Quantum nonlocality was first discovered by Einstein, Podolsky, and Rosen (EPR) in 1935 [1], including quantum entanglement, quantum steering, and Bell nonlocality. They formulated an apparent paradox of quantum theory (EPR paradox) and gave a “thought” experiment that argues the wave function description in quantum mechanics is incomplete.

Bell nonlocality originated from the Bell’s 1964 paper [2]. He found that when some entangled state is suitably measured, the probabilities for the outcomes violate an inequality, named the Bell inequality. This property of quantum states is the so-called Bell nonlocality and was reviewed by Brunner et al. [3] for the “behaviors” P(ab|xy) (correlations), a terminology introduced by Tsirelson (1993) [4], but not for quantum states.

The result obtained by Bell [2] was named Bell’s theorem, which states that quantum predictions are incompatible with a local hidden variable description and are a cornerstone of quantum theory and at the center of many quantum information processing protocols. Over the years, different perspectives on non-locality have been put forward, including different ways to detect non-locality and quantify it.

Usually, Bell nonlocality for quantum states is detected by violation of some of Bell’s inequalities, such as Clause-Horne-Shimony-Holt (CHSH) inequality for two qubits. A proof of nonlocality without inequalities for two particles had been given earlier by Heywood and Redhead [5], which was much simplified by Brown and Svetlichny [6]. Greenberger, Horne, and Zeilinger (GHZ) [7] gave a proof of nonlocality but without using inequalities, in which a minimum of three particles was required in their proof. Mermin [8] provided a simple unified form for the major no-hidden-variables theorems by two examples. Hardy in [9, 10] proposed the two-particle 2-dimensional 2-setting Hardy paradox and gave the maximum probability of Bell’s nonlocality. Hardy et al. [11] discovered the two-particle 2-dimensional k-setting Hardy paradox. Aravind [12] established a Bell’s theorem without inequalities and only two distant observers. Dong et al. obtained in [13] some methods for detecting Bell nonlocality based on the Hardy Paradox. Chen et al. [14] proved that Bell nonlocal states can be constructed from some steerable states. They also established in [15] a mapping criteria between nonlocality and steerability. Jiang et al. [16] proposed a generalized Hardy’s paradox, and Yang et al. [17] presented a stronger Hardy-type paradox based on the Bell inequality and its experimental test. Cao and Guo [18] introduced mathematically the Bell locality and the unsteerability of a bipartite state for a given measurement setting and established their characterizations.

Viewed as joint outcome probabilities (correlations) for a specific experimental configuration as a vector of a Euclidean space R t , Pironio [19] proved that a Bell inequality defining a facet of the polytope B of Bell local correlations can be lifted to one that also defines a facet of the more complex polytope, and established a formula for finding the affine dimension dim ( B ) of B .

By placing quantum possibilities within a wider context, Barrett et al. [20] investigated the polytope L of no-signaling correlations, which contains the quantum correlations as a proper subset, determined the vertices of L in the some special cases, and discussed how interconversions between different sorts of correlations may be achieved. They also considered some multipartite examples. Barrett et al. [21] introduced a version of the chained Bell inequality for an arbitrary number of measurement outcomes and use it to give a simple proof that the maximally entangled state of two d-dimensional quantum systems has no local component. Masanes et al. [22] considered nonlocality of n-partite correlations and identified a series of properties common to all theories that do not allow for superluminal signaling and predict the violation of Bell inequalities. They observed that intrinsic randomness, uncertainty due to the incompatibility of two observables, monogamy of correlations, impossibility of perfect cloning, privacy of correlations, and bounds in the shareability of some states are solely a consequence of the no-signaling principle and nonlocality. Loubenets [25] proved that the probabilistic description of an arbitrary multipartite correlation scenario admits a local quasi hidden variable (LqHV) simulation if and only if all joint probability distributions of this scenario satisfy the general nonsignaling condition formulated in [23, 24] using the notions of an LqHV model and a deterministic LqHV given by integrals rather than sums. Loubenets [26] also proved that the probabilistic description of any quantum multipartite correlation scenario with an arbitrary number of settings and outcomes at each site does admit an LqHV model. In an LqHV model given in [23, 24, 26], locality inherent to an LHV model is preserved but the basic concept of Kolmogorov’s probability model [27], a probability space, is replaced by a measure space with a normalized bounded real-valued measure not necessarily positive. M e ́ ndez, J. Urías [28] formulated the set of half-spaces describing the polytope of no-signaling probability states that are admitted by the most general class of Bell scenarios, presented a computational tool to solve the no-signaling description for the elements, which are the pure no-signaling boxes and the facets of Bell polytopes. Chaves and Budroni [29] introduced the concept of entropic nonsignaling correlations, and characterized and showed the relevance of these entropic correlations in a variety of different scenarios, ranging from typical Bell experiments to more refined descriptions such as bilocality and information causality. They applied the framework to derive the first entropic inequality testing genuine tripartite nonlocality in quantum systems of arbitrary dimension and also proved the first known monogamy relation for entropic Bell inequalities. Cope and Colbeck [30] found a series of Bell inequalities from no-signaling distributions by exploiting knowledge of the set of extremal no-signaling distributions. Eli et al. [31] characterized Bell nonlocality of bipartite correlations using tensor networks [32] and sparse recovery and proved that nonsignaling bipartite correlations can be described by local hidden variable models (LHVMs) governed by a quasi-probability distribution.

In the present paper, we continue to discuss nonsignaling and Bell nonlocality of n-partite correlations in order to generalize the Eli’s result to a multipartite case. Such correlations define the entries of a nonnegative tensor P of order 2n, which we call an n-partite correlation tensor (CT). In Section 2, we review some concepts and notations about tensors used later. In Section 3, n-partite nonsignaling correlation tensors are recalled and some observations are obtained. Also, correlation-type tensors are introduced as an extension of correlation tensors. In Section 4, a tensor-network decomposition of an n-partite nonsignaling CT is deduced using the singular-value-decomposition theorem of matrices and a decomposition lemma of row-stochastic matrices (RSMs) into a convex combination of {0, 1}-RSMs. In Section 5, we discuss Bell locality of an n-partite CT P and establish a relationship between Bell local CTs and nonsignaling ones.

In what follows, we use the notation [m] = {1, 2, …, m} for every positive integer m.

Let e A = { | e i 〉 } i = 1 m and f B = { | f j 〉 } j = 1 n be orthonormal bases for H A and H B , respectively. Then e A ⊗ f B ≔ { | e i 〉 ⊗ | f j 〉 } ( i , j ) ∈ [ m ] × [ n ] forms an orthonormal basis for H A B . Thus, every state ρ ^AB of the system AB can be represented as ρ A B = ∑ i , j , k , ℓ ρ i j k ℓ | e i 〉 | f j 〉 〈 e k | 〈 f ℓ | . (2.1) This implies that every state ρ ^AB is determined by a set of complex coefficients ρ _ijkℓ labeled by four indices i, j, k and ℓ, which defines a complex tensor T _ρ = [[ρ _ijkℓ ]] of order 4.

Generally, a complex tensor is a multi-dimensional array of complex numbers and the order (rank) of a tensor is the number of indices [34]. Equivalently, we refer to a complex (or real) tensor of order k as a function T from an index set D _T = [d ₁] × [d ₂] ×⋯ × [d _k ] into C (or R ), denoted by T = [ [ T i 1 i 2 … i k ] ] , where T i 1 i 2 … i k = T ( i 1 , i 2 , … , i k ) , the value of the function T at (i ₁, i ₂, …, i _k ), called the (i ₁, i ₂, …, i _k )-entry of T. We also call such a T a (d ₁, d ₂, …, d _k )-dimensional tensor of order k, or a rank-k tensor over D _T . Thus, a rank-0 tensor is a scalar x, a d-dimensional tensor of order 1 is a d-dimensional vector (v ₁, v ₂, …, v _d ), and an (m, n)-dimensional tensor of order 2 is just an m × n matrix [A _ij ].

Two tensors A and B are said to be equal, denoted by A = B, if they are equal as functions, having the same domain of definition D and taking the same values at each index (i ₁, i ₂, …, i _k ) in D. A and B are said to contractive if they share at least one index. The contraction of A and B is the tensor A⋄B whose entries are the sum over all the possible values of the repeated indices of A and B. For instance, when A = [[A _ij ]] and B = [[B _ik ]] are tensors over [m] × [n] and [m] × [p], respectively, they are contractive with the contraction C = [[C _jk ]] where C j k = ∑ i = 1 m A i j B i k . That is, A⋄B = C, which is just the matrix product of matrices A ^T and B. In this case, B and A are also contractive with the contraction D = [[D _kj ]] where D k j = ∑ i = 1 m B i k A i j , which is just the matrix product of matrices B ^T and A. Generally, A⋄B ≠ B⋄A.

Furthermore, the outer product (also called the tensor product) A ⊗B of two tensors A and B is the tensor whose entries are the products of entries of A and B. Say, when A = [[A _ijk ]] and B = [[B _xyzuv ]] are tensors over D _A and D _B , respectively, the outer product of A and B is the tensor A ⊗ B = [ [ A i j k B x y z u v ] ] = [ [ A i j k B x y z u v ] ] i j k x y z u v , which is a rank-8 tensor over D _A × D _B . And that of A and B reads B ⊗ A = [ [ B x y z u v A i j k ] ] = [ [ B x y z u v A i j k ] ] x y z u v i j k , which is a rank-8 tensor over D _B × D _A . Generally, A ⊗B ≠ B ⊗A.

A tensor-network decomposition of a bipartite nonsignaling correlation was given in [31]. The following technical Lemma 4.2 was proved and used there. We rewrite it and give an alternative proof. To do so, let us recall a result proved by Li et al. [33], which implies that the set of all extreme points of the set CSM(m,n) of all nonnegative column-stochastic matrices (CSMs) of order m × n are exactly m ⁿ {0, 1}-CSMs of order m × n. Since an m × n nonnegative matrix A = [a _ij ] is column-stochastic, i.e., ∑ _i a _ij = 1 for all j, if and only if its transpose A ^T = [a _ji ] is row-stochastic, we get immediately the following result.

Lemma 4.1. The set of all extreme points of the set RSM(m,n) of all nonnegative row-stochastic matrices (RSMs) of order m × n are exactly n ^m {0, 1}-RSMs of order m × n: R k = r i j k ≔ δ j , J k i k = 1,2 , ⋯ , n m , where { J 1 , J 2 , ⋯ , J n m } are the set of all maps from [m] into [n].

The following lemma was given in [31]. Here, we give a detailed proof based on Lemma 4.1.

Lemma 4.2. [31] An m × o matrix M = [M _ij ] has constant row sums ∑ j = 1 o M i j = C (independent of i) if and only if it can be decomposed as M = ∑ k = 1 o m c k δ j , J k i ,  i.e.,  M i j = ∑ k = 1 o m c k δ j , J k i ∀ i , j , (4.1) with ∑ k = 1 o m c k = C , where \{J _{k : k} ∈ [o ^m ]} denotes the set of all maps from [m] into [o]. If all M _ij ≥ 0, then we can choose all c _k ≥ 0.

Proof. The sufficiency is clear. To prove the necessity, we assume that M = [M _ij ] has the desired property. Put a = min{M _ij : i ∈ [m], j ∈ [o]} and use 1 m × o to denote the m × o matrix whose entries are all 1. Then M − a 1 m × o becomes a nonnegative matrix with constant row sums C − oa ≥ 0.

Case 1. C − oa = 0. In this case, M = a 1 m × o . Since o − 1 1 m × o is a row stochastic matrix, Lemma 3.1 implies that it can be written as a convex combination of (0, 1)-RSMs: o − 1 1 m × o = ∑ k = 1 o m b k δ j , J k i , (4.2) where ∑ k = 1 o m b k = 1 and all b _k ≥ 0. Thus, M = a o × o − 1 1 m × o = ∑ k = 1 o m a o b k δ j , J k i with ∑ k = 1 o m a o b k = C . Clearly, all aob _k ≥ 0 if all M _ij ≥ 0.

Case 2. C − oa > 0. In this case, ( C − o a ) − 1 ( M − a 1 m × o ) is a row stochastic matrix and so it can be written as a convex combination of (0, 1)-RSMs (Lemma 3.1): C − o a − 1 M − a 1 m × o = ∑ k = 1 o m d k δ j , J k i , (4.3) where ∑ k = 1 o m d k = 1 and all d _k ≥ 0. It follows from Eq. 4.3 that M = a 1 m × o + C − a o ∑ k = 1 o m d k δ j , J k i = a 1 m × o + ∑ k = 1 o m C − o a d k δ j , J k i = ∑ k = 1 o m c k δ j , J k i , where c _k = oab _k + (C − oa)d _k being of sum C. Clearly, when all M _ij ≥ 0, we have a ≥ 0 and so all c _k ≥ 0. The proof is completed.

Theorem 4.1. For any n ∈ N + , an NSCTT T = [[T(a ₁⋯a _n |x ₁⋯x _n )]] of order 2n can be decomposed as T a 1 ⋅ ⋅ ⋅ a n | x 1 ⋅ ⋅ ⋅ x n = ∑ k 1 = 1 N 1 ⋯ ∑ k n = 1 N n q k 1 k 2 ⋯ k n δ a 1 , J k 1 1 x 1 ⋅ ⋅ ⋅ δ a n , J k n n x n (4.4) for all x _i ∈ [m _i ], a _i ∈ [o _i ](i = 1, 2,⋯n), where q k 1 k 2 ⋯ k n ∈ R , and J k i ( i ) k i = 1 N i denotes the set of all maps from [m _i ] into [ o i ] ( N i = o i m i , i = 1,2 , ⋯ , n ) .

Proof. When n = 1, to get the decomposition of an NSCTT T, let us consider T = [[T(a ₁|x ₁)]] as a matrix with (x ₁, a ₁)-entry T(a ₁|x ₁). Using Lemma 4.2 yields that T a 1 | x 1 = ∑ k 1 = 1 N 1 q k 1 δ a 1 , J k 1 1 x 1 , ∀ x 1 , a 1 , (4.5) where N 1 = o 1 m 1 and { J k 1 ( 1 ) : k 1 ∈ [ N 1 ] } denotes the set of all maps from [m ₁] into [o ₁].

Suppose that for some n ≥ 1, a decomposition (4.4) exists for any n-partite NSCTT T .

Let T = [[T(a ₁⋯a _n+1|x ₁⋯x _n+1)]] be an (n + 1)-partite NSCTT. To consider T as a matrix, we choose bijections α ₁ : [m ₁] × [o ₁] → [m ₁ o ₁], α 2 : m 2 × ⋯ × m n + 1 × o 2 × ⋯ × o n + 1 → m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 and then define an (m ₁ o ₁) × (m ₂ o ₂⋯m _n+1 o _n+1) matrix T ̄ = [ T ̄ i 1 i 2 ] with (i ₁, i ₂)-entry T ̄ i 1 i 2 = T ( a 1 ⋅ ⋅ ⋅ a n + 1 | x 1 ⋅ ⋅ ⋅ x n + 1 ) if i 1 = α 1 x 1 , a 1 , i 2 = α 2 x 2 , ⋅ ⋅ ⋅ , x n + 1 , a 2 , ⋅ ⋅ ⋅ , a n + 1 . Let T ̄ = U S V be a singular value decomposition (SVD) of T ̄ where U = [U _iλ ] and V = [V _λj ] are some real orthogonal matrices of orders m ₁ o ₁ and m ₂ o ₂⋯m _n+1 o _n+1, respectively, and S is one of the following matrices: Σ 0 m 1 o 1 < m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 ; Σ 0 m 1 o 1 > m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 ; Σ m 1 o 1 = m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 , where Σ is a nonnegative diagonal matrix. Without loss of generality, we assume that S is the first case. Thus, Σ = diag d 1 , d 2 , ⋯ , d m 1 o 1 , d λ > 0 1 ≤ λ ≤ r , d λ = 0 r < λ ≤ m 1 o 1 where r = r a n k ( T ̄ ) . Put A x 1 a 1 ( λ ) = U i λ if i = α ₁(x ₁, a ₁), 1 ≤ λ ≤ m ₁ o ₁; and B x 2 ⋯ x n + 1 a 2 ⋯ a n + 1 ( λ ) = V λ j if j = α ₂(x ₂,⋯,x _n+1, a ₂,⋯,a _n+1), 1 ≤ λ ≤ m ₂ o ₂⋯m _n+1 o _n+1. Then the SVD T ̄ = U S V of T ̄ yields that T a 1 ⋅ ⋅ ⋅ a n + 1 | x 1 ⋅ ⋅ ⋅ x n + 1 = T ̄ i j = ∑ λ = 1 m 1 o 1 d λ A x 1 a 1 λ B x 2 ⋯ x n + 1 a 2 ⋯ a n + 1 λ , (4.6) The nonsignaling condition on T implies that ∑ λ = 1 m 1 o 1 ∑ a 1 = 1 o 1 A x 1 a 1 λ d λ B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ = ∑ λ = 1 m 1 o 1 ∑ a 1 = 1 o 1 A x 1 ′ a 1 λ d λ B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ (4.7) for all x 1 , x 1 ′ , x 2 , a 2 , ⋯ , x n + 1 , a n + 1 and for each k = 2, 3,⋯, n + 1, ∑ λ = 1 m 1 o 1 A x 1 a 1 λ d λ ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ = ∑ λ = 1 m 1 o 1 A x 1 a 1 λ d λ ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ′ ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ (4.8) for all x 1 , x 2 , ⋯ , x k , x k ′ , ⋯ , x n + 1 , a j ( j ≠ k ) . By writing B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 = B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 1 , ⋯ , B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 T , which is a vector in R m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 and letting f x 1 λ = d λ ∑ a 1 = 1 o 1 A x 1 a 1 λ , 1 ≤ λ ≤ m 1 o 1 ; 0 , m 1 o 1 < λ ≤ m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 , f x 1 = f x 1 1 , f x 1 2 , ⋯ , f x 1 m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 T , which is a vector in R m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 , we get from (4.7) that 〈 f x 1 , B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 〉 = 〈 f x 1 ′ , B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 〉 for all x ₂,⋯,x _n+1, a ₂,⋯,a _n+1 and x 1 , x 1 ′ . Since the column vectors of the unitary matrix V form an orthonormal basis B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 : x j ∈ m j , a j ∈ o j 2 ≤ j ≤ n + 1 for R m 2 o 2 ⋅ ⋅ ⋅ m n + 1 o n + 1 , we conclude that f x 1 = f x 1 ′ , i.e., d λ ∑ a 1 = 1 o 1 A x 1 a 1 λ = d λ ∑ a 1 = 1 o 1 A x 1 ′ a 1 λ , ∀ x 1 , x 1 ′ ∈ m 1 , λ = 1,2 , ⋯ , m 1 o 1 . Since d _λ > 0 for each λ = 1, 2,⋯, r, we obtain ∑ a 1 = 1 o 1 A x 1 a 1 λ = ∑ a 1 = 1 o 1 A x 1 ′ a 1 λ , ∀ x 1 , x 1 ′ ∈ m 1 , λ = 1,2 , ⋯ , r . (4.9) Using Lemma 4.2 yields that for each λ = 1, 2,⋯, r, A x 1 a 1 λ = ∑ k 1 = 1 N 1 c k 1 λ δ a 1 , J k 1 1 x 1 , ∀ a 1 ∈ o 1 , x 1 ∈ m 1 , (4.10) where N 1 = o 1 m 1 , { J 1 ( 1 ) , J 2 ( 1 ) , ⋯ , J N 1 ( 1 ) } is the set of all maps from [m ₁] into [o ₁]. Similarly, by writing A x 1 a 1 = A x 1 a 1 1 , A x 1 a 1 2 , ⋯ , A x 1 a 1 m 1 o 1 T ≡ A x 1 a 1 λ λ = 1 m 1 o 1 ∈ R m 1 o 1 , and for fixed 2 ≤ k ≤ n + 1, x _j ∈ [m _j ](j ≠ k), a _j ∈ [o _j ](j ≠ k), letting g x k λ = d λ ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ 1 ≤ λ ≤ o 1 m 1 , g x k = g x k 1 , ⋯ , g x k m 1 o 1 T ∈ R m 1 o 1 , we get from (4.8) that ⟨ g x k , A x 1 a 1 ⟩ = ⟨ g x k ′ , A x 1 a 1 ⟩ for all x ₁, a ₁ and x k , x k ′ . Since the row vectors A x 1 a 1 ’s of the unitary matrix U form an orthonormal basis A x 1 a 1 x 1 ∈ [ m 1 ] , a 1 ∈ [ o 1 ] for R m 1 o 1 , we conclude that g x k = g x k ′ , i.e., d λ ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ = d λ ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ′ ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ for all x k , x k ′ . Since d _λ > 0 for each λ = 1, 2,⋯, r, we obtain ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ = ∑ a k = 1 o k B x 2 ⋅ ⋅ ⋅ x k ′ ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ (4.11) for all x k , x k ′ , λ = 1,2 , ⋯ , r . This shows that Q λ ≔ [ [ Q λ a 2 ⋅ ⋅ ⋅ a n + 1 | x 2 ⋅ ⋅ ⋅ x n + 1 ] ] = [ [ B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ ] ] defines an n-partite NSCTT. It follows from the assumption of induction that B x 2 ⋅ ⋅ ⋅ x n + 1 a 2 ⋅ ⋅ ⋅ a n + 1 λ = ∑ k 2 = 1 N 2 ⋯ ∑ k n + 1 = 1 N n + 1 q k 2 ⋅ ⋅ ⋅ k n + 1 λ δ a 2 , J k 2 2 x 2 ⋯ δ a n + 1 , J k n + 1 n + 1 x n + 1 (4.12) for all x _i ∈ [m _i ], a _i ∈ [o _i ](i = 2,⋯, n + 1). Now, we obtain from Eqs 4.6, 4.10, 4.12 that T a 1 ⋅ ⋅ ⋅ a n + 1 | x 1 ⋅ ⋅ ⋅ x n + 1 = ∑ k 1 = 1 N 1 ⋅ ⋅ ⋅ ∑ k n + 1 = 1 N n + 1 q k 1 ⋅ ⋅ ⋅ k n + 1 δ a 1 , J k 1 1 x 1 ⋅ ⋅ ⋅ δ a n + 1 , J k n + 1 n + 1 x n + 1 for all x _i ∈ [m _i ], a _i ∈ [o _i ](i = 1, 2,⋯, n + 1), where q k 1 ⋅ ⋅ ⋅ k n + 1 = ∑ λ = 1 r d λ c k 1 λ q k 2 ⋅ ⋅ ⋅ k n + 1 λ . This shows that a decomposition (4.4) exists for an (n + 1)-partite NSCTT T. The proof is completed.