∈ e∏C_e\{k_1,⋯,k_N-2}^m_e|+⟩^⊗ 2_d=|+⟩^⊗ 2_d.

The product involves all the hyperedges containing {k_1,⋯,k_N-2}, i.e., the cardinalities of these hyperedges are larger or equal to N-2. As is shown in the previous paragraph, hyperedges whose cardinalities are larger than N-2 must have 0 multiplicity, thus contributing to identity factors. So Eq. (<ref>) can be reduced to

    C_∅^m_{k_1,k_2,⋯,k_N-2}|+⟩^⊗ 2_d=|+⟩^⊗ 2_d,

indicating that m_{k_1,k_2,⋯,k_N-2}=0. Generally, if |e|=N-2, m_e=0.

Similarly, for all the e that satisfy |e|=N-3,N-2,⋯,0, m_e=0.

So if |H_d⟩=|+⟩ ^⊗ N_d, m_e=0 (∀ e∈ 2^V), i.e., E=∅.


With these lemmas, we can prove the following theorems.
theoremTheorem


Suppose H_d'=(V, E') and H_d=(V, E), then |H_d'⟩ = |H_d⟩ if and only if E'=E.


(i)“if”. By definition, in terms of representing d-level hypergraph states, a multi-hypergraph corresponds to a unique d-level hypergraph state.

(ii)“only if”. For e ∈2^V, denote its multiplicity corresponding to H_d' as m_e', then |H'_d⟩=e∏C_e^m'_e|+⟩^⊗ N_d.
If |H_d'⟩ = |H_d⟩,

    |+⟩^⊗ N_d=e∏C_e^m'_e-m_e|+⟩^⊗ N_d.

According to Lemma <ref>, this equation holds if and only if for all e, m'_e-m_e=0, i.e., E'=E.


Theorem <ref> indicates that distinct multi-hypergraphs correspond to distinct quantum states, assuming that the systems are both N-qudit systems.

An important entanglement property of qudit hypergraph states is revealed in the following theorem.



If one part of a multi-hypergraph is connected with the other part, then these two corresponding subsystems are entangled.


Suppose H_d=(V,E), divide V into two parts, one is called the control part (C={c_1,c_2,⋯,c_|C|}) and the other is called the target (T={t_1,t_2,⋯,t_|T|}), satisfying C∪ T=V and C∩ T=∅. Accordingly, we can define 3 sub-multisets of E, i.e., E_C, E_T and Λ. E_C (E_T) constitutes of all the elements in E that are subsets of C (T); Λ consists of all the elements in E that contains vertices in C and T simultaneously. If Λ≠∅, C and T are connected through hyperedges in Λ. 

Define the multi-hypergraphs H_d^C=(C,E_C) and H_d^T=(T,E_T), then

    |H_d⟩ =C_∅^-m_∅ϵ∈Λ∏ C_ϵ^m_ϵ |H_d^C⟩ |H_d^T⟩,

where |H_d^C⟩=∏_e'∈ E_C C_e'^m_e'|+⟩^⊗ |C|_d and |H_d^T⟩=∏_e”∈ E_T C_e”^m_e”|+⟩^⊗ |T|_d (notice that the multiplicity of each hyperedge in E_C, E_T and Λ is the same as the one in E).
Expanding |H_d^C⟩ in the computational basis explicitly, one has

    |H_d^C⟩=1/√(d)^|C|i_c_1,⋯,i_c_|C|=0d-1∑e^i ϕ(i_c_1,⋯ ,i_c_|C|)|i_c_1⋯ i_c_|C|⟩.

According to Lemma <ref>, all the C_es in Eq. (<ref>) can be expressed in a form like Eq. (<ref>), so

    |H_d⟩=C_∅^m_∅/√(d)^|C|i_c_1,⋯,i_c_|C|=0d-1∑|i_c_1⋯ i_c_|C|⟩' f̂(i_c_1,⋯ ,i_c_|C|)|H_d^T⟩,

where |i_c_1⋯ i_c_|C|⟩'=e^i ϕ(i_c_1,⋯ ,i_c_|C|)|i_c_1⋯ i_c_|C|⟩ and f̂(i_c_1,⋯ ,i_c_|C|) is some composite hyperedge transformation.

If |H_d⟩ is a product state, all f̂(i_c_1,i_c_2,⋯,i_c_|C|)|H_d^T⟩ (∀ i_c_1,⋯,i_c_|C|∈{0,1,⋯,d-1}) must be parallel (Lemma <ref>), i.e.,

    f̂(i_c_1,⋯ ,i_c_|C|)|H_d^T⟩   =e^i δ(i_c_1,⋯ ,i_c_|C|)f̂(0,⋯ ,0)|H_d^T⟩
       =e^i δ(i_c_1,⋯ ,i_c_|C|)|H_d^T⟩.

Divide every ϵ in Λ into c_ϵ and t_ϵ, where c_ϵ=ϵ∩ C and t_ϵ=ϵ∩ T, then (Lemma <ref>),

    f̂(1,⋯,1)|H_d^T⟩=ϵ∈Λ∏ C_t_ϵ^m_ϵ|H_d^T⟩.

So

    ϵ∈Λ∏ C_t_ϵ^m_ϵ|H_d^T⟩=e^i δ(1,⋯ ,1)|H_d^T⟩=C_∅^z|H_d^T⟩,

where z∈{0,1,⋯,d-1}, thus

    C_∅^d-zϵ∈Λ∏ C_t_ϵ^m_ϵ|+⟩^⊗ |T|_d=|+⟩^⊗ |T|_d.

This equation cannot be true because of Lemma <ref>. So |H_d⟩ cannot be a product state in the form like |ψ⟩ _C|ϕ⟩ _T, i.e., the two parts are entangled.


Theorem <ref> offers us an ability to knowing the entanglement structure of a qudit hypergraph state by reading the connectivity property of the multi-hypergraph. With the result in this theorem, we have the following two corollaries.

corollaryCorollary

If a multi-hypergraph H_d is connected, then |H_d⟩ is genuinely entangled.


If H_d is connected, divide it into arbitrary two parts, then the two parts are connected through some hyperedges. According to Theorem <ref>, these two parts are entangled. As the division is arbitrary, |H_d⟩ is non-biseparable, i.e., it is genuinely entangled.




Suppose an unconnected multi-hypergraph H_d is composed of several blocks (H_d^(i)) that are not connected to each other, and each one is a connected multi-hypergraph or possesses only one vertex, then each |H_d^(i)⟩ that possesses more than one vertex is a genuinely entangled state, and different blocks are not entangled with each other.


Different blocks are not connected to each other, so they are not entangled (Eq. (<ref>)). For connected H_d^(i), because |H_d^(i)⟩ is also a qudit hypergraph state, it is genuinely entangled (Corollary <ref>).


Corollary <ref> and Corollary <ref> enable multi-hypergraph a useful tool for visualizing the entanglement of its corresponding qudit hypergraph state.





§ RELATIONSHIP AMONG QUDIT HYPERGRAPH STATES AND SOME WELL-KNOWN STATE CLASSES


In this section, we will discuss relationships among qudit hypergraph states and some well-known state classes, i.e., generalized real equally weighted states, qudit graph states, and stabilizer states. 



 §.§ Qudit hypergraph states and generalized real equally weighted states


The real equally weighted states are the quantum states that all the coefficients in the computational basis are real and with equal absolute value. For example, real equally weighted states describing N-qubit systems can all be represented in the form 

    |ψ(f,N)⟩=1/2^N/2∑_i_1,⋯,i_N=0^1 (-1)^f(i_1,⋯,i_N)|i_1 ⋯ i_N⟩,

where f(i_1,⋯,i_N)∈ℤ_2. By interpretating -1 as ω_2, the generalized real equally weighted states (GREWSs) can be expressed as

    |ψ(f,N)⟩_d=1/d^N/2∑_i_1,⋯,i_N=0^d-1ω_d^f(i_1,⋯,i_N)|i_1 ⋯ i_N⟩,

in which f(i_1,⋯,i_N)∈ℤ_d.

It has been demonstrated in the literature that qubit hypergraph states are equivalent to real equally weighted states <cit.>. For the qudit case, it would be interesting to investigate whether a similar relationship exists. From the definition of qudit hypergraph states, we can see that every N-qudit hypergraph state can be expressed in the form of Eq. (<ref>), i.e., all qudit hypergraph states are GREWSs. For specific N and d, the total number of GREWSs is d^d^N, while in total there are only d^2^N qudit hypergraph states (There are d^2^N such multi-hypergraphs in total and Theorem <ref> shows that the states and multi-hypergraphs have a one-to-one correspondence).  Only if d=2, d^d^N=d^2^N, otherwise, d^d^N > d^2^N. This indicates that if d>2, the set of qudit hypergraph states is a proper subset of GREWSs. This relationship is different from the qubit case (See Fig. <ref>). 











 §.§ Relationship among qudit hypergraph states, qudit graph states, and stabilizer states


Qudit hypergraph state is a generalization of qudit graph state, so qudit graph states form a subclass of qudit hypergraph states. According to Theorem <ref>, two qudit hypergraph states are equal only if their corresponding multi-hypergraphs are the same. Generally speaking, a multi-hypergraph can have hyperedges with cardinalities larger than 2, which is different from that of multigraphs. Therefore, in general, a qudit hypergraph state is not a qudit graph state.

Stabilizer states of N-qudit systems are the common eigenstates with eigenvalue 1 of N independent elements in the Pauli group 𝒢_N^(d) <cit.>, where 𝒢_N^(d) is the N-fold product of 𝒢^(d), and 𝒢^(d)=ł{ω_d^a X^b Z^c |a, b, c ∈ℤ_d}̊ (X and Z are the qudit Pauli operators defined by Eq. (<ref>)). 
According to this definition, qudit graph states are all stabilizer states, because there are N independent stabilizers that can be expressed in the form g_k=X_k n:{k,n}∈ E∏Z_n^d-m_{k,n}, i.e., g_k∈𝒢_N^(d) (k∈{1,2,⋯,N}). As for the relationship between qudit hypergraph states and stabilizer states, we illustrate the result in the following proposition.
propositionProposition

A qudit hypergraph state is a stabilizer state if and only if the cardinalities of the hyperedges are all no more than 2.


The stabilizer group of |H_d⟩ is generated by {g_k=X_ke:k∈ e∏C_e\{k}^d-m_e|k=1,2,⋯,N}. If the cardinalities of the hyperedges are all no more than 2, then ∀ e, k, C_e\{k} is ω_d or a Z operator. Thus in this case, |H_d⟩ must be a stabilizer state. If some hyperedge in H_d has cardinality larger than 2 (suppose the vertex k is included by such a hyperedge), then g_k∉𝒢_N^(d). The reason is as follows. If g_k ∈𝒢_N, then X_k^-1 g_k ∈𝒢_N. Define a new qudit hypergraph state that |H_d(k)⟩=e:k∈ e∏C_e\{k}^d-m_e|+⟩_d^⊗ N, then it must be a product state. If a hyperedge e satisfies |e|>2, H_d(k) possesses a hyperedge e\{k} satisfying |e\{k}|≥2, which means that some vertices in H_d(k) are connected by e\{k}. According to Theorem <ref>, such a qudit hypergraph state cannot be a product state, which is contrary to that |H_d(k)⟩ is a product state. So only if the cardinalities of all the hyperedges are no more than 2 can |H_d⟩ be a stabilizer state. 

According to Proposition <ref>, a qudit hypergraph state that is also a stabilizer state at the same time may not be a qudit graph state (See Fig. <ref>). It may also be a qudit graph state operated by some generalized local Pauli operations.

To summarize, the relationship among qudit hypergraph states, qudit graph states and stabilizer states can be expressed in Fig. <ref>, which is very similar to the qubit case studied in Ref. <cit.>. 







§ QUDIT HYPERGRAPH STATES' RESPONSES TO LOCAL OPERATIONS AND MEASUREMENTS


In this section, we will consider how qudit hypergraph states response to the generalized Z, X operations, and generalized Z, X measurements <cit.>.
The initial states are all assumed to be |H_d⟩ and the final states are all denoted as |ψ_f⟩.

The local unitary operator Z_k, which acts upon the kth vertex, can also be interpreted as the hyperedge operation C_{k}. So when Z_k is
implemented on |H_d⟩,

    |ψ_f⟩   =C_{k}|H_d⟩,

which is also a qudit hypergraph state. Denote the corresponding multi-hypergraph as H_d'=(V,E'), then the associated multiplicity function is

    m_e'=[        m_e     e≠{k},; m_{k}+1  d     e={k}. ]

In pictorial representation, the multiplicity of the hyperedge {k} increases by 1 (when m_{k}=d-1, the hyperedge cancels, due to Eq. (<ref>)) while the multiplicities of all the other hyperedges do not change.

When X_k is implemented upon |H_d⟩, the final state is (see the detailed calculation in Appendix <ref>)

    |ψ_f⟩=e:k ∈ e∏C_e\{k}^m_e|H_d⟩,

which is also a d-level hypergraph state. Denote the corresponding multi-hypergraph as H_d'=(V,E'), then the associated multiplicity function is

    m_e'=[           m_e         k∈ e,; m_e+m_e∪{k} d        k ∉ e. ]

Equations (<ref>) and (<ref>) indicate that the multi-hypergraph corresponding to the final state is the one that the multiplicities of all the hyperedges in the form e\{k} are added by m_e (k∈ e) with respect to that of H_d (may also subtract a multiple of d in order to keep m_e∈{0,1,⋯,d-1}), while the multiplicities of the other hyperedges do not change.

The cyclic group generated by {X_k, .Z_k|k∈ V} preserves the set structure of qudit hypergraph states. For any element in the group, the action on the qudit hypergraph state can be represented by adding related hyperedges to the hypergraph.

For a general d-level system, Z and X operators are unitary but usually non-Hermitian. As Z and X both possess orthonormal eigenvectors, one can still define von Neumann measurements of Z and X by associating each eigenvector with a real value. After the measurement, the measured qudit collapses to an eigenstate of Z (X). Correspondingly, the system composed of the remaining qudits collapses to a new state.

Suppose we measure Z_k and the vertex collapses to |i_k⟩ (denote that Π̂_i_k=|i_k⟩⟨ i_k|), then the whole system collapses to

    Π̂_i_k|H_d⟩=   (e':k∉ e'∏C_e'^m_e')Π̂_i_k(e:k∈ e∏ C_e^m_e)|+⟩^⊗ N_d.

While when k∈ e,

    Π̂_i_kC_e=Π̂_i_kj_k=0d-1∑Π̂_j_kC_e\{k}^j_k=C_e\{k}^i_kΠ̂_i_k,

so

    Π̂_i_ke:k∈ e∏C_e^m_e|+⟩ ^⊗ N_d=   e:k∈ e∏C_e\{k}^i_k m_eΠ̂_i_k|+⟩ ^⊗ N_d 
    
    =   1/√(d)|i_k⟩e:k∈ e∏C_e\{k}^i_k m_e|+⟩ ^⊗ N-1_d.

The final state of the unmeasured part is

    |ψ_f⟩=(e':k∉ e'∏C_e'^m_e')(e:k∈ e∏C_e\{k}^i_k m_e)|+⟩ ^⊗ N-1_d,

which is also a qudit hypergraph state. Denote the multi-hypergraph as H_d'=(V',E'), where V'={1,2,⋯,k-1,k+1,⋯,N} and E' is the multiset of the hyperedges of H_d', then the associated multiplicity function is

    m_e'= m_e +i_k m_e∪{k} d.

If the measurement breaks some hyperedges, then we can get the remaining hypergraph through the following steps: (i) Delete the measured vertex; (ii) Multiply the multiplicities of the broken hyperedges by i_k (the measurement result); (iii) Make the “multiplicities” valid by subtracting some multiple of d.

In general, for a 2-level hypergraph state, after a local Pauli-X_k measurement, the unmeasured part collapses to a state that does not correspond to a hypergraph <cit.>. For the more general d-level hypergraph states, the generalized X_k measurements cannot maintain the structure of the set of d-level hypergraph states either.

The situations demonstrated in this section are interesting and important because qudit hypergraph states' responses to basic local unitary operations and measurements have potential applications in quantum codes and quantum error correction.




§ BELL NON-LOCALITY OF QUDIT HYPERGRAPH STATES AND THE EXPERIMENTAL DETECTION


The exhibition of non-locality by graph states and qubit hypergraph states
is very important and even necessary in many quantum information tasks.
Behind such investigation is the challenging problem of the non-locality of multipartite
entangled states in quantum information theory. It has been proven
that all entangled pure states are non-local, no matter how many particles
there are and how many dimensions each particle contains <cit.>.
 In particular, a scheme of non-locality exhibition
was provided in an operational manner in Ref. <cit.>. The idea is that any two particles
can be measured to violate Clauser-Horne-Shimony-Holt (CHSH) inequality <cit.>, in
assistance of measuring the rest particles. Below we discuss how it
works in the scenario of qudit-hypergraph states.




 §.§ Nonlocality exhibition by the CHSH inequality


For clarity of the discussion, we consider a multi-hypergraph H_N,d,m=(V, E), in which V={1,2,⋯,N} and
E={V,V,⋯,V} (|E|=m), then the corresponding quantum state is

    |H_N,d,m⟩    =C_V^m|+⟩_d ^⊗ N
        =1/√(d^N)∑_i_1,⋯,i_N=0^d-1ω_d^m i_1⋯ i_N|i_1⋯ i_N⟩.

Without losing generality, let the vertices 3,4,⋯,N
assist the vertices 1 and 2 in exhibiting Bell non-locality
by violating CHSH inequality [Notice that for the state |H_N,d,m⟩, the
exchange of any pair of vertices does not change the state.]. The assistance can be done by projecting the
vertices to their respective |+⟩_d. After this operation, the state of the rest part (composed of vertices 1 and 2)
becomes

    |H_N,d,m^(2)⟩ =𝒩/d^N-1∑_i_1,i_2=0^d-1Ω_i_1i_2|i_1i_2⟩,

where 𝒩 is the normalization factor, Ω_i_1i_2=∑_i_3,⋯,i_N=0^d-1ω^m i_1i_2⋯ i_N_d forming the d× d matrix Ω. 


We state that the remaining two vertices are entangled. The proof can
be done through analyzing the rank of Ω.
Assume |H_N,d,m^(2)⟩ is separable, the rank of Ω should be 1.
But the upper-left 2 × 2 submatrix (i_1,i_2∈{0,1}) of Ω

    Ω̃=(
    [                         d^N-2                         d^N-2;                         d^N-2 ∑_i_3,⋯,i_N=0^d-1ω^m i_3⋯ i_N;                               ]),

has a non-zero determinant, therefore rank(Ω)≥2 <cit.>, indicating that the rest two
vertices are entangled.

For analyzing the entanglement property and Bell non-locality, it
is more convenient to transform |H_N,d,m^(2)⟩ to its Schmidt form

    |H_N,d,m^(2)⟩ =∑_μ=0^d-1c_μ|μ⟩_1 |μ⟩_2,

where 

c_μ are the Schmidt coefficients, 
and |μ⟩_1 and |μ⟩_2 are the Schmidt bases for vertex 1 and 2, respectively.
The entanglement of |H_N,d,m^(2)⟩ implies that there are more than 1 non-trivial
term in the right hand side of Eq. (<ref>). Finally, we can measure vertex 1 on the settings S_1=σ_z and T_1=σ_x, and vertex 2 on the settings S_2=σ_zcos2t+σ_xsin2t and T_2=σ_zcos2t-σ_xsin2t, where σ_z=|0⟩⟨ 0|-|1⟩⟨ 1|, σ_x=|0⟩⟨ 1|+|1⟩⟨ 0| on respective basis |μ⟩_1 and |μ⟩_2, and tan2t=2c_0c_1. The measurement results will disclose the nonlocality by violating the following CHSH inequality <cit.>

    C= | E(S_1S_2||+⟩_d ^⊗ N-2)+E(S_1T_2||+⟩_d ^⊗ N-2)
    
    +E(T_1S_2||+⟩_d ^⊗ N-2)-E(T_1T_2||+⟩_d ^⊗ N-2) |    ≤2.

More precisely, the left hand side of the above inequality can achieve 2 √(1+4 c_0^2 c_1^2/(c_0^2 + c_1^2)^2), such that the lhv bound 2 is violated.




 §.§ The prime-dimensional case


Specially, when the dimension of the qudits is prime (d∈ℙ), Ω_i_1i_2 has a simple analytic form

    Ω_i_1i_2=[             d^N-2     i_1=0  i_2=0,; d^N-2-d (d-1)^N-3   i_1≠ 0  i_2≠ 0. ]

In this case, the Schmidt form of |H_N,d,m^(2)⟩ is

    |H_N,d,m^(2)⟩=x_+|0⟩_1|0⟩_2+x_-|1⟩_1|1⟩_2/√(x_+^2+x_-^2),

where

    x_±=λ±√(λ^2+4(d-λ))/2,

and

    |0⟩_k   =1/N_+((x_+ -λ+1)|0⟩+∑_i=1^d-1|i⟩),
    |1⟩_k   =1/N_-((x_- -λ+1)|0⟩+∑_i=1^d-1|i⟩),




with N_±=√((x_± -λ+1)^2+d-1), k∈{1,2} and 
λ=d-(d-1)^N-2/d^N-3.
The Schmidt number of |H_N,d,m^(2)⟩ is 2, which indicates that the entanglement of vertices 1 and 2 is equivalent to the entanglement of two qubits. The results in the previous paragraph can be applied here directly except that here tan2t=2x_+x_- /(x_+^2 + x_-^2). Explicitly, in this case the left-hand side of Eq. (<ref>) can violate the CHSH inequality by an amount of 2 √(1+4 x_+^2 x_-^2/(x_+^2 + x_-^2)^2).

Figure <ref> reveals the violation of CHSH inequality for various combinations of d (d∈ℙ) and N in this measurement scheme. Here, C is always greater than 2, indicating that this measurement scheme can reveal the nonclassical correlation between the vertices.
When d is fixed and N is large (see (a) in Fig. <ref>), the matrix elements of the normalized Ω  are nearly equal, i.e., the normalized quantum state of the remaining vertices is approximately |+⟩_d ^⊗ 2, thus C approaches 2 when N goes to infinity. When N is fixed and d increases (see (b) in Fig. <ref>), |H_N,d,m^(2)⟩ approaches (|0⟩∑_i=1^d-1|i⟩+∑_i=1^d-1|i⟩|0⟩)/√(2(d-1)), which is equivalent to a 2-qubit maximally entangled state, thus C approaches 2√(2) when d goes to infinity.





 §.§ Discussion


Remarkably, the above scheme of exhibiting the nonlocality of multipartite quantum systems is potentially applicable in the current use of qudit hypergraph states and conventional qubit graph states. In fact, it can be, and in some cases has been, used in practice with current technology. In the case of entanglement verification, it involves only two measurement settings at each side, where the measurement settings of assistant qudits never change. Besides, the CHSH inequality can always reveal the “strong” nonlocality, in the sense that the entanglement between arbitrary faraway two qudits can be revealed, as long as the two are connected by other vertices and edges. All these features make the above scheme rather experimentally friendly.

For example, the entanglement verification of cluster states (a special class of graph states) generated by cold-atom lattices is a necessary work for the future use in quantum computing. However, the detection of the entanglement in large-scale cluster states is always a challenging problem <cit.>.  From a practical perspective, the CHSH scheme discussed in this section can also be used as an entanglement witness for cluster states, especially for the long-distance entanglement. That is, one can always choose two interested particles (connected by other particles with C-phase operations), and test the entanglement correlation between them, no matter how far the two particles are. 

Another example is its application in quantum networks <cit.>, in which thousands of users complete a quantum information task via a multipartite entangled state.
A typical task is the so-called third-man quantum cryptography in which generation of a cryptographic key is controlled by a third operator who decides whether to activate the key generation or not <cit.>. 


The scheme we discussed thus exactly offers an operational way to analyze the security of the third-man quantum cryptography. 

An important problem in qudit hypergraph states we did not discuss is the genuine multipartite entanglement. In particular, the relationship between the classification of multipartite entanglement and the property of hypergraphs deserves to be studied deeply, and the triple entanglement case has been discussed in <cit.>. As an analog, the concept of genuine multipartite nonlocality is also put forwarded in <cit.>. However, despite its significance in the theoretical study, its applications in quantum information processing need further study.



§ CONCLUSIONS

In this work, we have proposed a large class of quantum states named qudit hypergraph states in which every vertex of the multi-hypergraph represents a d-level quantum system. We have investigated the operational definition of these states and studied their stabilizers, which possess potential applications in quantum codes and quantum computation. It is shown that generalized local X, Z operations, and Z measurements can transform a qudit hypergraph state to another one.

The multi-hypergraphs and qudit hypergraph states have a one-to-one correspondence, and the entanglement of the qudit hypergraph states can be directly illustrated by the structure of their corresponding multi-hypergraphs. If a multi-hypergraph (or part of it) is connected, the corresponding quantum system (the quantum system corresponding to the connected part) is genuinely entangled. Such entanglement leads to potential exhibition of Bell non-locality. As an example, we showed how to obtain the violation of Bell inequality in N-uniform qudit hypergraph states. The method is also applicable to other qudit hypergraph states and general N-qudit quantum states. 

We also study the relationship among qudit hypergraph states and some important state classes. As for the real equally weighted states, we generalized them to the qudit case according to their form in the computational basis. It is shown that only in the 2-level case are the two state classes (“generalized real equally weighted states” and “qudit hypergraph states”) the same; otherwise, qudit hypergraph states are a subclass of “generalized real equally weighted states”. 
The relationship among qudit hypergraph states, qudit graph states and stabilizer states are discussed. Our results demonstrate that qudit graph states are a common subclass of qudit hypergraph states and stabilizer states. What is more, the union of these two state classes contains more than qudit graph states, which is very similar to the qubit case.


Nevertheless, much work is still needed to be done for the potential properties and applications of qudit hypergraph states. It is known that the set of qubit hypergraph states is the same as the set of real equally weighted states, which is a class of quantum states having important applications in quantum algorithms. Qudit hypergraph states form a subclass of generalized real equally weighted states.  In this sense, it is highly probable that qudit hypergraph states also have important applications in quantum algorithms. It has been shown in the literature that the unique entanglement form and Bell non-locality of qubit hypergraph states have important applications in quantum metrology and novel quantum computation schemes. It is worth further study to see whether the qudit hypergraph states have similar applications. In this paper, we have focused on the simplest definition of entanglement (a quantum state is entangled if it can not be written as a tensor product of two state vectors), while in fact there are much more comprehensive content in the study of multipartite entanglement, for example, equivalent classes of multipartite entanglement. The discussion of such issues in the context of qudit hypergraph states is not only interesting by itself but also essential for the future applications.


We thank Y. Y. Zhao, Y. Liu and Y. L. Zheng for the helpful discussions, and F. E. S. Steinhoff for his nice comments. F. L. X. and Z. B. C. were supported by the National Natural Science Foundation of China (Grant No. 61125502) and the CAS. Y. Z. Z., W. F. C., and K. C. were supported by the National Natural Science Foundation of China (Grants No. 11575174) and the CAS.



Note added: During the preparation of our manuscript, we notice that there is a paper which proposed qudit hypergraph states in a different manner and discussed some different issues <cit.>.




§ DERIVATION OF EQUATION (<REF>)

If k ∉ e, C_e X_k C_e^†=X_k.

If k ∈ e, for simplicity of the discussion, assume that k=1 and e={1,⋯ ,n}, then C_e=∑_i_1=0^d-1Π̂_i_1 C_e\{1}^i_1 (Lemma <ref>), thus

    C_e X_1 C_e^†
    
    =   i_1,j_1=0d-1∑Π̂_i_1(|0⟩⟨ 1|+|1⟩⟨2|+⋯ +|d-1⟩⟨ 0|)
    Π̂_j_1C_e\{1}^i_1-j_1
    
    =   |0⟩⟨ 1|C_e\{1}^-1+|1⟩⟨ 2|C_e\{1}^-1+⋯ +|d-1⟩⟨ 0|C_e\{1}^-1
    
    =   X_1C_e\{1}^-1=X_1C_e\{1}^†.

Generally, C_e X_k C_e^†=X_k C_e\{k}^†.

Let X_k pass over all C_e, then we have

    (∏ _e∈ E C_e^m_e) X_k (∏ _e'∈ E C_e'^m_e')^† =X_ke:k∈ e∏(C_e\{k}^†)^m_e,

which is exactly what is demonstrated in Eq. (<ref>).





§ DERIVATION OF EQUATION (<REF>)


For the vertex e={1,2,⋯,n}, denote that Π̂_i_2 ⋯ i_n=|i_2 ⋯  i_n⟩⟨ i_2 ⋯  i_n|, then

    X_1 C_e 
    
    =   X_1i_2,⋯ ,i_n=0d-1∑Z_1^i_2 ⋯  i_nΠ̂_i_2 ⋯ i_n
    
    =   i_2,⋯ ,i_n=0d-1∑ω ^i_2 ⋯  i_n Z_1^i_2 ⋯  i_n X_1 Π̂_i_2 ⋯ i_n
    
    =   i_2,⋯ ,i_n=0d-1∑Z_1^i_2 ⋯  i_nΠ̂_i_2 ⋯ i_nj_2,⋯ ,j_n=0d-1∑ω ^j_2 ⋯ j_nΠ̂_j_2 ⋯ j_nX_1
    
    =   C_eC_e\{1}X_1.


Generally, if k∈ e, X_k C_e=C_e C_e\{k}X_k, so

    |ψ_f⟩
    
    =   X_k (∏C_e^m_e)|+⟩ ^⊗ N_d
    
    =   (e:k∉ e∏C_e^m_e) X_k (e':k∈ e'∏C_e'^m_e') |+⟩ ^⊗ N_d
    
    =   (e:k∉ e∏C_e^m_e) (e':k∈ e'∏C_e'^m_e'C_e'\{k}^m_e') X_k|+⟩ ^⊗ N_d
    
    =   (∏ C_e^m_e)(e':k∈ e'∏C_e'\{k}^m_e')|+⟩ ^⊗ N_d
    
    =   (e':k∈ e'∏C_e'\{k}^m_e')(∏ C_e^m_e)|+⟩ ^⊗ N_d
    
    =   e:k ∈ e∏C_e\{k}^m_e|H_d⟩.











apsrev4-1




