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European Quantum Electronics Conference (EQEC)

Iterative algorithm for the detection and quantification of entanglement

Jaroslav Řeháček, Z. Hradil

Published Nov 15, 2004• 0 citations
Publisher

Abstract

A set of extremal equations for relative entropy E is derived. Solving these equations by means of repeated convergent iterations is shown . The extremal equations indicate that there is a structure of quantum measurement associated with the extremal solution. The separable measurement obtained in this way specifies the extremal separable state and, significantly, provides the optimal entanglement witness operator revealing the possible entanglement of the input states.

DOI
10.1109/eqec.2003.1314257

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A set of extremal equations for relative entropy E is derived. Solving these equations by means of repeated convergent iterations is shown . The extremal equations indicate that there is a structure of quantum measurement associated with the extremal solution. The separable measurement obtained in this way specifies the extremal separable state and, significantly, provides the optimal entanglement witness operator revealing the possible entanglement of the input states.
We introduce an approach which allows a full structural and quantitative analysis of multipartite entanglement. The sets of states with different structures are convex and nested. Hence, they can be distinguished from each other using appropriate measurable witnesses. We derive equations for the construction of optimal witnesses and discuss general properties arising from our approach. As an example, we formulate witnesses for a 4-cluster state and perform a full quantitative analysis of the entanglement structure in the presence of noise and losses. The strength of the method in multimode continuous variable systems is also demonstrated for a dephased Greenberger-Horne-Zeilinger-type state.
Quantum entanglement lies at the heart of quantum mechanics and quantum information processing. In this work, we show a framework where entangled states play the role of witnesses. We extend the notion of entanglement witnesses, developing a hierarchy of witnesses for classes of observables. This hierarchy captures the fact that entangled states act as witnesses for detecting entanglement witnesses and separable states act as witnesses for the set of non-block-positive Hermitian operators. Indeed, more hierarchies of witnesses exist. We introduce the concept of finer and optimal entangled states. These definitions not only give an unambiguous and non-numeric quantification of entanglement and an alternative perspective on edge states but also answer the open question of what the remainder of the best separable approximation of a density matrix is. Furthermore, we classify all entangled states into disjoint families with optimal entangled states at its heart. This implies that we can focus only on the study of a typical family with optimal entangled states at its core when we investigate entangled states. Our framework also assembles many seemingly different findings with simple arguments that do not require lengthy calculations.
We present an approach to estimate the operational distinguishability between an entangled state and any separable state directly from measuring an entanglement witness. We show that this estimation also implies bounds on a variety of other well-known entanglement quantifiers. This approach for entanglement estimation is then extended to both the measurement-device-independent scenario and the fully device-independent scenario, where we obtain nontrivial but suboptimal bounds. The procedure requires no numerical optimization and is easy to compute. It offers ways for experimenters to not only detect, but also quantify, entanglement from the standard entanglement witness procedure.