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arXiv:1705.10248 · International Journal of Quantum Information

Entanglement witnesses for families of the bound entangled three-qubit states

Y. Akbari-Kourbolagh, M. Azhdargalam

Published Apr 1, 2018• 0 citations
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Abstract

Entanglement witnesses are of special interest for distinguishing between separable and entangled states. For three-qubit states, a family of optimal entanglement witnesses has been introduced by Jafarizadeh et al. [Eur. Phys. J. D 50 (2008) 107]. We give an alternative demonstration for some elements of this family and show that they are able to detect bound entangled states. To illustrate this fact, we consider two families of bound entangled three-qubit states, introduced by Kay [Phys. Rev. A 83 (2011) 020303] and Kye [J. Phys. A, Math. Theor. 48 (2015) 235303], and show that these witnesses detect the entanglement of all states of these two families.

DOI
10.1142/s0219749918500259

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•Jan 1, 2004•3 citations
Entanglement witnesses are of special interest for distinguishing between separable and entangled states. For three-qubit states, a family of optimal entanglement witnesses has been introduced by Jafarizadeh et al. [Eur. Phys. J. D 50 (2008) 107]. We give an alternative demonstration for some elements of this family and show that they are able to detect bound entangled states. To illustrate this fact, we consider two families of bound entangled three-qubit states, introduced by Kay [Phys. Rev. A 83 (2011) 020303] and Kye [J. Phys. A, Math. Theor. 48 (2015) 235303], and show that these witnesses detect the entanglement of all states of these two families.
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.
In 2008, the conjecture that structural physical approximations to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, in [K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, arXiv:1210.1088v3], Ha and Kye proposed a decomposable entanglement witness whose SPA is entangled and argued that it is optimal. In this note, which is based on a comment to the latter work [R. Augusiak et al., Comment on "Separable states with unique decompositions", arXiv:1304.2040v1], we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking optimality of entanglement witnesses developed already in [M. Lewenstein et al., Phys. Rev. A 62, 052310 (2000)], however, hardly exploited so far in the literature.
This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.