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:2501.08858 · Physical review. E

Information geometry of transitions between quantum nonequilibrium steady states

Artur M. Lacerda, Laetitia P. Bettmann, John Goold

Published Jul 16, 2025• 6 citations
arXivHTMLPDF

Abstract

In a transition between nonequilibrium steady states, the entropic cost associated with the maintenance of steady-state currents can be distinguished from that arising from the transition itself through the concepts of excess (housekeeping) entropy flux and adiabatic (nonadiabatic) entropy production. The thermodynamics of this transition is embodied by the Hatano-Sasa relation. In this letter, we show that for a slow transition between quantum nonequilibrium steady states the nonadiabatic entropy production is, to leading order, given by the path action with respect to a Riemannian metric in the parameter space, which can be connected to the Kubo-Mori-Bogoliubov quantum Fisher information. We then demonstrate how to obtain minimally dissipative paths by solving the associated geodesic equation and illustrate the procedure with a simple example of a three-level maser. Furthermore, by identifying the quantum Fisher information with respect to time as a metric in state space, we derive an upper bound on the excess entropy flux that holds for arbitrarily fast processes.

DOI
10.1103/9f6l-d766

Related papers

Information-geometric bounds on nonequilibrium relaxation in quantum Markov dynamics

Xiao-Kan Guo, Zhiqiang Huang

In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.

quant-pharXiv:2609.09599•Sep 10, 2026•0 citations

Information geometry approach to quantum stochastic thermodynamics

Laetitia P. Bettmann, John Goold

Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.

arXiv:2409.06083•Physical review. E•Jan 16, 2025•21 citations

Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

A. T. Rezakhani, Damian F. Abasto, Daniel A. Lidar +1 more

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

arXiv:1004.0509•Physical Review A•Jul 20, 2010•83 citations

Thermodynamic length in open quantum systems

Matteo Scandi, Martí Perarnau-Llobet

The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on two paradigmatic examples: an Ising chain and a two-level system interacting with a bosonic bath with different spectral densities.

arXiv:1810.05583•Quantum•Oct 24, 2019•114 citations

Nonequilibrium Quantum Impurities: From Entropy Production to Information Theory

Pankaj Mehta, Natan Andrei

Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This Letter discusses these issues in the context of quantum impurity models. We use simple thermodynamic arguments to define the rate of entropy production sigma and show that sigma has a simple information-theoretic interpretation in terms of nonequilibrium distribution functions. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady state. We conclude by applying these ideas to the resonance level model and the Kondo model.

arXiv:0705.4479•Physical Review Letters•Feb 26, 2008•22 citations

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

Carlo Cafaro, Paul M. Alsing

We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su(2;C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external time-dependent magnetic field that specifies the su(2;C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

arXiv:2002.02244•Physical review. E•Feb 10, 2020•23 citations
In a transition between nonequilibrium steady states, the entropic cost associated with the maintenance of steady-state currents can be distinguished from that arising from the transition itself through the concepts of excess (housekeeping) entropy flux and adiabatic (nonadiabatic) entropy production. The thermodynamics of this transition is embodied by the Hatano-Sasa relation. In this letter, we show that for a slow transition between quantum nonequilibrium steady states the nonadiabatic entropy production is, to leading order, given by the path action with respect to a Riemannian metric in the parameter space, which can be connected to the Kubo-Mori-Bogoliubov quantum Fisher information. We then demonstrate how to obtain minimally dissipative paths by solving the associated geodesic equation and illustrate the procedure with a simple example of a three-level maser. Furthermore, by identifying the quantum Fisher information with respect to time as a metric in state space, we derive an upper bound on the excess entropy flux that holds for arbitrarily fast processes.
In this paper, we study the quantum information-geometric structure underlying short-time nonequilibrium relaxation in quantum Markov dynamics, and derive bounds on the nonequilibrium correction to the short-time relaxation curvature. These bounds generalize the information-geometric structure underlying Auconi's classical nonequilibrium relaxation inequality to quantum Markov dynamics. We consider the von Neumann relative entropy as quantum divergence, and parametrize perturbations through the Kubo-Mori map, which converts the second-order expansion of the relative entropy into the Bogoliubov-Kubo-Mori (BKM) inner product at all temperatures. For a quantum Markov semigroup with full-rank stationary state, the induced tangent-space generator admits a canonical decomposition into a BKM-symmetric and a BKM-antisymmetric part, and the leading nonequilibrium curvature correction takes the exact form of the expectation of the commutator between the dissipative and the transport part of the generator. This correction is bounded by the product of a BKM-symmetric dissipative activity and a transport-sector activity. The bound is verified on a noncommuting qubit model and a driven-dissipative qutrit, and the full chain is confirmed numerically on a two-dimensional Fokker-Planck steady state. In the high-temperature and overdamped limit the quantum formula reduces to the position-space Fokker-Planck expression and reproduces Auconi's entropy-production bound with the correct temperature factor.
Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.
We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity, we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions that generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second-order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.
The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on two paradigmatic examples: an Ising chain and a two-level system interacting with a bosonic bath with different spectral densities.
Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This Letter discusses these issues in the context of quantum impurity models. We use simple thermodynamic arguments to define the rate of entropy production sigma and show that sigma has a simple information-theoretic interpretation in terms of nonequilibrium distribution functions. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady state. We conclude by applying these ideas to the resonance level model and the Kondo model.
We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su(2;C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external time-dependent magnetic field that specifies the su(2;C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.