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arXiv:quant-ph/0604184 · Physical Review A

Classification of multipartite entanglement containing infinitely many kinds of states

Lin Chen, Yixin Chen, Yu-Xue Mei

Published Nov 22, 2006• 23 citations
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Abstract

We give a further investigation of the range criterion and low-to-high rank generating mode introduced in Chen and Chen [Phys. Rev. A 73, 052310 (2006)], which can be used for the classification of $2\ifmmode\times\else\texttimes\fi{}M\ifmmode\times\else\texttimes\fi{}N$ states under reversible local filtering operations. By using these techniques, we entirely classify the family of $2\ifmmode\times\else\texttimes\fi{}4\ifmmode\times\else\texttimes\fi{}4$ states, which actually contains infinitely many kinds of states. The classifications of true entanglement of $2\ifmmode\times\else\texttimes\fi{}(M+3)\ifmmode\times\else\texttimes\fi{}(2M+3)$ and $2\ifmmode\times\else\texttimes\fi{}(M+4)\ifmmode\times\else\texttimes\fi{}(2M+4)$ systems are briefly listed, respectively.

DOI
10.1103/physreva.74.052331

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Classification of multipartite entanglement in 2×M×N2\times{M}\times{N}2×M×N quantum system: range criterion and a systematic method
We develop a range criterion which is a sufficient and necessary condition satisfied between two states transformable to each other under reversible stochastic local operations assisted by classical communication (SLOCC). Based on this criterion, we can effectively discriminate any two states in 2timesMtimesN2\\times{M}\\times{N}2timesMtimesN quantum system and we propose a systematic method for seeking all kinds of states in any true 2timesMtimesN2\\times{M}\\times{N}2timesMtimesN system. This method can greatly reduce the influence of parameters and it shows the potentially regular structure of inequivalent classes. The classification in 2times3timesN2\\times3\\times{N}2times3timesN system is explicitly described.
We propose a range criterion that is a sufficient and necessary condition satisfied by two pure states transformable into each other under reversible stochastic local operations assisted by classical communication. We also provide a systematic method for seeking all kinds of true entangled states in the $2\ifmmode\times\else\texttimes\fi{}M\ifmmode\times\else\texttimes\fi{}N$ 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 tripartite systems.
The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. K\"ok\'enyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the local operations and classical communication convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some ``unified'' notion of entanglement.
The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure $N\text{\ensuremath{-}}$partite entangled states in terms of polynomial $\mathrm{SL}(d,\mathrm{C}){}^{\ensuremath{\bigotimes}N}$ invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through ``tangle patterns.'' This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.
The coefficient matrix is an efficient tool in entanglement classification under stochastic local operation and classical communication. In this work, we take all the ranks of the coefficient matrices into account in the method of entanglement classification, and give the entanglement classification procedure for arbitrary-dimensional multipartite pure states under stochastic local operation and classical communication. As a main application, we study the entanglement classification for quantum systems in the Hilbert space $\mathcal{H}={\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ in terms of the ranks of the coefficient matrices.