Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources

This paper considers the question: How much work $\mathcal{W}$ is extracted when a quantum system S undergoes a cyclic thermodynamic process? The answer depends on details such as the duration of the process; whether or not the system exchanges energy with heat baths along the way; how the system is driven during the process; and the system’s initial state, ${\widehat{\rho }}_i$. We are specifically interested in the potential thermodynamic consequences of energy coherences—non-zero matrix elements $\langle m\left|{\widehat{\rho }}_i\right|n\rangle$ for eigenstates of different energies—in the initial state. The thermodynamic utility of such coherences has been investigated in recent years [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19], using a variety of approaches. Of particular relevance to the present paper, Kammerlander and Anders [9], using the definition of work [20,21] that we will use, have argued that if ${\widehat{\rho }}_i$ contains coherences in the system’s energy basis, then more work can be extracted than would be possible in the absence of coherences. In this sense, quantum energy coherences represent a thermodynamic resource.

It seems natural to view the presence of energy coherences in ${\widehat{\rho }}_i$ as a uniquely quantum thermodynamic resource, with no classical counterpart—in much the same way that superpositions of qubit states represent a quantum computational resource unavailable to classical computers [22]. We will argue otherwise. We will identify a classical analogue of quantum energy coherences, namely energy-shell inhomogeneities in the initial classical phase space distribution ${\rho }_i(\Gamma )$. We will show that the presence of such inhomogeneities in ${\rho }_i(\Gamma )$ allows more work to be extracted than would be possible in their absence. Thus, both quantum energy coherences and classical energy-shell inhomogeneities can be viewed as thermodynamic resources from which work can be extracted. We will further argue that for systems that support a well-defined semiclassical limit, a fair comparison reveals that equal amounts of work can be extracted from the two resources. We therefore conclude that quantum energy coherences do not provide a quantum “thermodynamic advantage”, as the same gain can be obtained from classical energy-shell inhomogeneities.

In Section 2, we introduce the framework and notation we will use to study a quantum system undergoing a cyclic thermodynamic process, in the presence of a thermal reservoir, and we analyze the work that can be extracted from energy coherences during such a process. In Section 3, we introduce the analogous classical framework and analyze the work that can be extracted from energy-shell inhomogeneities. In Section 4, we argue that when a fair comparison is made, the maximum amount of work that can be extracted in the quantum case is the same as that in the classical case. In Section 5, we extend these results to a broader class of processes. We conclude with a brief discussion in Section 6.

Throughout this paper, we will adopt an ensemble perspective, in which the state of an open quantum system is specified by a density matrix $\widehat{\rho }$, and the state of a classical system is specified by a phase space distribution $\rho (\Gamma )$ rather than a phase point $\Gamma$.

Let S denote a quantum system of interest, and $\widehat{H}$ its Hamiltonian. We consider the following situation, illustrated schematically in Figure 1: S is prepared in an initial state ${\widehat{\rho }}_i$ at time $t=0$, then from $t=0$ to $\tau$ it evolves in time as its Hamiltonian is varied according to a schedule, or protocol, $\widehat{H}(t)$. We take this process to be cyclic, in the sense that where ${\widehat{H}}_0$ is a fixed reference Hamiltonian. We then ask the question: How much work is extracted during this cyclic process?

We assume the reference Hamiltonian ${\widehat{H}}_0$ has a discrete, non-degenerate spectrum with eigenstates $\left|n\right.〉$ and eigenvalues ${ϵ}_n$. The assumption of non-degeneracy ensures an unambiguously defined energy basis in which coherence can be considered. It further implies that no operators commute with ${\widehat{H}}_0$, aside from ones that are functions of ${\widehat{H}}_0$ itself: for some scalar function $k(·)$ of a single variable.

During the cyclic process described above, the system is in contact with a thermal bath B, at temperature ${\beta }^{-1}$. As a result, the evolution of S is not unitary, rather, we will say that S evolves under isothermal dynamics. This terminology is not meant to suggest that the system’s temperature is constant, or even well-defined, merely that the system is in contact with a bath whose bulk temperature ${\beta }^{-1}$ is well-defined. We will not specify the equations of motion for the system, as our discussion will be relatively insensitive to the exact dynamics used to model the system’s evolution. However, we will demand that the isothermal dynamics of S satisfy the following thermodynamically motivated conditions: (1) if $\widehat{H}$ is held fixed then the system relaxes to the canonical equilibrium state, and (2) the dynamics support a generalized second law linking suitably defined notions of free energy and work.

More precisely, condition (1) means that if $\widehat{H}$ is fixed, then the isothermal dynamics cause the system to relax to the equilibrium state where are the partition function and free energy associated with this state. (The superscript q stands for “quantum” and distinguishes this case from the classical setup that will be introduced later. The dependence of $Z^q$ and ${\mathcal{F}}^{q,\mathrm{eq}}$ on $\beta$ is notationally suppressed.) We assume this relaxation occurs over a finite characteristic timescale ${\tau }_{rel}$. As a consequence, if the system Hamiltonian is varied quasistatically, then the state of S tracks the instantaneous equilibrium state: $\widehat{\rho }(t)=\widehat{\pi }(t)$, where $\widehat{\pi }(t)$ is the canonical state associated with $\widehat{H}(t)$. In this quasistatic limit, the system’s evolution is isothermal in the strong sense of the word: its temperature is well-defined and constant at all times. A system that evolves under a detailed balanced Lindblad master equation satisfies condition (1) [23].

By condition (2), we mean that the system obeys a generalized second law where the work extracted, non-equilibrium free energy, internal energy, and entropy are respectively defined by the following functional and functions of $\widehat{\rho }(t)$ and $\widehat{H}(t)$: For convenience, as in Equation (5), we will often use the shorthand $\mathcal{X}(t)\equiv \mathcal{X}(\widehat{\rho }(t),\widehat{H}(t))$, or the even more concise ${\mathcal{X}}_i=\mathcal{X}(0)$ and ${\mathcal{X}}_f=\mathcal{X}(\tau )$, where $\mathcal{X}$ stands for ${\mathcal{F}}^q$, ${\mathcal{U}}^q$, or ${\mathcal{S}}^q$, or the classical counterparts of these quantities, defined below in Section 3.

Equations (7)–(9) generalize familiar equilibrium notions [24] of free energy, internal energy, and entropy to non-equilibrium states $\widehat{\rho }$ [25]. They reduce to the usual equilibrium values when $\widehat{\rho }=\widehat{\pi }$. The bound given by Equation (5) is not restricted to transitions between equilibrium states, and has been derived using a variety of approaches for modeling the dynamics of a quantum system in contact with a thermal reservoir, see, e.g., Refs. [26,27,28,29,30]. Note that we follow engineering convention and work extraction is positive. While Equation (6) should be interpreted as the average work extracted from an ensemble, fluctuations will not be considered in this paper, hence we will simply refer to Equation (6) as extracted work.

The non-equilibrium free energy defined by Equation (7) can equivalently be written as where $\widehat{\pi }$ and ${\mathcal{F}}^{q,\mathrm{eq}}$ are given by Equations (3) and (4), and is the quantum relative entropy, or Kullback–Leibler divergence [31], between arbitrary states ${\widehat{\rho }}_1$ and ${\widehat{\rho }}_2$. For a cyclic process, as defined above, Equation (5) becomes where ${\widehat{\rho }}_{i,f}$ are the states of the system at $t=0,\tau$, and ${\widehat{\pi }}_0$ is the equilibrium state associated with the reference Hamiltonian ${\widehat{H}}_0$. Although relative entropy $D({\widehat{\rho }}_1|{\widehat{\rho }}_2)$ is not a proper distance measure, it vanishes when ${\widehat{\rho }}_1={\widehat{\rho }}_2$ and is strictly positive otherwise, and can be viewed as quantifying the degree to which ${\widehat{\rho }}_1$ differs from ${\widehat{\rho }}_2$. In this sense, Equation (12) implies that the extracted work is bounded from above by the degree to which the system is brought closer to the equilibrium state ${\widehat{\pi }}_0$, during the cyclic process. This interpretation is in agreement with the intuition, from classical thermodynamics, that non-equilibrium states represent a thermodynamic resource: work can be extracted by cleverly facilitating a system’s evolution toward equilibrium.

We take Equation (6) as our definition of work for several reasons. First, it is an established notion of thermodynamic work in quantum systems [20,21,32]. Moreover, it agrees with the notion of average work derived from the quantum work (quasi)distribution in Ref. [33], which satisfies a fluctuation theorem. Finally, this definition closely resembles those used in classical stochastic thermodynamics [34,35] and, as we will see in later sections, it allows us to establish connections with results from classical statistical physics. For the special case of isolated quantum systems, the definition given by Equation (6) is called “untouched work” in Ref. [36]. We will not discuss here how (or whether) Equation (6) connects to the traditional thermodynamic concept of raising a mass against gravity, or otherwise delivering energy to a work reservoir [24]; this question involves subtle issues related to backaction as well as potential quantum coherences in the work reservoir.

We note that other definitions of work are also commonly used in quantum thermodynamics, particularly when fluctuations in work are of interest. For instance, defining a work distribution according to the two-time energy measurement protocol [37,38,39] leads to a mean value that differs from Equation (6) whenever the initial state ${\widehat{\rho }}_i$ has non-vanishing energy coherences. Additionally, some definitions of work developed in quantum resource theory [40] have a so-called work-locking property [41] which prevents the extraction of work from coherence. These resource theory definitions, which explicitly model the heat bath and demand that work be transferred deterministically, also differ from Equation (6).

Now, imagine a classical system with N degrees of freedom and phase space variables Adopting (as in the quantum case) an ensemble perspective, let the system’s state at time t be described by a phase space density $\rho (\Gamma ,t)$. We will consider a thermodynamic process in which the system begins in a state $\rho (\Gamma ,0)={\rho }_i(\Gamma )$, then evolves from $t=0$ to $\tau$ as its Hamiltonian is varied according to a cyclic protocol $H(\Gamma ,t)$, with where $H_0(\Gamma )$ specifies a reference Hamiltonian; see Figure 2. We assume that no observables commute with $H_0(\Gamma )$ under the Poisson bracket, except those that are functions of $H_0$: for some function $k(·)$ (compare with Equation (2)). This assumption implies that energy is the only non-trivially conserved quantity along all trajectories $\Gamma (t)$ obeying Hamiltonian dynamics $d\Gamma /dt=\{\Gamma ,H_0\}$. (This conclusion follows from the identity $(d/dt)A(\Gamma (t))=\{A,H_0\}$, which applies to any observable $A(\Gamma )$ and any trajectory $\Gamma (t)$ obeying $d\Gamma /dt=\{\Gamma ,H_0\}$). This is a necessary but not sufficient condition for the dynamics to be ergodic on constant-energy surfaces in phase space—an assumption often made in statistical physics. (Roughly speaking, ergodicity means that a generic Hamiltonian trajectory of energy E visits all regions of the surface $H_0=E$, given sufficient time.) For our purposes, we do not need the assumption of ergodicity, only the weaker assumption given by Equation (24).

If the system were thermally isolated, then its state $\rho (\Gamma ,t)$ would evolve under the Liouville equation, $\partial \rho /\partial t=\{H,\rho \}$. However, we assume that the system is in contact with a thermal bath as it undergoes the cyclic process, hence its evolution follows classical isothermal dynamics, rather than Hamiltonian dynamics. As in the quantum case, we will not specify the equations of motion that describe the isothermal dynamics, but we will make the following assumptions.

(1) If the system’s Hamiltonian is held fixed, then the isothermal dynamics drive the system to the equilibrium state with partition function and free energy Here, h is a constant with dimensions of action that ensures the argument of the logarithm is dimensionless. We choose h to coincide with Planck’s constant as this will facilitate comparisons of quantum and classical work extraction in Section 4. We assume this relaxation takes place over a finite timescale ${\tau }_{\mathrm{rel}}$. As a consequence, if $H(\Gamma ,t)$ is varied quasistatically, then the system’s state follows the instantaneous equilibrium state, $\rho (\Gamma ,t)=\pi (\Gamma ,t)$.

(2) When the system evolves over a time interval $0\le t\le \tau$ under isothermal dynamics and a time-dependent Hamiltonian $H(\Gamma ,t)$, it obeys a generalized second law with

Unlike the quantum von Neumann entropy (9) which is always non-negative, the classical Shannon differential (or continuous) entropy (31) can become arbitrarily negative for probability distributions that are highly concentrated in phase space, as we will see in Section 4.1. (For brevity, we will henceforth refer to the Shannon differential entropy simply as the Shannon entropy.) As with the quantum bound (Equation (5)), Equation (27) is not restricted to transitions between equilibrium states, and has been derived under a variety of modeling approaches, see, e.g., Refs. [26,27,28,29,30,42,43,44].

The classical non-equilibrium free energy (29) can be rewritten as where is the classical relative entropy or Kullback–Leibler divergence. Thus, for a cyclic process, Equation (27) becomes where ${\rho }_i(\Gamma )=\rho (\Gamma ,0)$ and ${\rho }_f(\Gamma )=\rho (\Gamma ,\tau )$ are the system’s initial and final states, and ${\pi }_0(\Gamma )$ is the equilibrium state for the reference Hamiltonian. As in the quantum case (Equation (12)), the right side of Equation (34) provides a measure of the degree to which the process brings the system closer to equilibrium.

We have seen that the maximum work extracted in the quantum case, subject to the isoenergetic constraint, $\mathrm{diag}{\widehat{\rho }}_i=\mathrm{diag}{\widehat{\rho }}_f$, is achieved by quasistatically removing all energy coherences from the system’s initial state: ${\widehat{\rho }}_i\to \mathrm{diag}{\widehat{\rho }}_i$. Similarly, the maximum work extracted in the classical case is achieved by quasistatically removing all energy-shell inhomogeneities. The optimized work values ${\mathcal{W}}^{q🟉}$ and ${\mathcal{W}}^{c🟉}$ are given by Equations (21) and (47). The close similarity between these results supports our view that classical energy-shell inhomogeneities are thermodynamic counterparts of quantum energy coherences. Both are resources that can be leveraged to extract work.

While the expressions for ${\mathcal{W}}^{q🟉}$ and ${\mathcal{W}}^{c🟉}$ are nearly identical, it still remains to compare them quantitatively. Ideally, we would like to compare the values of ${\mathcal{W}}^{q🟉}$ and ${\mathcal{W}}^{c🟉}$ for a given quantum reference Hamiltonian ${\widehat{H}}_0$ and initial state ${\widehat{\rho }}_i$, and appropriately defined classical counterparts $H_0(\Gamma )$ and ${\rho }_i(\Gamma )$. To this end, throughout this section and the next we assume that ${\widehat{H}}_0$ is a function of position and momentum operators $({\widehat{x}}_1,\cdots {\widehat{x}}_N)$ and $({\widehat{p}}_1,\cdots {\widehat{p}}_N)$, and we further assume that ${\widehat{H}}_0$ has a well-defined counterpart $H_0(\Gamma )$. This condition is satisfied, for instance, by Hamiltonians of the kinetic-plus-potential form ${\widehat{H}}_0=K({\widehat{p}}_1,\cdots {\widehat{p}}_N)+V({\widehat{x}}_1,\cdots {\widehat{x}}_N)$, for which $H_0(\Gamma )$ is obtained by replacing momentum and position operators with classical momentum and position variables.

Identifying a correspondence between quantum and classical states $\widehat{\rho }$ and $\rho (\Gamma )$ is trickier. Common approaches that map density operators into phase space distributions [45,46] suffer from undesirable properties. For instance, neither the Wigner [47] nor Husimi [48] function representation of the quantum thermal state corresponds to the classical thermal phase space distribution. Additionally, the Wigner function in general can become negative while the Husimi function depends on the choice of coherent states.

To circumvent such issues, we will compare quantum and classical energy distributions rather than individual states. Instead of focusing on the maximum work that can be extracted from a particular initial state, we will consider the maximum work that can be extracted given a particular initial energy distribution. We begin by defining energy equivalence classes in Section 4.1, then in Section 4.2 and Section 4.3 we compare maximum work values for corresponding quantum and classical energy equivalence classes.

In the previous sections, we have imposed the isoenergetic constraint, namely that the initial and final energy distributions are identical (Equations (13) and (42)). Let us now drop this constraint and pose the following question. For a quantum or classical system described by an initial Hamiltonian ${\widehat{H}}_0$ or $H_0(\Gamma )$, in the presence of a thermal bath at temperature ${\beta }^{-1}$, what is the maximum work that can be extracted during a cyclic process if the energy distribution of the initial state is determined by a given function $\kappa (·)$?

In the quantum case, we first let ${\mathcal{W}}^{q†}({\widehat{\rho }}_i)$ denote the maximum work extracted for a given initial state ${\widehat{\rho }}_i$—this quantity is analogous to ${\mathcal{W}}^{🟉}({\widehat{\rho }}_i)$ (Section 2) but without the constraint $\mathrm{diag}{\widehat{\rho }}_f=\mathrm{diag}{\widehat{\rho }}_i$. From Equations (5) and (10) and the non-negativity of the Kullback–Leibler divergence, we have where the inequality on the first line is valid for any final state ${\widehat{\rho }}_f$, and ${\mathcal{F}}_0^{q,\mathrm{eq}}\equiv {\mathcal{F}}^{q,\mathrm{eq}}({\widehat{H}}_0)$. As shown in Appendix A, the bound obtained in Equation (73) is saturated by the protocol where $\lambda \equiv t/\tau$ and the process is quasistatic: $\tau \to \infty$. (Note that there is no quench at $t=\tau$.) Since the bound can be saturated, and ${\mathcal{W}}^{q†}({\widehat{\rho }}_i)$ was defined as the maximum work that can be extracted, we simply write