Finding a maximally correlated state - Simultaneous Schmidt decomposition of bipartite pure states
We consider a bipartite mixed state of the form, $ρ=\sum_{α, β=1}^{l}a_{αβ} | ψ_α> < ψ_ β}| $, where $| ψ_α>$ are normalized bipartite state vectors, and matrix $(a_{αβ})$ is positive semidefinite. We provide a necessary and sufficient condition for the state $ρ$ taking the form of maximally correlated states by a local unitary transformation. More precisely, we give a criterion for simultaneous Schmidt decomposability of $| ψ_α>$ for $α=1,2,..., l$. Using this criterion, we can judge completely whether or not the state $ρ$ 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.