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arXiv:0710.2642 · Physical Review A

Comment on “Stochastic local operations and classical communication invariant and the residual entanglement for n qubits”

Xin-Wei Zha, Haiyang Song, Ming‐Liang Hu

Published May 6, 2008• 1 citation
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Abstract

In a recent paper [D. Li, X. Li, H. Huang, and X. Li, Phys. Rev. A 76, 032304 (2007)], Li et al. proposed the definition of the residual entanglement for general $n$ qubits by means of the stochastic local operations and classical communication. Here we argue that their definition is not suitable for the case of odd-$n$ qubits.

DOI
10.1103/physreva.77.056301

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In a recent paper [D. Li, X. Li, H. Huang, and X. Li, Phys. Rev. A 76, 032304 (2007)], Li et al. proposed the definition of the residual entanglement for general nnn qubits by means of the stochastic local operations and classical communication. Here we argue that their definition is not suitable for the case of odd-nnn qubits.
In D. Li, X. Li, H. Huang, and X. Li [Phys. Rev. A 76, 032304 (2007)], we defined the residual entanglement for odd nnn qubits. In this reply, we improve the definition by averaging the original residual entanglement with respect to each qubit, which makes the defined quantity invariant under any permutation of all the qubits.
We find an invariant for nnn qubits and propose residual entanglement for nnn qubits by means of the invariant. Thus, we establish a relation between stochastic local operations and classical communication (SLOCC) entanglement and residual entanglement. Invariant and residual entanglement can be used for SLOCC entanglement classification of nnn qubits.
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.