Equivalence of 2-tensors and 3-tensors under Local Unitary and General Linear Groups

: In quantum information, pure states correspond to tensors. Two states equivalent under local unitary group or local general linear group can be used for the same quantum-information tasks, and it is thus an essential problem to determine whether two tensors are equivalent. In this paper, we determine the two sorts of equivalence for 2-tensors in terms of the Schmidt decomposition and singular value decomposition of matrices. We also extend the equivalence to some 3-tensors, proving that Schmidt decomposition is unachievable for 3-tensors. However, we present a method to express 3-tensors using the logic behind Schmidt decomposition. Furthermore, we generalized the scenario to higher dimensions and discussed the characteristic behaviors of different groups in tensor transformations.


Introduction
Quantum information science is an emerging and swiftly evolving field, with quantum entanglement being its main component of interest.Using local quantum operations and classical communication (LOCC), one can consolidate partially entangled pairs of particles into fewer pairs [1].Under these same operations, classification of pure states of multipartite entanglement can be simplified using asymptotic transformations [2].Two pure states of a bipartite system can be transformed locally, with the most efficient method identified by the authors in Ref [3].A novel method to compute the entanglement of formation for a pair of qubits employs local quantum operations combined with classical communications (LQCC).This approach relates to the multiplication using a Lorentz matrix and incorporates a normalization [4].Equivalence classes in sets of entangled states are defined using invertible local transformations.A W state is remarkable in that it retains bipartite entanglement for three qubits [5].In Ref [6], authors found a classification methods for three qubits based on five entanglement parameters.In Ref [7], the authors examined the classification of four-qubit pure states using stochastic local operations and classical communications (SLOCC).The findings in Ref [8] expanded this classification to include multipartite states, encompassing infinitely many types of states that are inequivalent under SLOCC.Multidimensional determinants, also referred to as "hyperdeterminants" can be used to establish how local actions affect inequivalent multipartite entangled classes, providing classification of these states [9].In Ref [10], the authors introduce a criterion for determining the feasibility of transforming between two pure states via SLOCC.This standard can also categorize 2 × M × N systems.In Ref [11], unitary equivalence of quantum states is conditioned using the von Neumann entropy.This also uses the partial trace operation, a generalisation of the probability concept of marginal distributions for multipartite quantum systems, and the strong subadditivity (SSA) property, relating von Neumann entropies of quantum systems, are explained in linear algebra terms in Ref [12].In Ref [13], authors introduce the Greenberger-Horne-Zeilinger (GHZ) states.Furthermore, Ref [5] proves that GHZ states and W states are not mutually convertible under SLOCC, as their minimal product decompositions have different numbers of terms.
In this paper, we present applications of the Kronecker products on high-dimension tensors.We start by tackling the following problem.We construct two 2-tensors, We also generalize the problem to higher-dimension 2-tensors and 3-tensors.In particular, we find some 3-tensors which are not equivalent under LU transformation.

Preliminaries to Matrices
In this section, we introduce the preliminary knowledge and facts used in this paper.In Sec.2.1, we introduce the different types of matrices.In Sec.2.2, we review some basic matrix operations, namely addition, multiplication, and exponentiation.In Sec.2.3, we present matrix transposition.In Sec.2.4, we discuss the unitary matrix.In Sec.2.5, we examine symmetric matrices.In Sec.2.6, we introduce the Kronecker product.In Sec.2.7, we present singular value decomposition (SVD), and in Sec.2.8, Schmidt decomposition.

Types of Matrices
We refer to the row matrix as (  1  2 ⋯   ).
We refer to the column matrix as We refer to the zero matrix as ( We refer to the diagonal matrix as ( ) .
We refer to the identity matrix (also denoted as I) as ( ) .

Basic Matrix Operations
It is only possible to add matrices of the same size, i.e. two matrices A and B can only be added if  = (  ) × and  = (  ) × .Then, we have  +  =  = (  ) × where Matrix addition satisfies the following properties: + ( + ) = ( + ) + When performing matrix multiplication, the first matrix's columns should match the number of rows in the second matrix.For  = (  ) × and  = (  ) × , we have  =  = (  ) × where Matrix multiplication satisfies the following properties: ( + ) =  + , ( + ) =  +  (7) is a scalar matrix Matrix exponentiation can only be applied to square matrices.For = (  ) × , we have and The following properties also hold when  = :

Matrix Transposition
Matrix transposition consists of switching the rows and columns of a matrix and is denoted by T .

Unitary Matrices
We denote  as the unitary matrix.The unitary matrix satisfies  •  † = 1  , where  † = ( * )  and  * denotes the complex conjugate matrix of .

Symmetric Matrices
A square matrix  = (  ) × is said to be symmetric if  =   , i.e.   =   for i,j = 1, 2, ... n.On the other hand,  is said to be asymmetric/antisymmetric/skew symmetric if  = −  .In this case, all numbers on the diagonal must be 0.
A Hermitian matrix is a square matrix  = (  ) × such that   =   * for any i,j.
Here is an interesting problem using symmetric matrices: Given the row matrix  = ( 1 ,  2 , . . .,   )  satisfying    = 1, and the identity matrix of size n, , if we define the matrix  =  − 2  , prove that  is symmetric and that   = .
The proof goes as follows:

Kronecker Product
The Kronecker product, sometimes referred to as the direct product or tensor product, is a method for multiplying matrices that don't meet the criteria for standard matrix multiplication.
Given matrix A of size  ×  and matrix B of size  × , their resultant matrix will have dimensions  ×  and is computed as follows: The Kronecker product satisfies the following properties :

Results
Within this section, we present the primary findings of our study.In Sec.3.1, we apply the Kronecker product to 2-tensors in  2 ⨂  2 .Then, in Sec.3.2, we generalize the conclusion to higherdimension tensors.In Sec.3.3, we show that a Schmidt decomposition for a 3-tensor is impossible to achieve.In Sec.3.4, we follow with a generalization of the question in 3.1 to higher-dimension tensors.In.Sec.3.5, we discussed the two groups, (2, 2, 2) and (4) × (2) , illustrating how they transform distinct tensors.

Applications of the Kronecker Product on 2-Tensors
We can consider the following problem to better understand the use of the Kronecker product: We construct two 2-tensors, ) and , , ,  are complex numbers.
Can we find an element  in (2) × (2) such that  1 =  2 , where (2) is the 2 × 2 unitary group?If yes, we say that  1 and  2 are equivalent under local unitary (LU) group.
We can look at the norm of each side of the equation This gives us the equation It is possible to prove a bijection (an isomorphism between two sets) between elements of the following two sets: Using the norm obtained for the RHS, we have that  2 corresponds to (  0 0  ).
From this, and using SVD, we observe that Hence, the necessary condition for the existence of a matrix  satisfying the problem is that The unitary matrix  which can convert  1 to  2 can be written as Similarly, if || = || and || = ||, the unitary matrix  which can convert  1 to  2 can be written as To conclude, we have presented the conditions by which two 2-tensors  1 and  2 are LU equivalent.

Applications of the Kronecker Product on 2-Tensors with Higher Dimensions
It is possible to generalize the question in Sec.
If we consider the case where We want to find  such that ( ⨂ ) 1 =  2 .Then, we can write  as ) ⨂   .
Here, () is set of all  ×  invertible matrices, () is the set of all  ×  unitary matrices, so () is a proper subset of ().
We can see that .

Schmidt
For example, we consider the 3-tensor We claim that the decomposition in (33) is the Schmidt decomposition for |W> taken as a 2-tensor in the bipartite space being the tensor product of  2 and  2 ⨂  2 .We compute that Furthermore, we show that Hence, the two vectors |0> ⨂ |0> and are orthonormal.Therefore, we have proven the claim below (33).

Conclusion
We have established two types of equivalences for 2-tensors by applying local unitary groups and general linear groups.This equivalence is further extended to include certain 3-tensors.However, due to the lack of Schmidt decomposition in 3-tensors (or more generally, n-tensors when n > 2), this extension becomes significantly complex.A key question arising from this study is to explore how the number of parameters in 3-tensors can be reduced under these two established equivalences.We also attempt to expand the discussion on this issue to higher dimensions, discussing the distinct characteristics displayed by specific groups in this process.Furthermore, it remains to be determined whether the unitary groups spanning two systems can eliminate more parameters than the multiplication of two general linear groups.