Unitary local invariance

Abstract

We address unitary local (UL) invariance of bipartite pure states. Given a bipartite state $|\Psi >>={\sum }_{ij}{\psi }_{ij}|i{>}_1\otimes |j{>}_2$ the complete characterization of the class of local unitaries $U_1\otimes U_2$ for which $U_1\otimes U_2|\Psi >>=|\Psi >>$ is obtained.The two relevant parameters are the rank of the matrix $\Psi$, $[\Psi {]}_{ij}={\psi }_{ij}$, and the number of its equal singular values, {\em i.e.} the degeneracy of the eigenvalues of the partial traces of $|\Psi >>$.

arXiv commentRevised version. Accepted for publication Int. J. Quant. Inf