We shortly discuss the entanglement and generalized Bell bases in quantum information theory for bipartite quantum systems. Using complex Hadamard matrices, we are able to construct a large class of bases for bipartite systems analogous to the well-known Bell basis for qubits. Moreover, this approach can further be used to construct generalized Bell bases for multipartite systems with interesting properties.

For treating mixed states we adopt a tool, complex-valued simultaneous diagonalization for a set of matrices, which deeply roots in the matrix theory. Surprisingly, this simultaneous diagonalization problem is connected to the classification of maximally correlated states and inspires the construction of generalized Bell bases by Hadamard matirces. Besides these, there may be other potential applications of this technique for treating tensor spaces.