We present a partition of quantum observables in an open quantum system that is inherited from the division of the underlying Hilbert space or configuration space. It is shown that this partition leads to the definition of an inhomogeneous continuity equation for generic, non-local observables. This formalism is employed to describe the local evolution of the von Neumann entropy of a system of independent quantum particles out of equilibrium. Crucially, we find that all local fluctuations in the entropy are governed by an entropy current operator, implying that the production of entanglement entropy is not measured by this partitioned entropy. For systems linearly perturbed from equilibrium, it is shown that this entropy current is equivalent to a heat current, provided that the system-reservoir coupling is partitioned symmetrically. Finally, we show that any other partition of the coupling leads directly to a divergence of the von Neumann entropy. Thus, we conclude that Hilbert-space partitioning is the only partition of the von Neumann entropy that is consistent with the laws of thermodynamics.
We present a partition of quantum observables in an open quantum system that is inherited from the division of the underlying Hilbert space or configuration space. It is shown that this partition leads to the definition of an inhomogeneous continuity equation for generic, non-local observables. This formalism is employed to describe the local evolution of the von Neumann entropy of a system of independent quantum particles out of equilibrium. Crucially, we find that all local fluctuations in the entropy are governed by an entropy current operator, implying that the production of entanglement entropy is not measured by this partitioned entropy. For systems linearly perturbed from equilibrium, it is shown that this entropy current is equivalent to a heat current, provided that the system-reservoir coupling is partitioned symmetrically. Finally, we show that any other partition of the coupling leads directly to a divergence of the von Neumann entropy. Thus, we conclude that Hilbert-space partitioning is the only partition of the von Neumann entropy that is consistent with the laws of thermodynamics.
Local properties of a quantum system are defined as the expectation values of its observables in a microstate of some complete set of commuting observables. An equation for the time evolution of local properties is obtained for any system whose statistical operator (density matrix) evolves by unitary transformation in accordance with the von Neumann equation. The formalism is applied to the example of a system of one particle. In this case the local properties are fields, the time-evolution equation is an equation of continuity with source terms. For constants of motion the source terms vanish, giving equations of continuity for the fields. For each scalar field a flux vector for its transport current is defined. For momentum, a stress tensor is obtained. The effects on local properties of realization of a latent ensemble of the statistical operator (an entropy-increasing mechanism recently proposed to explain approach to equilibrium) are also considered; a non-negative local entropy production is identified, as well as a discontinuous redistribution of local properties among microstates.
A key question in the thermodynamics of open quantum systems is how to partition thermodynamic quantities such as entropy, work, and internal energy between the system and its environment. We show that the only partition under which entropy is nonsingular is based on a partition of Hilbert space, which assigns half the system-environment coupling to the system and half to the environment. However, quantum work partitions nontrivially under Hilbert-space partition, and we derive a work sum rule that accounts for quantum work at a distance. All state functions of the system are shown to be path independent once this nonlocal quantum work is properly accounted for. Our results are illustrated with application to a driven resonant level strongly coupled to a reservoir.
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
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.
The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: ? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.
The von Neumann entropy rate for open quantum systems is, in general, written in terms of entropy production and entropy flow rates, encompassing the second law of thermodynamics. For memoryless open quantum systems, whose evolutions correspond to one-parameter quantum dynamical semigroups, we find a decomposition of the infinitesimal generator of the dynamics that unveils the fluctuation-dissipation structure of these evolutions. Thus, the fluctuation part of this decomposition allows to relate the von Neumann entropy rate with the divergence-based quantum Fisher information, at any time. Applied to continuous-in-time quantum Gaussian channels, our decomposition leads to the quantum analog of the generalized classical de Bruijn identity, thus expressing the quantum fluctuation-dissipation relation in that kind of channels. Finally, from this perspective, we analyze how stationarity arises.