Weak Schmidt decomposition and generalized Bell basis related to Hadamard matrices

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.