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arXiv:1003.4774 · Quantum Information and Computation

Relationship between the n-tangle and the residual entanglement of even n qubits

Xiangrong Li, Dafa Li

Published Nov 1, 2010• 9 citations
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Abstract

We show that for even $n$ qubits,$\ n$-tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the $n$-tangle is not the residual entanglement for any even $n\geq 4$\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence $C_{1(2...n)}$. The condition implies that the concurrence $C_{1(2...n)}$ is always positive for any entangled states while the $n$-tangle vanishes for some entangled states. We argue that for even $n$\ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the $n$-tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any $n$ qubits while the $n$-tangle is multiplicative for some product states.

DOI
10.26421/qic10.11-12-11

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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.
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