Finding a maximally correlated state: Simultaneous Schmidt decomposition of bipartite pure states

Abstract

We consider a bipartite mixed state of the form $\ensuremath{\rho}={\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=1}^{l}{a}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩⟨{\ensuremath{\psi}}_{\ensuremath{\beta}}⟩$, where $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ are normalized bipartite state vectors, and matrix $({a}_{\ensuremath{\alpha}\ensuremath{\beta}})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $\ensuremath{\rho}$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $\ensuremath{\mid}{\ensuremath{\psi}}_{\ensuremath{\alpha}}⟩$ for $\ensuremath{\alpha}=1,2,\dots{},l$. Using this criterion, we can judge completely whether or not the state $\ensuremath{\rho}$ is equivalent to the maximally correlated state, in which the distillable entanglement is given by a simple formula. For generalized Bell states, this criterion is written as a simple algebraic relation between indices of the states. We also discuss the local distinguishability of the generalized Bell states that are simultaneously Schmidt decomposable.