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Probing quantum nonlocality of bipartite entangled qutrits by generalising Wigner's argument

Debarshi Das, Shounak Datta, Suchetana Goswami, A. S. Majumdar, Dipankar Home

Published May 28, 2016• 0 citations
OpenAlex

Abstract

Bell-type local realist inequalities are developed for probing quantum nonlocality of bipartite entangled qutrit states by generalizing Wigner's argument that was originally formulated for the bipartite qubit singlet state. This treatment is based upon assuming the existence of the overall joint probability distributions for the measurement outcomes pertaining to the relevant trichotomic observables, satisfying the locality condition, and yielding the measurable marginal probabilities. We first show that such generalized Wigner inequalities (GWI) are violated by quantum mechanics (QM) for both the bipartite qutrit isotropic and singlet states, thereby demonstrating their nonlocality. The efficacy of GWIs are also probed by comparing their QM violation with that obtained for other types of local realist inequalities that have been used for probing nonlocality of entangled bipartite qutrits. This is done in two different ways, (i) by employing unsharp measurements, and (ii) by incorporating white noise in the isotropic and singlet qutrit states. We find that in both these cases, GWIs outperform the two other types of local realist inequalities considered here in terms of their QM violation, by being (i) more robust against the usharpness of measurement, and (ii) more tolerant to the white noise incorporated in the states considered.

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Bell-type local realist inequalities are developed for probing quantum nonlocality of bipartite entangled qutrit states by generalizing Wigner's argument that was originally formulated for the bipartite qubit singlet state. This treatment is based upon assuming the existence of the overall joint probability distributions for the measurement outcomes pertaining to the relevant trichotomic observables, satisfying the locality condition, and yielding the measurable marginal probabilities. We first show that such generalized Wigner inequalities (GWI) are violated by quantum mechanics (QM) for both the bipartite qutrit isotropic and singlet states, thereby demonstrating their nonlocality. The efficacy of GWIs are also probed by comparing their QM violation with that obtained for other types of local realist inequalities that have been used for probing nonlocality of entangled bipartite qutrits. This is done in two different ways, (i) by employing unsharp measurements, and (ii) by incorporating white noise in the isotropic and singlet qutrit states. We find that in both these cases, GWIs outperform the two other types of local realist inequalities considered here in terms of their QM violation, by being (i) more robust against the usharpness of measurement, and (ii) more tolerant to the white noise incorporated in the states considered.
Wigner's argument inferring a Bell-type inequality for the Einstein--Podolsky--Rosen--Bohm entangled state is generalized here for any NNN-partite state. This is based on assuming for the relevant dichotomic observables the existence of the overall joint probability distributions, satisfying the locality condition, which would yield the measurable marginal probabilities. For any NNN, such a generalized Wigner inequality (GWI) is violated by quantum mechanics for all pure entangled states. The efficacy of GWI is probed, comparing with the Seevinck--Svetlichny multipartite Bell-type inequality, by calculating threshold visibilities for the quadripartite Greenberger--Horne--Zeilinger, Cluster, and WWW states that determine their respective robustness with respect to the quantum-mechanical violation of GWI in the presence of white noise.
Distinct from Bell's approach, Wigner derived a form of local realist (LR) inequality which is quantum-mechanically violated for a bipartite maximally entangled state. Subsequently, this approach was generalized to obtain a multipartite LR inequality. However, the violation of such a generalized Wigner inequality does not guarantee nonlocality between all possible different bipartitions of the multipartite system. In the present paper, this limitation has been overcome by formulating a further generalization of Wigner's approach through the derivation of a set of LR inequalities with respect to all different bipartitions of a NNN-partite system. Quantum-mechanical violations of all individual LR inequalities belonging to such a set would rigorously certify multipartite nonlocality by also providing a finer characterization of the nature of multipartite nonlocality in the following sense. The quantum-mechanical violation of any given inequality of our complete set of LR inequalities would enable identification of the corresponding bipartition which exhibits nonlocality. This is in contrast to other multipartite LR inequalities such as the Svetlichny inequality or its generalization that cannot be used to detect whether there is any particular bipartition which is nonlocally correlated. The efficacy of the scheme developed in this paper is illustrated for the tripartite and quadripartite states.
We have recently shown that for the special case of a bipartite system with binary inputs and outputs there exist equalities in local theories which are violated by quantum theory. The amount of white noise tolerated by these equalities are twice that of inequalities. In this paper we will first introduce an inequality in bipartite qutrits systems which, if non-maximally entangled state is used instead of maximally entangled state, is violated more strongly by quantum theory. Hence reproducing the results obtained in the literature. We will then prove that our equalities in this case are violated by quantum theory too, and they tolerate much more white noise than inequalities.
While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality. In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.
Bell inequalities reveal when quantum correlations cannot be explained by any local hidden-variable model, and the CHSH inequality is the most fundamental test of this phenomenon. While entangled two-qubit states can violate CHSH inequalities with suitable local measurements, the situation is more subtle for higher-dimensional systems when the allowed observables are physically restricted. In this work, we study bipartite qutrit systems in which each party is restricted to spin-1 observables S(u)=uxSx+uySy+uzSzS(u)=u_xS_x+u_yS_y+u_zS_zS(u)=ux​Sx​+uy​Sy​+uz​Sz​, with u∈R3u\in\mathbb R^3u∈R3 a unit vector. Hanotel and Loubenets conjectured that nonseparable pure states of two qutrits do not violate the CHSH inequality under this measurement restriction. We prove the stronger result that every bipartite state on C3⊗C3\mathbb C^3\otimes\mathbb C^3C3⊗C3, pure or mixed, satisfies the CHSH inequality for all choices of local spin-1 observables. Thus the absence of CHSH violation is not a property of special states, but a structural limitation of the spin-1 measurement family. Our result clarifies a distinction between two-qutrit entanglement and CHSH nonlocality under natural spin-component measurements.
The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from d=2d=2d=2 up to d=10d=10d=10. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension ddd grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviors. Furthermore, in the latter approach the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison to maximally entangled states, so no anomaly is observed. Finally, the nonlocality of states with nonmaximal Schmidt rank is investigated.