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

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.