Necessary and Sufficient Conditions for Local Unitary Equivalence of Multi-qubit States

TL;DR

This paper establishes exact criteria for when two multi-qubit states are equivalent under local unitary transformations, using their reduced states and symmetries, enabling a clear protocol for LU-equivalence determination.

Contribution

It provides necessary and sufficient conditions for LU-equivalence of multi-qubit states and introduces a practical protocol for testing this equivalence.

Findings

Derived conditions for LU-equivalence of pure and mixed states

Identified local symmetries and cyclic unitary operators

Proposed a straightforward LU-equivalence decision protocol

Abstract

We derive necessary and sufficient conditions for the LU-equivalence of two general (pure or mixed) $n$-qubit states as well as we determine the local unitary operators connecting them. Almost all relevant information is contained in the $1$-qubit reduced matrices of the multiqubit states under investigation Our technique relies on identifying {\it ab initio} all local symmetries and the corresponding local cyclic unitary operators. To derive the above conditions we use the reference forms of the multiqubit states whose definition requires the diagonalization of the 1-qubit reduced matrices. Based on those conditions we propose a straightforward protocol to decide wether or not two $n$-qubit states are LU-equivalent.