*Article* **Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources**

**Andrew Smith 1, Kanupriya Sinha 2,3 and Christopher Jarzynski 1,4,5,\***


**Abstract:** Quantum energy coherences represent a thermodynamic resource, which can be exploited to extract energy from a thermal reservoir and deliver that energy as work. We argue that there exists a closely analogous classical thermodynamic resource, namely, energy-shell inhomogeneities in the phase space distribution of a system's initial state. We compare the amount of work that can be obtained from quantum coherences with the amount that can be obtained from classical inhomogeneities, and find them to be equal in the semiclassical limit. We thus conclude that coherences do not provide a unique thermodynamic advantage of quantum systems over classical systems, in situations where a well-defined semiclassical correspondence exists.

**Keywords:** quantum thermodynamics; quantum coherence; work extraction

**1. Introduction**

This paper considers the question: How much work 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, *ρ* ˆ *i*. We are specifically interested in the potential thermodynamic consequences of *energy coherences*—non-zero matrix elements *m*|*ρ*<sup>ˆ</sup>*i*|*n*- for eigenstates of different energies—in the initial state. The thermodynamic utility of such coherences has been investigated in recent years [1–19], 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 *ρ*ˆ*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 *ρ*ˆ*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 *ρi*(Γ). We will show that the presence of such inhomogeneities in *ρi*(Γ) 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

**Citation:** Smith, A.; Sinha, K.; Jarzynski, C. Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources. *Entropy* **2022**, *24*, 474. https://doi.org/10.3390/e24040474

Academic Editor: Ronnie Kosloff

Received: 8 March 2022 Accepted: 24 March 2022 Published: 29 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

<sup>1</sup> Department of Physics, University of Maryland, College Park, MD 20742, USA; andrew.maven.smith@gmail.com

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 *ρ*ˆ, and the state of a classical system is specified by a phase space distribution *ρ*(Γ) rather than a phase point Γ.

#### **2. Quantum Setup and Notation**

Let *S* denote a quantum system of interest, and *H*ˆ its Hamiltonian. We consider the following situation, illustrated schematically in Figure 1: *S* is prepared in an initial state *ρ*ˆ*i* at time *t* = 0, then from *t* = 0 to *τ* it evolves in time as its Hamiltonian is varied according to a schedule, or *protocol*, *H*ˆ (*t*). We take this process to be cyclic, in the sense that

$$
\hat{H}(0) = \hat{H}(\pi) = \hat{H}\_0 \tag{1}
$$

where *H*ˆ 0 is a fixed *reference Hamiltonian*. We then ask the question: How much work is extracted during this cyclic process?

**Figure 1.** Schematic illustration of the quantum process described in the text. The system begins in state *ρ*<sup>ˆ</sup>*i*, then evolves in contact with a thermal bath to a final state *ρ*ˆ*f* as the Hamiltonian is driven through a cycle from *H*ˆ (0) = *H*ˆ 0 to *H*ˆ (*τ*) = *H*ˆ 0. We impose the constraint diag *ρ*ˆ*i* = diag *ρ*ˆ*f* , which indicates that the initial and final energy distributions are identical, while the coherences may differ.

We assume the reference Hamiltonian *H*ˆ 0 has a discrete, non-degenerate spectrum with eigenstates |*n*- and eigenvalues *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 *H*ˆ 0, aside from ones that are functions of *H*ˆ 0 itself:

$$[\mathcal{K}, \hat{H}\_0] = 0 \text{ iff } \hat{\mathcal{K}} = k(\hat{H}\_0) \tag{2}$$

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 *β*−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 sugges<sup>t</sup> that the system's temperature is constant, or even well-defined, merely that the system is in contact with a bath whose bulk temperature *β*−<sup>1</sup> 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 *H*ˆ 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 *H* ˆ is fixed, then the isothermal dynamics cause the system to relax to the equilibrium state

$$
\hat{\pi} = \frac{1}{Z^{\emptyset}} e^{-\beta \hat{H}} \tag{3}
$$

where

$$Z^q(\hat{H}) = \text{Tr} \, e^{-\beta \hat{H}} \quad , \quad \mathcal{F}^{q, \text{eq}}(\hat{H}) = -\beta^{-1} \ln Z^q(\hat{H}) \tag{4}$$

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 *Zq* and F*q*,eq on *β* is notationally suppressed.) We assume this relaxation occurs over a finite characteristic timescale *τrel*. As a consequence, if the system Hamiltonian is varied quasistatically, then the state of *S* tracks the instantaneous equilibrium state: *ρ*<sup>ˆ</sup>(*t*) = *<sup>π</sup>*<sup>ˆ</sup>(*t*), where *π*<sup>ˆ</sup>(*t*) is the canonical state associated with *H*ˆ (*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

$$\mathcal{W}^{\eta} \le -\Delta \mathcal{F}^{\eta} = \mathcal{F}^{\eta}(0) - \mathcal{F}^{\eta}(\tau) \tag{5}$$

where the work *extracted*, non-equilibrium free energy, internal energy, and entropy are respectively defined by the following functional and functions of *ρ*<sup>ˆ</sup>(*t*) and *H*ˆ (*t*):

$$\mathcal{W}^q[\not p(t), \not H(t)] \quad = \quad - \int\_0^\tau \text{Tr}\left[\frac{d\not H}{dt}\not p\right] dt \tag{6}$$

$$\mathcal{F}^{\eta}(\mathfrak{H}, \hat{\mathcal{H}}) \quad = \quad \mathcal{U}^{\eta} - \mathcal{S}^{\eta} / \mathcal{B} \tag{7}$$

U*<sup>q</sup>*(*ρ*ˆ, *H*ˆ ) = Tr[*H*<sup>ˆ</sup> *ρ*ˆ] (8)

$$\mathcal{S}^q(\not p) \quad = \ -\text{Tr}[\not p \ln \not p] \ge 0. \tag{9}$$

For convenience, as in Equation (5), we will often use the shorthand X (*t*) ≡ X (*ρ*<sup>ˆ</sup>(*t*), *H*ˆ (*t*)), or the even more concise X*i* = X (0) and X*f* = X (*τ*), where X stands for F*q*, U*q*, or 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 *ρ*ˆ [25]. They reduce to the usual equilibrium values when *ρ*ˆ = *π*ˆ. 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. [26–30]. 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

$$\mathcal{F}^{\mathbb{q}}(\boldsymbol{\beta}, \boldsymbol{\hat{H}}) = \mathcal{F}^{\mathbb{q}, \text{eq}}(\boldsymbol{\hat{H}}) + \boldsymbol{\beta}^{-1}D(\boldsymbol{\beta}|\boldsymbol{\hat{\pi}}) \tag{10}$$

where *π*ˆ and F*q*,eq are given by Equations (3) and (4), and

$$D(\hat{\rho}\_1|\hat{\rho}\_2) = \text{Tr}[\hat{\rho}\_1(\ln \hat{\rho}\_1 - \ln \hat{\rho}\_2)] \ge 0 \tag{11}$$

is the quantum relative entropy, or Kullback–Leibler divergence [31], between arbitrary states *ρ*ˆ1 and *ρ*ˆ2. For a cyclic process, as defined above, Equation (5) becomes

$$\mathcal{W}^q \le \mathcal{S}^{-1} \left[ D(\mathfrak{H}\_i|\hat{\pi}\_0) - D(\mathfrak{H}\_f|\hat{\pi}\_0) \right] \tag{12}$$

where *ρ*ˆ*i*, *f* are the states of the system at *t* = 0, *τ*, and *π*ˆ 0 is the equilibrium state associated with the reference Hamiltonian *H*ˆ 0. Although relative entropy *<sup>D</sup>*(*ρ*ˆ1|*ρ*ˆ2) is not a proper distance measure, it vanishes when *ρ*ˆ1 = *ρ*ˆ2 and is strictly positive otherwise, and can be viewed as quantifying the degree to which *ρ*ˆ1 differs from *ρ*ˆ2. 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 *π*ˆ 0, during the cyclic process. This interpretation is in agreemen<sup>t</sup> 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 [37–39] leads to a mean value that differs from Equation (6) whenever the initial state *ρ*ˆ*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).
