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arXiv:1605.00238 · Journal of Mathematical Physics

Local unitary equivalence of quantum states and simultaneous orthogonal equivalence

Naihuan Jing, Min Yang, Hui Zhao

Published Jun 1, 2016• 10 citations
arXivHTMLPDF

Abstract

The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂd1 ⊗ ℂd2 are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.

DOI
10.1063/1.4954230

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The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂd1 ⊗ ℂd2 are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.
Abstract In this paper, we generalize the major results from Jing-Yang-Zhao’s paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established a correspondence between local unitary (LU) equivalence and simultaneous orthogonal (SO) equivalence of bipartite quantum states, and then used this correspondence to show that the problem of determining LU equivalence of bipartite states reduces to checking trace identities. In particular, we extend both results to tripartite quantum states. We are able to do this by utilizing a hypermatrix algebra framework and by applying a powerful generalization of Specht’s criterion proved in Futorny-Horn-Sergeichuk’s paper "Specht’s Criterion for Systems of Linear Mappings." With our established hypermatrix algebra framework and the aforementioned generalization of Specht’s criterion, it is apparent that our results can be extended to arbitrary multipartite quantum states, however there are some practical limitations which are explored and discussed towards the end.
In this brief report, we consider the equivalence between two sets of m+1m+1m+1 bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree mmm matrix polynomials are unitarily equivalent; i.e. UAiV†=BiUA_iV^\dagger=B_iUAi​V†=Bi​ for 0≤i≤m0\leq i\leq m0≤i≤m where UUU and VVV are unitary and (Ai,Bi)(A_i, B_i)(Ai​,Bi​) are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices UUU and VVV.
Local unitary equivalence plays a crucial role in quantifying and categorizing the entanglement of quantum states. When it comes to two isospectral diagonal bipartite quantum states, determining whether they are locally unitary equivalent turns out to be a rather challenging problem. Intuitively speaking, we conjecture that for two diagonal quantum states, local unitary equivalence and local permutation equivalence between them are the same. By taking into account the cosets of the fixed point subgroup and the tensor decomposability of permutation matrices, we present the classifications of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text] under local permutation equivalence, respectively. Furthermore, relying on the classification results of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text], we demonstrate that if two diagonal quantum states are not locally permutation equivalent, then they cannot be locally unitary equivalent. This offers significant backing for the conjecture that two general bipartite diagonal quantum states are locally unitary equivalent if and only if they are locally permutation equivalent.
We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary UUU s.t UAlU∗=BlUA_lU^*=B_lUAl​U∗=Bl​, l∈{1,...,p}l \in \{1,...,p\}l∈{1,...,p}, where AlA_lAl​ and BlB_lBl​ are nxnnxnnxn complex matrices. We observe that the problem is simplest when UUU is diagonal, where we see that the `paths' in the graph defined by non-zero elements of AlA_lAl​ and BlB_lBl​ determine the solution. Inspired by this we generalize this to the case when UUU is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of Al,BlA_l,B_lAl​,Bl​ which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of nnn steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence (S...U...Eq) problem where we solve for U,VU,VU,V in UAlV∗=BlUA_lV^*=B_lUAl​V∗=Bl​, Al,BlA_l,B_lAl​,Bl​ being mxnmxnmxn Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of O(pn4)O(pn^4)O(pn4). This work finds application in Quantum Evolution, Quantum gate design and Simulation. The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.
In this brief report, we consider the equivalence between two sets of m + 1 bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree m matrix polynomials are unitarily equivalent; i.e. UAiV† = Bi for 0 ≤ i ≤ m where U and V are unitary and (Ai, Bi) are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices U and V.