Quantum Entanglement And Classification Of Quantum States Under Local Unitary Transformation

A key distinguishing feature of quantum theory is the possibility of entanglement among the subsystems. The significance of this phenomenon has been shown in several important achievements of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, considerable efforts have been taken to understand and characterize its properties recently. For pure states, the separability is quite well understood. For mixed states, the partial transposition condition criterion can determine whether two qubits state or a qubit-qutrit state is entangled or not. The criterion fails, however, for other higher dimensional or multipartite states.An important property of quantum entanglement is that the entanglement of a quantum state remains invariant under local unitary transformations (LUT) on the subsystems. Therefore, the invariants of local unitary transformations have special importance. A complete set of invariants can be used to judge the equivalence of two states under LUT. There have been many works on this subject, however till now, we have the complete set of invariants only for some simple cases, for example, the mixed states of two qubits [33]. For general bipartite system [34], a complete set of invariants for generic density matrices with full rank has been presented. There are also some partial results about the system with three or more subsystems [32, 20]. Generally even for pure states, one does not know how to find all the invariants.We first investigate the equivalence of quantum states under LUT by finding the complete set of invariants to characterize the equivalence. We extend the result in [34] to the general bipartite systems, by presenting some ancillary invariants, from which we can solve the problem...