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Physical Review A

Homogeneous and genuine Bell inequalities

F. De Zela

Published Feb 4, 2009• 1 citation

Abstract

We discuss homogeneous and inhomogeneous Bell inequalities, following Santos's classification. According to it, homogeneous inequalities entail only coincidence probabilities, whereas inhomogeneous inequalities entail coincidence probabilities together with single probabilities or with numbers. Because of technical limitations, all performed tests of Bell inequalities have been based on homogeneous inequalities whose derivation required additional assumptions besides realism and locality, thereby losing their genuine character. Here we derive, starting from the Clauser-Horne inequality, a homogeneous inequality that was at the basis of an experimental test performed some years ago by Torgerson et al. [Phys. Rev. A 51, 4400 (1995)]. We show that its derivation does not require anything but realism and locality, contrary to what has been previously assumed. It can thus be considered a genuine Bell inequality, appropriate for testing local realism. Similar, homogeneous inequalities can be analogously derived. They constitute a promising family that is likely to serve as a basis for loophole-free tests of local realism. The existence of such a family proves false the assertion that all genuine Bell inequalities must be inhomogeneous.

DOI: 10.1103/physreva.79.022102

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We discuss homogeneous and inhomogeneous Bell inequalities, following Santos's classification. According to it, homogeneous inequalities entail only coincidence probabilities, whereas inhomogeneous inequalities entail coincidence probabilities together with single probabilities or with numbers. Because of technical limitations, all performed tests of Bell inequalities have been based on homogeneous inequalities whose derivation required additional assumptions besides realism and locality, thereby losing their genuine character. Here we derive, starting from the Clauser-Horne inequality, a homogeneous inequality that was at the basis of an experimental test performed some years ago by Torgerson et al. [Phys. Rev. A 51, 4400 (1995)]. We show that its derivation does not require anything but realism and locality, contrary to what has been previously assumed. It can thus be considered a genuine Bell inequality, appropriate for testing local realism. Similar, homogeneous inequalities can be analogously derived. They constitute a promising family that is likely to serve as a basis for loophole-free tests of local realism. The existence of such a family proves false the assertion that all genuine Bell inequalities must be inhomogeneous.

Abstract This chapter presents a detailed discussion of Bell inequalities and their experimental tests. Starting from the Aharonov-Bohm version of the Einstein-Podolsky-Rosen (EPR) paradox, we review EPR’s criteria for realism and completeness and Bell’s locality condition, before deriving the Clauser-Horne-Shimony-Holt (CHSH) inequality. This is followed by an exposition on Bell’s original inequality and other Bell inequalities by Wigner, as well as Clauser and Horne. We then discuss the milestones in the experimental violation of Bell inequalities, from the experiment of Freedman and Clauser in the 1970s, to the efforts of Aspect et al. in the 1980s, the experiments by the teams of Zeilinger and Gisin in the 1990s, and culminating the loophole-free tests in 2015. Finally we turn to a brief discussion of the interpretations of quantum mechanics, focusing on contrasting the realist views held by, e.g., Einstein and Bell, and the informational views with proponents such as Brukner and Zeilinger

In a local realist model, physical properties are defined prior to and independent of measurement and no physical influence can propagate faster than the speed of light. Proper experimental violation of a Bell inequality would show that the world cannot be described with such a model. Experiments intended to demonstrate a violation usually require additional assumptions that make them vulnerable to a number of ``loopholes.'' In both pulsed and continuously pumped photonic experiments, an experimenter needs to identify which detected photons belong to the same pair, giving rise to the coincidence-time loophole. Here, via two different methods, we derive Clauser-Horne- and Eberhard-type inequalities that are not only free of the fair-sampling assumption (thus not being vulnerable to the detection loophole), but also free of the fair-coincidence assumption (thus not being vulnerable to the coincidence-time loophole). Both approaches can be used for pulsed as well as for continuously pumped experiments. Moreover, as they can also be applied to already existing experimental data, we finally show that a recent experiment [Giustina et al., Nature (London) 497, 227 (2013)] violated local realism without requiring the fair-coincidence assumption.

A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of

\sqrt 2

, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.

A fundamental Bell-type inequality is derived by referring solely to laws of logic, tertium non datur and the transitivity axiom, that are used to express the local realism assumption. The general form of the achieved inequality makes it a root of other more specific Bell-type theorems involving additional mathematical formalism and experimental conditions. Its universality is demonstrated explicitly by showing that the most famous theorems provided by Bell, Clauser, Horne, Shimony and Holt (CHSH), Clauser and Horne (CH), Eberhard, d'Espagnat and Maccone are direct corollaries of the general inequality. The method to retrieve Bell's inequalities of different kinds, as being based on a single scheme of reasoning, unifies diversity of proofs presented in the literature. Additionally, some of the known inequalities can be easily achieved in a more general form compared to their original versions and still suitable, or possibly even more convenient, for experimental investigations. By extracting local realism in the form of an explicit logical propositions and making it a cornerstone of a systematic reasoning leading to the universal Bell's inequality, the paper offers a synthetic perspective on Bell-type theorems and their logical basis.

Statistical tests are needed to determine experimentally whether a hypothetical theory based on local realism can be an acceptable alternative to quantum mechanics. It is impossible to rule out local realism by a single test, as often claimed erroneously. The “strength” of a particular Bell inequality is measured by the number of trials that are needed to invalidate local realism at a given confidence level. Various versions of Bell's inequality are compared from this point of view. It is shown that Mermin's inequality for Greenberger-Horne-Zeilinger states requires fewer tests than the Clauser-Horne-Shimony-Holt inequality or than its chained variants applied to a singlet state, and also than Hardy's proof of nonlocality.