Optimal unambiguous discrimination of two subspaces as a case in mixed-state discrimination

Abstract

We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ can be any state in the subspace ${S}_{1}$, or $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩$ can be any state in the subspace ${S}_{2}$, and our task is to determine in which of the subspaces the state of our quantum system lies. We do not want to make any error, which means that our procedure will sometimes fail if the subspaces are not orthogonal. This is a special case of the unambiguous discrimination of mixed states. We present the positive operator valued measures that solve this problem and several applications of this procedure, including the discrimination of multipartite states without classical communication.