We derive thermodynamic speed limits for isolated quantum systems undergoing finite-time unitary driving. The key ingredient is a set of generalized second-law inequalities: for the Gibbs entropy and for several thermodynamically motivated observational entropies, the entropy production is bounded from below by a function of the Vu--Saito quantum Wasserstein distance between the initial and final coarse-grained states. These inequalities yield lower bounds on the operation time in terms of the average entropy-production rate. We apply the framework to system--bath coarse-graining, local-energy coarse-graining, and diagonal entropy, and illustrate the resulting bounds with numerical and analytically solvable examples. Our results provide a thermodynamic characterization of finite-time state transformations in isolated quantum systems and clarify how coarse-grained entropy production constrains macroscopic reachability.
We derive thermodynamic speed limits for isolated quantum systems undergoing finite-time unitary driving. The key ingredient is a set of generalized second-law inequalities: for the Gibbs entropy and for several thermodynamically motivated observational entropies, the entropy production is bounded from below by a function of the Vu--Saito quantum Wasserstein distance between the initial and final coarse-grained states. These inequalities yield lower bounds on the operation time in terms of the average entropy-production rate. We apply the framework to system--bath coarse-graining, local-energy coarse-graining, and diagonal entropy, and illustrate the resulting bounds with numerical and analytically solvable examples. Our results provide a thermodynamic characterization of finite-time state transformations in isolated quantum systems and clarify how coarse-grained entropy production constrains macroscopic reachability.
The quantum speed limit for open quantum systems in the non-Markovian regime is analyzed. We provide a lower bound for the time required to transform an initial state to a final state in terms of thermodynamic quantities such as the energy fluctuation, entropy production rate, and dynamical activity. Such bound was already analyzed for Markovian evolution satisfying detailed balance condition. Here we generalize this approach to deal with arbitrary evolution governed by a time-local generator. Our analysis is illustrated by three paradigmatic examples of qubit evolution: amplitude damping, pure dephasing, and the eternally non-Markovian evolution.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
Quantum speed limit (QSL) for open quantum systems in the non-Markovian regime is analyzed. We provide the lower bound for the time required to transform an initial state to a final state in terms of thermodynamic quantities such as the energy fluctuation, entropy production rate and dynamical activity. Such bound was already analyzed for Markovian evolution satisfying detailed balance condition. Here we generalize this approach to deal with arbitrary evolution governed by time-local generator. Our analysis is illustrated by three paradigmatic examples of qubit evolution: amplitude damping, pure dephasing, and the eternally non-Markovian evolution.
We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. We find two coarse grainings, one using measurements of local particle numbers and then total energy, and the second using local energy measurements, which lead to an entropy that is defined outside of equilibrium, is in accord with the thermodynamic entropy for equilibrium systems, and reaches the thermodynamic entropy in the long-time limit, even in genuinely isolated quantum systems. This answers the long-standing conceptual problem, as to which entropy is relevant for the formulation of the second thermodynamic law in closed quantum systems. This entropy could be in principle measured, especially now that experiments on such systems are becoming feasible.
We investigate fundamental connections between thermodynamics and quantum information theory. First, we show that the operational framework of thermal operations is nonequivalent to the framework of Gibbs-preserving maps, and we comment on this gap. We then introduce a fully information-theoretic framework generalizing the above by making further abstraction of physical quantities such as energy. It is technically convenient to work with and reproduces known results for finite-size quantum thermodynamics. With our framework we may determine the minimal work cost of implementing any logical process. In the case of information processing on memory registers with a degenerate Hamiltonian, the answer is given by the max-entropy, a measure of information known from quantum information theory. In the general case, we obtain a new information measure, the "coherent relative entropy", which generalizes both the conditional entropy and the relative entropy. It satisfies a collection of properties which justifies its interpretation as an entropy measure and which connects it to known quantities. We then present how, from our framework, macroscopic thermodynamics emerges by typicality, after singling out an appropriate class of thermodynamic states possessing some suitable reversibility property. A natural thermodynamic potential emerges, dictating possible state transformations, and whose differential describes the physics of the system. The textbook thermodynamics of a gas is recovered as well as the form of the second law relating thermodynamic entropy and heat exchange. Finally, noting that quantum states are relative to the observer, we see that the procedure above gives rise to a natural form of coarse-graining in quantum mechanics: Each observer can consistently apply the formalism of quantum information according to their own fundamental unit of information.
We investigate the detailed properties of observational entropy, introduced by \ifmmode \check{S}\else \v{S}\fi{}afr\'anek et al. [Phys. Rev. A 99, 010101 (2019)] as a generalization of Boltzmann entropy to quantum mechanics. This quantity can involve multiple coarse grainings, even those that do not commute with each other, without losing any of its properties. It is well-defined out of equilibrium, and for some coarse grainings it generically rises to the correct thermodynamic value even in a genuinely isolated quantum system. The quantity contains several other entropy definitions as special cases, it has interesting information-theoretic interpretations, and mathematical properties---such as extensivity and upper and lower bounds---suitable for an entropy. Here we describe and provide proofs for many of its properties, discuss its interpretation and connection to other quantities, and provide numerous simulations and analytic arguments supporting the claims of its relationship to thermodynamic entropy. This quantity may thus provide a clear and well-defined foundation on which to build a satisfactory understanding of the second thermodynamical law in quantum mechanics.