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arXiv:1708.08497 · New Journal of Physics

Shortcuts to adiabaticity applied to nonequilibrium entropy production: an information geometry viewpoint

Kazutaka Takahashi

Published Oct 23, 2017• 22 citations
arXivHTMLPDF

Abstract

We apply the method of shortcuts to adiabaticity to nonequilibrium systems. For unitary dynamics, the system Hamiltonian is separated into two parts. One of them defines the adiabatic states for the state to follow and the nonadiabatic transitions are prevented by the other part. This property is implemented to the nonequilibrium entropy production and we find that the entropy is separated into two parts. The separation represents the Pythagorean theorem for the Kullback–Leibler divergence and an information-geometric interpretation is obtained. We also study a lower bound of the entropy, which is applied to derive a trade-off relation between time, entropy and state distance.

DOI
10.1088/1367-2630/aa9534

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We apply the method of shortcuts to adiabaticity to nonequilibrium systems. For unitary dynamics, the system Hamiltonian is separated into two parts. One of them defines the adiabatic states for the state to follow and the nonadiabatic transitions are prevented by the other part. This property is implemented to the nonequilibrium entropy production and we find that the entropy is separated into two parts. The separation represents the Pythagorean theorem for the Kullback–Leibler divergence and an information-geometric interpretation is obtained. We also study a lower bound of the entropy, which is applied to derive a trade-off relation between time, entropy and state distance.
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.
We reveal that the information geometry of entropy production exhibits an intrinsic anisotropy, providing a unified origin for the inequivalent geometric constraints obeyed by its different components. Using a hierarchical projection structure of quantum relative entropy, we derive an exact orthogonal decomposition of entropy production into three distinct geometric contributions associated with the deviation of the Gibbs projection temperature from the reference temperature, the deviation of the environment from its Gibbs projection state, and system--environment correlations. We demonstrate that these contributions obey fundamentally different geometric bounds: the first reduces to a classical Kullback--Leibler divergence and admits no universal upper bound in terms of the trace distance and the system and environment dimensions alone, whereas the latter two admit rigorous distance-based bounds with dimension-dependent factors. The framework also applies to classical Markovian dynamics described solely in terms of the system state, for which the system--environment correlation contribution is absent from the decomposition. Combining this geometric decomposition with dynamical speed limits, we obtain rigorous bounds on entropy production expressed solely in terms of physically accessible quantities, such as the mean energy and Hamiltonian variance.
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.
We analyze the production of entropy along nonequilibrium processes in quantum systems coupled to generic environments.First, we show that the entropy production due to final measurements and the loss of correlations obeys a fluctuation theorem in detailed and integral forms.Second, we discuss the decomposition of the entropy production into two positive contributions, adiabatic and nonadiabatic, based on the existence of invariant states of the local dynamics.Fluctuation theorems for both contributions hold only for evolutions verifying a specific condition of quantum origin.We illustrate our results with three relevant examples of quantum thermodynamic processes far from equilibrium.
The second law does more than set the direction of thermodynamic evolution: it endows nonequilibrium transformations with an underlying geometry. In this work, we provide a unified geometric description of entropy production in classical and quantum thermodynamics based on Wasserstein-2 structures arising from gradient flows of free energy. We review how relaxation to equilibrium, in overdamped diffusions, discrete detailed-balanced Markov chains, and dissipative Lindblad dynamics, can be formulated as a gradient flow on the space of states. The associated Wasserstein-2 distance bounds entropy production, yielding a finite-time refinement of the second law. We extend this framework beyond purely dissipative dynamics by introducing generalized Wasserstein-2 metrics that incorporate conservative (Hamiltonian) dynamics in both classical inertial systems and open quantum systems, yielding intrinsic distances that exactly characterize minimal entropy production under fixed dissipative mobilities. We establish equivalence bounds between purely dissipative and Hamiltonian-dissipative geometries, explicitly quantifying how inertial or coherent dynamics can reduce dissipation. Finally, when restricted to equilibrium distributions, we recover the thermodynamic length of linear response-including the quantum thermodynamic length-thereby linking optimal transport, thermodynamic length, and counterdiabatic protocols within a single geometric framework. All in all, our results extend the Riemannian program of thermodynamics further from equilibrium and provide a geometric foundation for optimal protocols beyond the overdamped setting.
We present, from an information theoretic viewpoint, an analysis of phase transitions and critical phenomena in quantum systems. Our study is based on geometrical considerations within the Riemannian space of thermodynamic parameters that characterize the system. A metric for the space can be derived from an appropriate definition of distance between quantum states. For this purpose, we consider generalized α-divergences that include the standard Kullback–Leibler relative entropy. The use of other measures of information distance is taken into account, and the thermodynamic stability of the system is discussed from this geometric perspective.