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We study the mathematical structures and relations among some quantities inthe theory of quantum entanglement, such as separability, weak Schmidtdecompositions, Hadamard matrices etc.. We provide an operational method toidentify the Schmidt-correlated states by using weak Schmidt decomposition. Weshow that a mixed state is Schmidt-correlated if and only if its spectraldecomposition consists of a set of pure eigenstates which can be simultaneouslydiagonalized in weak Schmidt decomposition, i.e. allowing for complex-valueddiagonal entries. For such states, the separability is reduced to theorthogonality conditions of the vectors consisting of diagonal entriesassociated to the eigenstates, which is surprisingly related to the so-calledcomplex Hadamard matrices. Using the Hadamard matrices, we provide a variety ofgeneralized maximal entangled Bell bases.

Weak Schmidt decomposition and generalized Bell basis related to Hadamard matrices