Local unitary invariants for multipartite states

TL;DR

This paper develops a set of local unitary invariants for multipartite quantum states, enabling the classification of states under local transformations, with specific results for pure and mixed states.

Contribution

It introduces new invariants based on singular values and traces for pure and mixed states, providing necessary and sufficient conditions for local unitary equivalence in certain cases.

Findings

Invariants based on singular values for pure states

Trace-based invariants for mixed states

Necessary and sufficient conditions for full-ranked mixed states

Abstract

We study the invariants of arbitrary dimensional multipartite quantum states under local unitary transformations. For multipartite pure states, we give a set of invariants in terms of singular values of coefficient matrices. For multipartite mixed states, we propose a set of invariants in terms of the trace of coefficient matrices. For full ranked mixed states with nondegenerate eigenvalues, this set of invariants is also the necessary and sufficient conditions for the local unitary equivalence of such two states.