Abstract This study investigates the emergence of macroscopic classical behavior from quantum foundations via the entropic Leggett--Garg inequality. We introduce a geometric framework for deriving entropic Leggett--Garg inequalities with higher-order temporal correlations and demonstrate their advantages over conventional formulations. Numerical analyses show that entropic Leggett--Garg inequalities offer a robust and complementary criterion to standard approaches, providing a transparent information theoretic interpretation that facilitates the characterization of coherent quantum processes. By applying the WKB approximation, we prove that violations for maximally mixed states remain bounded by a constant in the macroscopic limit, indicating that macrorealism dominates in generic parameter regimes. We further explain previously reported maximal violations at specific parameter regimes as a consequence of the breakdown of the WKB approximation. Our findings indicate that quantum and classical descriptions remain macroscopically incompatible, while violations persist only in fine-tuned regimes, clarifying the conditions for detecting macroscopic quantum phenomena.
Abstract This study investigates the emergence of macroscopic classical behavior from quantum foundations via the entropic Leggett--Garg inequality. We introduce a geometric framework for deriving entropic Leggett--Garg inequalities with higher-order temporal correlations and demonstrate their advantages over conventional formulations. Numerical analyses show that entropic Leggett--Garg inequalities offer a robust and complementary criterion to standard approaches, providing a transparent information theoretic interpretation that facilitates the characterization of coherent quantum processes. By applying the WKB approximation, we prove that violations for maximally mixed states remain bounded by a constant in the macroscopic limit, indicating that macrorealism dominates in generic parameter regimes. We further explain previously reported maximal violations at specific parameter regimes as a consequence of the breakdown of the WKB approximation. Our findings indicate that quantum and classical descriptions remain macroscopically incompatible, while violations persist only in fine-tuned regimes, clarifying the conditions for detecting macroscopic quantum phenomena.
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.
We formulate entropic Leggett-Garg inequalities, which place constraints on the statistical outcomes of temporal correlations of observables. The information theoretic inequalities are satisfied if macrorealism holds. We show that the quantum statistics underlying correlations between time-separated spin component of a quantum rotor mimics that of spin correlations in two spatially separated spin-s particles sharing a state of zero total spin. This brings forth the violation of the entropic Leggett-Garg inequality by a rotating quantum spin-s system in a similar manner as does the entropic Bell inequality [S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 61, 662 (1988)] by a pair of spin-s particles forming a composite spin singlet state.
For multilevel spin systems, robustness of the quantum mechanical (QM) violation of macrorealism (MR) with respect to coarse-grained measurements is investigated using three different necessary conditions of MR, namely, the Leggett-Garg inequality (LGI), Wigner's form of the Leggett-Garg inequality (WLGI), and the condition of no-signaling in time (NSIT). It is shown that for dichotomic sharp measurements, in the asymptotic limit of spin, the algebraic maxima of the QM violations of all these three necessary conditions of MR are attained. Importantly, the QM violations of all these persist in that limit even for arbitrary unsharp measurements, i.e., for any nonzero value of the sharpness parameter characterizing the degree of fuzziness of the relevant measurements. We also find that, when different measurement outcomes are clubbed into two groups for the sake of dichotomizing the outcomes, the asymmetry or symmetry in the number of outcomes in the two groups, signifying the degree of coarse graining of measurements, has a crucial role in discerning quantum violation of MR. The results clearly demonstrate that classicality does not emerge in the asymptotic limit of spin, whatever be the unsharpness and degree of coarse graining of the measurements.
While quantum theory has been tested to an incredible degree on microscopic scales, quantum effects are seldom observed in our everyday macroscopic world. The curious results of applying quantum mechanics to macroscopic objects are perhaps best illustrated by Erwin Schrödinger's famous thought experiment, where a cat can be put into a superposition state of being both dead and alive. Obviously, these quantum predictions are in stark contradiction to our common experience. Even with plenty of theoretical explanations put forward to explain this discrepancy, a large number of questions about the frontier between the quantum and the classical world remain unanswered. To distinguish between classical and quantum behavior, two fundamental concepts inherent to classical physics have been established over the years: The world view of local realism limits the power of classical experiments to establish correlations over space, while the world view of macroscopic realism (or macrorealism) restricts temporal correlations. Necessary conditions for both world views have been formulated in the form of Bell and Leggett-Garg inequalities, and Bell inequalities have been shown to be violated by quantum mechanics through increasingly conclusive experiments. Furthermore, many challenging steps towards convincing violations of macrorealism have been taken in a number of recent experiments. In the first part of this thesis, conditions for macrorealism are analyzed in detail. Two necessary conditions for macrorealism, the original Leggett-Garg inequality and the recently proposed no-signaling in time condition, are presented. It is then shown that a combination of no-signaling in time conditions is not only necessary but also sufficient for the existence of a macrorealistic description. Finally, an operational formulation of no-signaling in time, in terms of positive-operator valued measurements and Hamiltonians, is derived. In the next part, we argue that these results lead to a suitable definition of classical behavior. In particular, we provide a formalism to judge the classicality of measurements and time evolutions. We then proceed to apply it to a number of exemplary measurement operators and Hamiltonians. Finally, we argue for the importance of spontaneously realized Hamiltonians in our intuition of classical behavior. Next, differences between local realism and macrorealism are analyzed. For this purpose, the probability polytopes for spatially and temporally separated experiments are compared, and a fundamental difference in the power of quantum mechanics to build both types of correlations is discovered. This result shows that Fine's theorem, which states that a set of Bell inequalities is necessary and sufficient for local realism, is not transferable to macrorealism. Thus, (Leggett-Garg) inequalities are in principle not well-suited for tests of macrorealism, as they can never form a necessary and sufficient condition, and unnecessarily restrict the violating parameter space. No-signaling in time is both better suited and more strongly motivated from the underlying physical theory. In the final part of this thesis, a concrete experimental setup for implementing quantum experiments with macroscopic objects is proposed. It consists of a superconducting micro-sphere in the Meißner state, which is levitated by magnetic fields. Through its expelled magnetic field, the sphere's center-of-mass motion couples to a superconducting quantum circuit. Properly tuned, ground state cooling can be realized, since the sphere's motion is extremely well isolated from the surrounding environment. This setup therefore is a promising candidate for the observation of quantum effects in macroscopic systems.
Leggett and Garg defined macrorealism by positing that a macroscopic system will exist in a well-defined state at all times, and that this state can be measured without disturbing it. They also formulated a class of inequalities which hold under macrorealism but can be violated by quantum mechanics. The degree to which quantum systems can violate these inequalities is restricted by the temporal Tsirelson bounds. Recently variants of Leggett-Garg inequalities which can be violated to the algebraic maximum 3 for a sufficient large number of measurement times have been proposed. In this work we report an experimental test of quantum violation of the variants of Leggett-Garg inequalities beyond the temporal Tsirelson bounds, up to $2.248\ifmmode\pm\else\textpm\fi{}0.039$ in five-time measurement scenarios for a photonic qubit. Our results clearly show the advantages of the modified measurement scenarios.
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 no 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 no generic connection.