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arXiv:2104.04398 · Physical Review A

Leggett-Garg inequalities and decays of unstable systems

Francesco Giacosa, Giuseppe Pagliara

Published Nov 29, 2021• 4 citations
arXivHTMLPDF

Abstract

We apply the Leggett-Garg inequalities (LGIs) to the cases of classical and quantum unstable systems. For classical systems the two assumptions of macroscopic realism and noninvasive measurements imply that the three-measurement string ${K}_{3}$ is identically equal to 1. Also, for quantum-mechanical systems, for which the two assumptions are, in general, not valid, we find that ${K}_{3}=1$ for purely exponential decays (${K}_{3}\ensuremath{\le}1$ is the general LGI). On the other hand, the necessary deviations from the exponential decay law at short and long times predicted by quantum mechanics lead to values of ${K}_{3}\ensuremath{\ne}1$. Moreover, a strict violation ${K}_{3}>1$ typically occurs at short times. Thus, we conclude that experiments in which such deviations from the exponential decay law have been observed should also have in their data violations of the LGIs.

DOI
10.1103/physreva.104.052225

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We apply the Leggett-Garg inequalities (LGIs) to the cases of classical and quantum unstable systems. For classical systems the two assumptions of macroscopic realism and noninvasive measurements imply that the three-measurement string K3{K}_{3}K3​ is identically equal to 1. Also, for quantum-mechanical systems, for which the two assumptions are, in general, not valid, we find that K3=1{K}_{3}=1K3​=1 for purely exponential decays (K3\ensuremath≤1{K}_{3}\ensuremath{\le}1K3​\ensuremath≤1 is the general LGI). On the other hand, the necessary deviations from the exponential decay law at short and long times predicted by quantum mechanics lead to values of K3\ensuremath≠1{K}_{3}\ensuremath{\ne}1K3​\ensuremath=1. Moreover, a strict violation K3>1{K}_{3}>1K3​>1 typically occurs at short times. Thus, we conclude that experiments in which such deviations from the exponential decay law have been observed should also have in their data violations of the LGIs.
Leggett-Garg inequality (LGI) is used to detect whether the macroscopic superposition state exists. Classical systems obey the LGI, but quantum systems may violate it. In the multilevel system, different state update rules determine different upper bounds of LGI. In this work, we experimentally test the LGI in a three-level trapped-ion system under the L\"uders and the von Neumann state update rules, respectively. The maximum observed value of the Leggett-Garg correlator under the von Neumann state update rule is ${K}_{3}=1.739\ifmmode\pm\else\textpm\fi{}0.014$, which demonstrates a violation of the L\"uders bound by 17 standard deviations and is by far the most significant violation under the specific model. The method used in our experiment could also be used in other multilevel experiments in the trapped-ion system.
As is well known, the macroscopic realism and the noninvasive measurability together lead to Leggett—Garg inequalities violated by quantum mechanics. We consider tests of the Leggett—Garg type with use of the q-entropies. For all q ≥ 1, quantum mechanics predicts violations of an entire family of q-entropic inequalities of the Leggett—Garg type. Violations are exemplified with a quantum spin-s system. In general, entropic Leggett—Garg inequalities give only necessary conditions that some probabilistic model is compatible with the macrorealism in the broader sense. The presented q-entropic inequalities allow to widen a class of situations, in which an incompatibility with the macrorealism can be tested. In the considered example, both the strength and range of violations are somehow improved by varying q. We also examine q-entropic inequalities of the Leggett—Garg type in the case of detection inefficiencies, when the no-click event may occur in each observation. With the use of the q-entropic inequalities, the required amount of efficiency may be reduced.
Leggett-Garg inequalities are tests of macroscopic realism that can be violated by quantum mechanics. In this letter, we realise photonic Leggett-Garg tests on a three-level system and implement measurements that admit three distinct measurement outcomes, rather than the usual two. In this way we obtain violations of three- and four-time Leggett-Garg inequalities that are significantly in excess of those obtainable in standard Leggett-Garg tests. We also report violations the quantum-witness equality up to the maximum permitted for a three-outcome measurement. Our results highlight differences between spatial and temporal correlations in quantum mechanics.
Leggett-Garg inequalities are tests of macroscopic realism that can be violated by quantum mechanics. In this letter, we realise photonic Leggett-Garg tests on a three-level system and implement measurements that admit three distinct measurement outcomes, rather than the usual two. In this way we obtain violations of three- and four-time Leggett-Garg inequalities that are significantly in excess of those obtainable in standard Leggett-Garg tests. We also report violations the quantum-witness equality up to the maximum permitted for a three-outcome measurement. Our results highlight differences between spatial and temporal correlations in quantum mechanics.
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.
Leggett-Garg inequalities (LGIs) have been proposed in order to assess how far the predictions of quantum mechanics defy "macroscopic realism." With LGIs, correlations of measurements performed on a single system at different times are described. We report on an experiment that demonstrates the violation of an LGI with neutrons. The final measured value of the Leggett-Garg correlator K=1.120±0.007(stat)±0.019(sys), obtained in a neutron interferometric experiment, is clearly above the limit K=1 predicted by macrorealistic theories. The experimental results are analyzed within the framework of dynamical theory of neutron diffraction, evidently reproducing the obtained values.