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Quantum Information and Computation

Deciding unitary equivalence between matrix polynomials and sets of bipartite quantum states

Eric Chitambar, Carl A. Miller, Yaoyun Shi

Published Sep 1, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.5555/2230936.2230942

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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.
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We consider the problem of deciding if some multiparty entangled pure state can be converted, with a nonzero success probability, into a given bipartite pure state shared between two specified parties through local quantum operations and classical communication. We show that this question is equivalent to the well-known computational problem of deciding if a multivariate polynomial is identically zero. Efficient randomized algorithms developed to study the latter can thus be applied to our question. As a result, a given transformation is possible if and only if it is generically attainable by a simple randomized protocol.
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