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} | \psi_{\alpha}> < \psi_ \beta}| $, where $| \psi_{\alpha}>$ are normalized bipartite state vectors, and matrix $(a_{\alpha \beta})$ 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.