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arXiv:1210.1754 · Physical Review A

Nonlocality and entanglement for symmetric states

Zizhu Wang, Damian Markham

Published Jan 4, 2013• 20 citations
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Abstract

In this paper, building on some recent progress combined with numerical techniques, we shed some light on how the nonlocality of symmetric states is related to their entanglement properties and on their potential usefulness in quantum information processing. We use semidefinite programming techniques to devise a device-independent classification of three 4-qubit states into two classes inequivalent under local unitaries and permutation of systems. We study nonlocal properties when the number of parties grows large for two important classes of symmetric states: $W$ states and GHZ states, showing that they behave differently under the inequalities we consider. We also discuss the monogamy arising from the nonlocal correlations of symmetric states. We show that although monogamy in a strict sense is not guaranteed for all symmetric states, strict monogamy is achievable for all Dicke states when the number of parties goes to infinity.

DOI
10.1103/physreva.87.012104

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In this paper, building on some recent progress combined with numerical techniques, we shed some light on how the nonlocality of symmetric states is related to their entanglement properties and on their potential usefulness in quantum information processing. We use semidefinite programming techniques to devise a device-independent classification of three 4-qubit states into two classes inequivalent under local unitaries and permutation of systems. We study nonlocal properties when the number of parties grows large for two important classes of symmetric states: WWW states and GHZ states, showing that they behave differently under the inequalities we consider. We also discuss the monogamy arising from the nonlocal correlations of symmetric states. We show that although monogamy in a strict sense is not guaranteed for all symmetric states, strict monogamy is achievable for all Dicke states when the number of parties goes to infinity.
This thesis is about the nonlocal properties of permuation symmetric states and the potential usefulness of such properties in quantum information processing. The nonlocality of almost all symmetric states, except Dicke states is shown by constructing an n-party Hardy paradox. With the help of the Majorana representation, suitable measurement settings can be chosen for these symmetric states which satisfy the paradox. An extended CH inequality can be derived from the probabilistic conditions of the paradox. The inequality is shown to be violated by all symmetric states. The nonlocality properties and entanglement properties of symmetric states are also discussed and compared, natbly with respect to persistency and monogamy. It is shown that te degeneracy of some symmetric states is linked to the persistency, which provides a way to use device independent tests to separte nonlocality classes. It is also shown that the inequalities used to show the nonlocality of all symmetric states are not strictly monogamous.A new inequality for Dicke states is shown to be monogamous when the number of parties goes to infinity. But all these inequalites can not detect genuine nonlocality. Applications of nonlocality to communication complexity and Bayesian game theory are also discussed.
In this Letter we study the nonlocal properties of permutation symmetric states of n qubits. We show that all these states are nonlocal, via an extended version of the Hardy paradox and associated inequalities. Natural extensions of both the paradoxes and the inequalities are developed which relate different entanglement classes to different nonlocal features. Belonging to a given entanglement class will guarantee the violation of associated Bell inequalities which see the persistence of correlations to subsets of players, whereas there are states outside that class which do not violate.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
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We analyze entanglement and nonlocal properties of the convex set of symmetric NNN-qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability---which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.
The future progress of semi-device independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite dimensional quantum resources. In this work, we characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. In the bipartite case, we find that the first level of the hierarchy returns non-trivial bounds in all cases considered, allowing to study nonlocality scenarios with four measurement settings on one side and twelve (12) on the other in a normal desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than two, hence certifying three-dimensional tripartite entanglement in a device independent way. Finally, we show how the new method can be trivially modified to detect non-separable measurements in two-qubit scenarios.