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arXiv:1106.4082 · Quantum Information and Computation

Rank-based SLOCC classification for odd nnn qubits

Xiangrong Li, Dafa Li

Published Jul 1, 2011• 2 citations
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Abstract

We study the entanglement classification under stochastic local operations and classical communication (SLOCC) for odd n-qubit pure states. For this purpose, we introduce the rank with respect to qubit i for an odd n-qubit state. The ranks with respect to qubits 1,2, ... n give rise to the classification of the space of odd $n$ qubits into 3^n families.

DOI
10.26421/qic11.7-8-10

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We study the entanglement classification under stochastic local operations and classical communication (SLOCC) for odd n-qubit pure states. For this purpose, we introduce the rank with respect to qubit i for an odd n-qubit state. The ranks with respect to qubits 1,2, ... n give rise to the classification of the space of odd nnn qubits into 3^n families.
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 solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for all multipartite symmetric states in the general N-qubit case. For this purpose, we introduce 2 parameters playing a crucial role, namely, the diversity degree and the degeneracy configuration of a symmetric state. Those parameters give rise to a simple method of identifying operational families of SLOCC entanglement classes of all symmetric N-qubit states, where the number of families grows as the partition function of the number of qubits.
We construct coefficient matrices of size ${2}^{\ensuremath{\ell}}$ by ${2}^{n\ensuremath{-}\ensuremath{\ell}}$ associated with pure nnn-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure nnn-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.
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.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
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.