Integral quantum fluctuation theorems under measurement and feedback control

Quantum Fluctuation Theorem for Arbitrary Measurement and Feedback Schemes

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.

Thermodynamic bound on quantum state discrimination

We show that the second law of thermodynamics poses a restriction on how well we can discriminate between quantum states. By examining an ideal gas with a quantum internal degree of freedom undergoing a cycle based on a proposal by Peres, we establish a nontrivial upper bound on the attainable accuracy of quantum state discrimination. This thermodynamic bound, which relies solely on the linearity of quantum mechanics and the constraint of no work extraction, matches Holevo's bound on accessible information, but is looser than the Holevo-Helstrom bound. The result gives more evidence on the disagreement between thermodynamic entropy and von Neumann entropy and places potential limitations on proposals beyond quantum mechanics.

Stochastic Entropy Production: Fluctuation Relation and Irreversibility Mitigation in Non-unital Quantum Dynamics

In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in Phys Rev E 92:032129 (2015), we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in Phys Rev Res 2:033250 (2020), is analyzed in this context.

Quantum ergotropy and quantum feedback control

We study the energy extraction from and charging to a finite-dimensional quantum system by general quantum operations. We prove that the changes in energy induced by unital quantum operations are limited by the ergotropy and charging bounds for unitary quantum operations. This implies that, in order to break the ergotropy bound for unitary quantum operations, one needs to perform a quantum operation with feedback control. We also show that the ergotropy bound for unital quantum operations, applied to initial thermal equilibrium states, is tighter than the inequality representing the standard second law of thermodynamics without feedback control.

Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines

We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.

Fluctuation Theorem for Information Thermodynamics of Quantum Correlated Systems

We establish a fluctuation theorem for an open quantum bipartite system that explicitly manifests the role played by quantum correlation. Generally quantum correlations may substantially modify the universality of classical thermodynamic relations in composite systems. Our fluctuation theorem finds a non-equilibrium parameter of genuinely quantum nature that sheds light on the emerging quantum information thermodynamics. Specifically we show that the statistics of quantum correlation fluctuation obtained in a time-reversed process can provide a useful insight into addressing work and heat in the resulting thermodynamic evolution. We illustrate these quantum thermodynamic relations by two examples of quantum correlated systems.

Necessity of feedback control for the quantum Maxwell demon in a finite-time steady feedback cycle

We revisit quantum Maxwell demon in thermodynamic feedback cycle in the steady-state regime. We derive a generalized version of the Clausius inequality for a finite-time steady feedback cycle with a single heat bath. It is shown to be tighter than previously known ones, and allows us to clarify that feedback control is necessary to violate the standard Clausius inequality.

Quantum Fokker-Planck Master Equation for Continuous Feedback Control

Measurement and feedback control are essential features of quantum science, with applications ranging from quantum technology protocols to information-to-work conversion in quantum thermodynamics. Theoretical descriptions of feedback control are typically given in terms of stochastic equations requiring numerical solutions, or are limited to linear feedback protocols. Here we present a formalism for continuous quantum measurement and feedback, both linear and nonlinear. Our main result is a quantum Fokker-Planck master equation describing the joint dynamics of a quantum system and a detector with finite bandwidth. For fast measurements, we derive a Markovian master equation for the system alone, amenable to analytical treatment. We illustrate our formalism by investigating two basic information engines, one quantum and one classical.

Quantum Fluctuation Theorem under Quantum Jumps with Continuous Measurement and Feedback

While the fluctuation theorem in classical systems has been thoroughly generalized under various feedback control setups, an intriguing situation in quantum systems, namely under continuous feedback, remains to be investigated. In this work, we derive the generalized fluctuation theorem under quantum jumps with continuous measurement and feedback. The essence for the derivation is to newly introduce the operationally meaningful information, which we call quantum-classical-transfer (QC-transfer) entropy. QC-transfer entropy can be naturally interpreted as the quantum counterpart of transfer entropy that is commonly used in classical time series analysis. We also verify our theoretical results by numerical simulation and propose an experiment-numerics hybrid verification method. Our work reveals a fundamental connection between quantum thermodynamics and quantum information, which can be experimentally tested with artificial quantum systems such as circuit quantum electrodynamics.

Experimental Assessment of Entropy Production in a Continuously Measured Mechanical Resonator

The information on a quantum process acquired through measurements plays a crucial role in the determination of its nonequilibrium thermodynamic properties. We report on the experimental inference of the stochastic entropy production rate for a continuously monitored mesoscopic quantum system. We consider an optomechanical system subjected to continuous displacement Gaussian measurements and characterize the entropy production rate of the individual trajectories followed by the system in its stochastic dynamics, employing a phase-space description in terms of the Wigner entropy. Owing to the specific regime of our experiment, we are able to single out the informational contribution to the entropy production arising from conditioning the state on the measurement outcomes. Our experiment embodies a significant step towards the demonstration of full-scale control of fundamental thermodynamic processes at the mesoscopic quantum scale.

Information fluctuation theorem for an open quantum bipartite system

We study an arbitrary nonequilibrium dynamics of a quantum bipartite system coupled to a reservoir. For its characterization, we present a fluctuation theorem (FT) that explicitly addresses the quantum correlation of subsystems during the thermodynamic evolution. To our aim, we designate the local and the global states altogether in the time-forward and the time-reversed transition probabilities. In view of the two-point measurement scheme, only the global states are subject to measurements whereas the local states are used only as an augmented information on the composite system. We specifically derive a FT in such a form that relates the entropy production of local systems in the time-forward transition to the change of quantum correlation in the time-reversed transition. This also leads to a useful thermodynamic inequality and we illustrate its advantage by an example of an isothermal process on Werner states.

Thermodynamic uncertainty relations including measurement and feedback

Thermodynamic uncertainty relations quantify how the signal-to-noise ratio of a given observable is constrained by dissipation. Fluctuation relations generalize the second law of thermodynamics to stochastic processes. We show that any fluctuation relation directly implies a thermodynamic uncertainty relation, considerably increasing their range of applicability. In particular, we extend thermodynamic uncertainty relations to scenarios which include measurement and feedback. Since feedback generally breaks time-reversal invariance, the uncertainty relations involve quantities averaged over the forward and the backward experiment defined by the associated fluctuation relation. This implies that the signal-to-noise ratio of a given experiment can in principle become arbitrarily large as long as the corresponding backward experiment compensates, e.g., by being sufficiently noisy. We illustrate our results with the Szilard engine as well as work extraction by free energy reduction in a quantum dot.

Detailed Fluctuation Relation for Arbitrary Measurement and Feedback Schemes

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.

Information Gain and Loss for a Quantum Maxwell's Demon

We use continuous weak measurements of a driven superconducting qubit to experimentally study the information dynamics of a quantum Maxwell's demon. We show how information gained by a demon who can track single quantum trajectories of the qubit can be converted into work using quantum coherent feedback. We verify the validity of a quantum fluctuation theorem with feedback by utilizing information obtained along single trajectories. We demonstrate, in particular, that quantum backaction can lead to a loss of information in imperfect measurements. We furthermore probe the transition between information gain and loss by varying the initial purity of the qubit.

Information-to-work conversion by Maxwell's demon in a superconducting circuit quantum electrodynamical system

Information thermodynamics bridges information theory and statistical physics by connecting information content and entropy production through measurement and feedback control. Maxwell’s demon is a hypothetical character that uses information about a system to reduce its entropy. Here we realize a Maxwell’s demon acting on a superconducting quantum circuit. We implement quantum non-demolition projective measurement and feedback operation of a qubit and verify the generalized integral fluctuation theorem. We also evaluate the conversion efficiency from information gain to work in the feedback protocol. Our experiment constitutes a step toward experimental studies of quantum information thermodynamics in artificially made quantum machines.

Maxwell’s demon is a hypothetical character that uses information about a system to reduce its entropy, highlighting the link between information and thermodynamic entropies. Here the authors experimentally realise a Maxwell's demon controlling a quantum system and explore how it affects thermodynamic laws.

Quantum Jarzynski equality of measurement-based work extraction

Many studies of quantum-size heat engines assume that the dynamics of an internal system is unitary and that the extracted work is equal to the energy loss of the internal system. Both assumptions, however, should be under scrutiny. In the present paper, we analyze quantum-scale heat engines, employing the measurement-based formulation of the work extraction recently introduced by Hayashi and Tajima [M. Hayashi and H. Tajima, arXiv:1504.06150]. We first demonstrate the inappropriateness of the unitary time evolution of the internal system (namely, the first assumption above) using a simple two-level system; we show that the variance of the energy transferred to an external system diverges when the dynamics of the internal system is approximated to a unitary time evolution. Second, we derive the quantum Jarzynski equality based on the formulation of Hayashi and Tajima as a relation for the work measured by an external macroscopic apparatus. The right-hand side of the equality reduces to unity for "natural" cyclic processes but fluctuates wildly for noncyclic ones, exceeding unity often. This fluctuation should be detectable in experiments and provide evidence for the present formulation.

Einstein-Podolsky-Rosen paradox implies a minimum achievable temperature

This work examines the thermodynamic consequences of the repeated partial projection model for coupling a quantum system to an arbitrary series of environments under feedback control. This paper provides observational definitions of heat and work that can be realized in current laboratory setups. In contrast to other definitions, it uses only properties of the environment and the measurement outcomes, avoiding references to the "measurement" of the central system's state in any basis. These definitions are consistent with the usual laws of thermodynamics at all temperatures, while never requiring complete projective measurement of the entire system. It is shown that the back action of measurement must be counted as work rather than heat to satisfy the second law. Comparisons are made to quantum jump (unravelling) and transition-probability based definitions, many of which appear as particular limits of the present model. These limits show that our total entropy production is a lower bound on traditional definitions of heat that trace out the measurement device. Examining the master equation approximation to the process at finite measurement rates, we show that most interactions with the environment make the system unable to reach absolute zero. We give an explicit formula for the minimum temperature achievable in repeatedly measured quantum systems. The phenomenon of minimum temperature offers an explanation of recent experiments aimed at testing fluctuation theorems in the quantum realm and places a fundamental purity limit on quantum computers.

Experimental Rectification of Entropy Production by Maxwell's Demon in a Quantum System

Maxwell's demon explores the role of information in physical processes. Employing information about microscopic degrees of freedom, this "intelligent observer" is capable of compensating entropy production (or extracting work), apparently challenging the second law of thermodynamics. In a modern standpoint, it is regarded as a feedback control mechanism and the limits of thermodynamics are recast incorporating information-to-energy conversion. We derive a trade-off relation between information-theoretic quantities empowering the design of an efficient Maxwell's demon in a quantum system. The demon is experimentally implemented as a spin-1/2 quantum memory that acquires information, and employs it to control the dynamics of another spin-1/2 system, through a natural interaction. Noise and imperfections in this protocol are investigated by the assessment of its effectiveness. This realization provides experimental evidence that the irreversibility in a nonequilibrium dynamics can be mitigated by assessing microscopic information and applying a feed-forward strategy at the quantum scale.

Nonequilibrium equalities in absolutely irreversible processes

We generalize nonequilibrium integral equalities to situations involving absolutely irreversible processes for which the forward-path probability vanishes and the entropy production diverges, rendering conventional integral fluctuation theorems inapplicable. We identify the mathematical origins of absolute irreversibility as the singularity of probability measure. We demonstrate the validity of the obtained equalities for several models.

Extracting work from magnetic-field-coupled Brownian particles

Thermodynamics of the magnetic-field-coupled Brownian particles is studied. We show that in the presence of the magnetic field, work can be extracted from the reservoir even when the measurement operation and the potential change operation are applied in different spatial directions. In particular, we show that more work can be extracted if the measurements are applied in two different directions simultaneously. In all these cases, we show that the generalized second law involving the measurement information and potential change is satisfied. In addition, we show how the continuous potential change and measurement position affect the work extraction.