Matrix Tensor Product Approach to the Equivalence of Multipartite States under Local Unitary Transformations

Quantum entangled states are playing fundmental roles in quantum information processing such as quantum computation, quantum teleportation, dense coding, quantum cryptographic schemes quantum error correction, entanglement swapping, and remote state preparation (RSP) etc.. However the theory of quantum entanglement is still far from being satisfied. To quantify the degree of entanglement a number of entanglement measures have been proposed for bipartite states. Most of these proposed measures of entanglement involve extremizations which are difficult to handle analytically. For multipartite case how to give a well defined measure is still under discussion. For general mixed states till now we don’t even have an operational criterion to verify whether a state is separable or not. As a matter of fact, the degree of entanglement of a multipartite quantum system remains invariant under local unitary transformations of every subsystems. Therefore the quantum states can be classified according to the local unitary transformations. Nevertheless an explicit picture of the orbits (geometry and topology) under such transformations is not yet ready. We even don’t have a general (operational) criterion to verify if two mixed states are equivalent or not under local unitary transformations.