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arXiv:quant-ph/0309193 · Journal of Physics A Mathematical and General

d-outcome measurement for a nonlocality test

Wonmin Son, Jinhyoung Lee, M S Kim

Published Nov 25, 2004• 20 citations
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Abstract

For the purpose of a nonlocality test, we propose a general correlation observable of two parties by utilizing local d -outcome measurements with SU ( d ) transformations and classical communications. Generic symmetries of the SU ( d ) transformations and correlation observables are found for the test of nonlocality. It is shown that these symmetries dramatically reduce the number of numerical variables, which is important for numerical analysis of nonlocality. A linear combination of the correlation observables, which is reduced to the Clauser–Horne–Shimony-Holt (CHSH) Bell's inequality for two outcome measurements, leads to the Collins–Gisin–Linden–Massar–Popescu (CGLMP) nonlocality test for d -outcome measurement. As a system to be tested for its nonlocality, we investigate a continuous-variable (CV) entangled state with d measurement outcomes. It allows the comparison of nonlocality based on different numbers of measurement outcomes on one physical system. In our example of the CV state, we find that a pure entangled state of any degree violates Bell's inequality for d (⩾2) measurement outcomes when the observables are of SU ( d ) transformations.

DOI
10.1088/0305-4470/37/49/009

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For the purpose of a nonlocality test, we propose a general correlation observable of two parties by utilizing local d -outcome measurements with SU ( d ) transformations and classical communications. Generic symmetries of the SU ( d ) transformations and correlation observables are found for the test of nonlocality. It is shown that these symmetries dramatically reduce the number of numerical variables, which is important for numerical analysis of nonlocality. A linear combination of the correlation observables, which is reduced to the Clauser–Horne–Shimony-Holt (CHSH) Bell's inequality for two outcome measurements, leads to the Collins–Gisin–Linden–Massar–Popescu (CGLMP) nonlocality test for d -outcome measurement. As a system to be tested for its nonlocality, we investigate a continuous-variable (CV) entangled state with d measurement outcomes. It allows the comparison of nonlocality based on different numbers of measurement outcomes on one physical system. In our example of the CV state, we find that a pure entangled state of any degree violates Bell's inequality for d (⩾2) measurement outcomes when the observables are of SU ( d ) transformations.
Bell inequality tests based on high-dimensional entanglement usually require measurements that can resolve multiple possible outcomes. However, the implementation of high-dimensional multi-outcome measurements is often only emulated via a collection of ``click or no-click'' measurements. This reduction of multi-outcome measurements to binary-outcome measurements opens a loophole in high-dimensional tests Bell inequalities which can be exploited by local hidden variable models [Tavakoli et al., Phys. Rev. A 111, 042433 (2025)]. Here, we close this loophole by using four-dimensional photonic path-mode entanglement and multi-outcome detection. We test both the well-known Collins-Gisin-Linden-Massar-Popescu inequality and a related Bell inequality tailored for maximally entangled states in high-dimension. We observe violations that are large enough to also rule out any quantum model based on entanglement of lower dimension, thereby demonstrating genuinely high-dimensional nonlocality free of the binarisation loophole.
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.
Recently, it was demonstrated by Son et al., Phys. Rev. Lett. 102, 110404 (2009), that a separable bipartite continuous-variable quantum system can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality via operationally local transformations. Operationally local transformations are parametrized only by local variables; however, in order to allow violation of the CHSH inequality, a maximally entangled ancilla was necessary. The use of the entangled ancilla in this scheme caused the state under test to become dependent on the measurement choice one uses to calculate the CHSH inequality, thus violating one of the assumptions used in deriving a Bell inequality, namely, the free will or statistical independence assumption. The novelty in this scheme however is that the measurement settings can be external free parameters. In this paper, we generalize these operationally local transformations for multipartite Bell inequalities (with dichotomic observables) and provide necessary and sufficient conditions for violation within this scheme. Namely, a violation of a multipartite Bell inequality in this setting is contingent on whether an ancillary system admits any realistic local hidden variable model (i.e., whether the ancilla violates the given Bell inequality). These results indicate that violation of a Bell inequality performed on a system does not necessarily imply that the system is nonlocal. In fact, the system under test may be completely classical. However, nonlocality must have resided somewhere, this may have been in the environment, the physical variables used to manipulate the system or the detectors themselves provided the measurement settings are external free variables.
To reproduce in a local hidden variables theory correlations that violate Bell inequalities, communication must occur between the parties. We show that the amount of violation of a Bell inequality imposes a lower bound on the average communication needed to produce these correlations. Moreover, for every probability distribution there exists an optimal inequality for which the degree of violation gives the minimal average communication. As an example, to produce using classical resources the correlations that maximally violate the Clauser-Horne-Shimony-Holt inequality, $\sqrt{2}\ensuremath{-}1\ensuremath{\simeq}0.4142$ bits of communication are necessary and sufficient. For Bell tests performed on two entangled states of dimension d> 3d>~3d> 3 where each party has the choice between two measurements, our results suggest that more communication is needed to simulate outcomes obtained from certain nonmaximally entangled states than maximally entangled ones.