We show that the quantum bound for temporal correlations in a Leggett-Garg test, analogous to the Tsirelson bound for spatial correlations in a Bell test, strongly depends on the number of levels N that can be accessed by the measurement apparatus via projective measurements. We provide exact bounds for small N that exceed the known bound for the Leggett-Garg inequality, and we show that in the limit N→∞ the Leggett-Garg inequality can be violated up to its algebraic maximum.

arXiv:1309.3678Physical Review LettersJul 29, 2014138 citations

arXiv:2005.04241 · Physical Review Research

Ticking-clock performance enhanced by nonclassical temporal correlations

Costantino Budroni, Giuseppe Vitagliano, Mischa P. Woods

Published Jul 15, 2021• 12 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.3.033051

Related papers

Least Variable Quantum Counting Processes

Bita Olamaei, Florian Meier, Costantino Budroni +2 more

Counting processes provide a fundamental description of stochastic events ranging from photon detection to clock ticks. A central question is how accurately such events can be timed when only finite memory resources are available. Here, we investigate this problem within a general framework of finite-dimensional classical and quantum counting processes. We derive a rigorous finite-memory variance bound obeyed by every classical $d$ -state counting process, which is tight and saturated by a discrete Erlang-type ladder process. Through numerical optimization, we identify quantum counting processes that violate this classical bound, achieving smaller first-tick fluctuations than any classical process with the same memory size and mean tick time. For the qubit case, we further derive an analytical large-mean bound within a single-Kraus no-tick family, showing that the quantum advantage persists asymptotically within this class. The optimized quantum processes exhibit coherent conditioned dynamics and approach a continuous-time quantum-jump description as the mean increases. Our results establish a finite-memory quantum advantage in temporal precision and connect discrete-time counting processes with continuous-time quantum timekeeping.Counting processes provide a fundamental description of stochastic events ranging from photon detection to clock ticks. A central question is how accurately such events can be timed when only finite memory resources are available. Here, we investigate this problem within a general framework of finite-dimensional classical and quantum counting processes. We derive a rigorous finite-memory variance bound obeyed by every classical $d$-state counting process, which is tight and saturated by a discrete Erlang-type ladder process. Through numerical optimization, we identify quantum counting processes that violate this classical bound, achieving smaller first-tick fluctuations than any classical process with the same memory size and mean tick time. For the qubit case, we further derive an analytical large-mean bound within a single-Kraus no-tick family, showing that the quantum advantage persists asymptotically within this class. The optimized quantum processes exhibit coherent conditioned dynamics and approach a continuous-time quantum-jump description as the mean increases. Our results establish a finite-memory quantum advantage in temporal precision and connect discrete-time counting processes with continuous-time quantum timekeeping.

quant-pharXiv:2608.26240Aug 28, 20260 citations

Ticking-clock performance enhanced by nonclassical temporal correlations

Costantino Budroni, Giuseppe Vitagliano, Mischa P. Woods

We investigate the role of nonclassical temporal correlations in enhancing the performance of ticking clocks in a discrete-time scenario. We show that the problem of optimal models for ticking clocks is related to the violation of Leggett-Garg-type temporal inequalities formulated in terms of, possibly invasive, sequential measurements, but on a system with a bounded memory capacity. Ticking clocks inspire the derivation of a family of temporal inequalities showing a gap between classical and quantum correlations, despite involving no input. We show that quantum ticking-clock models achieving accuracy beyond the classical bound are also those violating Leggett-Garg-type temporal inequalities for finite sequences and we investigate their continuous-time limit. Interestingly, we show that optimal classical clock models in the discrete-time scenario do not have a well-defined continuous-time limit, a feature that is absent in quantum models.We investigate the role of nonclassical temporal correlations in enhancing the performance of ticking clocks in a discrete-time scenario. We show that the problem of optimal models for ticking clocks is related to the violation of Leggett-Garg-type temporal inequalities formulated in terms of, possibly invasive, sequential measurements, but on a system with a bounded memory capacity. Ticking clocks inspire the derivation of a family of temporal inequalities showing a gap between classical and quantum correlations, despite involving no input. We show that quantum ticking-clock models achieving accuracy beyond the classical bound are also those violating Leggett-Garg-type temporal inequalities for finite sequences and we investigate their continuous-time limit. Interestingly, we show that optimal classical clock models in the discrete-time scenario do not have a well-defined continuous-time limit, a feature that is absent in quantum models.

Jan 1, 20217 citations

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.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:1907.02879Physical Review AMar 17, 202113 citations

Canonical Leggett-Garg inequality: Nonclassicality of temporal quantum correlations under energy constraint

Titas Chanda, Tamoghna Das, Shiladitya Mal +2 more

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.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.

arXiv:1805.04665Physical Review AAug 29, 201815 citations

Memory cost of temporal correlations

Costantino Budroni, Gabriel Fagundes, Matthias Kleinmann

Abstract A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.Abstract A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.

arXiv:1902.06517New Journal of PhysicsAug 19, 201925 citations

Temporal Quantum Correlations and Leggett-Garg Inequalities in Multilevel Systems

Costantino Budroni, Clive Emary

arXiv:1309.3678Physical Review LettersJul 29, 2014138 citations

d

-state counting process, which is tight and saturated by a discrete Erlang-type ladder process. Through numerical optimization, we identify quantum counting processes that violate this classical bound, achieving smaller first-tick fluctuations than any classical process with the same memory size and mean tick time. For the qubit case, we further derive an analytical large-mean bound within a single-Kraus no-tick family, showing that the quantum advantage persists asymptotically within this class. The optimized quantum processes exhibit coherent conditioned dynamics and approach a continuous-time quantum-jump description as the mean increases. Our results establish a finite-memory quantum advantage in temporal precision and connect discrete-time counting processes with continuous-time quantum timekeeping.

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.