  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.
