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Journal of the Physical Society of Japan

Jarzynski Equality Modified in the Linear Feedback System

Youhei Fujitani, Hiroyuki Suzuki

Published Oct 12, 2010• 45 citations
Publisher

Abstract

It is known that one can derive the Jarzynski equality in a stochastic process of a classical system by assuming the local detailed balance. We study how the equality is modified in the linear feedback system. There, measurement is performed on the state following a linear Langevin equation, the measured variable is linear to the state variable with white sensor noise, the Kalman filter estimates the state by utilizing measured values in the past, and a linear regulator controls the state dynamics. Although a stochastic process produced by this dynamics is non-Markovian because of the feedback loop, it is known in the control theory that a Markov process for the estimation can be separated from the whole process. To the exponent in the Jarzynski equality, we find an additional term, whose average gives the mutual information between state variables and measured variables in the Markov process for the estimation. The resultant equality holds whether the gain is optimal or not.

DOI
10.1143/jpsj.79.104003

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It is known that one can derive the Jarzynski equality in a stochastic process of a classical system by assuming the local detailed balance. We study how the equality is modified in the linear feedback system. There, measurement is performed on the state following a linear Langevin equation, the measured variable is linear to the state variable with white sensor noise, the Kalman filter estimates the state by utilizing measured values in the past, and a linear regulator controls the state dynamics. Although a stochastic process produced by this dynamics is non-Markovian because of the feedback loop, it is known in the control theory that a Markov process for the estimation can be separated from the whole process. To the exponent in the Jarzynski equality, we find an additional term, whose average gives the mutual information between state variables and measured variables in the Markov process for the estimation. The resultant equality holds whether the gain is optimal or not.
In this study, we rederive the fluctuation theorems in the presence of feedback by assuming the known Jarzynski equality and detailed fluctuation theorems. We find that both the classical and quantum systems can be analyzed using a similar treatment in terms of state space trajectories. We first briefly reproduce the already known work theorems for a classical system in order to show its equivalence with the quantum treatment. We then extend the treatment to arrive at new results, namely the generalizations of Seifert's entropy production theorem and the Hatano–Sasa fluctuation theorem, in the presence of feedback. We have also derived the extended version of the Tasaki–Crooks fluctuation theorem for a quantum particle in the presence of multiple loop feedback. For deriving the extended quantum fluctuation theorems, we have considered open systems. No assumption is made on the nature of environment and the strength of system–bath coupling. However, it is assumed that the measurement process involves classical errors.
We derive a general information–theoretic equality for a system undergoing two projective measurements separated by a general temporal evolution. The equality implies the non-negativity of the mutual information between the measurement outcomes of the earlier and later projective measurements. We show that it also contains the Jarzynski relation between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. Our result elucidates the information–theoretic underpinning of thermodynamics and explains why the Jarzynski relation holds identically both quantumly as well as classically.
Abstract In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.
State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.
There are two related theorems which hold even in far from equilibrium, namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states the existence of symmetry of fluctuation of entropy production, while the Jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes. On the other hand, the relationship between these theorems was investigated by Crooks [Phys. Rev. E 60, 2721 (1999)] for the classical stochastic systems. In this paper, we derive quantum analogues of fluctuation theorem and Jarzynski equality in terms of microscopic reversibility. In other words, the quantum analog of the work by Crooks is presented. Also, for the quasiclassical Langevin system, microscopically reversible condition is confirmed.
Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.