Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2504.10587 · Physical Review Research

Restoring the second law to classical-quantum dynamics

Isaac Layton, Harry J. D. Miller

Published Apr 7, 2026• 1 citation
arXivHTMLPDF

Abstract

All physical theories should obey the second law of thermodynamics. However, existing proposals to describe the dynamics of hybrid classical-quantum systems either violate the second law or lack a proof of its existence. Here, we rectify this by studying classical-quantum dynamics that are (1) linear and completely positive and (2) preserve the thermal state of the classical-quantum system. We first prove that such dynamics necessarily satisfy the second law. We then show how these dynamics may be constructed, proposing dynamics that generalize the standard Langevin and Fokker-Planck equations for classical systems in thermal environments to include backreaction from a quantum degree of freedom. Deriving necessary and sufficient conditions for completely positive, linear, and continuous classical-quantum dynamics to satisfy detailed balance, we find this property satisfied by our dynamics. To illustrate the formalism and its applications, we introduce two models. The first is an analytically solvable model of an overdamped classical system coupled to a quantum two-level system, which we use to study the total entropy production in both quantum system and classical measurement apparatus during a quantum measurement. The second describes an underdamped classical-quantum oscillator system subject to friction, which we numerically demonstrate exhibits thermalization in the adiabatic basis, showing the relevance of our dynamics for the mixed classical-quantum simulation of molecules.

DOI
10.1103/kqy2-1vmx

Related papers

Quantum detailed balance conditions and fluctuation relations for thermalizing quantum dynamics

Mehdi Ramezani, Fabio Benatti, Roberto Floreanini B +3 more

Quantum detailed balance conditions and quantum fluctuation relations are two important concepts in the dynamics of open quantum systems: both concern how such systems behave when they thermalize because of interaction with an environment. We prove that for thermalizing quantum dynamics the quantum detailed balance conditions yield validity of a quantum fluctuation relation (where only forward-time dynamics is considered). This implies that to have such a quantum fluctuation relation (which in turn enables a precise formulation of the second law of thermodynamics for quantum systems) it suffices to fulfill the quantum detailed balance conditions. We, however, show that the converse is not necessarily true; indeed, there are cases of thermalizing dynamics which feature the quantum fluctuation relation without satisfying detailed balance. We illustrate our results with three examples.

arXiv:1810.09174•Physical review. E•Nov 5, 2018•14 citations

Objective trajectories in hybrid classical-quantum dynamics

Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda +1 more

Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.

arXiv:2011.06009•Quantum•Jan 3, 2023•31 citations

Stochastic Thermodynamics at the Quantum-Classical Boundary: A Self-Consistent Framework Based on Adiabatic-Response Theory

Joshua Eglinton, Federico Carollo, Igor Lesanovsky +1 more

Microscopic thermal machines promise to play an important role in future quantum technologies. Making such devices widely applicable will require effective strategies to channel their output into easily accessible storage systems like classical degrees of freedom. Here, we develop a self-consistent theoretical framework that makes it possible to model such quantum-classical hybrid devices in a thermodynamically consistent manner. Our approach is based on the assumption that the quantum part of the device is subject to strong decoherence and dissipation induced by a thermal reservoir. Due to the ensuing separation of time scales between slowly evolving classical and fast relaxing quantum degrees of freedom, the dynamics of the hybrid system can be described by means of adiabatic-response theory. We show that, upon including fluctuations in a minimally consistent way, the resulting equations of motion can be equipped with a first and second law, both on the ensemble level and on the level of individual trajectories of the classical part of the system, where thermodynamic quantities like heat and work become stochastic variables. As an application of our theory, we work out a physically transparent model of a quantum-classical hybrid engine, whose working system consists of a chain of Rydberg atoms, which is confined in an optical cavity and driven by periodic temperature variations. We demonstrate through numerical simulations that the engine can sustain periodic oscillations of a movable mirror, which acts as a classical load, against external friction and extract the full distributions of input heat and output work. By making the statistics of thermodynamic processes in quantum-classical hybrid systems accessible without the need to further specify a measurement protocol, our work contributes to bridging the long-standing gap between classical and quantum stochastic thermodynamics.

arXiv:2404.10118•Quantum•Sep 26, 2024•2 citations

Boltzmann to Lindblad: Classical and Quantum Approaches to Out-of-Equilibrium Statistical Mechanics

Stefano Giordano, Giuseppe Florio, Giuseppe Puglisi +2 more

Open quantum systems play a central role in contemporary nanoscale technologies, including molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and complete positivity. In this work, we develop a framework that addresses this issue by extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein-Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while satisfying the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, thus fully justifying the classical construction. Moreover, we find that the friction coefficients must satisfy the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism offers a versatile tool for deriving quantum versions of the thermodynamic laws and is directly applicable to a wide class of nonequilibrium nanoscale systems.

quant-pharXiv:2512.11613•Dec 15, 2025•0 citations

Boltzmann to Lindblad: classical and quantum approaches to out-of-equilibrium statistical mechanics

Stefano Giordano, Giuseppe Florio, Giuseppe Puglisi +2 more

Abstract Open quantum systems play a central role in current nanoscale technologies, such as molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and with the requirement of complete positivity of quantum evolution. In this work we develop a framework that addresses this issue by systematically extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein–Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while fulfilling the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, fully justifying the classical construction. Moreover, we find that the friction coefficients must adhere to the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism developed here presents a thermodynamically consistent and completely positive quantum extension of classical stochastic mechanics. It offers a versatile tool for deriving quantum versions of thermodynamic laws and is directly applicable to a broad class of non-equilibrium nanoscale systems of current theoretical and technological interest.

Journal of Statistical Mechanics Theory and Experiment•Mar 1, 2026•0 citations

Quantum thermodynamics of continuous feedback control

Kacper Prech, Joël Aschwanden, Patrick P. Potts

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.

arXiv:2505.16615•Physical Review A•Oct 14, 2026•0 citations
All physical theories should obey the second law of thermodynamics. However, existing proposals to describe the dynamics of hybrid classical-quantum systems either violate the second law or lack a proof of its existence. Here, we rectify this by studying classical-quantum dynamics that are (1) linear and completely positive and (2) preserve the thermal state of the classical-quantum system. We first prove that such dynamics necessarily satisfy the second law. We then show how these dynamics may be constructed, proposing dynamics that generalize the standard Langevin and Fokker-Planck equations for classical systems in thermal environments to include backreaction from a quantum degree of freedom. Deriving necessary and sufficient conditions for completely positive, linear, and continuous classical-quantum dynamics to satisfy detailed balance, we find this property satisfied by our dynamics. To illustrate the formalism and its applications, we introduce two models. The first is an analytically solvable model of an overdamped classical system coupled to a quantum two-level system, which we use to study the total entropy production in both quantum system and classical measurement apparatus during a quantum measurement. The second describes an underdamped classical-quantum oscillator system subject to friction, which we numerically demonstrate exhibits thermalization in the adiabatic basis, showing the relevance of our dynamics for the mixed classical-quantum simulation of molecules.
Quantum detailed balance conditions and quantum fluctuation relations are two important concepts in the dynamics of open quantum systems: both concern how such systems behave when they thermalize because of interaction with an environment. We prove that for thermalizing quantum dynamics the quantum detailed balance conditions yield validity of a quantum fluctuation relation (where only forward-time dynamics is considered). This implies that to have such a quantum fluctuation relation (which in turn enables a precise formulation of the second law of thermodynamics for quantum systems) it suffices to fulfill the quantum detailed balance conditions. We, however, show that the converse is not necessarily true; indeed, there are cases of thermalizing dynamics which feature the quantum fluctuation relation without satisfying detailed balance. We illustrate our results with three examples.
Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.
Microscopic thermal machines promise to play an important role in future quantum technologies. Making such devices widely applicable will require effective strategies to channel their output into easily accessible storage systems like classical degrees of freedom. Here, we develop a self-consistent theoretical framework that makes it possible to model such quantum-classical hybrid devices in a thermodynamically consistent manner. Our approach is based on the assumption that the quantum part of the device is subject to strong decoherence and dissipation induced by a thermal reservoir. Due to the ensuing separation of time scales between slowly evolving classical and fast relaxing quantum degrees of freedom, the dynamics of the hybrid system can be described by means of adiabatic-response theory. We show that, upon including fluctuations in a minimally consistent way, the resulting equations of motion can be equipped with a first and second law, both on the ensemble level and on the level of individual trajectories of the classical part of the system, where thermodynamic quantities like heat and work become stochastic variables. As an application of our theory, we work out a physically transparent model of a quantum-classical hybrid engine, whose working system consists of a chain of Rydberg atoms, which is confined in an optical cavity and driven by periodic temperature variations. We demonstrate through numerical simulations that the engine can sustain periodic oscillations of a movable mirror, which acts as a classical load, against external friction and extract the full distributions of input heat and output work. By making the statistics of thermodynamic processes in quantum-classical hybrid systems accessible without the need to further specify a measurement protocol, our work contributes to bridging the long-standing gap between classical and quantum stochastic thermodynamics.
Open quantum systems play a central role in contemporary nanoscale technologies, including molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and complete positivity. In this work, we develop a framework that addresses this issue by extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein-Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while satisfying the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, thus fully justifying the classical construction. Moreover, we find that the friction coefficients must satisfy the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism offers a versatile tool for deriving quantum versions of the thermodynamic laws and is directly applicable to a wide class of nonequilibrium nanoscale systems.
Abstract Open quantum systems play a central role in current nanoscale technologies, such as molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and with the requirement of complete positivity of quantum evolution. In this work we develop a framework that addresses this issue by systematically extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein–Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while fulfilling the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, fully justifying the classical construction. Moreover, we find that the friction coefficients must adhere to the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism developed here presents a thermodynamically consistent and completely positive quantum extension of classical stochastic mechanics. It offers a versatile tool for deriving quantum versions of thermodynamic laws and is directly applicable to a broad class of non-equilibrium nanoscale systems of current theoretical and technological interest.
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.