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      Entanglement criteria and full separability of multi-qubit quantum states

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          Abstract

          We introduce an entanglement criterion to exclude full separability of quantum states. We present numerical evidence that the criterion is necessary and sufficient for the class of GHZ diagonal three-qubit states and estimate the volume of bound entangled states within this class. Finally, we extend our approach to bound entangled states which are not GHZ diagonal.

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          Quantum entanglement

          Ryszard Horodecki, Karol Horodecki, Pawel Horodecki … (2007)
          All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory}. But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding. However, it appeared that this new resource is very complex and difficult to detect. Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying. In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations. They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon. A basic role of entanglement witnesses in detection of entanglement is emphasized.
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            Detecting multipartite entanglement

            Federico Spedalieri, Andrew C. Doherty, Pablo A Parrilo (2004)
            We discuss the problem of determining whether the state of several quantum mechanical subsystems is entangled. As in previous work on two subsystems we introduce a procedure for checking separability that is based on finding state extensions with appropriate properties and may be implemented as a semidefinite program. The main result of this work is to show that there is a series of tests of this kind such that if a multiparty state is entangled this will eventually be detected by one of the tests. The procedure also provides a means of constructing entanglement witnesses that could in principle be measured in order to demonstrate that the state is entangled.
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              Complete Separability and Fourier representations of n-qubit states

              Arthur O. Pittenger, Morton Rubin (1999)
              Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of \(n\)-qubit states whose densities all satisfy the Peres condition.
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                Author and article information

                Journal
                Publication date Created: 20 September 2010
                Publication date Updated: 2011-01-01
                Article
                DOI: 10.1016/j.physleta.2010.11.032
                ArXiV ID: 1009.3782
                SO-VID: b78212c2-e4b1-49d1-b60e-3a598e83a2a8
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Journal reference Phys. Lett. A 375, 406 (2011)
                Comments 11 pages, no figures, v2: small changes
                Categories quant-ph

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