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

Physical Review E

Entropy increase in K -step Markovian and consistent dynamics of closed quantum systems

Jochen Gemmer B, Robin Steinigeweg

Published Apr 7, 2014• 24 citations
Publisher

Abstract

We consider sequences of measurements implemented by positive operator valued measures (POVMs). Starting from the assumption that these sequences may be described as consistent and Markovian, even and especially for closed quantum systems, we identify properties of the equilibrium state that coincide with the properties of typical pure quantum states. We define a physical entropy that converges against the standard entropies in the approach to equilibrium. Furthermore, strict limits to its possible decrease are derived on the basis of Renyi entropies. It is demonstrated that Landauer's principle follows directly from these limits. Since the above assumptions are rather strong, we exemplify the fact that they may nevertheless apply by checking them numerically for some transition paths in a concrete model.

DOI
10.1103/physreve.89.042113

Related papers

Transient fluctuation theorem in closed quantum systems

C. Bartsch, Jochen Gemmer B

Our point of departure are the unitary dynamics of closed quantum systems as generated from the Schrödinger equation. We focus on a class of quantum models that typically exhibit roughly exponential relaxation of some observable within this framework. Furthermore, we focus on pure state evolutions. An entropy in accord with Jaynes principle is defined on the basis of the quantum expectation value of the above observable. It is demonstrated that the resulting deterministic entropy dynamics are in a sense in accord with a transient fluctuation theorem. Moreover, we demonstrate that the dynamics of the expectation value are describable in terms of an Ornstein-Uhlenbeck process. These findings are demonstrated numerically and supported by analytical considerations based on quantum typicality.

arXiv:1112.5292•Europhysics Letters (EPL)•Dec 1, 2011•15 citations

Entropy for quantum pure states and quantum H theorem

Xizhi Han, Biao Wu

We construct a complete set of Wannier functions that are localized at both given positions and momenta. This allows us to introduce the quantum phase space, onto which a quantum pure state can be mapped unitarily. Using its probability distribution in quantum phase space, we define an entropy for a quantum pure state. We prove an inequality regarding the long-time behavior of our entropy's fluctuation. For a typical initial state, this inequality indicates that our entropy can relax dynamically to a maximized value and stay there most of time with small fluctuations. This result echoes the quantum H theorem proved by von Neumann [Zeitschrift für Physik 57, 30 (1929)]. Our entropy is different from the standard von Neumann entropy, which is always zero for quantum pure states. According to our definition, a system always has bigger entropy than its subsystem even when the system is described by a pure state. As the construction of the Wannier basis can be implemented numerically, the dynamical evolution of our entropy is illustrated with an example.

arXiv:1406.6527•Physical Review E•Jun 5, 2015•23 citations

Remarks concerning an Entropy-like Quantity $S_q$ especially for Quantum Systems with a Particle Picture; Model with pure Point Spectra

Siegfried Schlieder

For quantum mechanical systems an entropy-like quantity $S_q$ is defined. $S_q$ can differ from the usually defined entropy $S$ and $S_q$ may increase with time for an isolated system. The essential condition for the difference between $S$ and $S_q$ is the assumption that the set {\bf A} of observables which can be represented by a measurement is a proper subset of the set of selfadjoint operators. The underlying idea is made visible in the case of particle systems with non-trivial scattering. The model-character of the reasoning comes from the fact that continuous spectra are replaced by point-spectra. So it seems evident, that no direct connection exists between $S_q$ and the Sinai-Kolmogorov-Entropy at least in this model with pure point-spectra.

arXiv:quant-ph/9801040•Jan 20, 1998•0 citations

Nonequilibrium Dynamics of Renyi Entropy for Bosonic Many-Particle Systems

Ahana Chakraborty, Rajdeep Sensarma

We propose a new field theoretic method for calculating Renyi entropy of a subsystem of many interacting bosons without using replica methods. This method is applicable to dynamics of both open and closed quantum systems starting from arbitrary initial conditions. Our method identifies the Wigner characteristic of a reduced density matrix with the partition function of the whole system with a set of linear sources turned on only in the subsystem, and uses this to calculate the subsystem's Renyi entropy. We use this method to study the evolution of Renyi entropy in a noninteracting open quantum system starting from an initial Fock state. We find a relation between the initial state and final density matrix which determines whether the entropy shows nonmonotonic behavior in time. For non-Markovian dynamics, we show that the entropy approaches its steady-state value as a power law with exponents governed by nonanalyticities of the bath. We illustrate that this field-theoretic method can be used to study large bosonic open quantum systems.

arXiv:1810.10545•Physical Review Letters•Nov 12, 2021•8 citations

On the non-equilibrium thermodynamics of quantum systems

Stefano Marcantoni

A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.

•Feb 6, 2018•1 citation

ENTROPY INCREASE AND K-SYSTEMS IN QUANTUM THEORY

Walter E Thirring

It is shown that the relative entropy converges to its equilibrium value when the states converge strongly. This can happen only for subalgebras for which the clustering is uniform. It happens for K-systems for which this uniformity in clustering has the maximal extent which is possible.

International Journal of Modern Physics B•Apr 1, 1990•0 citations
We consider sequences of measurements implemented by positive operator valued measures (POVMs). Starting from the assumption that these sequences may be described as consistent and Markovian, even and especially for closed quantum systems, we identify properties of the equilibrium state that coincide with the properties of typical pure quantum states. We define a physical entropy that converges against the standard entropies in the approach to equilibrium. Furthermore, strict limits to its possible decrease are derived on the basis of Renyi entropies. It is demonstrated that Landauer's principle follows directly from these limits. Since the above assumptions are rather strong, we exemplify the fact that they may nevertheless apply by checking them numerically for some transition paths in a concrete model.
Our point of departure are the unitary dynamics of closed quantum systems as generated from the Schrödinger equation. We focus on a class of quantum models that typically exhibit roughly exponential relaxation of some observable within this framework. Furthermore, we focus on pure state evolutions. An entropy in accord with Jaynes principle is defined on the basis of the quantum expectation value of the above observable. It is demonstrated that the resulting deterministic entropy dynamics are in a sense in accord with a transient fluctuation theorem. Moreover, we demonstrate that the dynamics of the expectation value are describable in terms of an Ornstein-Uhlenbeck process. These findings are demonstrated numerically and supported by analytical considerations based on quantum typicality.
We construct a complete set of Wannier functions that are localized at both given positions and momenta. This allows us to introduce the quantum phase space, onto which a quantum pure state can be mapped unitarily. Using its probability distribution in quantum phase space, we define an entropy for a quantum pure state. We prove an inequality regarding the long-time behavior of our entropy's fluctuation. For a typical initial state, this inequality indicates that our entropy can relax dynamically to a maximized value and stay there most of time with small fluctuations. This result echoes the quantum H theorem proved by von Neumann [Zeitschrift für Physik 57, 30 (1929)]. Our entropy is different from the standard von Neumann entropy, which is always zero for quantum pure states. According to our definition, a system always has bigger entropy than its subsystem even when the system is described by a pure state. As the construction of the Wannier basis can be implemented numerically, the dynamical evolution of our entropy is illustrated with an example.
Remarks concerning an Entropy-like Quantity SqS_qSq​ especially for Quantum Systems with a Particle Picture; Model with pure Point Spectra
For quantum mechanical systems an entropy-like quantity SqS_qSq​ is defined. SqS_qSq​ can differ from the usually defined entropy SSS and SqS_qSq​ may increase with time for an isolated system. The essential condition for the difference between SSS and SqS_qSq​ is the assumption that the set {\bf A} of observables which can be represented by a measurement is a proper subset of the set of selfadjoint operators. The underlying idea is made visible in the case of particle systems with non-trivial scattering. The model-character of the reasoning comes from the fact that continuous spectra are replaced by point-spectra. So it seems evident, that no direct connection exists between SqS_qSq​ and the Sinai-Kolmogorov-Entropy at least in this model with pure point-spectra.
We propose a new field theoretic method for calculating Renyi entropy of a subsystem of many interacting bosons without using replica methods. This method is applicable to dynamics of both open and closed quantum systems starting from arbitrary initial conditions. Our method identifies the Wigner characteristic of a reduced density matrix with the partition function of the whole system with a set of linear sources turned on only in the subsystem, and uses this to calculate the subsystem's Renyi entropy. We use this method to study the evolution of Renyi entropy in a noninteracting open quantum system starting from an initial Fock state. We find a relation between the initial state and final density matrix which determines whether the entropy shows nonmonotonic behavior in time. For non-Markovian dynamics, we show that the entropy approaches its steady-state value as a power law with exponents governed by nonanalyticities of the bath. We illustrate that this field-theoretic method can be used to study large bosonic open quantum systems.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.