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arXiv:1605.05846 · Physical Review A

Bounding the persistency of the nonlocality of W states

Péter Diviánszky, Réka Trencsényi, Erika Bene, Tamás Vértesi

Published Apr 18, 2016• 9 citations
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Abstract

The nonlocal properties of the $W$ states are investigated under particle loss. By removing all but two particles from an $N$-qubit $W$ state, the resulting two-qubit state is still entangled. Hence, the $W$ state has high persistency of entanglement. We ask an analogous question regarding the persistency of nonlocality [see N. Brunner and T. V\'ertesi, Phys. Rev. A 86, 042113 (2012)]. Namely, we inquire what is the minimal number of particles that must be removed from the $W$ state so that the resulting state becomes local. We bound this value in function of $N$ qubits by considering Bell nonlocality tests with two alternative settings per site. In particular, we find that this value is between $2N/5$ and $N/2$ for large $N$. We also develop a framework to establish bounds for more than two settings per site.

DOI
10.1103/physreva.93.042113

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The nonlocal properties of the WWW states are investigated under particle loss. By removing all but two particles from an NNN-qubit WWW state, the resulting two-qubit state is still entangled. Hence, the WWW state has high persistency of entanglement. We ask an analogous question regarding the persistency of nonlocality [see N. Brunner and T. V'ertesi, Phys. Rev. A 86, 042113 (2012)]. Namely, we inquire what is the minimal number of particles that must be removed from the WWW state so that the resulting state becomes local. We bound this value in function of NNN qubits by considering Bell nonlocality tests with two alternative settings per site. In particular, we find that this value is between 2N/52N/52N/5 and N/2N/2N/2 for large NNN. We also develop a framework to establish bounds for more than two settings per site.
In a recent work Brunner and Vertesi [Phys. Rev. A 86, 042113 (2012)] the question of persistency of entanglement and nonlocality of multiparty systems under particle loss has been addressed. This question is of immense importance considering the practical realization of the information theoretic tasks which make use of the power of quantum correlations. But in the multipartite scenario more interesting cases arise since subsystems can also possess genuineness in correlation which is prevalently inequivalent to the bipartite scenario. In this paper we investigate the persistency of such genuine correlations under particle loss. Keeping in mind the practical importance, considerable attention has been devoted to find the multiparty states which exhibit maximal persistency of genuine correlations.
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Understanding the quantitative relation between entanglement and Bell nonlocality is a long-standing open problem of fundamental and practical interest. Here, we tackle this problem in a general Bell scenario. {We observe that lying in the center of quantifying these properties are two minimal distances: one from a state to separable states (entanglement), and the other from a correlation to local correlations (nonlocality).} We find that these two distances can be related to each other -- the minimal correlation distance provides a lower bound for the minimal state distance, which allows us to derive nontrivial bounds on many entanglement measures with an arbitrary nonlocal correlation. Moreover, with the on-hand structural knowledge of entanglement and nonlocality in the (n,2,2)(n, 2, 2)(n,2,2) Bell scenario, we refine our estimate significantly.
Understanding the quantitative relation between entanglement and Bell nonlocality is a long-standing open problem of fundamental and practical interest. Here, we tackle this problem in a general Bell scenario. We observe that lying in the center of quantifying these properties are two minimal distances: one from a state to separable states (entanglement), and the other from a correlation to local correlations (nonlocality). We find that these two distances can be related to each other—the minimal correlation distance provides a lower bound for the minimal state distance, which allows us to derive nontrivial bounds on many entanglement measures with an arbitrary nonlocal correlation. Moreover, with the on-hand structural knowledge of entanglement and nonlocality in the ( n , 2, 2) Bell scenario, we refine our estimate significantly.
Quantum nonlocality, rooted in the quantum state discrimination task, contributes to the development of the theory of local distinguishability of quantum states. Recently, Zhu et al. [Phys. A (Amsterdam, Neth.) 624, 128956 (2023)] found that a nonlocal set of orthogonal product states (OPSs) can be perfectly identified by local operations and classical communication after removing a specific quantum state in a bipartite quantum system. They called this phenomenon minimal nonlocality. It is obvious that minimal nonlocality effectively reflects the lower boundary of the elements contained in a nonlocal set of OPSs. In this paper, we generalize the methods of constructing nonlocal sets with minimal nonlocality from bipartite systems to multipartite systems and from orthogonal product states to entangled states. First, we give a method to construct a set of OPSs with minimal nonlocality in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and then we extend it to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system for $d\ensuremath{\ge}4$ and $n\ensuremath{\ge}3$. In addition, we give a method to construct a nonlocal set of orthogonal states with entanglement relations in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and we prove that this set has minimal nonlocality, which can also be extended to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system.
An experimental setup for testing quantum nonlocality of NNN qubits is proposed. This method is a generalization of the optical setup proposed by Banaszek and W\'odkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. The quantum nonlocality of NNN qubits can be obtained through its violation of NNN-qubit Bell inequalities. The correlation function measured in the experiment is described by the Wigner function. The effect of inefficient detector is also considered.