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arXiv:2509.03166 · Physical Review A

Assessing the dynamical assumptions in Tsirelson inequality tests of nonclassicality in harmonic oscillators

Arush Garg, J. J. Halliwell, Taejas Venkataraman

Published Apr 30, 2026• 0 citations
arXivHTMLPDF

Abstract

Macrorealism posits that a system possesses definite properties at all times and that we can discover these properties, in principle, without disturbing the system's subsequent behavior. The Leggett-Garg inequalities are derived under these assumptions and are readily violated by standard quantum mechanics, thereby providing a scheme to test whether demonstrably macroscopic systems can exhibit quantum coherence. Unfortunately, Leggett-Garg tests suffer from the difficult to avoid clumsiness loophole: the difficulty of proving that sequential measurements have not inadvertently disturbed the system. The recently uncovered Tsirelson inequality is derived from the simple dynamical assumption of uniform precession, obeyed by many classical systems, and requires only single-time measurements. However, Tsirelson inequality violations could be explained by a macrorealistic system that merely breaks the dynamical assumption, rather than genuine quantum behavior. By carrying out a quantum-mechanical analysis of the Tsirelson inequality in the harmonic oscillator, we develop a protocol to rule out this possibility by assessing generalized conditions of uniform precession. We show that various measures of uniform precession, some of which are related to Leggett-Garg quantities, are satisfied well enough that the presence of quantum-mechanical interference terms must be implied. We derive several incidental mathematical results relating to violating states of Tsirelson's inequality, concerning dwell time, crossing number, and probability currents, and also consider a group-theoretic analysis of the Tsirelson operator.

DOI
10.1103/qb27-86s1

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Macrorealism posits that a system possesses definite properties at all times and that we can discover these properties, in principle, without disturbing the system's subsequent behavior. The Leggett-Garg inequalities are derived under these assumptions and are readily violated by standard quantum mechanics, thereby providing a scheme to test whether demonstrably macroscopic systems can exhibit quantum coherence. Unfortunately, Leggett-Garg tests suffer from the difficult to avoid clumsiness loophole: the difficulty of proving that sequential measurements have not inadvertently disturbed the system. The recently uncovered Tsirelson inequality is derived from the simple dynamical assumption of uniform precession, obeyed by many classical systems, and requires only single-time measurements. However, Tsirelson inequality violations could be explained by a macrorealistic system that merely breaks the dynamical assumption, rather than genuine quantum behavior. By carrying out a quantum-mechanical analysis of the Tsirelson inequality in the harmonic oscillator, we develop a protocol to rule out this possibility by assessing generalized conditions of uniform precession. We show that various measures of uniform precession, some of which are related to Leggett-Garg quantities, are satisfied well enough that the presence of quantum-mechanical interference terms must be implied. We derive several incidental mathematical results relating to violating states of Tsirelson's inequality, concerning dwell time, crossing number, and probability currents, and also consider a group-theoretic analysis of the Tsirelson operator.
The rise of quantum information theory has lent new relevance to experimental tests for non-classicality, particularly in controversial cases such as adiabatic quantum computing superconducting circuits. The Leggett-Garg inequality is a "Bell inequality in time" designed to indicate whether a single quantum system behaves in a macrorealistic fashion. Unfortunately, a violation of the inequality can only show that the system is either (i) non-macrorealistic or (ii) macrorealistic but subjected to a measurement technique that happens to disturb the system. The "clumsiness" loophole (ii) provides reliable refuge for the stubborn macrorealist, who can invoke it to brand recent experimental and theoretical work on the Leggett-Garg test inconclusive. Here, we present a revised Leggett-Garg protocol that permits one to conclude that a system is either (i) non-macrorealistic or (ii) macrorealistic but with the property that two seemingly non-invasive measurements can somehow collude and strongly disturb the system. By providing an explicit check of the invasiveness of the measurements, the protocol replaces the clumsiness loophole with a significantly smaller "collusion" loophole. © 2011 Springer Science+Business Media, LLC.
I study several aspects of tests of macrorealism (MR), which for a given data set serves to give a quantitative signal of the presence of a specific notion of non-classical behaviour. The insufficiency of classical understanding underpins both the paradoxes of quantum mechanics, its future technological promise, and so these tests are of interest both foundationally and pragmatically. I derive generalisations of the Leggett-Garg (LG) inequalities and Fine's theorem, which together establish the necessary and sufficient conditions for macrorealism. First, I extend these conditions to tests involving an arbitrary number of measurement times. Secondly, I generalise them beyond the standard dichotomic variable, to systems described by many-valued variables. I also perform a quantum mechanical analysis examining the interplay of different conditions of MR. I then develop the theoretical framework to support tests of macrorealism in continuous variable systems, where I define variables based on coarse-grainings of position. I calculate temporal correlators for general bound systems, and analyse LG violations within the quantum harmonic oscillator (QHO), in its energy eigenstates and coherent states. I analyse the precise physical mechanisms underpinning the violations in terms of probability currents, Bohm trajectories. Staying within continuous variable systems, we outline a different approach to meeting the invasiveness requirement of LG tests. Reasoning that we may approximately non-invasively measure whether a particle crosses the axis, we measure an object which is related to the standard correlators, and derive a set of macrorealistic inequalities for these modified correlators. We demonstrate violations of these modified LG inequalities for several states within the QHO.
The Leggett-Garg (LG) inequalities were introduced to test for the possible presence of macroscopic quantum coherence. Since such effects may be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and also to more general bound systems, using a single dichotomic variable QQQ given by the sign of the oscillator position. We present a simple method to calculate the temporal correlation functions appearing in the LG inequalities for any bound system for which the eigenspectrum is (exactly or numerically) known. We apply this result to the quantum harmonic oscillator for a variety of experimentally accessible states, namely, energy eigenstates, and superpositions thereof. For the subspace of states spanned by only the ground state and first excited state, we readily find substantial regions of parameter space in which the LG inequalities at two, three, and four times can each be independently violated or satisfied. We also find that the violations persist, although are reduced, when the sign function defining QQQ is smeared to reflect experimental imprecision. For higher-energy eigenstates, we find that LG violations diminish, showing the expected classicalization. With a QQQ defined using a more general type of position coarse graining, we find two-time LG violations even in the ground state. We also show that two-time LG violations in a Gaussian state are readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the approach, we repeat much of the LG analysis for the Morse potential, finding qualitatively similar physical results.
Macrorealism formalizes the seemingly intuitive notion that, in contrast with the principles of quantum mechanics, a physical system can be in a definite state at any given time and moreover its dynamical evolution is independent of the measurements performed on it. In this study, we carry out a comparative analysis between three-time Leggett--Garg-type inequalities and the conditions of no-signaling-in-time and arrow-of-time for macrorealism within the context of meson oscillations. Our findings indicate that, under given initial conditions, no violations of Leggett--Garg inequalities are observed. However, no-signaling-in-time conditions are found to be violated, thereby revealing the impossibility of applying a macrorealistic description to the physics of meson oscillations.
Leggett-Garg (LG) tests for macrorealism were originally designed to explore quantum coherence on the macroscopic scale. Interference experiments and systems modeled by harmonic oscillators provide useful examples of situations in which macroscopicity has been approached experimentally and are readily turned into LG tests for a single dichotomic variable QQQ. Applying this approach to the double-slit experiment in which a noninvasive measurement at the slits is included, we exhibit LG violations. We find that these violations are always accompanied by destructive interference. The converse is not true in general and we find that there are nontrivial regimes in which there is destructive interference but the two-time LG inequalities are satisfied, which implies that it is in fact often possible to assign (indirectly determined) probabilities for the interferometer paths. Similar features have been observed in recent work involving a LG analysis of a Mach-Zehnder interferometer and we compare with those results. We extend the analysis to the triple-slit experiment again finding LG violations, and we also exhibit examples of some surprising relationships between LG inequalities and NSIT conditions that do not exist for dichotomic variables. For the simple harmonic oscillator, we find an analytically tractable example showing a two-time LG violation with a gaussian initial state, echoing in simpler form recent results of Bose et al. [Phys. Rev. Lett. 120, 210402 (2018)].
Macrorealistic description governing the classical realm is known to get uprooted in the quantum domain. The Leggett-Garg inequality [A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)] proposes to probe the macrorealistic limit emerging from the quantum scenario. It places a bound on the linear combinations of temporal correlations of observables measured sequentially at different time intervals. Violation of the Leggett-Garg inequality, up to the so-called temporal Tsirelson bound (TTB), has been realized in the quantum domain. It is important to understand whether quantum theory permits larger violation beyond the TTB and up to the algebraic maximum value of the Leggett-Garg inequality. To this end, it has been shown [C. Budroni and C. Emary, Phys. Rev. Lett. 113, 050401 (2014)] that a three-term Leggett-Garg inequality can be violated beyond the TTB 3/23/23/2 in the time evolution of N>2N>2N>2 level quantum systems by choosing suitable measurement schemes and state-update rules. Furthermore, it is revealed that the algebraic maximum value 3 can be realized in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Their measurement scheme rules out violation of the Leggett-Garg inequality beyond the TTB by a qubit. Being genuinely quantum in nature, we ask whether such enhanced violation of the Leggett-Garg inequality could be witnessed in a qubit itself. Here, we show that a qubit undergoing time evolution generated by a non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian indeed violates the three-term Leggett-Garg inequality beyond the TTB and up to its algebraic maximum value 3.