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

We consider a bipartite mixed state of the form rho=Sigma(alpha,beta=1)(l)a(alphabeta)\psi(alpha)][psi(beta)>, where \psi(alpha)> are normalized bipartite state vectors, and matrix (a(alphabeta)) 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 \psi(alpha)> for alpha=1,2,...,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.