On classes of local unitary transformations

TL;DR

This paper establishes a correspondence between local unitary equivalence classes of density matrices and double cosets of unitary groups, revealing that these classes depend only on eigenvalue multiplicities, not their specific values.

Contribution

It introduces a novel framework linking local unitary classes of states to double cosets, providing a new perspective on their structure independent of eigenvalues.

Findings

Classes depend only on eigenvalue multiplicities

Interrelationship among classes is eigenvalue-independent

Homogeneous space interpretation of classes

Abstract

We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed.