The Parametric Symmetry and Numbers of the Entangled Class of 2 \times M \times N System

We present in the work two intriguing results in the entanglement classification of pure and true tripartite entangled state of $2\times M\times N$ under stochastic local operation and classical communication. (i) the internal symmetric properties of the nonlocal parameters in the continuous entangled class; (ii) the analytic expression for the total numbers of the true and pure entangled class $2\times M \times N$ states. These properties help people to know more of the nature of the $2\times M\times N$ entangled system.Comment: 12 pages, 5 figure

Range criterion and classification of true entanglement in 2×M×N2\times{M}\times{N} system

We propose a range criterion which is a sufficient and necessary condition satisfied by two pure states transformable with each other under reversible stochastic local operations assisted with classical communication. We also provide a systematic method for seeking all kinds of true entangled states in the $2\times{M}\times{N}$ system, and can effectively distinguish them by means of the range criterion. The efficiency of the criterion and the method is exhibited by the classification of true entanglement in some types of the tripartite systems.Comment: 7 papes, no figure, Revtex. This is the published versio

Multipartite entanglement in 2 x 2 x n quantum systems

We classify multipartite entangled states in the 2 x 2 x n (n >= 4) quantum system, for example the 4-qubit system distributed over 3 parties, under local filtering operations. We show that there exist nine essentially different classes of states, and they give rise to a five-graded partially ordered structure, including the celebrated Greenberger-Horne-Zeilinger (GHZ) and W classes of 3 qubits. In particular, all 2 x 2 x n-states can be deterministically prepared from one maximally entangled state, and some applications like entanglement swapping are discussed.Comment: 9 pages, 3 eps figure

Classification of multipartite entangled states by multidimensional determinants

We find that multidimensional determinants "hyperdeterminants", related to entanglement measures (the so-called concurrence or 3-tangle for the 2 or 3 qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the $n \geq 4$ qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.Comment: revtex4, 10 pages, 4 eps figures with psfrag; v2 title changed, 1 appendix added, to appear in Phys. Rev.

Classification of multipartite entanglement containing infinitely many kinds of states

We give a further investigation of the range criterion and Low-to-High Rank Generating Mode (LHRGM) introduced in \cite{Chen}, which can be used for the classification of $2\times{M}\times{N}$ states under reversible local filtering operations. By using of these techniques, we entirely classify the family of $2\times4\times4$ states, which actually contains infinitely many kinds of states. The classifications of true entanglement of $2\times(M+3)\times(2M+3)$ and $2\times(M+4)\times(2M+4)$ systems are briefly listed respectively.Comment: 11 pages, revte

Classifying N-qubit Entanglement via Bell's Inequalities

All the states of N qubits can be classified into N-1 entanglement classes from 2-entangled to N-entangled (fully entangled) states. Each class of entangled states is characterized by an entanglement index that depends on the partition of N. The larger the entanglement index of an state, the more entangled or the less separable is the state in the sense that a larger maximal violation of Bell's inequality is attainable for this class of state.Comment: 4 pages, 3 figure