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

By Tzu-Chieh Wei, P. Goldbart. 2003

617 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/dfad8798075966f95e6a57e6ab92d43c58982c88

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Otfried Gühne
2. Rosario Fazio
3. Luigi Amico
4. A. Osterloh
5. Géza Tóth
6. Vlatko Vedral
7. Yu-Ran Zhang
8. Yi Peng

In Atomic and Molecular Physics, and Optics

1. Otfried Gühne
2. Rosario Fazio
3. Luigi Amico
4. A. Osterloh
5. Géza Tóth
6. Vlatko Vedral
7. Heng Fan
8. Ming‐Liang Hu

Citations in the corpus

• Entanglement detection (2008, background): “forW -type states [263,291] or certain graph states [292,293].”
• Conjugate gradient algorithms for best rank‐1 approximation of tensors (2007, background): “Step1: For the particular choice of X an analytical solution is known [3].”
• Riemannian Optimization on Tensor Products of Grassmann Manifolds: Applications to Generalized Rayleigh-Quotients (2010, background): “Here, the distance of a pure state (tensor) to the set of all product states (rank-1 tensors) provides a geometric measure for entanglement [6, 23, 34].”
• Entanglement in many-body systems (2007, background): “As discussed in detail in (Wei and Goldbart, 2003), the previous definition is an entanglement monotone if the convexroof extension to mixed states is taken.”
• Quantum coherence and geometric quantum discord (2017, methodology): “The motivation for this approach is very similar to the geometric measure of entanglement first introduced by Shimony (1995) and further extended by Wei and Goldbart (2003).”