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Journal of Physics A Mathematical and Theoretical

Various formulations of inequivalent Leggett–Garg inequalities

Swati Kumari, A. K. Pan

Published Dec 3, 2020• 3 citations
Publisher

Abstract

Abstract In two-party, two-input and two-output measurement scenario only relevant Bell’s inequality is the Clauser–Horne–Shimony–Holt (CHSH) form. They also provide the necessary and sufficient conditions (NSCs) for local realism. Any other form, such as, Clauser–Horne and Wigner forms reduce to the CHSH one. The standard Leggett–Garg inequalities, proposed for testing incompatibility between macrorealism and quantum theory, are often considered to be the temporal analog of CHSH inequalities. However, they do not provide the NSCs for macrorealism. There is thus scope of formulating new macrorealist inequalities inequivalent and stronger than the standard Leggett–Garg inequalities. In this paper, we propose two different classes of macrorealistic inequalities. A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory, and the other class of inequalities are inequivalent to the all the other formulations of Leggett–Garg inequalities both in macrorealist model and in quantum theory. The latter class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett–Garg inequalities fails to do so. We extend the formulations of inequivalent Leggett–Garg inequalities to the four-time and two-time measurement scenarios. Further, we provide a brief discussion about the alternative formulation of macrorealism known as the no-signaling in time (NSIT) conditions.

DOI
10.1088/1751-8121/abd077

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Abstract In two-party, two-input and two-output measurement scenario only relevant Bell’s inequality is the Clauser–Horne–Shimony–Holt (CHSH) form. They also provide the necessary and sufficient conditions (NSCs) for local realism. Any other form, such as, Clauser–Horne and Wigner forms reduce to the CHSH one. The standard Leggett–Garg inequalities, proposed for testing incompatibility between macrorealism and quantum theory, are often considered to be the temporal analog of CHSH inequalities. However, they do not provide the NSCs for macrorealism. There is thus scope of formulating new macrorealist inequalities inequivalent and stronger than the standard Leggett–Garg inequalities. In this paper, we propose two different classes of macrorealistic inequalities. A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory, and the other class of inequalities are inequivalent to the all the other formulations of Leggett–Garg inequalities both in macrorealist model and in quantum theory. The latter class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett–Garg inequalities fails to do so. We extend the formulations of inequivalent Leggett–Garg inequalities to the four-time and two-time measurement scenarios. Further, we provide a brief discussion about the alternative formulation of macrorealism known as the no-signaling in time (NSIT) conditions.
In two-party, two-input and two-output measurement scenario only relevant Bell's inequality is the Clauser-Horne-Shimony-Holt (CHSH) form. They also provide the necessary and sufficient conditions for local realism. Any other form, such as, Clauser-Horne and Wigner forms reduce to the CHSH one. Standard Leggett-Garg inequalities are often considered to be the temporal analog of CHSH inequalities. But, they do not provide the necessary and sufficient conditions for macrorealism. There is thus scope of formulating new macrorealist inequalities different and stronger than the standard Leggett-Garg inequalities for testing compatibility between the macrorealism and quantum theory. In this paper, we propose three different classes of macrorealistic inequalities in three-time scenario; (i) The ones equivalent to standard Leggett-Garg inequalities in both macrorealist model and in quantum theory. (ii) A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory (iii) Another class of inequalities which are inquivalent to the all formulations of Leggett-Garg inequalities both in macrorealist model and in quantum theory. This class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett-Garg inequalities fail to do so. We extend the formulations of the class (ii) inequalities to the four-time and two-time measurement scenario. Further, we provide a brief discussion about the alternate formulation of macrorealism which was derived based on the no-signaling in time conditions.
It remains an open question how a realist view of the macroscopic world emerges from a quantum formalism. For testing the macrorealism in the quantum domain, an interesting approach was put forward by Leggett and Garg in 1985, by formulating a suitable inequality valid for any macrorealistic theory. Recently, by following the Wigner idea of local realist inequality, a probabilistic version of standard Leggett-Garg inequalities has also been proposed. While the Wigner form of local realist inequalities is equivalent to the two-party, two-measurements and two-outcomes CHSH inequalities, in this paper we provide a generic proof to demonstrate that the Wigner form of Leggett-Garg inequalities is not only inequivalent to the standard ones, but also stronger than the latter. This is demonstrated by quantifying the amount of disturbance caused by a priori measurement to the subsequent measurements. In this connection, the relation between LGIs and another formulation of macrorealism known as no-signaling in time is examined.
We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable QQQ. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly studied test, are only necessary conditions for macrorealism, but, building on earlier work J. J. Halliwell, Phys. Rev. A 93, 022123 (2016), it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by J. Kofler and C. Brukner, Phys. Rev. A 87, 052115 (2013) and L. Clemente and J. Kofler, Phys. Rev. A 91, 062103 (2015), which ensure that all probabilities for QQQ at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of noninvasive measurability and so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; in contrast, the NSIT tests entail sequential measurements of incompatible quantities and are primarily tests for noninvasiveness.
Tests of local realism and macrorealism have historically been discussed in very similar terms: Leggett-Garg inequalities follow Bell inequalities as necessary conditions for classical behavior. Here, we compare the probability polytopes spanned by all measurable probability distributions for both scenarios and show that their structure differs strongly between spatially and temporally separated measurements. We arrive at the conclusion that, in contrast to tests of local realism where Bell inequalities form a necessary and sufficient set of conditions, no set of inequalities can ever be necessary and sufficient for a macrorealistic description. Fine's famous proof that Bell inequalities are necessary and sufficient for the existence of a local realistic model, therefore, cannot be transferred to macrorealism. A recently proposed condition, no-signaling in time, fulfills this criterion, and we show why it is better suited for future experimental tests and theoretical studies of macrorealism. Our work thereby identifies a major difference between the mathematical structures of local realism and macrorealism.
Standard Leggett-Garg inequalities (SLGIs) were formulated for testing incompatibility between the classical world view of macrorealism and quantum mechanics. In recent times, various other formulations such as the Wigner form of Leggett-Garg inequalities (WLGIs), entropic Leggett-Garg inequalities (ELGIs), and the no-signaling-in-time (NSIT) condition have also been proposed. It was also recently argued that nonono set of SLGIs can provide the necessary and sufficient conditions for macrorealism, but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulations of Leggett-Garg inequalities (LGIs) for testing macrorealism pertaining to the two different unsharp measurements. While the violations of WLGIs are more robust than SLGIs and ELGIs for spin positive operator-valued measures (POVMs), here we demonstrate that for the case of biased POVMs the quantum violations of both SLGIs and ELGIs provide the same robustness as WLGIs. Importantly, the violations of all formulations of LGIs can be achieved for any nonzero value of unsharpness parameter. We have also studied the connection between LGIs and NSIT conditions. Further, we investigate the role of the joint measurability of the POVMs in the violation of LGIs and find that there is nonono generic connection.
In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility between macrorealism and quantum mechanics. In this paper, we point out that, based on the same assumptions of macrorealism that are used to derive Leggett-Garg inequalities (LGIs), there is a scope of formulating another class of inequalities different from the standard LGIs. By considering the three-time measurement scenario in a dichotomic system, we first propose an interesting variant of the standard LGIs and show that its quantum violation is larger than the standard LGIs. By extending this formulation to the nnn-time measurement scenario, we found that the quantum violations of variants of the LGIs for a qubit system increase with nnn, and, for a sufficiently large nnn, an algebraic maximum can be reached. We then compare the violations of the standard and variants of the LGIs in an unsharp measurement scenario and show that, for any arbitrary nnn, the violation of the latter is more robust to unsharpness than the former. Furthermore, we examine the relation between the quantum violations of the variants of the LGIs and an another formulation macrorealism, known as no-signaling in time conditions.