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arXiv:0712.1681 · Physical Review A

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

Dafa Li, Xiangrong Li, Hongtao Huang, Xinxin Li

Published May 6, 2008• 4 citations
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Abstract

In D. Li, X. Li, H. Huang, and X. Li [Phys. Rev. A 76, 032304 (2007)], we defined the residual entanglement for odd $n$ 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.

DOI
10.1103/physreva.77.056302

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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.
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.
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.
Coffman et al. [Phys. Rev. A 61, 052306 (2000)] introduced the residual entanglement for three qubits. In this paper, we present the entanglement measure τ(ψ) for even n qubits; for odd n qubits, we propose the residual entanglement τ(i)(ψ) with respect to qubit i and the odd n-tangle R(ψ) by averaging the residual entanglement with respect to each qubit. In this paper, we show that these measures are LU-invariant, entanglement monotones, invariant under permutations of the qubits, and multiplicative in some cases.
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.