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arXiv:quant-ph/0011060 · Physical Review A

Optimal tests of quantum nonlocality

Itamar Pitowsky, Karl Svozil

Published Jun 7, 2001• 160 citations
arXivHTMLPDF

Abstract

We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.

DOI
10.1103/physreva.64.014102

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We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.
A problem in quantum information theory is to find the experimental setup that maximizes the nonlocality of correlations with respect to some suitable measure such as the violation of Bell inequalities. There are however some complications with Bell inequalities. First and foremost it is unfeasible to determine the whole set of Bell inequalities already for a few measurements and thus unfeasible to find the experimental setup maximizing their violation. Second, the Bell violation suffers from an ambiguity stemming from the choice of the normalization of the Bell coefficients. An alternative measure of nonlocality with a direct information-theoretic interpretation is the minimal amount of classical communication required for simulating nonlocal correlations. In the case of many instances simulated in parallel, the minimal communication cost per instance is called nonlocal capacity, and its computation can be reduced to a convex-optimization problem. This quantity can be computed for a higher number of measurements and turns out to be useful for finding the optimal experimental setup. Focusing on the bipartite case, we present a simple method for maximizing the nonlocal capacity over a given configuration space and, in particular, over a set of possible measurements, yielding the corresponding optimal setup. Furthermore, we show that there is a functional relationship between Bell violation and nonlocal capacity. The method is illustrated with numerical tests and compared with the maximization of the violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed by qutrits turns out to be even stronger if the nonlocal capacity is employed as a measure of nonlocality.
We present a family of Bell inequalities involving only two measurement settings of each party for N>2N>2N>2 qubits. Our inequalities include all the standard ones with fewer than NNN qubits and thus give a natural generalization. It is shown that all the Greenberger-Horne-Zeilinger states violate the inequalities maximally, with an amount that grows exponentially as ${2}^{(N\ensuremath{-}2)∕2}$. The inequalities are also violated by some states that do satisfy all the standard Bell inequalities. Remarkably, our results yield in an efficient and simple way an implementation of nonlocality tests of many qubits favorably within reach of the well-established technology of linear optics.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
We show that the rich structure of multipartite entanglement can be tested following a device-independent approach. Specifically we present Bell inequalities for distinguishing between different types of multipartite entanglement, without placing any assumptions on the measurement devices used in the protocol, in contrast with usual entanglement witnesses. We first address the case of three qubits and present Bell inequalities that can be violated by W states but not by Greenberger-Horne-Zeilinger states, and vice versa. Next, we devise 'subcorrelation Bell inequalities' for any number of parties, which can provably not be violated by a broad class of multipartite entangled states (generalizations of Greenberger-Horne-Zeilinger states), but for which violations can be obtained for W states. Our results give insight into the nonlocality of W states. The simplicity and robustness of our tests make them appealing for experiments.