Necessary and sufficient conditions for local unitary equivalence of multiqubit states

Abstract

We derive the necessary and sufficient conditions for the local unitary (LU) equivalence of two general (pure or mixed) $n$-qubit states and determine the local unitary operators connecting them. Almost all relevant information is contained in the one-qubit reduced matrices of the multiqubit states under investigation. Our technique relies on identifying 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 one-qubit reduced matrices. Based on those conditions, we propose a straightforward protocol to decide whether or not two $n$-qubit states are LU equivalent.