Deciding Unitary Equivalence Between Matrix Polynomials and Sets of Bipartite Quantum States

TL;DR

This paper introduces a randomized polynomial-time algorithm to determine whether two sets of bipartite quantum states are equivalent under local unitary transformations, linking the problem to matrix polynomial unitary equivalence.

Contribution

It provides the first efficient algorithm for deciding local unitary equivalence of bipartite quantum states via matrix polynomial equivalence.

Findings

Algorithm solves the problem with high success probability.

Outputs explicit unitary transformations U and V.

Applicable to arbitrary pairs of rectangular matrices.

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. $UA_iV^\dagger=B_i$ for $0\leq i\leq m$ where $U$ and $V$ are unitary and $(A_i, B_i)$ 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 $V$.