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Precision enhancement via symmetry breaking in quantum Markov process

Yoshihiko Hasegawa

Published Jan 18, 2021• 1 citation
OpenAlex

Abstract

Symmetry breaking with respect to the time reversal characterizes entropy production. This viewpoint induces us to regard thermodynamic uncertainty relation (TUR), stating that higher precision can be achieved at the cost of higher entropy production, as a relation between precision and the extent of symmetry breaking. In this Letter, we formalize a TUR as symmetry breaking in quantum Markov processes, where the extent of symmetry breaking constitutes the thermodynamic cost. We derive a TUR for quantum continuous measurement, which holds for any conceivable continuous measurement. We demonstrate that counting- and current-type TURs can be obtained as particular cases of the obtained bound.

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Symmetry breaking with respect to the time reversal characterizes entropy production. This viewpoint induces us to regard thermodynamic uncertainty relation (TUR), stating that higher precision can be achieved at the cost of higher entropy production, as a relation between precision and the extent of symmetry breaking. In this Letter, we formalize a TUR as symmetry breaking in quantum Markov processes, where the extent of symmetry breaking constitutes the thermodynamic cost. We derive a TUR for quantum continuous measurement, which holds for any conceivable continuous measurement. We demonstrate that counting- and current-type TURs can be obtained as particular cases of the obtained bound.
Entropy production characterizes irreversibility. This viewpoint induces us to regard thermodynamic uncertainty relation, stating that higher precision can be achieved at the cost of higher entropy production, as a relation between precision and the extent of irreversibility. In this Letter, we show that the precision of arbitrary Hermitian observables has a lower bound comprising the Loschmidt echo, which represents the irreversibility of quantum dynamics. Considering the continuous measurement in quantum Markov processes, the obtained relation can be viewed as a lower bound for the precision of counting observables, which is a quantum analog of thermodynamic uncertainty relation. Moreover, applying the obtained relation to the information scrambling, we obtain an upper bound for the out-of-time-order correlator.
We derive a thermodynamic uncertainty relation (TUR) under quantum continuous measurement and feedback control. By incorporating the quantum-classical-transfer entropy, which quantifies the information gained by continuous measurement, we show that the precision of currents is constrained by information-thermodynamic costs such as the entropy production and information gain. Our result shows that information gain has the potential to enhance the precision of currents beyond the bounds set by the conventional TUR. We illustrate the bound with a driven two-level system under continuous measurement and feedback, demonstrating that feedback achieves higher precision of currents while suppressing the entropy production.
Entropy production characterizes irreversibility. This viewpoint allows us to consider the thermodynamic uncertainty relation, which states that a higher precision can be achieved at the cost of higher entropy production, as a relation between precision and irreversibility. Considering the original and perturbed dynamics, we show that the precision of an arbitrary counting observable in continuous measurement of quantum Markov processes is bounded from below by the Loschmidt echo between the two dynamics, representing the irreversibility of quantum dynamics. When considering particular perturbed dynamics, our relation leads to several thermodynamic uncertainty relations, indicating that our relation provides a unified perspective on classical and quantum thermodynamic uncertainty relations.
The thermodynamic uncertainty relation (TUR) is a fundamental principle in nonequilibrium thermodynamics that relates entropy production to fluctuations in a system, establishing a trade-off between the precision of an observable and the thermodynamic cost. Investigating TUR violations challenges classical thermodynamic limits, offering the potential for improved precision-entropy trade-offs, which is crucial for enhancing performance and optimization in quantum technologies. In this work, we investigate the thermodynamic uncertainty relation within a quantum collisional model, which offers the advantage of discretizing interactions into successive collisions with auxiliaries, allowing for precise tracking of dynamics and the incorporation of memory effects and non-Markovian behavior. We consider three types of dynamics in the collisional model: one is Markovian evolution, achieved by taking the continuous-time limit and imposing the stability condition, while the other two are non-Markovian dynamics—one arising from increasing the collision time between the system and the auxiliaries, and the other from incorporating interactions between the auxiliaries. For the Markovian dynamics, we examine the classical and quantum TUR bounds in the nonequilibrium steady-state regime, and also the finite-time TUR bound. We identify two distinct regimes of classical TUR violation: in some cases, the maximum violation occurs in the steady state, while in others, it is necessarily transient—appearing only at early times and vanishing with further evolution. For the two non-Markovian approaches, we find that both the degree and type of non-Markovianity crucially affect TUR violations. The second approach shows more pronounced violations during transient times, while the first approach has much stronger violations in the steady-state regime for a certain parameter window.
Fluctuations are intrinsic to microscopic systems and impose fundamental limits on nonequilibrium precision, as captured by the thermodynamic uncertainty relation (TUR), which links current fluctuations to entropy production. While feedback control is expected to further suppress fluctuations, its role within the TUR framework has remained unclear, particularly in quantum systems where control is inherently information-driven. In this Letter, we consider open quantum systems weakly coupled to a thermal environment, in which quantum jumps are continuously monitored, and Markovian feedback is applied. Using quantum mutual information to quantify the information contribution induced by feedback, we derive a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information. Our results uncover how feedback control suppresses fluctuations together with thermodynamic cost and establish a fundamental precision bound imposed by information-based control. As an application, we analyze a quantum clock model and demonstrate that the clock precision can be enhanced by feedback control in the presence of a single thermal reservoir.
In quantum thermodynamics, entropy production is usually defined in terms of the quantum relative entropy between two states. We derive a lower bound for the quantum entropy production in terms of the mean and variance of quantum observables, which we refer to as a thermodynamic uncertainty relation (TUR) for the entropy production. In the absence of coherence between the states, our result reproduces classic TURs in stochastic thermodynamics. For the derivation of the TUR, we introduce a lower bound for a quantum generalization of the χ^{2} divergence between two states and discuss its implications for stochastic and quantum thermodynamics, as well as the limiting case where it reproduces the quantum Cramér-Rao inequality.