Introducing one-shot work into fluctuation relations

Consider a classical system that is coupled to a heat bath and driven externally. Due to the probabilistic nature of thermalization, the amount of heat transferred between the system and the bath in any given trial cannot be predicted. Hence the amount of work done by the drive, in any given trial, cannot be predicted. The protocol can be associated with a work distribution P(W), the probability density associated with some trial's costing an amount W of work. In equilibrium thermodynamics, P(W) peaks tightly at the average value $W=\langle W\rangle$. This value suffices to describe the work performed in each trial. In more general, nonequilibrium, thermodynamics, the average does not suffice. Yet thermodynamic state variables related to averages (temperature, free energy, etc) are used in nonequilibrium thermodynamics. Fluctuation relations link these variables to probability distributions over work or heat. We will focus mostly on continuous variables W, which have been used in classical and quantum contexts⁴ (e.g., [6, 8, 10, 15]).

One such fluctuation relation is Crooks' theorem [2]. Though originally derived in a classical setting, it has been shown to govern quantum processes [3–12, 14, 15]. Crooks' theorem describes the fluctuations in the work expended on systems subject to a time-changing Hamiltonian H(λ_t) in the presence of a heat bath. An experimenter can change the external scalar parameter λ_t during the time interval $t\in [-\tau ,\tau ]$ by performing work. We denote the bath's inverse temperature by $\beta =1/{k}_{{\rm{B}}}T$. The external driving can be performed in either a forward or a reverse direction. The forward process begins at time −τ, when the system occupies the thermal state ${{\rm{e}}}^{-\beta {H}_{-\tau }}/{Z}_{-\tau }$ of the initial Hamiltonian ${H}_{-\tau }\equiv H({\lambda }_{-\tau })$. The reverse process begins at time τ, when the system occupies the thermal state ${{\rm{e}}}^{-\beta {H}_{\tau }}/{Z}_{\tau }$ associated with the final Hamiltonian ${H}_{\tau }\equiv H({\lambda }_{\tau })$ of the forward protocol. (${Z}_{\pm \tau }$ denote normalization factors.)

Suppose that an agent implements the protocol in both directions many times, measuring the work invested in each forward trial and the work extracted from each reverse. Two probability distributions encapsulate these measurements: ${P}_{\mathrm{fwd}}(W)$ denotes the probability that some forward trial will require work W (or the probability per unit work, if ${P}_{\mathrm{fwd}}$ denotes a probability density), and ${P}_{\mathrm{rev}}(-W)$ denotes the probability that some reverse trial will output work W.

If the system's interactions with the bath are Markovian and microscopically reversible [33], and if the initial state is thermal, the work probability distributions satisfy Crooks' theorem [2],

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{\mathrm{fwd}}(W)}{{P}_{\mathrm{rev}}(-W)}={{\rm{e}}}^{\beta (W-{\rm{\Delta }}F)}.\end{eqnarray} \tag{ 1 }$$

This ${\rm{\Delta }}F$ denotes the difference $F({{\rm{e}}}^{-\beta {H}_{\tau }}/{Z}_{\tau })-F({{\rm{e}}}^{-\beta {H}_{-\tau }}/{Z}_{-\tau })$ between the Helmholtz free energies of thermal states over the forward process's final and initial Hamiltonians.

Multiplying each side of Crooks' theorem by ${P}_{\mathrm{rev}}(-W){{\rm{e}}}^{-\beta W}$ and integrating over W yields Jarzynski's equality [1],

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta W}\rangle }_{\mathrm{fwd}}={{\rm{e}}}^{-\beta {\rm{\Delta }}F},\end{eqnarray} \tag{ 2 }$$

wherein ${\langle .\rangle }_{\mathrm{fwd}}$ denotes an expectation value calculated from ${P}_{\mathrm{fwd}}$. Applied to a work distribution P(W) constructed from simulations or experiments, Jarzynski's equality can be used to calculate the equilibrium quantity ${\rm{\Delta }}F$. Combined with Jensen's inequality, $\langle {{\rm{e}}}^{x}\rangle \geqslant {{\rm{e}}}^{\langle x\rangle }$, Jarzynski's equality implies a lower bound on the average work required to complete a trial. This bound, $\langle W\rangle \geqslant {\rm{\Delta }}F$, has been considered a statement of the Second Law of Thermodynamics [34].

The left-hand side (lhs) of Jarzynski's equality has been recognized as the characteristic function, or Fourier transform, of ${P}_{\mathrm{fwd}}(W)$ [6, 15]. If $u={\rm{i}}\beta$ denotes the variable conjugate to W, the characteristic function is

$$\begin{eqnarray}&&{\chi }_{\mathrm{fwd}}(\beta )\equiv {\mathcal{F}}\{{P}_{\mathrm{fwd}}(W)\}\equiv {\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{W}{{P}}_{\mathrm{fwd}}(W){{\rm{e}}}^{{\rm{i}}{uW}}={\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{W}{{P}}_{\mathrm{fwd}}(W){{\rm{e}}}^{-\beta W}.\end{eqnarray} \tag{ 3 }$$

In terms of the characteristic function, Jarzynski's equality reads

$$\begin{eqnarray}&&{\chi }_{\mathrm{fwd}}(\beta )={{\rm{e}}}^{-\beta {\rm{\Delta }}F}.\end{eqnarray} \tag{ 4 }$$

The reverse process corresponds to the characteristic function ${\chi }_{\mathrm{rev}}(\beta )\equiv {\displaystyle \int }_{-\infty }^{\infty }{\rm{d}}{W}{{P}}_{\mathrm{rev}}(W){{\rm{e}}}^{-\beta W}$, in terms of which ${\chi }_{\mathrm{rev}}(\beta )={{\rm{e}}}^{\beta {\rm{\Delta }}F}$.

Mean values do not necessarily reflect a system's typical behavior. Consider a system that must output at least some threshold amount of work to trigger another process. One such threshold is the activation energy required to begin a chemical reaction. The system might output below-threshold work usually but far-above-threshold work occasionally. The average work might exceed the threshold, but the second process is usually not triggered.

By spotlighting statistics other than the mean, one-shot information theory extends idealized protocols implemented $n\to \infty$ times to realistic finite-n protocols that might fail. Conventional statistical mechanics describes the optimal rate at which work can be extracted asymptotically. Consider transforming n copies of one equilibrium state into n copies of another quasistatically, in the presence of a temperature-T heat bath. In the asymptotic, or thermodynamic, limit as $n\to \infty$, the average work required per copy approaches the difference ${\rm{\Delta }}F$ between the states' free energies. The free energy depends on the Shannon entropy. In reality, states are transformed finitely many times, and realistic processes have probabilities δ of failing to accomplish their purposes. Finite-n work-consumption rates have been quantified with one-shot entropies [17, 20–23]. So have the efficiencies of finite-n data compression, randomness extraction, quantum key distribution, and hypothesis testing [16, 35–37].

One one-shot entropy is the order-$\infty$ Rényi entropy ${H}_{\infty }({\mathcal{P}})$, known also as the min-entropy. For any discrete probability distribution ${\mathcal{P}}$ whose greatest element is ${{\mathcal{P}}}^{\mathrm{max}}$,

$$\begin{eqnarray}&&{H}_{\infty }({\mathcal{P}})\equiv -\mathrm{log}({{\mathcal{P}}}^{\mathrm{max}}).\end{eqnarray} \tag{ 5 }$$

(All logarithms in this article are base-e.) We will discuss two popular models in which one-shot entropies are applied to thermodynamics: work-extraction games and thermodynamic resource theories.

2.2.1. Work-extraction game

In the work-extraction game described by Egloff et al [18], a player transforms a system in a state ρ, governed by a Hamiltonian ${H}_{\rho }$, into a state σ governed by ${H}_{\sigma }$ : $(\rho ,{H}_{\rho })\mapsto (\sigma ,{H}_{\sigma })$. For simplicity, we take a quasiclassical model such that states are assumed to commute with their Hamiltonians. The agent has access to a temperature-T heat bath.

The player should choose an optimal strategy to maximize the transformation's work output (or minimize the transformation's work cost). The strategy consists of a sequence of operations of two types: (1) without investing work, the player can couple the system to the bath in any manner modeled by a stochastic matrix that preserves the Gibbs state ${{\rm{e}}}^{-\beta {H}_{\rho }}/{Z}_{\rho }$. (Such thermalization models are discussed in appendix A.) (2) By investing or extracting work, the agent can shift the Hamiltonian's levels.

The primary result in [18] implies an upper bound on the work extractable (up to a probability δ of failure) during the transformation $(\rho ,{H}_{\rho })\mapsto (\sigma ,{H}_{\sigma })$. Egloff et al show that the optimal strategy has a probability $1-\delta$ of outputting at least the work

$$\begin{eqnarray}&&{w}_{\mathrm{best}}^{\delta }\left(\rho ,{H}_{\rho }\mapsto \sigma ,{H}_{\sigma }\right)=T\mathrm{log}\left(M\left(\displaystyle \frac{{G}^{T}(\rho )}{1-\delta }\ \parallel \ {G}^{T}(\sigma )\right)\right)\end{eqnarray} \tag{ 6 }$$

in each trial. G^T denotes Gibbs-rescaling relative to the temperature T. Gibbs-rescaling facilitates the comparison of the work values of states governed by different Hamiltonians. A state can have the capacity to perform work due to the state's information content (e.g., because the state is pure) and energy contents (e.g., because the state has weight on high energy levels). Gibbs-rescaling recasts the state's work capacity as entirely informational. This recasting enables us to compare the final and initial states, even though different Hamiltonians govern them. M denotes the relative mixedness, a measure of how much more mixed one state is, or how much less information-sourced work capacity a state has. Dissipative processes yield less than the optimal amount ${w}_{\mathrm{best}}^{\delta }$ of work. Hence equation (6) upper-bounds the amount that could be extracted with an arbitrary (possibly suboptimal) strategy.

2.2.2. Thermodynamic resource theories

Suppose we consider the behavior of a system evolving under a process that is driven by a single external parameter, and otherwise satisfies the conditions for Crooks' theorem to hold. For clarity in this article, we shall choose examples where the forward process tends to cost work to complete. We will upper-bound the amount of work required to complete a single trial of a process successfully, such that this bound is only exceeded with probability (as illustrated on the right-hand side (rhs) of figure 2).

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Definition 1. Each implementation of the forward protocol has a probability $1-\varepsilon$ of requiring no more work than the ${\boldsymbol{\varepsilon }}$-required work ${W}^{\varepsilon }$ that satisfies

$$\begin{eqnarray}&&{\displaystyle \int }_{-\infty }^{{W}^{\varepsilon }}{\rm{d}}{W}{{P}}_{\mathrm{fwd}}(W)=1-\varepsilon .\end{eqnarray} \tag{ 7 }$$

The trial has a probability $\varepsilon \in [0,1]$ of requiring more work than ${W}^{\varepsilon }$.

Similarly, we will lower-bound the amount of work extracted in the reverse process, for all but δ of the trials (illustrated on the lhs of figure 2).

Definition 2. Each implementation of the reverse protocol has a probability $1-\delta$ of outputting at least the $\delta$-extractable work ${w}^{\delta }$ that satisfies

$$\begin{eqnarray}&&{\displaystyle \int }_{{w}^{\delta }}^{\infty }{\rm{d}}{W}{{P}}_{\mathrm{rev}}(-W)=1-\delta .\end{eqnarray} \tag{ 8 }$$

The trial has a probability $\delta \in [0,1]$ of outputting less work than ${w}^{\delta }$.

The failure probability has two interpretations. Suppose, in the work-investment case, that an agent invests only the amount ${W}^{\varepsilon }$ of work in a forward trial. The external parameter ${\lambda }_{t}$ has a probability of failing to reach ${\lambda }_{\tau }$. Alternatively, suppose the agent invests all the work required to evolve ${\lambda }_{t}$ to ${\lambda }_{\tau }$. The agent has a probability of overshooting the 'work budget' ${W}^{\varepsilon }$. The failure probability δ associated with work extraction can be interpreted similarly.

Even if only forward trials have been performed, the reverse process's ${w}^{\delta }$ can be calculated from Crooks' theorem.

Lemma 1. Each reverse trial has a probability $1-\delta$ of outputting at least the amount ${w}^{\delta }$ of work that satisfies

$$\begin{eqnarray}&&{\chi }_{\mathrm{fwd}}^{\delta }(\beta )=(1-\delta ){{\rm{e}}}^{-\beta {\rm{\Delta }}F},\end{eqnarray} \tag{ 9 }$$

wherein

$$\begin{eqnarray}&&{\chi }_{\mathrm{fwd}}^{\delta }(\beta )\equiv {\displaystyle \int }_{{w}^{\delta }}^{\infty }{\rm{d}}{W}{{P}}_{\mathrm{fwd}}(W){{\rm{e}}}^{-\beta W}\end{eqnarray} \tag{ 10 }$$

generalizes the characteristic function ${\chi }_{\mathrm{fwd}}(\beta )$.

Proof. Upon multiplying each side of Crooks' theorem (equation (1)) by ${P}_{\mathrm{rev}}(-W){{\rm{e}}}^{-\beta W}$, we integrate from ${w}^{\delta }$ to infinity. The lhs equals ${\chi }_{\mathrm{fwd}}^{\delta }(\beta )$ by definition (equation (10)). The right-hand integral evaluates to $1-\delta$ by definition 2. □

We can calculate ${W}^{\varepsilon }$ from ${P}_{\mathrm{rev}}(-W)$ via Crooks' theorem in the same way:

Lemma 2. Each forward trial has a probability $1-\varepsilon$ of requiring no more work than the ${W}^{\varepsilon }$ that satisfies

$$\begin{eqnarray}&&{\chi }_{\mathrm{rev}}^{\varepsilon }(\beta )=(1-\varepsilon ){{\rm{e}}}^{\beta {\rm{\Delta }}F},\end{eqnarray} \tag{ 11 }$$

wherein

$$\begin{eqnarray}&&{\chi }_{\mathrm{rev}}^{\varepsilon }(\beta )\equiv {\displaystyle \int }_{-{W}^{\varepsilon }}^{\infty }{\rm{d}}{W}{{P}}_{\mathrm{rev}}(W){{\rm{e}}}^{-\beta W}\end{eqnarray} \tag{ 12 }$$

generalizes the characteristic function ${\chi }_{\mathrm{rev}}(\beta )$.

These one-shot generalizations extend Jarzynski's equality (equation (3)), rendering it more robust against unlikely (probability less than $\epsilon$) but highly expensive (work cost more than ${W}^{\varepsilon }$) fluctuations in work. The generalizations characterize every quantum or classical process that produces a work distribution governed by Crooks' theorem. An alternative proof of these general lemmata—specialized for quantum systems undergoing unitary evolution and hence producing discrete work distributions—is presented in appendix B.

We can use Crooks' theorem, via the above lemmata, to derive bounds on the one-shot required and extractable work. These bounds depend on characteristics of the work distributions:

Theorem 3. The work δ-extractable from each reverse trial satisfies

$$\begin{eqnarray}&&{w}^{\delta }\leqslant {\rm{\Delta }}F-\displaystyle \frac{1}{\beta }[{H}_{\infty }^{\beta }({P}_{\mathrm{fwd}})+\mathrm{log}(1-\delta )]\end{eqnarray} \tag{ 13 }$$

for $\delta \in [0,1)$, wherein we have defined

$$\begin{eqnarray}&&{H}_{\infty }^{\beta }(P)\equiv -\mathrm{log}({P}^{\mathrm{max}}/\beta )\end{eqnarray} \tag{ 14 }$$

for continuous work distributions.

Proof. Let ${P}_{\mathrm{fwd}}^{\mathrm{max}}$ denote the greatest value of ${P}_{\mathrm{fwd}}(W)$ : ${P}_{\mathrm{fwd}}^{\mathrm{max}}\geqslant {P}_{\mathrm{fwd}}(W)\ \forall W$. We can upper-bound the integral implicit in the ${\chi }_{\mathrm{fwd}}^{\delta }(\beta )$ of equation (9) in lemma 1:

$$\begin{eqnarray}(1-\delta ){{\rm{e}}}^{-\beta {\rm{\Delta }}F} & = & {\chi }_{\mathrm{fwd}}^{\delta }(\beta )\equiv {\displaystyle \int }_{{w}^{\delta }}^{\infty }{\rm{d}}{{WP}}_{\mathrm{fwd}}(W){{\rm{e}}}^{-\beta W}\\ & \leqslant & {P}_{\mathrm{fwd}}^{\mathrm{max}}{\displaystyle \int }_{{w}^{\delta }}^{\infty }{\rm{d}}W{{\rm{e}}}^{-\beta W}\\ & = & {{\rm{e}}}^{\mathrm{log}({P}_{\mathrm{fwd}}^{\mathrm{max}}/\beta )}{{\rm{e}}}^{-\beta {w}^{\delta }}\\ & = & {{\rm{e}}}^{-{H}_{\infty }^{\beta }({P}_{\mathrm{fwd}})-\beta {w}^{\delta }}.\end{eqnarray} \tag{ 15 }$$

Solving for ${w}^{\delta }$ yields inequality (13).□

Analogous statements, which we present without further proof, describe ${W}^{\varepsilon }$:

Theorem 4. The work -required during each forward trial is bounded by

$$\begin{eqnarray}&&{W}^{\varepsilon }\geqslant {\rm{\Delta }}F+\displaystyle \frac{1}{\beta }[{H}_{\infty }^{\beta }({P}_{\mathrm{rev}})+\mathrm{log}(1-\varepsilon )]\end{eqnarray} \tag{ 16 }$$

for failure probability $\varepsilon \in [0,1)$.

These theorems demonstrate our central claim: that one-shot statistical mechanics can be applied to settings governed by fluctuation theorems. We have related the one-shot work quantities ${W}^{\epsilon }$ and ${w}^{\delta }$ to the one-shot entropy ${H}_{\infty }^{\beta }$.

Operationally when one handles data arising from a simulation or experiment, one does not directly observe a work distribution. Rather, one obtains a list of values that could be divided into bins of finite range (i.e., presented as a histogram) to approximate the true probability distribution. As want to consider a theoretical bound that reflects the behavior of the underlying thermal process independently of the choice of binning, we consider the entropy expressed in terms of the probability density $P(W)$. For theorems 3 and 4 to be applicable, it must be the case that the experimental data can be fitted to a theoretical model—a weak assumption for any scientific experiment.

There are further considerations that must be taken into account when considering the entropy of a continuous distribution in contrast with the entropy of a histogram taken from that distribution. (Throughout the following paragraph, we refer with ${P}_{\mathrm{fwd}}$ and ${P}_{\mathrm{rev}}$ jointly as P.) For a histogram taken of $P(W)$, in the limit of small enough bin width dW, the probability of each bin is well approximated by the product $P(W){\rm{d}}W$ as evaluated at a point W inside that bin. However, in the limit of decreasing bin size, the probability of being in any particular bin will tend to zero, and the order-$\infty$ entropy, as previously defined, diverges: ${H}_{\infty }(P)={\mathrm{lim}}_{{\rm{d}}W\to 0}[-\mathrm{log}({P}^{\mathrm{max}}{\rm{d}}W)]\to \infty$. An unmodified ${H}_{\infty }$ is not a useful quantity in this limit. By this measure, there is an infinite amount of information in any continuous distribution, and hence it can not be used to quantify the amount by which some continuous distributions are more entropic than others. To circumvent this problem, we employ a type of renormalization. ${H}_{\infty }$ can be split into a finite part that varies with the distribution under consideration, and an infinite part that does not: ${H}_{\infty }=-\mathrm{log}({P}^{\mathrm{max}}{\rm{d}}W)=-\mathrm{log}({P}^{\mathrm{max}}{k}^{-1})-\mathrm{log}(k{\rm{d}}W)$, where k is an arbitrary factor with units of inverse energy chosen such that the argument of each logarithm is dimensionless. When one considers the difference in entropy between distributions, the latter term always cancels out. As such the quantity ${H}_{\infty }^{k}=-\mathrm{log}({P}^{\mathrm{max}}{k}^{-1})$, by omitting the latter term, defines the amount by which the continuous order-$\infty$ entropy differs from a reference distribution of uniform probability density k over width ${k}^{-1}$. Our choice here of

$$\begin{eqnarray}&&{H}_{\infty }^{\beta }(P)\equiv -\mathrm{log}({P}^{\mathrm{max}}/\beta )\end{eqnarray} \tag{ 17 }$$

amounts to comparing the entropy of $P(W)$ with that of a uniform distribution over range ${\beta }^{-1}$ with probability density β. We remark this technique is not unique to the order-$\infty$ Rényi entropy, but is also necessary if one wishes to arrive at the differential entropy as a limiting case of the discrete Shannon entropy⁵ .

Although any quantity with units of inverse energy could be used as k, β is a natural choice: $-\mathrm{log}\beta$ appears everywhere $\mathrm{log}{P}^{\mathrm{max}}$ appears in our calculations; any alternative choice of normalization k would result in the need for an additional correction term of $\mathrm{log}(k/\beta )$ in inequalities (13) and (16). (This extra term can be interpreted as how the entropy of the new reference distribution compares to that of the uniform distribution of range ${\beta }^{-1}$ and probability β).

The bounds in theorems 3 and 4 shed light on the physical contributions to ${W}^{\varepsilon }$. To a first approximation, the -required work equals ${\rm{\Delta }}F$, the work needed to complete the process quasistatically. The negative contribution from $\mathrm{log}(1-\varepsilon )$ accounts for the tradeoff between work and failure probability: the agent can lower the bound on the required work by accepting a higher failure probability .

Outside the quasistatic limit, the system leaves equilibrium, and W fluctuates from trial to trial. This fluctuation necessitates a protocol-specific correction ${H}_{\infty }^{\beta }({P}_{\mathrm{rev}})$. In cryptography applications, ${H}_{\infty }(P)$ quantifies the uniform randomness (and hence resources usable to ensure privacy) extractable from a distribution P [35]. A distribution might have more randomness, but ${H}_{\infty }$ quantifies the minimum value (being the lowest-valued Rényi entropy). In our result, ${H}_{\infty }^{\beta }({P}_{\mathrm{rev}})$ can be thought of as the uniform randomness intrinsic to the work distribution. ${H}_{\infty }^{\beta }({P}_{\mathrm{rev}})$ thus quantifies fluctuations in work, caused by irreversibility, that raise the lower bound on ${W}^{\varepsilon }$.

Whereas some one-shot results (e.g. [18, 22]) involve Rényi entropies of states, ${H}_{\infty }^{\beta }({P}_{\mathrm{fwd}})$ is an entropy of a work distribution. ${H}_{\infty }^{\beta }({P}_{\mathrm{fwd}})$ captures fluctuation information from all sources that might affect the work distribution. These sources include the initial state and the manner in which the protocol is executed (e.g., quickly or quasistatically). In contrast, an entropy evaluated on states encodes only some of this information. By containing an entropy of a work distribution, rather than an entropy of a state, the above results can be applied in a more general setting: they can be related to the output of any procedure, as opposed to the worth of a particular input state under a fixed procedure (usually taken to be the optimal one [18]). Consequently, our results remain independent of the work-extraction model used. Theorems 3 and 4 govern (semi)classical and quantum systems, so long the protocol produces a work distribution consistent with Crooks' theorem.

As theoretical limits, these bounds will always hold true. However, to be operationally useful for bounding ${w}^{\delta }$ (or ${W}^{\varepsilon }$) they must be applied to models where ${P}^{\mathrm{max}}$ can be upper-bounded following enough trials of the experiment. This is possible if the distribution $P(W)$ is smooth such that beyond a certain narrowness of bin width, any further division of each bin results in approximately equal probability densities.

Applying information about the protocol executed in one direction, we have used Crooks' theorem to infer about the opposite direction. This information tightens the bound on ${W}^{\varepsilon }$ when⁶ ${H}_{\infty }^{\beta }({P}_{\mathrm{rev}})\ \geqslant 0$, i.e., when

$$\begin{eqnarray}&&{P}^{\mathrm{max}}\lt \beta .\end{eqnarray} \tag{ 18 }$$

Systems described poorly by conventional statistical mechanics tend to satisfy inequality (18). Such systems' work distributions have significant spreads relative to the characteristic energy scale ${\beta }^{-1}$, such that the distribution lacks tall peaks. We will present DNA-hairpin experiments as an example.

3.4.1. Tightening a bound in the work-extraction game

Egloff et al calculate the optimal amount ${w}_{\mathrm{best}}^{\delta }$ of work δ-extractable from a state via the most efficient strategy [18]. Their calculation implies an upper bound on the work extractable via arbitrary strategies. By applying theorems 3 and 4 to the Egloff et al framework, we can tighten the bound for protocols that satisfy the assumptions used to derive Crooks' theorem and for which ${P}^{\mathrm{max}}\lt \beta$.

In the Egloff et al setting, we can consider a forward protocol that consists of two stages: first, the thermal state ${\gamma }_{-\tau }\equiv {{\rm{e}}}^{-\beta {H}_{-\tau }}/{Z}_{-\tau }$ transforms into some nonequilibrium state σ as the externally driven Hamiltonian changes. The system either can remain thermally isolated or can thermalize, provided that the thermalization satisfies detailed balance (see appendix A). The agent can choose one of many possible strategies, e.g., by alternating Hamiltonian changes and thermalizations or by thermally isolating the system throughout the first stage. Second, σ thermalizes to ${\gamma }_{\tau }\equiv {{\rm{e}}}^{-\beta {H}_{\tau }}/{Z}_{\tau }.$ This thermalization neither costs nor produces work. The entire protocol is encapsulated in $({\gamma }_{-\tau },{H}_{-\tau })\mapsto (\sigma ,{H}_{\tau })\mapsto ({\gamma }_{\tau },{H}_{\tau }).$ At the start of the reverse protocol, the system begins in the themal state of ${H}_{\tau }$. Under the time-reversed process (in which the drive is reversed, such that the Hamiltonian retraces its path through configuration space), the system is transformed into some nonequilibrium state $\sigma ^{\prime}$. Then the state thermalizes to the thermal state of ${H}_{\tau }$.

For protocols which fall into the above category, knowing about one protocol, we can bound the work extractable from, or the work cost of, the opposite protocol:

Corollary 5. The work δ-extractable from each implementation of the reverse protocol, in terms of the forward protocol's ${H}_{\infty }^{\beta }({P}_{\mathrm{fwd}})$, satisfies

$$\begin{eqnarray}&&{w}^{\delta }\leqslant \displaystyle \frac{1}{\beta }\left[\mathrm{log}M\left(\displaystyle \frac{{G}^{T}({\gamma }_{\tau })}{1-\delta }\ \parallel \ {G}^{T}({\gamma }_{-\tau })\right)-{H}_{\infty }^{\beta }({P}_{\mathrm{fwd}})\right]\end{eqnarray} \tag{ 19 }$$

for $\delta \in [0,1)$.

Corollary 6. The work -required during each implementation of the forward protocol, in terms of the reverse protocol's ${H}_{\infty }^{\beta }({P}_{\mathrm{rev}})$, satisfies

$$\begin{eqnarray}&&{W}^{\varepsilon }\geqslant \displaystyle \frac{1}{\beta }\left[\mathrm{log}M\left(\displaystyle \frac{{G}^{T}({\gamma }_{-\tau })}{1-\varepsilon }\ \parallel \ {G}^{T}({\gamma }_{\tau })\right)+{H}_{\infty }^{\beta }({P}_{\mathrm{rev}})\right]\end{eqnarray} \tag{ 20 }$$

for $\varepsilon \in [0,1)$.

The proofs appear in appendix C.2. Each corollary consists of a bound derived from [18] and an ${H}_{\infty }^{\beta }$ correction attributable to Crooks' theorem (introduced via theorems 3 and 4). The ${H}_{\infty }^{\beta }$ quantifies the protocol's suboptimality, caused by dissipation due to the protocol's speed [29]. Positive values of ${H}_{\infty }^{\beta }$ tighten the bounds. Hence incorporating information about the forward (reverse) process into the reverse (forward) bound improves the bound when the process deviates sufficiently from the quasistatic ideal.

3.4.2. Modeling fluctuation-relation problems with resource theories

A simple example involves the heat-exchanging portion of Landauer bit reset and its reverse, Szilárd work extraction. The set-up consists of a two-level system ${\mathcal{S}}$ governed by the Hamiltonian $H({\lambda }_{t})=E(t)| E\rangle \langle E|$. Suppose ${\mathcal{S}}$ exchanges energy with a heat bath whose temperature is $T=1/\beta$. At time $t=-\tau$, $E(t)=0$, and ${\mathcal{S}}$ is in thermal equilibrium, i.e., in the maximally mixed state $\rho (-\tau )=\frac{1}{2}(| 0\rangle \langle 0| +| E\rangle \langle E| )$. If ρ represents the location of a particle in a two-compartment box, the agent has no idea which compartment the particle occupies.

Transforming $\rho (-\tau )$ into a pure state—forcing the particle into one half of the box—is called bit reset, or Landauer erasure. Resetting the bit quasistatically costs, on average,

$$\begin{eqnarray}&&\langle W\rangle ={\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{{\rm{e}}}^{-\beta E}}{Z}{\rm{d}}E={k}_{{\rm{B}}}T\mathrm{log}2.\end{eqnarray} \tag{ 21 }$$

If the bit is reset in a finite time, $\langle W\rangle$ might exceed ${k}_{{\rm{B}}}T\mathrm{log}2$ [29]. Such a protocol has appeared in fluctuation contexts before [18, 19, 29] and has been realized experimentally (e.g., in a test of Jarzynski's equality by Brownian motion [42, 43]).

We define failure under the assumption that every started trial is completed: a forward trial fails if it consumes more work than the budgeted work ${W}^{\varepsilon }$. (Alternatively, one could define 'failure' under the assumption that too-costly trials would not be completed. A trial would be said to fail if the budgeted work were consumed but the bit had not been reset.)

Reversal of the bit reset amounts to Szilárd work extraction. Leó Szilárd envisioned the conversion of information into work in 1929 [27]. ${\mathcal{S}}$ begins thermally isolated, in the pure state $| 0\rangle$, and governed by $H({\lambda }_{t})=0$. During the first leg of Szilárd work extraction, the agent raises $E(t)$ to infinity. The raising costs no work because ${\mathcal{S}}$ occupies the lower level. In the second leg, ${\mathcal{S}}$ is coupled to the bath, then performs positive work as $E(t)$ decreases to zero. To be strictly the reverse of the bit reset presented above, we consider only the second leg as an implementation of the reverse protocol. The initial state, although it is a pure state, is thermal since only the lower energy level has non-zero occupation probability according to the Gibbs distribution. As such this work-extraction step can be linked to the bit reset via Crooks' theorem; their Hamiltonians are the time-reverse of each other, and each protocol starts in the appropriate thermal state.

These two processes, forming a forward-and-reverse pair governed Crooks' theorem, provide a natural example with which to test our one-shot results. We performed a Monte Carlo simulation of Landauer erasure and Szilárd work extraction. Details appear in appendix E. The simulation produced results consistent with theorem 3, as shown in figure 3.

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When single molecules are manipulated experimentally, 'fluctuations are relevant and deviations from the average behavior are observable' [32]. Some such experiments are known to obey fluctuation relations. We show that data from DNA-hairpin experiments used previously to test Crooks' theorem [30–32] agree with the one-shot results in section 3. The agreement suggests that one-shot statistical mechanics might shed light on similar single-molecule experiments and applications. Alternatively, such experiments might be used to test one-shot statistical mechanics.

A DNA hairpin is a short double helix of about 21 base pairs [30–32]. The helix's two strands are called legs. One end of one leg is attached to one end of the other leg, forming a shape like a hairpin's. The other end of each leg ends in a handle formed from DNA. To each handle is attached a polystyrene or silica bead. One bead remains anchored on a micropipette. The other is caught in an optical trap that exerts a force. During the forward protocol, these optical tweezers pull the legs apart, unzipping the DNA into one strand. The more quickly the hairpin is split (the greater the pulling speed), the more work is dissipated. During the reverse protocol, the helix is rezipped.

We combined work distributions provided by Ritort and Alemany [30, 32, 44] with theorem 4 (illustrated in figure 4). Each graph shows the work ${W}^{\varepsilon }$ that a given unzipping trial is -guaranteed to require, plotted against the failure probability . We converted the data (a list of work values) into a probability distribution by forming a histogram with 50 equally sized bins that span the range of work costs. Energy is given in units of ${k}_{{\rm{B}}}T$, such that $\beta =1$. For pulling speeds of 15, 60, and $180{\mathrm{nms}}^{-1}$, the binning resulted in distributions whose ${P}^{\mathrm{max}}$ = 0.465 β, 0.277 β, and 0.162 β respectively. In all three cases ${P}^{\mathrm{max}}\lt \beta$, such that the ${H}_{\mathrm{max}}$ term in theorem 4 tightens the bound.

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Whereas a work investment of about 300–400 ${k}_{{\rm{B}}}$ T is required to complete the procedure, the jittering between the light blue curve (the directly measured value of ${W}^{\varepsilon }$) and the dark blue curve (the value of ${W}^{\varepsilon }$ predicted from the reverse work distribution ${P}_{\mathrm{rev}}$ via lemma 2) is of a scale less than $5\;{k}_{{\rm{B}}}{\rm{T}}$. Hence Crooks' theorem interrelates the work probability distributions up to a discrepancy of around 1%, as argued in [30, 32]. The red curve remains below the dark blue and light blue curves, confirming that the one-shot lower bound in theorem 4 governs this experimental setting. The red curve remains close to—always within about $5{k}_{{\rm{B}}}T$ of (around $1\%$ of ${\rm{\Delta }}F$)—the light blue curve that represents directly simulated work investment. This agreement between theory and experiment suggests that the application of one-shot results may shed light on similar single-molecule experiments and on applications such as molecular motors, thermal ratchets, and nanoscale engines (e.g., [45–47]).

Let us justify our modeling of processes governed by Crooks' theorem with the work-extraction game in [18] and with resource theories. Different frameworks (Crooks' theorem, the game, and the resource theories) model interactions with heat baths differently. One step in an interaction can be represented by a stochastic matrix that has at least one of three properties: Gibbs-preservation (${\mathcal{G}}$), detailed balance (${\mathcal{D}}$), and thermalization (${\mathcal{T}}$). The relationships among these matrices are summarized in figure A1 and in the following statements:

• (i)  
  ${\mathcal{T}}\subset {\mathcal{G}}$: all thermalizing matrices are Gibbs-preserving (lemma 7), but not vice versa (lemma 8).
• (ii)  
  ${\mathcal{D}}\subset {\mathcal{G}}$: all detailed-balanced matrices are Gibbs-preserving (lemma 9), but not vice versa (lemma 10).
• (iii)  
  ${\mathcal{D}}\ne {\mathcal{T}}$, ${\mathcal{D}}\;\not\hspace{-2pt}{\subset }\;{\mathcal{T}}$, and ${\mathcal{T}}\;\not\hspace{-2pt}{\subset }\;{\mathcal{D}}$: obeying detailed balance is not equivalent to being thermalizing, and neither category is a subset of the other (lemma 11).
• (iv)  
  ${\mathcal{D}}\cap {\mathcal{T}}\ne \varnothing$: some matrices are detailed-balanced and themalizing (lemma 12).

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While proving these claims, we justify the inclusion of two example matrices, the partial swap and quasicycles, in figure A1.

The proofs contain the following notation: ${\mathcal{S}}$ denotes a quasiclassical system that evolves under a Hamiltonian H and that exchanges heat with a bath whose inverse temperature is β. By $\vec{s}=({s}_{1},{s}_{2},\ldots ,{s}_{d})$, we denote the state of ${\mathcal{S}}$. The vector's elements are the diagonal elements of a density matrix relative to the eigenbasis of H. The Gibbs state relative to H and to β is denoted by $\vec{g}$.

To prove some of the foregoing claims, we characterize thermalizing matrices with the Perron–Frobenius theorem [52]. The theorem governs irreducible aperiodic non-negative matrices M⁷ . Consider the eigenvalue λ of M that has the greatest absolute value. According to the Perron–Frobenius Theorem, λ is the only positive real eigenvalue of M, and λ is associated with the only non-negative eigenvector ${\vec{v}}_{\lambda }$ of M. Suppose that M is stochastic, such that $\lambda =1$. By the spectral decomposition theorem, ${\mathrm{lim}}_{n\to \infty }{M}^{n}\vec{s}={\vec{v}}_{\lambda }$. If ${\vec{v}}_{\lambda }=\vec{g}$, the matrix is thermalizing.

Lemma 7. All thermalizing matrices are Gibbs-preserving.

Proof. Let M denote a thermalizing matrix associated with the same Hamiltonian and β as $\vec{g}$. For all states $\vec{s}$ of ${\mathcal{S}}$,

$$\begin{eqnarray}&&\underset{n\to \infty }{\mathrm{lim}}{M}^{n}\vec{s}=\vec{g}.\end{eqnarray} \tag{ A5 }$$

To prove the lemma by contradiction, we suppose that M does not map $\vec{g}$ to itself: $\vec{g}\;\not\hspace{-2pt}{\mapsto }\;\vec{g}$.

Premultiplying equation (A5) by M generates

$$\begin{eqnarray}&&\underset{n\to \infty }{\mathrm{lim}}{{MM}}^{n}\vec{s}=M\vec{g}\ne \vec{g}.\end{eqnarray} \tag{ A6 }$$

This equation contradicts

$$\begin{eqnarray}&&\underset{n\to \infty }{\mathrm{lim}}{{MM}}^{n}\vec{s}=\underset{n\to \infty }{\mathrm{lim}}{M}^{n+1}\vec{s}=\underset{n\to \infty }{\mathrm{lim}}{M}^{n}\vec{s}=\vec{g}.\end{eqnarray} \tag{ A7 }$$

By the contrapositive, all thermalizing matrices are Gibbs-preserving. □

Lemma 8. Not all Gibbs-preserving matrices are thermalizing.

Proof. In general, this will be the case for matrices which have the Gibbs state as an eigenvector but do not otherwise satisfy the conditions of irreducibility or aperiodicity required for the Perron–Frobenius theorem to govern their behavior. To prove the lemma by example, we construct one Gibbs-preserving matrix that is not thermalizing. Consider a block-diagonal stochastic $N\times N$ matrix M. (Being block-diagonal, M is reducible.) Let M decompose into two submatrices: $M={M}_{1}\oplus {M}_{2}$. Let M₁ be defined on the first n₁ energy levels, and let M₂ be defined on the remaining n₂ energy levels.

Denote by $\vec{{g}_{1}}$ the Gibbs state associated with the first n₁ energies (and the partition function Z₁), and by g₂ the Gibbs state associated with the final n₂ energies (and the partition function Z₂). Suppose that

$$\begin{eqnarray}&&{\tilde{g}}_{1}\equiv {g}_{1}\oplus \underset{{n}_{2}}{(\underbrace{0,0,\ldots ,0})}\quad \mathrm{and}\quad {\tilde{g}}_{2}\equiv \underset{{n}_{1}}{(\underbrace{0,0,\ldots ,0})}\oplus {g}_{2}\end{eqnarray} \tag{ A8 }$$

are normalized probability eigenvectors of M, each associated with the unit eigenvalue. Every vector of the form

$$\begin{eqnarray}&&{\vec{\nu }}_{\alpha }=\alpha {\tilde{g}}_{1}+(1-\alpha ){\tilde{g}}_{2}\end{eqnarray} \tag{ A9 }$$

is also a normalized probability eigenvector of M associated with the unit eigenvalue.

The possible forms of ${\vec{\nu }}_{\alpha }$ form a family. One member of the family is the Gibbs state $\vec{g}$, which corresponds to $\alpha ={Z}_{1}/Z$ (wherein Z denotes the total partition function). Hence $\vec{g}\in {\{{\vec{\nu }}_{\alpha }\}}_{\alpha \in [0,1]}$ is an eigenvector of M, and M is Gibbs-preserving. However, $\vec{g}$ is not the only eigenvector associated with the unit eigenvalue. M does not evolve every initial state toward $\vec{g}$. In general, ${\mathrm{lim}}_{n\to \infty }{M}^{n}\vec{s}={\nu }_{\alpha }$, wherein α is the total occupation probability of the first n₁ energy levels of $\vec{s}$. As some states $\vec{s}$ correspond to $\alpha \ne {Z}_{1}/Z$ and to ${\nu }_{\alpha }\ne g$, M does not map every initial state to the Gibbs state. Hence M is not thermalizing. Our claim has been proved by example. □

Together, lemmas 7 and 8 imply the strict relation ${\mathcal{T}}\subset {\mathcal{G}}$.

Lemma 9. All matrices that obey detailed balance relative to the Hamiltonian H and the inverse temperature β preserve the Gibbs state $\vec{g}$ associated with H and β.

Proof. We will write the forms of the elements in an arbitrary detailed-balanced stochastic $N\times N$ matrix M. By performing matrix multiplication explicitly, we show that $M\vec{g}=\vec{g}$.

Let M_ij denote the element in the $i\mathrm{th}$ row and $j\mathrm{th}$ column of M. This element equals the probability that, upon beginning in the $j\mathrm{th}$ energy level, a system ${\mathcal{S}}$ transitions to the $i\mathrm{th}$ level. Let g_i denote the thermal population of level i (the $i\mathrm{th}$ element of $\vec{g}$).

Detailed balance and stochasticity constrain the relationships among the M_ij. By the definition of detailed balance (equation (A2)), the elements in the lower left-hand triangle of M are related to the elements in the upper right-hand triangle by

$$\begin{eqnarray}&&{M}_{{ji}}={M}_{{ij}}\displaystyle \frac{{g}_{j}}{{g}_{i}}\qquad \forall j\gt i.\end{eqnarray} \tag{ A10 }$$

The matrix has the form

$$\begin{eqnarray}M=\left(\begin{array}{cccc}{M}_{11} & {M}_{12} & {M}_{13} & \ldots \\ {M}_{12}\displaystyle \frac{{g}_{2}}{{g}_{1}} & {M}_{22} & {M}_{23} & \ldots \\ {M}_{13}\displaystyle \frac{{g}_{3}}{{g}_{1}} & {M}_{23}\displaystyle \frac{{g}_{3}}{{g}_{2}} & {M}_{33} & \ldots \\ \vdots & \vdots & \vdots & \ddots \end{array}\right).\end{eqnarray} \tag{ A11 }$$

Because M is stochastic, the elements in each column sum to one. This normalization condition fixes each diagonal element M_ii as a function of the other M_ij that occupy the same column:

$$\begin{eqnarray}&&{M}_{{ii}}=1-\displaystyle \sum _{j\lt i}\;{M}_{{ji}}-\displaystyle \sum _{j\gt i}\;{M}_{{ij}}\displaystyle \frac{{g}_{j}}{{g}_{i}}.\end{eqnarray} \tag{ A12 }$$

Using index notation, we ascertain how M transforms the Gibbs state $\vec{g}$:

$$\begin{eqnarray}\displaystyle \sum _{j}{M}_{{ij}}{g}_{j} & = & {M}_{{ii}}{g}_{i}+\displaystyle \sum _{j\lt i}\;{M}_{{ij}}{g}_{j}+\displaystyle \sum _{j\gt i}\;{M}_{{ij}}{g}_{j}\\ & = & {g}_{i}-\displaystyle \sum _{j\lt i}\;{M}_{{ji}}{g}_{i}-\displaystyle \sum _{j\gt i}\;{M}_{{ij}}\displaystyle \frac{{g}_{j}}{{g}_{i}}{g}_{i}+\displaystyle \sum _{j\lt i}\;{M}_{{ij}}{g}_{j}+\displaystyle \sum _{j\gt i}\;{M}_{{ij}}{g}_{j}\\ & = & {g}_{i}-\displaystyle \sum _{j\lt i}\;{M}_{{ij}}{g}_{j}-\displaystyle \sum _{j\gt i}\;{M}_{{ij}}{g}_{j}+\displaystyle \sum _{j\lt i}\;{M}_{{ij}}{g}_{j}+\displaystyle \sum _{j\gt i}\;{M}_{{ij}}{g}_{j}\\ & = & {g}_{i}\end{eqnarray} \tag{ A13 }$$

The second line follows from the substitution of equation (A12) for M_ii. The third line follows from the substitution of equation (A10) into the elements of first sum.

We have shown that $\vec{g}$ is an eigenvector of M that corresponds to the unit eigenvalue. An $N\times N$ matrix M that obeys detailed balance relative to H and β preserves the Gibbs state associated with H and β. □

Lemma 10. Not every Gibbs-preserving matrix for some Hamiltonian H and inverse temperature β satisfies detailed balance for H and β.

Proof. Gibbs-preservation only places a restriction on one eigenvalue of a matrix, and so there remains enough freedom to choose a matrix exhibiting this property, but not detailed balance. An example of such is a quasicycle, described in [20]. A quasicycle is a process that has a probability $P(i\mapsto (i+1)\mathrm{mod}N)\equiv {p}_{i}$ of evolving a system ${\mathcal{S}}$ that occupies energy eigenstate i to eigenstate $i+1$ and has a probability $1-{p}_{i}$ of keeping ${\mathcal{S}}$ in state i. All ${p}_{i}\gt 0$, and for at least one value of i, ${p}_{i}\lt 1$. The directed graph of a quasicycle forms a ring in which at least one node also has a loop to itself, corresponding to a value of $(1-{p}_{i})\gt 0$. An example appears in figure A2 . The probability that i evolves to j forms element M_ij of matrix M. The matrix fails to satisfy detailed balance if ${\mathcal{S}}$ has more than three energy eigenstates, because $P(i\mapsto (i+1)\mathrm{mod}N)$ is finite, though $P((i+1)\mathrm{mod}N\mapsto i)=0$.

We will show that, if the p_i assume certain values, the Gibbs state $\vec{g}$ is an eigenvector of M. Let level i = 1 correspond to the lowest energy eigenvalue. Solutions of the form

$$\begin{eqnarray}&&{p}_{i}=p\ \displaystyle \frac{{g}_{1}}{{g}_{i}},\end{eqnarray} \tag{ A14 }$$

wherein $p\in (0,1)$ denotes a free parameter in the range $(0,1)$, are Gibbs-preserving. Roughly, the greater the value of p, the more quickly the quasicycle is traversed.

To verify that equation (A14) describes a Gibbs-preserving matrix, we express the matrix multiplication $M\vec{g}$ in index form:

$$\begin{eqnarray}&&{(M\vec{g})}_{1}=(1-{p}_{1}){g}_{1}+{p}_{N}{g}_{N}\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}&&{(M\vec{g})}_{i}=(1-{p}_{i}){g}_{i}+{p}_{i-1}{g}_{i-1}\qquad i=2\ldots N.\end{eqnarray} \tag{ A16 }$$

Upon substituting in from equation (A14), we can simplify the equations to

$$\begin{eqnarray}&&{(M\vec{g})}_{1}={g}_{1}\end{eqnarray} \tag{ A17 }$$

$$\begin{eqnarray}&&{(M\vec{g})}_{i}={g}_{i}\qquad i=2\ldots N.\end{eqnarray} \tag{ A18 }$$

Hence $M\vec{g}=\vec{g}$, so M preserves Gibbs states. □

Together, lemmas 9 and 10 imply the strict relation ${\mathcal{D}}\subset {\mathcal{G}}$.

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Lemma 11. Obeying detailed balance is not equivalent to thermalizing: ${\mathcal{D}}\ne {\mathcal{T}}$, nor is one category a subset of the other.

Proof. We will show that the quasicycle matrix M described in the proof of lemma 10—a matrix that does not obey detailed balance—is thermalizing. Because M is stochastic by construction, its greatest eigenvalue equals one.

In addition to being stochastic, M is irreducible, aperiodic⁸ and non-negative. By the Perron–Frobenius Theorem, the greatest eigenvalue λ of M corresponds to the only non-negative eigenvector ${\vec{v}}_{\lambda }$ of M. This $\lambda =1$, because M is stochastic. As shown in the proof of lemma 10, ${\vec{v}}_{\lambda }=\vec{g}$. As explained below the proof of lemma 7, ${\mathrm{lim}}_{n\to \infty }{M}^{n}$ maps every vector $\vec{s}$ to $\vec{g}$: the matrices that represent quasicycles thermalize. M does not obey detailed balance, as discussed in the proof of lemma 9.

A simple example of a matrix that obeys detailed balance, but is not thermalizing, is the identity matrix. Less trivially, it is possible in general to engineer a block-diagonal matrix of the form given in lemma 8, and if each block obeys detailed balance, the matrix as a whole will also obey detailed balance (noting that $P(A\mapsto B)=P(B\mapsto A)=0$ trivially satisfies detailed balance). Such a matrix will not be thermalizing. Hence we see that thermalizing is not equivalent to obeying detailed balance: ${\mathcal{D}}\ne {\mathcal{T}}$, and neither category is a subset of the other. □

Lemma 12. Some matrices are detailed-balanced and thermalizing: ${\mathcal{D}}\cap {\mathcal{T}}\ne \varnothing$.

Proof. We can prove this lemma by example. After reviewing the form of the partial-swap matrix M, we show that M thermalizes, then show that M obeys detailed balance.

A partial-swap operation has some probability p of replacing the operated-on state $\vec{s}$ with a thermal state and a probability $1-p$ of preserving $\vec{s}$. If $\vec{s}$ denotes the state of an N-level system,

$$\begin{eqnarray}&&M=(1-p){{\mathbb{1}}}_{N}+{pG},\end{eqnarray} \tag{ A19 }$$

wherein ${{\mathbb{1}}}_{N}$ denotes the $N\times N$ identity and every column of the matrix G is the thermal state $\vec{g}$.

Let us prove that M thermalizes. M is stochastic, as it is the probabilistic combination of ${{\mathbb{1}}}_{N}$ and G, which are stochastic. If N is finite, G is positive; so when $p\gt 0$, M is positive. Positivity implies irreducibility and aperiodicity. Hence the Perron–Frobenius Theorem⁹ implies that M has just one non-negative eigenvector ${\vec{v}}_{\lambda }$ and that this eigenvector corresponds to $\lambda =1$. Direct multiplication shows $M\vec{g}=\vec{g}$. Thus, $\vec{g}={\vec{v}}_{\lambda }$ is the only non-negative eigenvector of M and corresponds to the largest eigenvalue. By the argument above lemma 7, M thermalizes.

To show that M obeys detailed balance, we compare the matrix elements that represent the probabilities of transitions between states i and j:

$$\begin{eqnarray}\displaystyle \frac{P(i\mapsto j)}{P(j\mapsto i)} & = & \displaystyle \frac{{M}_{{ji}}}{{M}_{{ij}}}\\ & = & \displaystyle \frac{(1-p){\delta }_{{ij}}+{{pg}}_{j}}{(1-p){\delta }_{{ij}}+{{pg}}_{i}}\\ & = & {{\rm{e}}}^{-\beta ({E}_{j}-{E}_{i})},\end{eqnarray} \tag{ A20 }$$

wherein ${\delta }_{{ij}}$ denotes the Kronecker delta. This equation recapitulates the definition of detailed balance (equation (A2)). Hence matrices—such as the partial swap—can obey detailed balance while thermalizing. □

Let us briefly review the bound presented by Egloff et al [18]. Consider the most efficient transformation $(\rho ,{H}_{\rho })\mapsto (\sigma ,{H}_{\sigma })$ that has a probability $1-\delta$ of failing. That is, one sacrifices the certainty that the transformation will succeed, in hopes of extracting more work than can be gained from a certain-to-succeed transformation. All work that the system can output is collected; none is wasted. The transformation has a probability $1-\delta$ of outputting at least the work

$$\begin{eqnarray}&&{w}_{\mathrm{best}}^{\delta }\left(\rho ,{H}_{\rho }\mapsto \sigma ,{H}_{\sigma }\right)=T\mathrm{log}\left(M\left(\displaystyle \frac{{G}^{T}(\rho )}{1-\delta }\ \parallel \ {G}^{T}(\sigma )\right)\right).\end{eqnarray} \tag{ C1 }$$

We will briefly overview the geometric definitions of Gibbs-rescaling (G^T) and the relative mixedness (M).

Let ρ have the spectral decomposition ${\displaystyle \sum }_{i=1}^{{d}_{\rho }}{r}_{i}| {E}_{i}\rangle \langle {E}_{i}|$ such that

$$\begin{eqnarray}&&{r}_{1}{{\rm{e}}}^{\beta {E}_{1}}\geqslant {r}_{2}{{\rm{e}}}^{\beta {E}_{2}}\geqslant \ldots \geqslant {r}_{{d}_{\rho }}{{\rm{e}}}^{\beta {E}_{{d}_{\rho }}}.\end{eqnarray} \tag{ C2 }$$

Consider the histogram that represents the r_i. Gibbs-rescaling ρ resizes each box in the histogram. The width of box i changes from unity to ${{\rm{e}}}^{-\beta {E}_{i}}$, and the box's height increases by a factor of ${{\rm{e}}}^{\beta {E}_{i}}$. Denote by ${h}_{\rho }^{T}(u)$ the height of the point, on the rescaled histogram, whose x-coordinate is $u\in [0,Z({H}_{\rho })]$. Integrating ${h}_{\rho }^{T}(u)$, we define the Gibbs-rescaled Lorenz curve as the set of points

$$\begin{eqnarray}&&\left\{\left(u,{L}_{\rho }^{T}(u)\right)\ \bigg| \ u\in \left[0,Z({H}_{\rho })\right]\right\},\quad \mathrm{wherein}\quad {L}_{\rho }^{T}(u)\equiv {\displaystyle \int }_{0}^{u}{h}_{\rho }^{T}(u){\rm{d}}u.\end{eqnarray} \tag{ C3 }$$

The (unscaled) Lorenz curve ${L}_{\rho }$ is equivalent to ${L}_{\rho }^{0}$. Upon Gibbs-rescaling ρ and σ, we can compare the states' resourcefulness even though different Hamiltonians govern the states.

To incorporate the failure probability into the curve, we stretch ${L}_{\rho }^{T}$ upward by a factor of $1/(1-\delta )$. The resulting curve, ${L}_{\rho }^{T,\delta }$, encodes more reliable resourcefulness than $(\rho ,{H}_{\rho })$ possesses, because extractable work trades off with the failure probability δ. Consider plotting ${L}_{\rho }^{T,\delta }$ on the same graph as ${L}_{\sigma }^{T}$. The curves are concave, bowing outward from the x-axis or stretching straight from $(0,0)$ to y = 1. Consider compressing ${L}_{\sigma }^{T}$ leftward. M denotes the inverse of the greatest factor by which ${L}_{\sigma }^{T}$ can compress without popping above ${L}_{\rho }^{T,\delta }$:

$$\begin{eqnarray}&&{L}_{\rho }^{T}(u)\geqslant {\left[M\left(\displaystyle \frac{{G}^{T}(\rho )}{1-\delta }\ \parallel \ {G}^{T}(\sigma )\right)\right]}^{-1}{L}_{\sigma }^{T}(u)\quad \forall u\in \left[0,\mathrm{max}\left(Z({H}_{\rho }),Z({H}_{\sigma })\right.\right].\end{eqnarray} \tag{ C4 }$$

Illustrations appear in [18]. While transforming $(\rho ,{H}_{\rho })$ into $(\sigma ,{H}_{\sigma })$, the player can extract no more work than $T\mathrm{log}M$: according to equation (C1),

$$\begin{eqnarray}&&{w}^{\delta }\left(\rho ,{H}_{\rho }\mapsto \sigma ,{H}_{\sigma }\right)\leqslant T\mathrm{log}\left(M\left(\displaystyle \frac{{G}^{T}(\rho )}{1-\delta }\ \parallel \ {G}^{T}(\sigma )\right)\right).\end{eqnarray} \tag{ C5 }$$

Theorem 5 strengthens the above inequality (C5) (in the appropriate parameter regime), and theorem 6 generalizes this to work investment. These theorems are proved below:

Theorem. The work δ-extractable from each Crooks-type reverse trial satisfies

$$\begin{eqnarray}&&{w}^{\delta }\leqslant \displaystyle \frac{1}{\beta }\left[\mathrm{log}M\left(\displaystyle \frac{{G}^{T}({\gamma }_{\tau })}{1-\delta }\ \parallel \ {G}^{T}({\gamma }_{-\tau })\right)-{H}_{\infty }({P}_{\mathrm{fwd}})\right]\quad \forall \;\delta \in [0,1).\end{eqnarray} \tag{ C6 }$$

Proof. During the reverse protocol, the state of a system ${\mathcal{S}}$ transforms as

$$\begin{eqnarray}&&({\gamma }_{\tau },{H}_{\tau })\mapsto (\sigma ,{H}_{-\tau })\mapsto ({\gamma }_{-\tau },{H}_{-\tau }),\end{eqnarray} \tag{ C7 }$$

wherein σ denotes some density operator that likely is not an equilibrium state.

The set of strategies for transforming from $({\gamma }_{\tau },{H}_{\tau })$ to $({\gamma }_{-\tau },{H}_{-\tau })$ contains all strategies that achieve this transformation via $(\sigma ,{H}_{-\tau })$. As such, when optimizing to maximize the work production allowing an error rate of δ,

$$\begin{eqnarray}{w}_{\mathrm{best}}^{\delta }({\gamma }_{\tau },{H}_{\tau }\mapsto {\gamma }_{-\tau },{H}_{-\tau }) & \geqslant & {w}_{\mathrm{best}}^{\delta }({\gamma }_{\tau },{H}_{\tau }\mapsto \sigma ,{H}_{-\tau })\\ & & +{w}_{\mathrm{best}}^{0}(\sigma ,{H}_{-\tau }\mapsto {\gamma }_{-\tau },{H}_{-\tau }).\end{eqnarray} \tag{ C8 }$$

The final term (which only involves thermalization) always succeeds, does not contribute either to the failure probability or the work cost, and hence can be eliminated. An arbitrary, possibly suboptimal, strategy from $({\gamma }_{\tau },{H}_{\tau })$ to $(\sigma ,{H}_{-\tau })$ that generates work ${w}^{\delta }$ will be bounded by this value:

$$\begin{eqnarray}&&{w}^{\delta }\leqslant {w}_{\mathrm{best}}^{\delta }({\gamma }_{\tau },{H}_{\tau }\mapsto {\gamma }_{-\tau },{H}_{-\tau }).\end{eqnarray} \tag{ C9 }$$

Hence, by equation (C1),

$$\begin{eqnarray}&&{w}^{\delta }\leqslant \displaystyle \frac{1}{\beta }\mathrm{log}\ M\left(\displaystyle \frac{{G}^{T}({\gamma }_{\tau })}{1-\delta }\ \parallel \ {G}^{T}({\gamma }_{-\tau })\right).\end{eqnarray} \tag{ C10 }$$

Let us calculate M. ${L}_{{\gamma }_{-\tau }}^{T}$ stretches straight from $(0,0)$ to $({Z}_{-\tau },1)$, whereas ${L}_{{\gamma }_{\tau }}^{T}/(1-\delta )$ stretches straight to $({Z}_{\tau },\frac{1}{1-\delta })$. Compressing ${L}_{{\gamma }_{\tau }}^{T}(u)/(1-\delta )$ leftward by a factor of ${M}^{-1}=\frac{{Z}_{\tau }(1-\delta )}{{Z}_{-\tau }}$ keeps the latter curve from dipping below ${L}_{{\gamma }_{-\tau }}^{T}(u)$. Hence

$$\begin{eqnarray}\displaystyle \frac{1}{\beta }\mathrm{log}\ M\left(\displaystyle \frac{{G}^{T}({\gamma }_{\tau })}{1-\delta }\parallel {G}^{T}({\gamma }_{-\tau })\right) & = & \displaystyle \frac{1}{\beta }\left[\mathrm{log}\left(\displaystyle \frac{{Z}_{-\tau }}{{Z}_{\tau }}\right)-\mathrm{log}(1-\delta )\right]\\ & = & {\rm{\Delta }}F-\displaystyle \frac{1}{\beta }\mathrm{log}(1-\delta ),\end{eqnarray} \tag{ C11 }$$

wherein ${\rm{\Delta }}F\equiv F({\gamma }_{\tau })-F({\gamma }_{-\tau })$. We substitute from equation (C11) into the inequality (13) derived from Crooks' theorem in theorem 3. □

Crooks' theorem introduces an ${H}_{\infty }$ into the bound, derived from [18], on extractable work. If ${P}_{\mathrm{fwd}}$ satisfies inequality (18), this ${H}_{\infty }$ strengthens the bound. A work cost can similarly be derived from [18], then enhanced with Crooks' theorem.

Each thermodynamic resource theory models energy-preserving transformations performed with a heat bath characterized by an inverse temperature β. To specify a state, one specifies a density operator and a Hamiltonian: $(\rho ,H)$. Sums of Hamiltonians will be denoted by ${H}_{1}+{H}_{2}\equiv {H}_{1}\otimes {\mathbb{1}}+{\mathbb{1}}\otimes {H}_{2}$.

Thermal operations can be performed for free. Each consists of three steps: (1) a Gibbs state relative to β and to any Hamiltonian ${H}_{\gamma }$ can be drawn from the bath:

$$\begin{eqnarray}&&(\gamma ,{H}_{\gamma }),\quad \mathrm{wherein}\quad \gamma \equiv \displaystyle \frac{{{\rm{e}}}^{-\beta {H}_{\gamma }}}{Z}.\end{eqnarray} \tag{ D1 }$$

(Below, the Gibbs state relative to β and to ${H}_{\gamma }$ will be denoted also by $\gamma ({H}_{\gamma })$.) (2) Any unitary U that conserves the total energy can be implemented, and (3) any subsystem A associated with its own Hamiltonian H_A can be discarded. Each thermal operation on $(\rho ,H)$ has the form

$$\begin{eqnarray}&&(\rho ,H)\mapsto \left({\mathrm{Tr}}_{A}(U[\rho \otimes \gamma ]{U}^{\dagger }),H+{H}_{\gamma }-{H}_{A}\right),\end{eqnarray} \tag{ D2 }$$

wherein $[U,H+{H}_{\gamma }]=0$.

Some resource-theory operations obey detailed balance and Markovianity. We can use resource theories to model processes governed by Crooks' theorem if we define a battery and a clock. Our model for the battery appears in [23] and resembles the model in [21]. The quasiclassical battery B has closely spaced energy levels and occupies an energy eigenstate:

$$\begin{eqnarray}&&{B}_{i}\equiv \left(| {B}_{i}\rangle \langle {B}_{i}| ,{H}_{B}\right),\quad \mathrm{wherein}\quad {H}_{B}\equiv \displaystyle \sum _{i}\;{E}_{{B}_{i}}| {B}_{i}\rangle \langle {B}_{i}| .\end{eqnarray} \tag{ D3 }$$

If ${E}_{{B}_{i}}$ is large, a work-costing (forward) process can transfer work from the battery to ${\mathcal{S}}$. If ${E}_{{B}_{i}}$ is small, a work-extraction (reverse) process can transfer work from ${\mathcal{S}}$ to the battery.

We model the evolution of H with a clock C that occupies a pure state $| {C}_{j}\rangle$ [20, 26]. The changing of $| {C}_{j}\rangle$, like the movement of a clock hand, models the passing of instants. In processes governed by Crooks' theorem, $H=H({\lambda }_{t})$. We discretize t such that the system's Hamiltonian is $H({\lambda }_{{t}_{j}})$ when the clock occupies the state $| {C}_{j}\rangle$. In the notation introduced earlier, ${t}_{1}=-\tau$, and ${t}_{n}=\tau$. The composite-system Hamiltonian

$$\begin{eqnarray}&&{H}_{\mathrm{tot}}\equiv \displaystyle \sum _{i=1}^{n}\;H({\lambda }_{{t}_{i}})\otimes \big| {C}_{i}\rangle \langle {C}_{i}| \otimes {\mathbb{1}}+{\mathbb{1}}\otimes {\mathbb{1}}\otimes \displaystyle \sum _{j}\;{E}_{{B}_{j}}| {B}_{j}\rangle \langle {B}_{j}| \end{eqnarray} \tag{ D4 }$$

remains constant.

Having defined the battery and clock, we define the work extractable from, and the work cost of, a protocol. Let ${E}_{{B}_{0}}=0$. The most work extractable from the reverse protocol equals the greatest ${E}_{{B}_{m}}$ for which some sequence of thermal operations evolves the state of ${\mathcal{S}}{CB}$ as

$$\begin{eqnarray}\gamma ({H}_{-\tau })\otimes | 1\rangle \langle 1| \otimes | 0\rangle \langle 0| & \mapsto & \rho ({t}_{2})\otimes | 2\rangle \langle 2| \otimes | {E}_{{B}_{2}}\rangle \langle {E}_{{B}_{2}}| \mapsto \ldots \\ & \mapsto & \gamma ({H}_{\tau })\otimes | m\rangle \langle m| \otimes | {E}_{{B}_{m}}\rangle \langle {E}_{{B}_{m}}| ,\end{eqnarray} \tag{ D5 }$$

wherein $\rho ({t}_{i})$ represents the state occupied by ${\mathcal{S}}$ at time t_i. The forward protocol's minimum work cost equals the least ${E}_{{B}_{n}}$ for which a sequence of thermal operations implements

$$\begin{eqnarray}&&\gamma ({H}_{\tau })\otimes | n\rangle \langle n| \otimes | {E}_{{B}_{n}}\rangle \langle {E}_{{B}_{n}}| \mapsto \ldots \mapsto \gamma ({H}_{-\tau })\otimes | 1\rangle \langle 1| \otimes | 0\rangle \langle 0| .\end{eqnarray} \tag{ D6 }$$

Results in [20] can be applied if all the states commute with their Hamiltonians. Horodecki and Oppenheim have calculated the maximum work yield, or minimum work cost, of any quasiclassical transformation $(\rho ,{H}_{\rho })\mapsto (\sigma ,{H}_{\sigma })$ by thermal operations. They have calculated also faulty transformations' work yields and work costs. A faulty transformation generates a state $({\sigma }^{\prime },{H}_{\sigma })$ that differs from the desired state. The discrepancy is quantified by the trace distance between the density operators:

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel \sigma -{\sigma }^{\prime }\parallel }_{1}\leqslant \epsilon \in [0,1].\end{eqnarray} \tag{ D7 }$$

According to [20], this can be interpreted as the probability that the process fails to accomplish its mission, similarly to and δ. Generalizations to nonclassical states appear in [38, 39].