Equivalence of Quantum States under Local Unitary Transformations

As one of the most striking features of quantum phenomena [1], quantum entanglement has been identified as a non-local resource for quantum information processing such as quantum computation [2, 3], quantum teleportation [4], dense coding [5], quantum cryptographic schemes [6]entanglement swapping [7], and remote state preparation (RSP) [8, 9, 10, 11] etc.. As a matter of fact, the degree of entanglement of two parts of a quantum system remains invariant under local unitary transformations of these parts. And two quantum states with the same degree of entanglement (say, entanglement of formation [12]) may be not equivalent under local unitary transformations [3]. It is important to classify and characterize the quantum states in terms of local unitary transformations. One way to deal with the problem is to find the complete set of invariants of local unitary transformations. Two states are equivalent under local unitary transformations if and only if they have the same values of all these invariants. The method developed in [13, 14], in principle, allows one to compute