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Annalen der Physik

Unraveling multipartite entanglement for an important family of mixed states

Marcus Huber, Jens Siewert

Published Jan 7, 2016• 0 citations
Publisher

Abstract

Schematic of the state space of an N-qubit system. The space is subdivided into N separability classes , (genuinely multipartite entangled), (biseparable), (fully separable). Buchholz et al. have exactly obtained the entanglement measure for a specific family of symmetric states. The geometric entanglement is the geometric distance of the state ρ from the set of k-separable states.

DOI
10.1002/andp.201500353

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Schematic of the state space of an N-qubit system. The space is subdivided into N separability classes , (genuinely multipartite entangled), (biseparable), (fully separable). Buchholz et al. have exactly obtained the entanglement measure for a specific family of symmetric states. The geometric entanglement is the geometric distance of the state ρ from the set of k-separable states.
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.
The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric N qubit states and provide the analytical expression for their Best Separable Approximation (BSA [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled one with maximal weight of the separable one.
To determine whether a given multipartite quantum state is separable with respect to some partition, we construct a family of entanglement measures {Rm(ρ)}. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Furthermore, we provide an analytically computable lower bound for {Rm(ρ)} via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for each intermediate separability.
We geometrically examine the entanglement in symmetric multiquibt states using the spin coherent states properties. We employ the Majorana representation to examine how coherent (polarized) and unpolarized states can be represented in the Bloch sphere and subsequently to quantify entanglement in multipartite systems. Entanglement can be viewed as a distance separate two points in Bloch sphere and how this notion can be extended for multipartite states in W class. This provides us with an entanglement measure for [Formula: see text]-states analogue to the notion of Concurrence.