  We study the mathematical structures and relations among some quantities in
the theory of quantum entanglement, such as separability, weak Schmidt
decompositions, Hadamard matrices etc.. We provide an operational method to
identify the Schmidt-correlated states by using weak Schmidt decomposition. We
show that a mixed state is Schmidt-correlated if and only if its spectral
decomposition consists of a set of pure eigenstates which can be simultaneously
diagonalized in weak Schmidt decomposition, i.e. allowing for complex-valued
diagonal entries. For such states, the separability is reduced to the
orthogonality conditions of the vectors consisting of diagonal entries
associated to the eigenstates, which is surprisingly related to the so-called
complex Hadamard matrices. Using the Hadamard matrices, we provide a variety of
generalized maximal entangled Bell bases.
