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arXiv:0710.3425 · Journal of Mathematical Physics

An entanglement measure for n qubits

Dafa Li, Xiangrong Li, Hongtao Huang, Xinxin Li

Published Jan 1, 2009• 17 citations
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Abstract

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.

DOI
10.1063/1.3050298

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Modern Physics Letters A•Sep 14, 2022•1 citation
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.
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 this paper, we generalize the residual entanglement to the case of multipartite states in arbitrary dimensions by making use of a different method. Through the introduction of a special entanglement measure, the residual entanglement of mixed states takes on a form that is more elegant than that in Ref. [7] used by Coffman et al. [Phys. Rev. A 61, 052306 (2000)]. The result obtained in this paper is different from the previous one given in Ref. [8] by Wong and Christensen [Phys. Rev. A 63, 044301 (2000)]. Several examples demonstrate that our present result is a good measurement of the multipartite entanglement. Furthermore, the original residual entanglement is a special case of our result.
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 this Brief Report we discuss entanglement of multiparticle quantum systems. We propose a potential measure of a type of entanglement of pure states of nnn qubits, the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}.$ For a system of two qubits the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is equal to the square of the concurrence, and for systems of three qubits it is equal to the ``residual entanglement.'' We show that the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is also equal to a generalization of the concurrence squared for even n,n,n, and use this fact to prove that the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is an entanglement monotone. However, the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is undefined for odd n>3.n>3.n>3. Finally, we propose a measure related to the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ for mixed-state systems of nnn qubits, and find an analytical formula for this measure for even n.n.n.
We show that for even nnn qubits, n\ n n-tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the nnn-tangle is not the residual entanglement for any even n≥4n\geq 4n≥4\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence C1(2...n)C_{1(2...n)}C1(2...n)​. The condition implies that the concurrence C1(2...n)C_{1(2...n)}C1(2...n)​ is always positive for any entangled states while the nnn-tangle vanishes for some entangled states. We argue that for even nnn\ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the nnn-tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any nnn qubits while the nnn-tangle is multiplicative for some product states.
Entanglement is an important resource for quantum technologies that allow us to go beyond the classical world. Quantum systems can be entangled in various ways. Also, the degree of entanglement between the subsystems of a quantum system can be calculated in different ways. In our recent paper [Prog. Theor. Exp. Phys. (2022)], we directly measure the three-tangle of a pure three-qubit quantum state. In this paper, we extend this method to pure N-qubit quantum states (N is any odd number greater than 1) by considering polynomial invariant of degree 4 as a measure of entanglement. In this algorithm, we use quantum gates and output probabilities to calculate polynomial invariant of degree 4. The importance of this method lies in the possibility of experimental implementation of quantum calculations in the language of quantum gates.