Local unitary equivalence of quantum states and simultaneous orthogonal equivalence

TL;DR

This paper explores the relationship between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence, providing new methods to determine equivalence using algebraic techniques.

Contribution

It establishes a connection between local unitary and simultaneous orthogonal equivalence and introduces four algorithms for testing local unitary equivalence of density matrices.

Findings

Proves local unitary equivalence corresponds to simultaneous similarity under orthogonal transformations.

Develops four algorithms based on trace identities, Weierstrass pencils, Albert determinants, and Smith normal forms.

Provides a theoretical framework to analyze quantum state equivalence through algebraic methods.

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 $\mathbb C^{d_1}\otimes \mathbb C^{d_2}$ are locally unitary equivalent in connection with trace identities, Weierstrass pencils, Albert determinants and Smith normal forms.