Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.15717 · Annals of Physics

Quantum coherence and Leggett-Garg inequality

A. Jalal, S. M. Fazeli, M. M. Ettefaghi

quant-phSubmitted Jun 14, 2026• Announced Jun 16, 2026• 0 citations
arXivHTMLPDF

Abstract

In this paper, we attempt to establish the relationship between quantum coherence and the violation of the Leggett-Garg inequality. In particular, employing the Lindblad equation, we obtain the pseudo-density matrix for a damping system to study the effect of environment interaction on the violation of this inequality in a two-state quantum system. It is shown that the violation of the Leggett-Garg inequality can be observed as long as temporal evolution does not induce decoherence. This statement is independent of the initial state of the system. Furthermore, similar to the Horodecki criterion for the CHSH inequality (R. Horodecki et al. Phys. Lett. {\bf A200}, 340), we study necessary and sufficient conditions for violating the Leggett-Garg inequality. Hereby, under the circumstance that the inequality violation occurs, an upper bound for the time interval between consecutive measurements with respect to the time scale of interaction with the environment (the relaxation time) is obtained.

Comments
23 pages, 4 figures. Accepted for publication in Annals of Physics

Related papers

Investigating Leggett-Garg inequality for a two level system under decoherence in a non-Markovian dephasing environment

Po‐Wen Chen, Md. Manirul Ali

Leggett-Garg inequalities (LGI) test the correlations of a single system measured at different times. Violation of LGI implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. We investigate the violation of the Leggett-Garg inequality for a two level system under decoherence in a non-Markovian dephasing environment. We discuss the non-Markovian dynamics of the violation of LGI at zero temperature and also at finite temperature for different structured environments. An enhanced quantum coherence is shown through the violation of Leggett-Garg inequality in the strong non-Markovian regime of the environment.

Scientific Reports•Aug 22, 2014•21 citations

Leggett-Garg Inequality for a Two-Level System under Decoherence

Nasim Shahmansoori, Afshin Shafiee

We consider a macroscopic quantum system subjected to an asymmetric double-well potential in a harmonic environment. By using a time-dependent approach, we calculate tunnelling probabilities for the system which contain oscillation effects. To show how one can decide between quantum mechanics and the implications of macrorealism assumptions, a given form of Leggett-Garg inequality is considered. The violation of this inequality occurs for a broader range of the system-environment interactions, compared to previous results obtained for two-level systems. Assuming that the coupling strength between the system and the environment can be controlled with time, one can see the violation even for strong decoherence effects. We also investigate the variation of the tilt/tunnelling parameters on the violation of Leggett-Garg inequality.

arXiv:1605.00394•Open Systems & Information Dynamics•Mar 1, 2018•1 citation

Leggett-Garg inequalities and temporal correlations for a qubit under PT -symmetric dynamics

H. S. Karthik, Akshata Shenoy H., A. R. Usha Devi

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 $3/2$ in the time evolution of $N>2$ 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 $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian indeed violates the three-term Leggett-Garg inequality beyond the TTB and up to its algebraic maximum value 3.

arXiv:1907.02879•Physical Review A•Mar 17, 2021•13 citations

Leggett–Garg inequality in Markovian quantum dynamics: role of temporal sequencing of coupling to bath

Sayan Ghosh, Anant Vijay Varma, Sourin Das

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.

arXiv:2110.10696•Journal of Physics A Mathematical and Theoretical•Mar 30, 2023•3 citations

The Leggett-Garg inequality of a dissipative cavity mode coupled to a zero-temperature environment

Hiroo Azuma, Masashi Ban

In the present paper, we investigate the Leggett-Garg inequality of the cavity mode weakly coupled to a zero-temperature environment. We assume that the boson system undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of violation of the inequality for both cases where systems are prepared initially in the coherent state $|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. We choose the displaced parity operators for observing the cavity mode at three equally spaced measurement times (spacing $\tau$) in the derivation of the inequality. These operators are characterized by a complex number $\beta$. We look for the optimum parameter $\beta$ that lets the upper bound of the inequality be maximum by numerical calculations. We compare the maximized upper bounds for both the initial states, the coherent and cat states. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the Leggett-Garg inequality if its time difference $\tau$ is in a specified range. Moreover, as we let the time difference $\tau$ approach zero, the optimized parameter $\beta$ diverges, that is $|\beta|\to\infty$, and the Leggett-Garg inequality reveals intense singularity.

•Jan 18, 2021•0 citations

Leggett-Garg inequality for qubits coupled to thermal environment

Marcin Łobejko, J. Łuczka, Jerzy Dajka

We study violation of the Leggett-Garg inequality for the correlator ${K}_{3}$ in two cases: (i) for a qubit weakly coupled to a thermal decohering (dissipative and/or dephasing) environment and (ii) for a pair of qubits with only one of them coupled to the environment. In the case of a single qubit, we identify conditions depending both on initial state preparation and a measurement protocol when violation of the ${K}_{3}$ inequality becomes dependent only on dephasing and independent of dissipation and temperature. In the case of two qubits, we have found the temperature-independent regime as well as the regime where all three components of the decoherence can control violation of the Leggett-Garg inequality.

Physical Review A•Apr 13, 2015•19 citations
In this paper, we attempt to establish the relationship between quantum coherence and the violation of the Leggett-Garg inequality. In particular, employing the Lindblad equation, we obtain the pseudo-density matrix for a damping system to study the effect of environment interaction on the violation of this inequality in a two-state quantum system. It is shown that the violation of the Leggett-Garg inequality can be observed as long as temporal evolution does not induce decoherence. This statement is independent of the initial state of the system. Furthermore, similar to the Horodecki criterion for the CHSH inequality (R. Horodecki et al. Phys. Lett. {\bf A200}, 340), we study necessary and sufficient conditions for violating the Leggett-Garg inequality. Hereby, under the circumstance that the inequality violation occurs, an upper bound for the time interval between consecutive measurements with respect to the time scale of interaction with the environment (the relaxation time) is obtained.
Leggett-Garg inequalities (LGI) test the correlations of a single system measured at different times. Violation of LGI implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. We investigate the violation of the Leggett-Garg inequality for a two level system under decoherence in a non-Markovian dephasing environment. We discuss the non-Markovian dynamics of the violation of LGI at zero temperature and also at finite temperature for different structured environments. An enhanced quantum coherence is shown through the violation of Leggett-Garg inequality in the strong non-Markovian regime of the environment.
We consider a macroscopic quantum system subjected to an asymmetric double-well potential in a harmonic environment. By using a time-dependent approach, we calculate tunnelling probabilities for the system which contain oscillation effects. To show how one can decide between quantum mechanics and the implications of macrorealism assumptions, a given form of Leggett-Garg inequality is considered. The violation of this inequality occurs for a broader range of the system-environment interactions, compared to previous results obtained for two-level systems. Assuming that the coupling strength between the system and the environment can be controlled with time, one can see the violation even for strong decoherence effects. We also investigate the variation of the tilt/tunnelling parameters on the violation of Leggett-Garg inequality.
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.
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.
In the present paper, we investigate the Leggett-Garg inequality of the cavity mode weakly coupled to a zero-temperature environment. We assume that the boson system undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of violation of the inequality for both cases where systems are prepared initially in the coherent state ∣α⟩|\alpha\rangle∣α⟩ and the cat state (∣α⟩+∣−α⟩)(|\alpha\rangle+|-\alpha\rangle)(∣α⟩+∣−α⟩). We choose the displaced parity operators for observing the cavity mode at three equally spaced measurement times (spacing τ\tauτ) in the derivation of the inequality. These operators are characterized by a complex number β\betaβ. We look for the optimum parameter β\betaβ that lets the upper bound of the inequality be maximum by numerical calculations. We compare the maximized upper bounds for both the initial states, the coherent and cat states. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the Leggett-Garg inequality if its time difference τ\tauτ is in a specified range. Moreover, as we let the time difference τ\tauτ approach zero, the optimized parameter β\betaβ diverges, that is ∣β∣→∞|\beta|\to\infty∣β∣→∞, and the Leggett-Garg inequality reveals intense singularity.
We study violation of the Leggett-Garg inequality for the correlator K3{K}_{3}K3​ in two cases: (i) for a qubit weakly coupled to a thermal decohering (dissipative and/or dephasing) environment and (ii) for a pair of qubits with only one of them coupled to the environment. In the case of a single qubit, we identify conditions depending both on initial state preparation and a measurement protocol when violation of the K3{K}_{3}K3​ inequality becomes dependent only on dephasing and independent of dissipation and temperature. In the case of two qubits, we have found the temperature-independent regime as well as the regime where all three components of the decoherence can control violation of the Leggett-Garg inequality.