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arXiv:quant-ph/0307219 · Physical Review A

Geometric measure of entanglement and applications to bipartite and multipartite quantum states

Tzu-Chieh Wei, Paul M. Goldbart

Published Oct 9, 2003• 744 citations
arXivHTMLPDF

Abstract

The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, 675 (1995); H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, 6787 (2001)], is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of three-qubit Greenberger-Horne-Zeilinger, W, and inverted-$W$ states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.

DOI
10.1103/physreva.68.042307

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The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, 675 (1995); H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, 6787 (2001)], is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of three-qubit Greenberger-Horne-Zeilinger, W, and inverted-WWW states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, p.675 (1995) and H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, p.6787 (2001)], is explored for bipartite and multipartite pure and mixed states. It is determined for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavior near the quantum critical line. Along the way, connections are pointed out between the geometric measure of entanglement, the Hartree approximation, entanglement witnesses, correlation functions, and the relative entropy of entanglement.
A natural way of quantifying the degree of entanglement for a pure quantum state is to compare how far this state is from the set of all unentangled pure states. This geometric measure of entanglement is explored for bipartite and multipartite pure and mixed states. It is determined analytically for arbitrary two‐qubit mixed states and for generalized Werner and isotropic states. It is also applied to certain multipartite mixed states, including two multipartite bound entangled states discovered by Smolin and Dür. Moreover, the geometric measure of entanglement is applied to the ground state of the Ising model in a transverse magnetic field. From this model the entanglement is shown to exhibit singular behavior at the quantum critical point.
The geometric measure of entanglement, originated by Shimony and by Barnum and Linden, is determined for a family of tripartite mixed states consisting of aribitrary mixtures of GHZ, W, and inverted-W states. For this family of states, other measures of entanglement, such as entanglement of formation and relative entropy of entanglement, have not been determined. The results for the geometric measure of entanglement are compared with those for negativity, which are also determined. The results for the geometric measure of entanglement and the negativity provide examples of the determination of entanglement for nontrivial mixed multipartite states.
We introduce a class of generalized geometric measures of entanglement. For pure quantum states of N\mathit{N}N elementary subsystems, they are defined as the distances from the sets of K\mathit{K}K-separable states (K=2,…,N)(K=2,\dots{},N)(K=2,…,N). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows discrimination among the different contributions. The extended measures are applied to the study of entanglement in different classes of N\mathit{N}N-qubit pure states. These classes include W\mathit{W}W and Greenberger-Horne-Zeilinger (GHZ) states, and their symmetric superpositions; symmetric multimagnon states; cluster states; and, finally, asymmetric generalized W\mathit{W}W-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between GHZ and W\mathit{W}W states, clarifying the original result of Wei and Goldbart that W\mathit{W}W states possess a larger global entanglement than GHZ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.
The geometric measure of quantum entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to the geometric measure by means of a system of equations. The entanglement eigenvalue for a mixed state is introduced. And we show that, for a given mixed state, its nearest disentangled mixed state is associated with its entanglement eigenvalue. Two numerical examples are used to demonstrate the effectiveness of the proposed method.