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arXiv:quant-ph/0701099 · Journal of Physics A Mathematical and Theoretical

Revised geometric measure of entanglement

Ya Cao, An Min Wang

Published Mar 14, 2007• 21 citations
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Abstract

We present a revised geometric measure of entanglement (RGME) which is just certain generalization of geometric measure of entanglement (GME). The revised version is an entanglement monotone. Some useful inequalities about RGME are deduced. For example, we give the formulae of RGME for the two-parameter class of states in a 2 ⊗ n quantum system, the two particles' high dimensional maximally entangled mixed state, the isotropic state including the n -particle d-level case and two multipartite bound entangled states. The results show there is a relation , and then RGME is an appropriate measure of entanglement.

DOI
10.1088/1751-8113/40/13/014

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We present a revised geometric measure of entanglement (RGME) which is just certain generalization of geometric measure of entanglement (GME). The revised version is an entanglement monotone. Some useful inequalities about RGME are deduced. For example, we give the formulae of RGME for the two-parameter class of states in a 2 ⊗ n quantum system, the two particles' high dimensional maximally entangled mixed state, the isotropic state including the n -particle d-level case and two multipartite bound entangled states. The results show there is a relation , and then RGME is an appropriate measure of entanglement.
Based on the revised geometric measure of entanglement (RGME) proposed by us [J. Phys. A: Math. Theor. 40 (2007) 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the generalized Smolin state (GSS) in the presence of noise and the two-mode squeezed thermal state. Moreover, we compare their RGME with geometric measure of entanglement (GME) and relative entropy of entanglement (RE). The results indicate RGME is an appropriate measure of entanglement. Finally, we define the Gaussian GME which is an entangled monotone.
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Quantifying quantum entanglement is a pivotal challenge in quantum information science, particularly for high-dimensional systems, due to its computational complexity. This thesis extends the geometric measure of entanglement (GME) to introduce and investigate a suite of GME-based entanglement monotones tailored for diverse quantum contexts, including pure states, subspaces, and mixed states. These monotones are applicable to both bipartite and multipartite systems, offering a unified framework for characterizing entanglement across various scenarios. Notably, the proposed monotones are adept at identifying entanglement with varying entanglement dimensionalities, making them particularly effective for detecting high-dimensional entanglement. To support practical computation, we develop a non-convex optimization framework that yields accurate upper bounds, complemented by semidefinite programming techniques to establish robust lower bounds. Together, these approaches provide a consistent and efficient computational methodology. This work advances both the theoretical understanding and algorithmic tools for entanglement quantification, contributing to the study of complex quantum correlations in entangled systems.
One of the biggest problems in the theory of quantum information is the quantification of amount of entanglement in an arbitrary multipartite mixed state. Different axiomatic and operational measures were proposed so far. In this work we will establish a connection between geometric measure of entanglement proposed in [Phys. Rev. A 68, 042307 (2003)] and measures based on fidelity. One result will be, that geometric and revised geometric measure of entanglement proposed in [J. Phys. A: Math. Theor. 40, 3507 (2007)] are equal. Further a useful expression for fidelity is derived.
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.