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

We consider a bipartite mixed state of the form $\rho ={\sum }_{\alpha ,\beta =1}^l{a}_{\alpha \beta }\mid {\psi }_{\alpha }⟩⟨{\psi }_{\beta }⟩$, where $\mid {\psi }_{\alpha }⟩$ are normalized bipartite state vectors, and matrix $\left(a_{\alpha \beta }\right)$ is positive semidefinite. We provide a necessary and sufficient condition for the state $\rho$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $\mid {\psi }_{\alpha }⟩$ for $\alpha =1,2,\dots ,l$. Using this criterion, we can judge completely whether or not the state $\rho$ 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.