The increase in average energy of a quantum system undergoing projective energy measurements is referred to as "quantum heat", which is always zero. In the framework of quantum stochastic thermodynamics, this is constructed as the average over the fluctuating quantum heat (FQH), defined as the increase in expected value of the Hamiltonian along two-point eigenstate trajectories. However, such a definition has two drawbacks: (i) if the initial state does not commute with the Hamiltonian and has degeneracies, the higher moments of the FQH will not be uniquely defined, and therefore it is arguable whether such a quantity is physically meaningful; (ii) the definition is operationally demanding as it requires full knowledge of the initial state. In the present manuscript we show that the FQH is an instance of conditional increase in energy given sequential measurements, the first of which is with respect to the eigen-decomposition of the initial state. By coarse-graining this initial measurement, first by only distinguishing between degenerate subspaces of the state, and finally by not distinguishing between any subspace at all, we provide two alternative definitions for the FQH, which we call the partially coarse-grained FQH and fully coarse-grained FQH, respectively. The partially coarse-grained FQH resolves issue (i), whereas the fully coarse-grained FQH resolves both (i) and (ii).
The increase in average energy of a quantum system undergoing projective energy measurements is referred to as "quantum heat", which is always zero. In the framework of quantum stochastic thermodynamics, this is constructed as the average over the fluctuating quantum heat (FQH), defined as the increase in expected value of the Hamiltonian along two-point eigenstate trajectories. However, such a definition has two drawbacks: (i) if the initial state does not commute with the Hamiltonian and has degeneracies, the higher moments of the FQH will not be uniquely defined, and therefore it is arguable whether such a quantity is physically meaningful; (ii) the definition is operationally demanding as it requires full knowledge of the initial state. In the present manuscript we show that the FQH is an instance of conditional increase in energy given sequential measurements, the first of which is with respect to the eigen-decomposition of the initial state. By coarse-graining this initial measurement, first by only distinguishing between degenerate subspaces of the state, and finally by not distinguishing between any subspace at all, we provide two alternative definitions for the FQH, which we call the partially coarse-grained FQH and fully coarse-grained FQH, respectively. The partially coarse-grained FQH resolves issue (i), whereas the fully coarse-grained FQH resolves both (i) and (ii).
A complete thermodynamic analysis of a quantum measurement process necessitates a method for obtaining higher moments of thermodynamic quantities, such as work, heat, and entropy production. While such a method is provided by the eigenstate trajectory approach of stochastic quantum thermodynamics, conceptual difficulties are encountered when the quantum system being measured is degenerate; an infinite possibility of eigen-decompositions of the quantum state results in infinitely many sets of eigenstate trajectories, thereby precluding the unique assignment of higher moments in general. In the present manuscript we provide a solution to this problem for the case of ideal projective measurements, by ``coarse-graining'' the eigenstate trajectories, and modifying the definition of stochastic energy change. We show that the resulting statistics of work, heat, and entropy production are always uniquely defined, and reduce to those given by the eigenstate trajectory method when the latter does not result in ambiguities. Our proposed methodology thus paves the way towards a novel approach for studying the thermodynamic properties of degenerate quantum systems.
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 concept of work is basic for statistical thermodynamics. To gain a fuller understanding of work and its (quantum) features, it needs to be represented as an average of a fluctuating quantity. Here I focus on the work done between two moments of time for a thermally isolated quantum system driven by a time-dependent Hamiltonian. I formulate two natural conditions needed for the fluctuating work to be physically meaningful for a system that starts its evolution from a nonequilibrium state. The existing definitions do not satisfy these conditions due to issues that are traced back to noncommutativity. I propose a definition of fluctuating work that is free of previous drawbacks and that applies for a wide class of nonequilibrium initial states. It allows the deduction of a generalized work-fluctuation theorem that applies for an arbitrary (out-of-equilibrium) initial state.
Quantum measurement is ultimately a physical process, resulting from an interaction between the measured system and a measurement apparatus. Considering the physical process of measurement within a thermodynamic context naturally raises the following question: how can the work and heat resulting from the measurement process be interpreted? In the present manuscript, we model the measurement process for an arbitrary discrete observable as a measurement scheme. Here, the system to be measured is first unitarily coupled with a measurement apparatus, and subsequently the apparatus is measured by a pointer observable, thus producing a definite measurement outcome. The work can therefore be interpreted as the change in internal energy of the compound of system-plus-apparatus due to the unitary coupling. By the first law of thermodynamics, the heat is the subsequent change in internal energy of this compound due to the measurement of the pointer observable. However, in order for the apparatus to serve as a stable record for the measurement outcomes, the pointer observable must commute with the Hamiltonian, and its implementation must be repeatable. Given these minimal requirements, we show that the heat will necessarily be a classically fluctuating quantity.
In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.
In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.