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arXiv:1807.05034 · Physical Review Letters

Detailed Fluctuation Relation for Arbitrary Measurement and Feedback Schemes

Patrick P. Potts, Peter Samuelsson

Published Nov 21, 2018• 40 citations
arXivHTMLPDF

Abstract

Fluctuation relations are powerful equalities that hold far from equilibrium. However, the standard approach to include measurement and feedback schemes may become inapplicable in certain situations, including continuous measurements, precise measurements of continuous variables, and feedback induced irreversibility. Here we overcome these shortcomings by providing a recipe for producing detailed fluctuation relations. Based on this recipe, we derive a fluctuation relation which holds for arbitrary measurement and feedback control. The key insight is that fluctuations inferable from the measurement outcomes may be suppressed by postselection. Our detailed fluctuation relation results in a stringent and experimentally accessible inequality on the extractable work, which is saturated when the full entropy production is inferable from the data.

DOI
10.1103/physrevlett.121.210603

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Fluctuation relations are powerful equalities that hold far from equilibrium. However, the standard approach to include measurement and feedback schemes may become inapplicable in certain situations, including continuous measurements, precise measurements of continuous variables, and feedback induced irreversibility. Here we overcome these shortcomings by providing a recipe for producing detailed fluctuation relations. Based on this recipe, we derive a fluctuation relation which holds for arbitrary measurement and feedback control. The key insight is that fluctuations inferable from the measurement outcomes may be suppressed by postselection. Our detailed fluctuation relation results in a stringent and experimentally accessible inequality on the extractable work, which is saturated when the full entropy production is inferable from the data.
Fluctuation relations are powerful equalities that hold far from equilibrium. However, the standard approach to include measurement and feedback schemes may become inapplicable in certain situations, including continuous measurements, precise measurements of continuous variables, and feedback induced irreversibility. Here we overcome these shortcomings by providing a recipe for producing detailed fluctuation relations. Based on this recipe, we derive a fluctuation relation which holds for arbitrary measurement and feedback control. The key insight is that fluctuations inferable from the measurement outcomes may be suppressed by post-selection. Our detailed fluctuation relation results in a stringent and experimentally accessible inequality on the extractable work, which is saturated when the full entropy production is inferable from the data.
Fluctuation theorems and the second law of thermodynamics are powerful relations constraining the behavior of out-of-equilibrium systems. While there exist generalizations of these relations to feedback controlled quantum systems, their applicability is limited, in particular when considering strong and continuous measurements. In this Letter, we overcome this shortcoming by deriving a novel fluctuation theorem, and the associated second law of information thermodynamics, which remain applicable in arbitrary feedback control scenarios. In our second law, the entropy production is bounded by the coarse-grained entropy production that is inferrable from the measurement outcomes, an experimentally accessible quantity that does not diverge even under strong continuous measurements. We illustrate our results by a qubit undergoing discrete and continuous measurement, where our approach provides a useful bound on the entropy production for all measurement strengths.
Abstract Here, we investigate the uncertainty of dynamical observables in classical systems manipulated by repeated measurements and feedback control; the precision should be enhanced in the presence of an external controller but limited by the amount of information obtained from the measurements. We prove that the entropy production and the information quantity constrain from below the fluctuation of arbitrary observables that are antisymmetric under time reversal. The information term is the sum of the mutual entropy production and the Kullback–Leibler divergence, which characterises the irreversibility of the measurement outcomes. The result holds for finite observation times and for both continuous- and discrete-time systems. We apply the derived relation to study the precision of a flashing Brownian ratchet.
The laws of thermodynamics are a cornerstone for describing nanoscale and open quantum systems. However, formulating these laws for systems under continuous feedback control and under experimentally relevant conditions is challenging. In this work, we lay out a formalism for the laws of thermodynamics in an open quantum system under continuous measurement and feedback described by a Quantum Fokker Planck Master Equation. We derive expressions for work, heat, and measurement-induced energy changes, and we investigate entropy production and fluctuation theorems. We illustrate our results with a continuous version of a measurement-driven Szilard engine, as well as a work extraction scheme in a two-level system under bang-bang control. Our results provide insights into the energetics as well as the irreversibility of classical and quantum systems under continuous feedback control.
Discrete quantum feedback control consists of a managed dynamics according to the information acquired by a previous measurement. Energy fluctuations along such dynamics satisfy generalized fluctuation relations, which are useful tools to study the thermodynamics of systems far away from equilibrium. Due to the practical challenge to assess energy fluctuations in the quantum scenario, the experimental verification of detailed fluctuation relations in the presence of feedback control remains elusive. We present a feasible method to experimentally verify detailed fluctuation relations for discrete feedback control quantum dynamics. Two detailed fluctuation relations are developed and employed. The method is based on a quantum interferometric strategy that allows the verification of fluctuation relations in the presence of feedback control. An analytical example to illustrate the applicability of the method is discussed. The comprehensive technique introduced here can be experimentally implemented at a microscale with the current technology in a variety of experimental platforms.
Fluctuation theorems are relations constraining the out-of-equilibrium fluctuations of thermodynamic quantities like the entropy production that were initially introduced for classical or quantum systems in contact with a thermal bath. Here we show, in the absence of thermal bath, the dynamics of continuously measured quantum systems can also be described by a fluctuation theorem, expressed in terms of a recently introduced arrow of time measure. This theorem captures the emergence of irreversible behavior from microscopic reversibility in continuous quantum measurements. From this relation, we demonstrate that measurement-induced wave-function collapse exhibits absolute irreversibility, such that Jarzynski-like equalities are violated, and that this property is inherent to quantum information acquisition. We apply our results to different continuous measurement schemes on a qubit: dispersive measurement, homodyne, and heterodyne detection of qubit's fluorescence.