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arXiv:1201.2041 · Journal of Physics A Mathematical and Theoretical

Monogamy of measurement-induced nonlocality

Ajoy Sen, Debasis Sarkar, Amit Bhar

Published Sep 19, 2012• 18 citations
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Abstract

Measurement-Induced NonLocality was introduced by Luo and Fu (Phys. Rev. Lett. \textbf{106}, 120401,(2011)) as a measure of nonlocality in a bipartite state. In this paper we will discuss monogamy property of measurement-induced nonlocality for some three- and four-qubit classes of states. Unlike discord, we find quite surprising results in this situation. Both the GHZ and W states satisfy monogamy relations in the three-qubit case, however, in general there are violations of monogamy relations in both the GHZ-class and W-class states. In case of four-qubit system, monogamy holds for most of the states in the generic class. Four qubit GHZ does not satisfy monogamy relation, but W-state does. We provide several numerical results including counterexamples regarding monogamy nature of measurement induced nonlocality. We will also extend our results of generalized W-class to n-qubit.

DOI
10.1088/1751-8113/45/40/405306

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Measurement-Induced NonLocality was introduced by Luo and Fu (Phys. Rev. Lett. \textbf{106}, 120401,(2011)) as a measure of nonlocality in a bipartite state. In this paper we will discuss monogamy property of measurement-induced nonlocality for some three- and four-qubit classes of states. Unlike discord, we find quite surprising results in this situation. Both the GHZ and W states satisfy monogamy relations in the three-qubit case, however, in general there are violations of monogamy relations in both the GHZ-class and W-class states. In case of four-qubit system, monogamy holds for most of the states in the generic class. Four qubit GHZ does not satisfy monogamy relation, but W-state does. We provide several numerical results including counterexamples regarding monogamy nature of measurement induced nonlocality. We will also extend our results of generalized W-class to n-qubit.
Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary m×nm\times nm×n dimensional bipartite density matrix ρρρ for the case where one of its reduced density matrix, ρaρ^{a}ρa, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, ρaρ^{a}ρa has ddd degenerate subspaces with dimension mi(mi≤m,i=1,2,...,d)m_{i} (m_{i} \leq m, i=1, 2, ..., d)mi​(mi​≤m,i=1,2,...,d). We show that according to the degeneracy of ρaρ^{a}ρa, if we expand ρρρ in a suitable basis, the evaluation of MIN for an m×nm\times nm×n dimensional state ρρρ, is degraded to finding the MIN in the mi×nm_{i}\times nmi​×n dimensional subspaces of state ρρρ. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary m×nm\times nm×n state ρρρ for which mi≤2m_{i}\leq 2mi​≤2, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. In the final, we explain the evaluation of MIN for 3×n3\times n3×n dimensional states in details.
The limitation on the shareability of quantum entanglement over several parties, the so-called monogamy of entanglement, is an issue that has received considerable attention from the quantum information community over the last decade. A natural question of interest in this connection is whether monogamy of correlations is true for correlations other than entanglement. This issue is examined here by choosing quantum deficit, proposed by A. K. Rajagopal and R. W. Rendell [Phys. Rev. A 66, 022104 (2002)], an operational measure of correlations. In addition to establishing the polygamous nature of the class of three-qubit symmetric pure states characterized by two distinct Majorana spinors (to which the WWW states belong), those with three distinct Majorana spinors [to which Greenberger-Horne-Zeilinger (GHZ) states belong] are shown to either obey or violate monogamy relations. While the generalized WWW states can be monogamous or polygamous, the generalized GHZ states exhibit monogamy with respect to quantum deficit. The issue of using monogamy conditions based on quantum deficit to witness the states belonging to stochastic local operations and classical communication (SLOCC) inequivalent classes is discussed in light of these results.
In this paper, using relative entropy, we study monogamous properties of measurement-induced nonlocality based on relative entropy. Depending on different measurement sides, we provide necessary and sufficient conditions for two types of monogamy inequalities. By the concept of nonlocality monogamy score, we find a necessary condition of the vanished nonlocality monogamy score for arbitrary three-party states. In addition, two types of necessary and sufficient conditions of the vanished nonlocality monogamy scores are obtained for any pure states. As an application, we show that measurement-induced nonlocality based on relative entropy can be viewed as a "nonlocality witness" to distinguish generalized GHZ states from the generalized W states.
Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary m×nm\times nm×n dimensional bipartite density matrix ρ\rhoρ for the case where one of its reduced density matrix, ρa\rho^{a}ρa, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, ρa\rho^{a}ρa has ddd degenerate subspaces with dimension mi(mi≤m,i=1,2,...,d)m_{i} (m_{i} \leq m , i=1, 2, ..., d)mi​(mi​≤m,i=1,2,...,d). We show that according to the degeneracy of ρa\rho^{a}ρa, if we expand ρ\rhoρ in a suitable basis, the evaluation of MIN for an m×nm\times nm×n dimensional state ρ\rhoρ, is degraded to finding the MIN in the mi×nm_{i}\times nmi​×n dimensional subspaces of state ρ\rhoρ. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary m×nm\times nm×n state ρ\rhoρ for which mi≤2m_{i}\leq 2mi​≤2, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. Finally, we explain the evaluation of MIN for 3×n3\times n3×n dimensional states in details.