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arXiv:2003.10201 · Annalen der Physik

Second‐Second Moments and Two Observers Testing Quantum Nonlocality

Adam Bednorz

Published Dec 8, 2020• 1 citation
arXivHTMLPDF

Abstract

Abstract It is shown that rejection of local realism in quantum mechanics can be tested by Bell‐type inequalities for two observers and low‐order moments of continuous and unbounded observables. It is proved that one requires three observables for each observer for a maximally entangled state and two observables for a non‐maximally entangled state and write down appropriate inequalities and show violation by quantum examples. Finding an example for quadratures or position and momentum is left as an open problem.

DOI
10.1002/andp.202000456

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Abstract It is shown that rejection of local realism in quantum mechanics can be tested by Bell‐type inequalities for two observers and low‐order moments of continuous and unbounded observables. It is proved that one requires three observables for each observer for a maximally entangled state and two observables for a non‐maximally entangled state and write down appropriate inequalities and show violation by quantum examples. Finding an example for quadratures or position and momentum is left as an open problem.
The quest for fundamental tests of quantum mechanics is an ongoing effort. We here address the question of what are the lowest possible moments needed to prove quantum nonlocality and noncontextuality without any further assumptions---in particular, without the often assumed dichotomy. We first show that second-order correlations can always be explained by a classical noncontextual local-hidden-variable theory. Similar third-order correlations also cannot violate classical inequalities in general, except for a special state-dependent noncontextuality. However, we show that fourth-order correlations can violate locality and state-independent noncontextuality. Finally we obtain a fourth-order continuous-variable Bell inequality for position and momentum, which can be violated and might be useful in Bell tests, closing all loopholes simultaneously.
We investigate the violation of Leggett's inequality for nonlocal realism using entangled coherent states and various types of local measurements. We prove mathematically the relation between the violation of the Clauser-Horne-Shimony-Holt form of Bell's inequality and Leggett's one when tested by the same resources. For Leggett inequalities, we generalize the nonlocal realistic bound to systems in Hilbert spaces larger than bidimensional ones and introduce an optimization technique that allows one to achieve larger degrees of violation by adjusting the local measurement settings. Our work describes the steps that should be performed to produce a self-consistent generalization of Leggett's original arguments to continuous-variable states.
Experimental measurements of entangled photons have enabled effective probing of foundational principles of physical law, Quantum Mechanics (QM) in particular. When it was noted that QM is non-local and precludes certain real properties, change in the description of basic phenomena was required. Bell's Inequality work clarified this conflict with Local-Realistic descriptions but left open whether locality, realism, or both should be abandoned. More recent investigations consider models that maintain forms of non-local realism as alternatives to QM. Problems with those are illustrated here, and support a non-local QM description of entangled photons having non-determinate properties.
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.
We derive a Bell inequality based on a generalized quasiprobability function, which is parametrized by one nonpositive real value. Two types of known Bell inequalities formulated in terms of the Wigner and the QQQ functions are included as limiting cases. We investigate violations of our Bell inequalities for single-photon entangled states and two-mode squeezed vacuum states when varying the detector efficiency. We show that the Bell inequality for the QQQ function allows the lowest detection efficiency for violations of local realism.
Quantum physics violates the basic assumptions of local realism. Particles can be entangled with each other in ways that allow correlations stronger than possible in classical physics. This was first realized by Einstein, Podolsky and Rosen (EPR) in 1935 and led them to conclude that quantum theory was incomplete. It took about 30 years before J. S. Bell found observable consequences of this phenomenon by showing how entangled states violate inequalities for two-particle correlations derived purely from the assumption that the world obeys local realism. Violations of Bell inequalities were first observed experimentally in the Aspect experiment, which will be discussed below. Such experiments are technically challenging and care is needed to avoid any loopholes in the experimental setup that would still permit the description of the experiments by a local realistic theory. Bell inequalities describe violations of local realism on average, meaning that each individual experimental outcome could occur in a local realistic theory. It is the statistics of these outcomes that violates local realism. In contrast, another class of quantum states, so-called Greenberger–Horne–Zeilinger (GHZ) states, which contain three particles, can be shown to violate local realism in each measurement outcome. We will also discuss this class of states and experiments by Zeilinger's group, which demonstrated this type of violation of local realism for the first time. Our aim here is not to discuss the exact relations between quantum entanglement, non-locality and realism, and their physical and philosophical implications. While this is a highly interesting topic which attracts a lot of attention from researchers, such an exploration would go beyond the scope of this book.