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Modern Physics Letters B

A class of bound entangled states violating the Breuer–Hall criterion

Wei Cheng, Fang Xu, Hua Li

Published Aug 10, 2014• 0 citations
Publisher

Abstract

A strong and computational entanglement criterion, i.e. the Breuer–Hall criterion, has been proposed independently by Breuer [Phys. Rev. Lett. 97 (2006) 080501; J. Phys. A: Math. Gen. 39 (2006) 11847] and Hall [J. Phys. A: Math. Gen. 39 (2006) 14119]. As the first explicit example, we show that the class of bound entangled states found by Fei et al. [Phys. Lett. A 352 (2006) 321] exactly violate the Breuer–Hall criterion. This example can be considered as one of the test platforms of a variety of entanglement criteria and may be helpful for us to develop more powerful entanglement criteria.

DOI
10.1142/s0217984914501644

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A strong and computational entanglement criterion, i.e. the Breuer–Hall criterion, has been proposed independently by Breuer [Phys. Rev. Lett. 97 (2006) 080501; J. Phys. A: Math. Gen. 39 (2006) 11847] and Hall [J. Phys. A: Math. Gen. 39 (2006) 14119]. As the first explicit example, we show that the class of bound entangled states found by Fei et al. [Phys. Lett. A 352 (2006) 321] exactly violate the Breuer–Hall criterion. This example can be considered as one of the test platforms of a variety of entanglement criteria and may be helpful for us to develop more powerful entanglement criteria.
D\"ur [Phys. Rev. Lett. 87, 230402 (2001)] constructed NNN-qubit bound entangled states which violate a Bell inequality for $N\ensuremath{\ge}8$, and his result was recently improved by showing that there exists an NNN-qubit bound entangled state violating the same Bell inequality if and only if $N\ensuremath{\ge}6$ [Phys. Rev. A 79, 032309 (2009)]. On the other hand, it has been also shown that the states which D\"ur considered violate Bell inequalities different from the inequality for $N\ensuremath{\ge}6$. In this paper, by employing different forms of Bell inequalities, in particular, a specific form of Bell inequalities with MMM settings of the measuring apparatus for sufficiently large MMM, we prove that there exists an NNN-qubit bound entangled state violating the MMM-setting Bell inequality if and only if $N\ensuremath{\ge}4$.
In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion.
Recently it was shown that if a given state fulfils the reduction criterion, it must also satisfy the known entropic inequalities. The natural question arises as to whether it is possible to derive some scalar inequalities stronger than the entropic ones, assuming that stronger criteria based on positive but not completely positive maps are satisfied. In the present paper we show that if certain conditions hold, the extended reduction criterion [H.-P. Breuer, Phys. Rev. Lett 97, 080501 (2006); W. Hall, J. Phys. A 40, 6183 (2007)] leads to some entropiclike inequalities, much stronger than their entropic counterparts. The comparison of the derived inequalities with other separability criteria shows that such an approach might lead to strong scalar criteria detecting both distillable and bound entanglement. In particular, in the case of SO(3)-invariant states it is shown that the present inequalities detect entanglement in regions, in which linear entanglement witnesses based on the extended reduction map fail. It should also be emphasized that in the case of $2\ensuremath{\bigotimes}d$ states the derived inequalities detect entanglement efficiently, while the extended reduction maps are useless, when acting on the qubit subsystem. Moreover, there is a natural way to construct a many-copy entanglement witnesses based on the derived inequalities so, in principle, there is a possibility of experimental realization. Some open problems and possibilities for further research are outlined.