Canonical Form and Separability of PPT States in C2 ⊗ CM ⊗ CN Composite Quantum Systems

Quantum entangled states have become one of the key resources in the rapidly expanding field of quantum information processing and computation [1, 2, 3, 4]. Nevertheless, the study of physical character and mathematical structure of the quantum entanglement is far from being satisfied. One even does not have a general criterion to judge if a quantum (mixed) state is entangled or not. For bipartite states a number of entanglement measures such as entanglement of formation and distillation [5, 6], negativity [7], von Neumann entropy and relative entropy [5, 8] have been proposed. However most proposed measures of entanglement involve extremizations which are difficult to handle analytically. For instance, explicit analytic formulae for entanglement of formation [9] have been found only for a pair of qubits system [10], and for some symmetric states [11] and a class of special states [12]. For multipartite systems there is no well defined measure of entanglement yet.