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Unitary equivalence of different bipartite entangled states under unitary transformations

Shuai Wang, Li-Li Hou, Xuefen Xu

Published Aug 22, 2014• 1 citation
Publisher
DOI
10.1016/j.ijleo.2014.06.078

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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.