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Scientific Reports

Legget-Garg inequality for a quantum model of ion channels

Paulina Trybek, Łukasz Machura, Jerzy Dajka

Published Jun 18, 2026• 0 citations
Publisher

Abstract

Temporal quantum correlations, revealed by violations of the Leggett-Garg inequalities (LGIs), offer a powerful framework to probe nonclassical dynamics in conceptual models, including those inspired by the living systems. Here, we examine a three-state representation of ionic transport through the selectivity filter, previously presented by Seifi et al. Using this model, we systematically investigate how different scenarios, such as initial state preparation, the specification of measurement basis, hopping dynamics, and environmental fluctuations, affect the emergence of temporal nonclassicality. The choice of measurement basis is found to play a decisive role in shaping temporal quantum correlations. Our analysis reveals that the strongest LGI violations occur for intermediate model configurations, which represent a short-lived state where a potassium ion is poised for release while still coupled to surrounding water sites, whereas entry and exit configurations exhibit weaker quantum signatures. Static fluctuations of the hopping parameters confine LGI violations to shorter time windows, while dynamic decoherence suppresses them irrespective of the hopping amplitude. Overall, our findings demonstrate that LGI-based analyses can identify stages where temporal quantum correlations are most significant, highlighting the intermediate knock-on configuration as a promising target for future studies of intrinsically quantum effects in ion-conduction phenomena.

DOI
10.1038/s41598-026-57853-z

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Temporal quantum correlations, revealed by violations of the Leggett-Garg inequalities (LGIs), offer a powerful framework to probe nonclassical dynamics in conceptual models, including those inspired by the living systems. Here, we examine a three-state representation of ionic transport through the selectivity filter, previously presented by Seifi et al. Using this model, we systematically investigate how different scenarios, such as initial state preparation, the specification of measurement basis, hopping dynamics, and environmental fluctuations, affect the emergence of temporal nonclassicality. The choice of measurement basis is found to play a decisive role in shaping temporal quantum correlations. Our analysis reveals that the strongest LGI violations occur for intermediate model configurations, which represent a short-lived state where a potassium ion is poised for release while still coupled to surrounding water sites, whereas entry and exit configurations exhibit weaker quantum signatures. Static fluctuations of the hopping parameters confine LGI violations to shorter time windows, while dynamic decoherence suppresses them irrespective of the hopping amplitude. Overall, our findings demonstrate that LGI-based analyses can identify stages where temporal quantum correlations are most significant, highlighting the intermediate knock-on configuration as a promising target for future studies of intrinsically quantum effects in ion-conduction phenomena.
Nonclassicality of temporal correlations pertaining to noncommutative sequential measurements is defined through the violation of macrorealistic inequalities, known as Leggett-Garg inequalities (LGI). We investigate the energy cost of the process associated with the Leggett-Garg test in the context of noiseless and Markovian noise for arbitrary initial states. We prove that in noiseless and in certain noisy scenarios, the maximal violations of LGI under the energy constraint occurs when the average energy of the process is equal to the negative of the energy of the initial state. Such a dependence of LGI on the choice of the initial state is not seen in the unconstrained case. Moreover, we find that in the presence of a moderate amount of Markovian noise, the amount of violation of LGI remains almost unaltered with a suitable choice of the evolution and dephasing operators in the neighborhood of the maximal violation line, thereby showing the robustness of temporal correlations under environmental effects.
The Leggett-Garg inequality (LGI) delineates a boundary between quantum and classical systems. While the temporal quantum correlations represented by LGIs have been extensively investigated in the Hermitian domain, the exploration of LGIs under non-Hermitian conditions remains scarce. Theoretical conjectures posit that nonunitary dynamics could transcend the boundary of LGIs set by Hermitian quantum mechanics, yet its empirical validation is lacking. Here, we demonstrate the enhanced violation of LGIs in a non-Hermitian system of a parity-time (PT\mathcal{PT}PT)-symmetric trapped-ion qubit. The upper bounds of both the third-order parameter K3{K}_{3}K3​ and fourth-order parameter K4{K}_{4}K4​ increase with dissipation and can reach the maximum value when the system approaches the exceptional point, where ${K}_{3}={C}_{21}+{C}_{32}\ensuremath{-}{C}_{31}$ is comprising three two-time correlations and ${K}_{4}={C}_{21}+{C}_{32}+{C}_{43}\ensuremath{-}{C}_{41}$ is comprising four correlations. We also find that the lower bounds of K3{K}_{3}K3​ and K4{K}_{4}K4​ exhibit distinctive behaviors, where the lower bound of K3{K}_{3}K3​ remains constant, but that of K4{K}_{4}K4​ depends on the target states and measurement operators. These findings reveal a pronounced correlation between the dissipative nature of a quantum system and its temporal correlations.
Abstract We study Leggett–Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We first find the analytic expression of LG parameter K 3 (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: t 1 to t 2 and t 2 to t 3. We then show that the maximum violation of LGI for all possible unital maps can never exceed well known Lüders bound of K 3 L u ¨ d e r s = 3 / 2 over the full parameter space. We further show that if the map for the time interval t 1 to t 2 is non-unitary unital then irrespective of the choice of the map for interval t 2 to t 3 we can never reach Lüders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from t 1 to t 2, then depending on the degree of decoherence induced by the unital map for the interval t 2 to t 3 we may or may not obtain Lüders bound. Specifically, we find that if the unital map for interval t 2 to t 3 leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound K 3 L u ¨ d e r s is not possible. Hence our findings not only establish a threshold for decoherence which will allow for K 3 = K 3 L u ¨ d e r s , but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Lüders bound.
Temporal Bell inequalities, or Leggett-Garg inequalities (LGIs), are studied for continuous-variable systems placed in a squeezed state. The importance of those systems lies in their broad applicability, which allows the description of many different physical settings in various branches of physics, ranging from cosmology to condensed matter physics and from optics to quantum information theory. Leggett-Garg inequality violations are explored and systematically mapped in squeezing parameter space. Configurations for which LGI violation occurs are found, but it is shown that no violation can be obtained if all squeezing angles vanish, contrary to what happens for the spatial Bell inequalities. We also assess the effect of decoherence on the detectability of such violations. Our study opens up the possibility of new experimental designs for the observation of LGI violation.
Abstract We investigate the nonclassicality of an open quantum system using Leggett–Garg inequalities (LGI) which test the correlations of a single system measured at different times. Violation of LGI implies nonclassical behavior of the open system. We investigate the violation of the Leggett–Garg inequality for a two level system (qubit) spontaneously decaying under a general non-Markovian dissipative environment. A robust nonclassical behavior is shown through the violation of LGI in the non-Markovian regime of the environment. Our results are exact as we have evaluated the two-time correlation functions exactly without using quantum regression theorem. We probe quantum signatures of the open system through the exact dynamics of Leggett–Garg inequalities for a wide range of system-environment parameters beyond Born–Markov regime.
The Leggett-Garg inequality (LGI) is a temporal analogue of Bell's inequality and provides a quantitative test of the nonclassicality of a system through its violation. We analytically investigate the violation of the LGI in the context of photon-graviton conversion in a magnetic field background, motivated by its potential applications to testing the nonclassicality of gravity. When gravitational perturbations are quantized as gravitons, the conversion of an initial single photon state gives rise to a superposition of photon and graviton states. We show that the temporal correlations obtained from successive projective measurements on the photon-graviton system violate the LGI. Observation of such a violation would provide a novel avenue for probing the quantum nature of gravity.