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arXiv:1401.6609

Classification of the Entangled States of $2 \times L \times M \times N$

Liang-Liang Sun, Junli Li, Cong‐Feng Qiao

Published Jan 26, 2014• 0 citations
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Abstract

We present a practical entanglement classification scheme for pure state in form of $2\times L\times M\times N$ under the stochastic local operation and classical communication (SLOCC), where every inequivalent class of the entangled quantum states may be sorted out according to its standard form and the corresponding transformation matrix. This provides a practical method for determining the interconverting matrix between two SLOCC equivalent entangled states, and classification examples for some $2\times 4\times M\times N$ systems are also presented.

DOI
10.48550/arxiv.1401.6609

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Classification of the Entangled States of 2×L×M×N2 \times L \times M \times N2×L×M×N
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Classification of the Entangled States of 2×L×M×N×H2\times L\times M\times N\times H2×L×M×N×H
In this work we propose a practical entanglement classification scheme for pure states of 2×L×M×N×H2\times L\times M\times N\times H2×L×M×N×H, under the stochastic local operation and classical communication (SLOCC), which generalizes the method explored in the entanglement classification of 2×L×M×N2\times L\times M\times N2×L×M×N to the five-partite system. The entangled states of 2×L×M×N×H2\times L\times M\times N\times H2×L×M×N×H system are first classified into different coarse-grained standard forms using matrix decompositions, and then fine-grained identification of two inequivalent entangled states with the same standard form are completed by using the matrix realignment technique. As an practical example, entanglement classes of the five-qubit system of 2×2×2×2×22\times 2\times 2\times 2\times 22×2×2×2×2 are presented.
We develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in
We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |ℓ,n> with ℓ=1,…,[n/2] are inequivalent under SLOCC.
We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite (nnn-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the 2⊗2⊗2⊗42 \otimes 2 \otimes 2 \otimes 42⊗2⊗2⊗4 system is discussed in terms of the generalized method, and 22 different SLOCC families are found.
We presented a general classification scheme for the tripartite $L\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ entangled system under stochastic local operation and classical communication. The whole classification procedure consists of two correlated parts: the simultaneous similarity transformation of a commuting matrix pair into a canonical form and the study of internal symmetry of parameters in the canonical form. Based on this scheme, a concrete example of entanglement classification for a $3\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ system is given.
The coefficient matrix is an efficient tool in entanglement classification under stochastic local operation and classical communication (SLOCC). 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 SLOCC. As a main application, we study the entanglement classification for quantum systems in the Hilbert space H = C2 ⊗ C2 ⊗ C2 ⊗ Cd in terms of the ranks of the coefficient matrices.