Stochastic thermodynamics based on incomplete information: generalized Jarzynski equality with measurement errors with or without feedback

In order to incorporate the measurements on the system, we expand the phase space to the phase space of measured and true trajectories (see figure 1). A stochastic path in this extended space is denoted by $({\bf{z}},{\bf{y}})$ and the probability of choosing such a path is simply denoted by ${ \mathcal P }[{\bf{z}},{\bf{y}}]$. The trajectory ${\bf{z}}$ of the system is the projection of the whole trajectory onto the z-subspace and the probability distribution of this true stochastic path is given by ${ \mathcal P }[{\bf{z}}]=\int { \mathcal D }[{\bf{y}}]{ \mathcal P }[{\bf{z}},{\bf{y}}]$. Equivalently, the measured trajectory ${\bf{y}}$ lives in the y-subspace and its probability distribution is ${ \mathcal P }[{\bf{y}}]=\int { \mathcal D }[{\bf{z}}]{ \mathcal P }[{\bf{z}},{\bf{y}}]$. Discretizing the time interval $[0,{t}_{f}]$ again into N time steps, the probability density of a path in the space of system and measured trajectories will be factorized as

$$\begin{eqnarray}&&{ \mathcal P }[{\bf{z}},{\bf{y}}]={p}_{{\lambda }_{0}}({z}_{0},{y}_{0}){p}_{{\lambda }_{1}}({z}_{0},{y}_{0}\to {z}_{1},{y}_{1})\ldots {p}_{{\lambda }_{N}}({z}_{N-1},{y}_{N-1}\to {z}_{N},{y}_{N}).\end{eqnarray} \tag{ 15 }$$

Our main assumptions are that the evolution of the system is independent of the measurement process and that the outcome of a measurement y_k only depends on the state of the system z_k at time t_k, i.e., we assume

$$\begin{eqnarray}&&{p}_{{\lambda }_{k}}({z}_{k-1},{y}_{k-1}\to {z}_{k},{y}_{k})={p}_{{\lambda }_{k}}({z}_{k-1}\to {z}_{k}){p}_{m}({y}_{k}| {z}_{k}).\end{eqnarray} \tag{ 16 }$$

This can be seen as a Markov assumption for the measurement apparatus, i.e., the previous measurement result ${y}_{k-1}$ does not influence the system evolution and the next measurement result. The conditional probability ${p}_{m}({y}_{k}| {z}_{k})$ quantifies the uncertainty of the measurement (see equations (13) and (14)).

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The measured work ${W}_{m}[{\bf{y}}]$ along a measurement trajectory ${\bf{y}}$ is defined as in equations (1) and (5) by interchanging ${\bf{z}}$ with ${\bf{y}}$ and is in general different from the true work $W=W[{\bf{z}}]$. Even on average it might be that $\langle {W}_{m}[{\bf{y}}]{\rangle }_{{\bf{y}}}\ne \langle W[{\bf{z}}]{\rangle }_{{\bf{z}}}$. Nevertheless, we assume that the Hamiltonian of the system is known to us and unchanged by the measurement; the only mistake is in the measurement outcome y (see [13] for the case of different Hamiltonians).

From an experimental point of view it only makes sense to consider the distribution of measured work and we may write the average of the exponential of measured work and free energy difference ${\rm{\Delta }}F$ as

$$\begin{eqnarray}{\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}F)}\rangle }_{{\bf{y}}} & = & \displaystyle \int { \mathcal D }[{\bf{y}}]\,{ \mathcal P }[{\bf{y}}]\,{{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}]-{\rm{\Delta }}F)}=\displaystyle \int { \mathcal D }[{\bf{y}}]{ \mathcal D }[{\bf{z}}]\,{ \mathcal P }[{\bf{z}},{\bf{y}}]\,{{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}]-{\rm{\Delta }}F)}\\ & = & \displaystyle \int { \mathcal D }[{\bf{y}}]{ \mathcal D }[{\bf{z}}]\,{ \mathcal P }[{\bf{z}}]\displaystyle \prod _{i=0}^{N}{p}_{m}({y}_{i}| {z}_{i}){{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}]-{\rm{\Delta }}F)},\end{eqnarray} \tag{ 17 }$$

where in the last step we used (16). Again the assumption of microreversibility (see equation (9)) allows us to write equation (17) as

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}F)}\rangle }_{{\bf{y}}}=\int { \mathcal D }[{\bf{y}}]{ \mathcal D }[{\bf{z}}]\,{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{i=0}^{N}{p}_{m}({y}_{i}| {z}_{i}){{\rm{e}}}^{-\beta {\rm{\Delta }}{{e}}^{\dagger }({z}_{0},{z}_{f})}{{\rm{e}}}^{\beta \delta {q}^{\dagger }[{{\bf{z}}}^{\dagger }]}{{\rm{e}}}^{-\beta {W}_{m}[{\bf{y}}]}.\end{eqnarray} \tag{ 18 }$$

Here, ${\rm{\Delta }}{e}^{\dagger }({z}_{0},{z}_{f})$ and $\delta {q}^{\dagger }[{{\bf{z}}}^{\dagger }]$ are the energy difference and the exchange of heat with the reservoir along the system's backward trajectory, respectively. The first law also holds for the backwards paths of the system, ${\rm{\Delta }}{e}^{\dagger }=-{\rm{\Delta }}e({z}_{0},{z}_{f})={W}^{\dagger }[{{\bf{z}}}^{\dagger }]+\delta {q}^{\dagger }[{{\bf{z}}}^{\dagger }]$ and assuming time-reversal symmetry of the measurement, ${p}_{m}({y}_{i}| {z}_{i})={p}_{m}({y}_{i}^{* }| {z}_{i}^{* })$, we can further simplify equation (18) to

$$\begin{eqnarray}{\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}F)}\rangle }_{{\bf{y}}} & = & \displaystyle \int { \mathcal D }[{\bf{y}}]{ \mathcal D }[{\bf{z}}]\,{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{i=0}^{N}{p}_{m}({y}_{i}^{* }| {z}_{i}^{* }){{\rm{e}}}^{-\beta {W}^{\dagger }[{{\bf{z}}}^{\dagger }]}{{\rm{e}}}^{\beta {W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }]}\\ & = & \displaystyle \int { \mathcal D }[{\bf{y}}]{ \mathcal D }[{\bf{z}}]\,{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }]{{\rm{e}}}^{\beta ({W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }]-{W}^{\dagger }[{{\bf{z}}}^{\dagger }])},\end{eqnarray} \tag{ 19 }$$

where we have used that the measured work is asymmetric under time reversal, ${W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }]=-{W}_{m}[{\bf{y}}]$, which directly follows from the corresponding property of the true work. Thus, one finally arrives at the following expression for the MJE:

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}F)}\rangle }_{{\bf{y}}}={\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}.\end{eqnarray} \tag{ 20 }$$

This expression results from a formal manipulation and is at this point, however, still explicitly dependent on the (backward) trajectories ${{\bf{z}}}^{\dagger }$ of the system and is therefore of limited practical use. Later on we will see how to overcome this difficulty for various examples were we use equation (20) as our formal starting point. Note that depending on the probability distribution $p({W}^{\dagger },{W}_{m}^{\dagger })$ an expansion in terms of the moments of the distribution could be also attempted.

As an important limiting case we immediately see that for a perfect measurement, ${p}_{m}({y}_{k}| {z}_{k})={\delta }_{{y}_{k},{z}_{k}}$, the measured work coincides with the work of the system, ${W}_{m}[{\bf{y}}]=W[{\bf{z}}]$, and the right-hand side becomes unity recovering the original JE (see equation (2)). Moreover, the right-hand side of equation (20) may also be equal to one if there is a certain symmetry in the driven system, such that ${W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }]={W}^{\dagger }[{{\bf{z}}}^{\dagger }]$ (see, e.g., section 3.2).

Finally, let us comment on recent work by García-García et al [13], who also derive a modified JE including measurement errors and which is equivalent to our result, equation (20). However, their point of view as well as the derivation differ from the present approach. García-García et al introduce the error $E[{\bf{z}},{\bf{y}}]=W[{\bf{z}}]-{W}_{m}[{\bf{y}}]$ of system and measured work and derive a fluctuation theorem for the joint distribution of the measured work and this error [13]:

$$\begin{eqnarray}&&\mathrm{ln}\displaystyle \frac{p^{\prime} ({W}_{m},E)}{p{^{\prime} }^{\dagger }(-{W}_{m},-E)}=\beta ({W}_{m}+E-{\rm{\Delta }}F).\end{eqnarray} \tag{ 21 }$$

From the latter relation, one can immediately derive equation (20). Thus, whereas all measurement errors in [13] are incorporated at the level of the final work distribution $p^{\prime} ({W}_{m},E)$, we start with a particular measurement model for the state of the system expressed in terms of ${p}_{m}({y}_{k}| {z}_{k})$. This is closer to a microscopic modeling of the situation because any measurement model for the system ${p}_{m}({y}_{k}| {z}_{k})$ will also yield a certain work distribution $p^{\prime} ({W}_{m},E)$, whereas for a given work distribution $p^{\prime} ({W}_{m},E)$ there might be many different measurement models (and even different systems) which yield the same $p^{\prime} ({W}_{m},E)$. Thus, our findings show a completely different path to derive fluctuation theorems in the presence of measurement errors. Whether our approach or the one of [13] is superior might depend strongly on the specific situation and the system under study.

In the following sections we examine two paradigmatic systems for which the right-hand side of equation (20) can be evaluated analytically, namely an OBP in a harmonic potential in section 3.2 and a TLS in section 3.3.

We consider the overdamped dynamics of a particle in a harmonic potential in one-dimension such that the Hamiltonian of the system is only given by the potential energy:

$$\begin{eqnarray}&&{H}_{\lambda (t)}(z)={V}_{\lambda (t)}(z)={f}_{\lambda (t)}{(z-{\mu }_{\lambda (t)})}^{2}.\end{eqnarray} \tag{ 22 }$$

The stiffness ${f}_{\lambda (t)}$ as well as the center of the potential ${\mu }_{\lambda (t)}$ can be altered in time by an external driving protocol $\lambda (t)$. To simulate the system dynamics we use the Langevin equation

$$\begin{eqnarray}&&\dot{z}(t)=-\beta {{DV}}_{\lambda (t)}^{\prime }(z)+\sqrt{2D}\xi (t)\end{eqnarray} \tag{ 23 }$$

with diffusion constant D, which is related to the friction constant γ by the Einstein relation $D={(\beta \gamma )}^{-1}$, and Gaussian white noise $\xi (t)$.

We specify our measurement model by assuming that the measured position of the particle y_i is normally distributed around the real position z_i with a standard deviation of ${\sigma }_{m}$,

$$\begin{eqnarray}&&{p}_{m}({y}_{i}| {z}_{i})=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{m}^{2}}}\exp \left(-\displaystyle \frac{{({z}_{i}-{y}_{i})}^{2}}{2{\sigma }_{m}^{2}}\right),\end{eqnarray} \tag{ 24 }$$

such that, if ${\sigma }_{m}\to 0$, the conditional probability becomes a Dirac distribution and the measured coordinate coincides with the true coordinate of the particle. Such a Gaussian measurement model might be a good approximation for a noisy measurement without systematic error (i.e., we have $\langle y\rangle =\langle z\rangle$) and simplifies a lot analytical calculations.

Note that the Langevin equation (23) now merely presents a convenient numerical tool. From the point of view of the observer, it has no objective reality unless ${\sigma }_{m}=0$. The correct state of knowledge of the observer would be indeed described by a stochastic Fokker–Planck equation [20, 21].

3.2.1. Continuous driving protocol

Evaluating the general expression, equation (20), for a continuous and piecewise differentiable (c.p.d.) driving protocol $\lambda (t)$ yields (see appendix A.1 for the derivation)

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}F)}\rangle }_{y}={{\rm{e}}}^{-{\sigma }_{m}^{2}\beta {\rm{\Delta }}f},\end{eqnarray} \tag{ 25 }$$

where ${\rm{\Delta }}f\equiv {f}_{\lambda ({t}_{f})}-{f}_{\lambda (0)}$. The right-hand side of the above equation equals unity for ${\sigma }_{m}=0$ corresponding to the original JE. Similarly, if we vary the width of the potential periodically such that ${f}_{\lambda (0)}={f}_{\lambda ({t}_{f})}$, then the original JE is also recovered. However, this attribute is, as far as we know, specific to the model of the OBP with c.p.d. driving protocol. In general the right-hand side will be different from one. Interestingly, shifting the center ${\mu }_{\lambda (t)}$ of the potential has no effect at all on the MJE. Furthermore, if we define an effective free energy, ${\rm{\Delta }}\tilde{F}\equiv {\rm{\Delta }}F+{\sigma }_{m}^{2}{\rm{\Delta }}f$, which may be interpreted as an additonal contribution due to the uncertainty of the measurements, a JE of the form ${\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}\tilde{F})}\rangle }_{y}=1$ holds.

3.2.2. Instantaneous change of driving protocol ('quench')

We also derive in appendix A.2 an analytic expression for the MJE for an instantaneous change of the system Hamiltonian at a time t_m (also called a 'quench'). We consider here that the position and the width of the parabola is altered at the same time and is constant before and after t_m. We find

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}_{m}-{\rm{\Delta }}F)}\rangle }_{y}=\displaystyle \frac{1}{\sqrt{1+2\beta {\sigma }^{2}\tfrac{{f}_{\lambda (0)}}{{f}_{\lambda ({t}_{f})}}{\rm{\Delta }}f}}\exp \left\{\displaystyle \frac{2{\beta }^{2}{\sigma }_{m}^{2}{f}_{\lambda (0)}^{2}{\rm{\Delta }}{\mu }^{2}}{1+2{\beta }^{2}{\sigma }_{m}^{2}\tfrac{{f}_{\lambda (0)}}{{f}_{\lambda ({t}_{f})}}{\rm{\Delta }}f}\right\},\end{eqnarray} \tag{ 26 }$$

where ${\rm{\Delta }}\mu \equiv {\mu }_{\lambda ({t}_{f})}-{\mu }_{\lambda (0)}$ is the difference of the center of the parabola before and after t_m ².

3.2.3. Numerics

In order to verify our findings we performed Brownian dynamics (BD) simulations and used the weighted ensemble path sampling algorithm [23], which shifts the computational resources towards the sampling of rare trajectories, which have the largest impact on the JE. It has been shown that this method is statistically exact for a broad class of Markovian stochastic processes [24]. Please note that we set $\beta \equiv 1$ for all simulations in this paper.

As a simple example we change both parameters of the potential continuously and linearly in time. We choose ${f}_{\lambda (t)}={f}_{\lambda (0)}+\alpha t$ and ${\mu }_{\lambda (t)}={\mu }_{\lambda (0)}+\alpha ^{\prime} t$. For this driving scheme we find very good agreement of BD simulation and the analytic expression, equation (25), which is presented in figure 2 (left).

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Furthermore, we compare equation (26) with simulation results where intially the Hamiltonian of the system is given by ${H}_{0}={f}_{\lambda (0)}{(z-{\mu }_{\lambda (0)})}^{2}$ and which is instantaneously changed to ${H}_{f}={f}_{\lambda ({t}_{f})}{(z-{\mu }_{\lambda ({t}_{f})})}^{2}$ at t_m. In figure 2 (right) we show the results of the BD simulation (marks) as well as the analytic expression (line) for different values of ${\sigma }_{m}$ verifying our findings also for a quench.

Consider a driven system consisting of two energy levels, a ground state with energy ${\varepsilon }_{\lambda (t)}(g)$ and an excited state with energy ${\varepsilon }_{\lambda (t)}(e)$, coupled to a heat bath with inverse temperature β. The master equation (ME) describing this system is

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}\left(\displaystyle \genfrac{}{}{0em}{}{{p}_{g}(t)}{{p}_{{e}}(t)}\right)={\rm{\Gamma }}\left(\begin{array}{cc}-{{\rm{e}}}^{-\beta {\omega }_{\lambda (t)}/2} & {{\rm{e}}}^{\beta {\omega }_{\lambda (t)}/2}\\ {{\rm{e}}}^{-\beta {\omega }_{\lambda (t)}/2} & -{{\rm{e}}}^{\beta {\omega }_{\lambda (t)}/2}\end{array}\right)\left(\displaystyle \genfrac{}{}{0em}{}{{p}_{g}(t)}{{p}_{{e}}(t)}\right).\end{eqnarray} \tag{ 27 }$$

Here, we denoted the energy gap of excited and ground state by ${\omega }_{\lambda (t)}\equiv {\varepsilon }_{\lambda (t)}(e)-{\varepsilon }_{\lambda (t)}(g)$ and ${p}_{g/e}(t)$ denotes the probability to find the system in the ground/excited state.

We measure the state of the system continuously with $(1-\eta )$ being the probability of measuring the state of the system correctly and consequently η of measuring it wrongly, i.e., we set ${p}_{m}({y}_{k}| {z}_{k})=(1-\eta ){\delta }_{{y}_{k},{z}_{k}}+\eta (1-{\delta }_{{y}_{k},{z}_{k}})$ with $\eta \in [0,1]$.

3.3.1. Continuous driving protocol

The MJE of the TLS, where the external control parameter $\lambda (t)$ is c.p.d., can be well approximated by (see appendix A.3 for the derivation)

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}^{m}-{\rm{\Delta }}F)}\rangle }_{{\bf{y}}}\approx \exp \left(-\eta \beta {\int }_{0}^{{t}_{f}}{\rm{d}}{t}\,{\dot{\omega }}_{{\lambda }^{\dagger }(t)}({p}_{{\rm{e}}}^{\dagger }(t)-{p}_{g}^{\dagger }(t))\right),\end{eqnarray} \tag{ 28 }$$

where ${p}_{g/e}^{\dagger }(t)$ denotes the probability that the system is in the ground/excited state in the backward process at time t, respectively. Furthermore, ${\omega }_{{\lambda }^{\dagger }(t)}$ denotes the energy gap of the TLS. We remark, that for a c.p.d. protocol with non-differentiable points at $0\lt {t}_{1}\lt \ldots \,\lt \,{t}_{K}\lt {t}_{f}$ we have to split the integral at the respective points as ${\int }_{0}^{{t}_{f}}{\rm{d}}{t}={\int }_{0}^{{t}_{1}}{\rm{d}}{t}+{\int }_{{t}_{1}}^{{t}_{2}}{\rm{d}}{t}\,+\,\cdots \,+\,{\int }_{{t}_{K}}^{{t}_{f}}{\rm{d}}{t}$. Moreover, equation (28) is exact up to first order in η. For higher orders (say ${\eta }^{k}$) we have to assume that ${ \mathcal P }[{z}_{{i}_{1}},\ldots ,{z}_{{i}_{k}}]\approx p({z}_{{i}_{1}})\ldots p({z}_{{i}_{k}})$ which seems to be remarkably well justified (see our numerical results below). In fact, though this result strictly holds only for slow driving, orders of ${\eta }^{k}$ for $k\gg 1$ become negligible since $\eta \in [0,1]$, hence, justifying our approximation. Furthermore, it is important to note that for the evaluation of the right-hand side of equation (28) we only need to solve for the average evolution of the system (as dictated by the master equation); it is not necessary to have access to higher order statistics.

3.3.2. Instantaneous change of driving protocol ('quench')

For a quench we assume that at t_m with $0\lt {t}_{m}\lt {t}_{f}$ the energy levels are shifted instantaneously and are held constant before and after. Then, the MJE is given by (see appendix A.4 for the derivation)

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}^{m}-{\rm{\Delta }}F)}\rangle }_{{\bf{y}}}=1-\eta [1-{p}_{g}^{\dagger }({t}_{m}){{\rm{e}}}^{\beta {\rm{\Delta }}{\omega }^{\dagger }}-{p}_{{e}}^{\dagger }({t}_{m}){{\rm{e}}}^{-\beta {\rm{\Delta }}{\omega }^{\dagger }}],\end{eqnarray} \tag{ 29 }$$

where ${\rm{\Delta }}{\omega }^{\dagger }\equiv {\omega }_{{\lambda }^{\dagger }({t}_{f})}-{\omega }_{{\lambda }^{\dagger }(0)}$ and ${\omega }_{{\lambda }^{\dagger }(t)}$ is defined as before. Note that both relations for the TLS (equation (28) and (29)) give the original JE for perfect measurement ($\eta =0$).

3.3.3. Numerics

To test these expression, we performed Monte Carlo (MC) simulations for different values of $\eta \in [0,0.3]$ for two driving schemes. First, the driving scheme varies the energy levels continuously and linearly in time, i.e., ${\omega }_{\lambda (t)}={\omega }_{0}+\alpha t$. In figure 3 (left) we plotted the left-hand side of equation (28) from MC simulations (marks) and the right-hand side from numerical integration of the associated ME of the backward protocol (line). As one can see, the approxiamtion of the MJE, equation (28), is in very good agreement with the simulation results for small values of η. Note that a value of $\eta =0.3$ corresponds to a very large error of the conditional probability ${p}_{m}({y}_{k}| {z}_{k})$ because for a value of $\eta =0.5$ the measurement becomes identical to infering the system state by a fair coin toss. We also test equation (29) where we change the driving protocol instantaneously, i.e., ${\omega }_{\lambda (t)}={\omega }_{0}+\alpha ^{\prime} {\rm{\Theta }}(t-{t}_{m})$. Here, we find perfect agreement of simulation (marks) and numerical integration (line), which is shown in figure 3 (right).

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Let us suppose we measure our system as we did without feedback control but at one instance in time, denoted t_m with $0\lt {t}_{m}\lt {t}_{f}$, the protocol is changed according to the measurement outcome y_m such that the protocol is fixed before t_m, i.e., $\lambda =\lambda (t)$ for $t\in [0,{t}_{m}]$ and is dependent on y_m after t_m, i.e., $\lambda =\lambda (t,{y}_{m})$ for $t\in ({t}_{m},{t}_{f}]$. The work applied to the system, which now depends on y_m, is given by

$$\begin{eqnarray}&&W[{\bf{z}}| {y}_{m}]={\int }_{0}^{{t}_{m}}{\rm{d}}{t}\,\dot{\lambda }(t)\displaystyle \frac{\partial {H}_{\lambda (t)}[{\bf{z}}(t)]}{\partial \lambda }+{\int }_{{t}_{m}}^{{t}_{f}}{\rm{d}}{t}\,\dot{\lambda }(t,{y}_{m})\displaystyle \frac{\partial {H}_{\lambda (t,{y}_{m})}[{\bf{z}}(t)]}{\partial \lambda ({y}_{m})}.\end{eqnarray} \tag{ 32 }$$

The same equation holds also for the measured work ${W}_{m}[{\bf{y}}| {y}_{m}]$ by interchanging ${\bf{z}}$ with ${\bf{y}}$ (keeping y_m). The probability of a path in phase space $({\bf{z}},{\bf{y}})$ under feedback control is denoted by ${{ \mathcal P }}_{\lambda ({y}_{m})}[{\bf{z}},{\bf{y}}]$ and we again assume that it factorizes into the probability density of the system trajectory ${{ \mathcal P }}_{\lambda ({y}_{m})}[{\bf{z}}]$, which now explicitely depends on y_m, and the conditional probabilities ${\prod }_{i}{p}_{m}({y}_{i}| {z}_{i})$ (see equation (16)). Then, the MJE with feedback control an be expressed as

$$\begin{eqnarray}{\langle {{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}| {y}_{m}]-{\rm{\Delta }}F({y}_{m}))}\rangle }_{{\bf{y}}} & = & \displaystyle \int { \mathcal D }[{\bf{z}}]{ \mathcal D }[{\bf{y}}]\,{{ \mathcal P }}_{\lambda ({y}_{m})}[{\bf{z}},{\bf{y}}]\,{{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}| {y}_{m}]-{\rm{\Delta }}F({y}_{m}))}\\ & = & \displaystyle \int { \mathcal D }[{\bf{z}}]{ \mathcal D }[{\bf{y}}]\,{{ \mathcal P }}_{\lambda ({y}_{m})}[{\bf{z}}]\displaystyle \prod _{i=0}^{N}{p}_{m}({y}_{i}| {z}_{i}){{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}| {y}_{m}]-{\rm{\Delta }}F({y}_{m}))}.\end{eqnarray} \tag{ 33 }$$

Note, that the difference in free energy does now also depend on the measurement outcome, i.e., ${\rm{\Delta }}F={\rm{\Delta }}F({y}_{m})$, because the Hamiltonian of the system at time t_f depends on y_m. Using again the condition of microreversiblity (see equation (9)) and assuming time-reversal symmetry of the conditional probabilities, ${p}_{m}({y}_{i}| {z}_{i})={p}_{m}({y}_{i}^{* }| {z}_{i}^{* })$, the following equation holds:

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}| {y}_{m}]-{\rm{\Delta }}F({y}_{m}))}\rangle }_{{\bf{y}}}\\ &&\quad =\displaystyle \int { \mathcal D }[{{\bf{z}}}^{\dagger }]{ \mathcal D }[{{\bf{y}}}^{\dagger }]\,{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{i=0}^{N}{p}_{m}({y}_{i}^{* }| {z}_{i}^{* }){{\rm{e}}}^{-\beta ({\rm{\Delta }}{{\rm{e}}}^{\dagger }({y}_{m}^{* })-\delta {q}^{\dagger }({y}_{m}^{* })+{W}_{m}[{\bf{y}}| {y}_{m}])}\\ &&\quad =\displaystyle \int { \mathcal D }[{{\bf{z}}}^{\dagger }]{ \mathcal D }[{{\bf{y}}}^{\dagger }]\,{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }]{{\rm{e}}}^{-W[{{\bf{z}}}^{\dagger }| {y}_{m}]}{{\rm{e}}}^{\beta {W}_{m}[{{\bf{y}}}^{\dagger }| {y}_{m}]}.\end{eqnarray} \tag{ 34 }$$

From equation (34) we immediately obtain the MJE in the presence of feedback control:

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{-\beta ({W}_{m}[{\bf{y}}| {y}_{m}]-{\rm{\Delta }}F({y}_{m}))}\rangle }_{{\bf{y}}}={\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }| {y}_{m}]-{W}^{\dagger }[{{\bf{z}}}^{\dagger }| {y}_{m}])}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }},\end{eqnarray} \tag{ 35 }$$

which looks remarkably similar to equation (20). Here, ${W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }| {y}_{m}]$ and ${W}^{\dagger }[{{\bf{z}}}^{\dagger }| {y}_{m}]$ are the measured and true work, respectively, in the backward process applying the time-reversed protocol ${\lambda }^{\dagger }(t,{y}_{m})$ according to the measurement outcome y_m in the forward process. We stress that we do not perform any feedback in the backward process equivalently to [25]. Analogously to the efficacy parameter γ (see equations (30) and (31)) we call the right-hand side of equation (35) measured efficacy parameter,

$$\begin{eqnarray}&&{\gamma }_{m}\equiv {\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }| {y}_{m}]-{W}^{\dagger }[{{\bf{z}}}^{\dagger }| {y}_{m}])}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }},\end{eqnarray} \tag{ 36 }$$

because the JE is evaluated using the measured trajectories. Note the subtle distinction between equations (30) and (35). Equation (30) starts with ${\langle \exp (-\beta (W-{\rm{\Delta }}F))\rangle }_{{\bf{z}}}$ which experimentally requires an error-free detector to evaluate it. We instead start with ${\langle \exp (-\beta (W-{\rm{\Delta }}F))\rangle }_{{\bf{y}}}$ which can be directly evaluated also with a faulty detector. Our final theoretical result (36) then depends on ${{\bf{z}}}^{\dagger }$ indeed. However, based on this definition we show below how to overcome this difficulty for various examples. Furthermore, note that a complementary analytical analysis confirming our results has been reported in [37] for the example of the Szilard engine.

In the limiting case of perfect measurement, ${p}_{m}({y}_{k}| {z}_{k})={\delta }_{{y}_{k},{z}_{k}}$, equation (35) simplifies to

$$\begin{eqnarray}{\langle {{\rm{e}}}^{-\beta {W}_{m}[{\bf{y}}| {y}_{m}]-{\rm{\Delta }}F({y}_{m})}\rangle }_{{\bf{y}}} & = & \displaystyle \int { \mathcal D }[{{\bf{z}}}^{\dagger }]{ \mathcal D }[{{\bf{y}}}^{\dagger }]\,{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger }]\,\displaystyle \prod _{i}{\delta }_{{y}_{i}^{* },{z}_{i}^{* }}{{\rm{e}}}^{-\beta ({W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }| {y}_{m}]-{W}^{\dagger }[{{\bf{z}}}^{\dagger }| {y}_{m}])}\\ =\displaystyle \int { \mathcal D }[{\bf{y}}]{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{y}}}^{\dagger }] & = & \displaystyle \int {{\rm{d}}{y}}_{N}^{* }\ldots \displaystyle \int {{\rm{d}}{y}}_{0}^{* }\,{p}_{{\lambda }_{N}^{* }({y}_{m})}({y}_{N}^{* })\,\ldots {p}_{{\lambda }_{0}^{* }}({y}_{1}^{* }\,\to \,{y}_{0}^{* }).\end{eqnarray} \tag{ 37 }$$

Due to normalization of conditional probabilites, it holds that the integrals of all ${y}_{k}^{* }$ with $k\lt m$ are equal to unity, hence,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \int {{\rm{d}}{y}}_{N}^{* }\ldots \displaystyle \int {{\rm{d}}{y}}_{0}^{* }\,{p}_{{\lambda }_{N}^{* }({y}_{m})}({y}_{N}^{* })\ldots {p}_{{\lambda }_{0}^{* }}({y}_{1}^{* }\,\to \,{y}_{0}^{* })\\ \quad =\displaystyle \int {{\rm{d}}{y}}_{N}^{* }\ldots \displaystyle \int {{\rm{d}}{y}}_{m}^{* }\,{p}_{{\lambda }_{N}^{* }({y}_{m})}({y}_{N}^{* })\,\ldots {p}_{{\lambda }_{m}^{* }}({y}_{m+1}^{* }\,\to \,{y}_{m}^{* })=\displaystyle \int {{\rm{d}}{y}}_{m}^{* }\,{p}_{{\lambda }_{m}^{* }({y}_{m})}({y}_{m}^{* })\\ \quad =\displaystyle \int {{\rm{d}}{z}}_{m}^{* }\,{p}_{{\lambda }^{\dagger }({z}_{m})}({z}_{m}^{* })\mathrm{.}\end{array}\end{eqnarray} \tag{ 38 }$$

Only in this case the efficacy γ and the measured efficacy ${\gamma }_{m}$ are the same as it should be.

However, for a measurement outcome y_m including errors, γ deviates from ${\gamma }_{m}$. The interpretation and physical significance of the difference between γ and ${\gamma }_{m}$ can be explained as follows: consider two observes Alice and Bob. Suppose that Alice measures the state of the system with a faulty detector whereas Bob measures the system with a perfect detector. Furthermore, suppose that only Alice performs the feedback control based on her measurement result at time t_m. Then, if Alice evaluates the JE of the work done on the system along her measured trajectories, she will observe the result ${\gamma }_{m}$. In contrast, Bob—given the correct system trajectories and knowledge about the feedback action of Alice and her faulty detector—is able to verify the standard Sagawa–Ueda relation with the efficacy parameter γ.

As an explicit example, for which we can evaluate the right-hand side of equation (35) analytically, we look again at an OBP in a harmonic potential (see section 3.2) and assume that the center of the potential is intially at ${\mu }_{\lambda (0)}=0$ and the width is ${f}_{\lambda (0)}$. Both parameters will be changed instantaneously at time t_m if the measured position at that time is ${y}_{m}\gt 0$, the position to ${\mu }_{\lambda ({t}_{f})}$ and the stiffness to ${f}_{\lambda ({t}_{f})}$. Otherwise, for ${y}_{m}\lt 0$, the potential remains unchanged. For this specific example equation (35) can be evaluated explicitly and we obtain (see appendix A.5)

$$\begin{eqnarray}&&{\gamma }_{m}=\displaystyle \frac{1}{2}\left(1+\displaystyle \frac{1}{\sqrt{1+\tfrac{\kappa }{{f}_{\lambda ({t}_{f})}}{\rm{\Delta }}f}}{\rm{erfc}}\left[-{\mu }_{{t}_{f}}\sqrt{\displaystyle \frac{\beta {f}_{\lambda ({t}_{f})}(1+\kappa )}{1+\tfrac{\kappa }{{f}_{\lambda ({t}_{f})}}{\rm{\Delta }}f}}\right]\exp \left\{\displaystyle \frac{\kappa \beta {f}_{\lambda (0)}{\mu }_{{t}_{f}}^{2}}{1+\tfrac{\kappa }{{f}_{\lambda ({t}_{f})}}{\rm{\Delta }}f}\right\}\right),\end{eqnarray} \tag{ 39 }$$

where $\kappa \equiv 2\beta {f}_{\lambda (0)}{\sigma }_{m}^{2}$.

For the special case of only altering ${f}_{\lambda (t)}$ and keeping the position of the parabola fixed, i.e., ${\mu }_{\lambda ({t}_{f})}={\mu }_{\lambda (0)}$, equation (39) reduces to

$$\begin{eqnarray}&&{\gamma }_{m}=\displaystyle \frac{1}{2}\left[1+{\left(1+2\beta \displaystyle \frac{{f}_{\lambda (0)}}{{f}_{\lambda ({t}_{f})}}{\rm{\Delta }}f{\sigma }_{m}^{2}\right)}^{-1/2}\right].\end{eqnarray} \tag{ 40 }$$

On the other hand, if the stiffness is held constant, ${f}_{\lambda ({t}_{f})}={f}_{\lambda (0)}=f$, but the parabola is shifted, we find

$$\begin{eqnarray}&&{\gamma }_{m}=\displaystyle \frac{1}{2}(1+{{\rm{e}}}^{2{(f\beta {\mu }_{{t}_{f}}{\sigma }_{m})}^{2}}{\rm{erfc}}[-{\mu }_{{t}_{f}}\sqrt{f\beta (1+2f\beta {\sigma }_{m}^{2})}]).\end{eqnarray} \tag{ 41 }$$

We have varified equations (39)–(41) by perfoming BD simulations for various driving schemes (not shown here) and will discuss the paradigmatic model of an 'information ratchet' [25] in the next paragraph in more detail also showing numerical results.

4.2.1. Information ratchet

The Brownian particle is initially in thermal equilibrium in the harmonic potential with center ${\mu }_{0}$. We then measure the position of the particle y_m at time t_m and perform the following feedback scheme: if ${y}_{m}\geqslant {\mu }_{0}+L$ with $L\gt 0$ being constant, we shift the center of the potential ${\mu }_{t\gt {t}_{m}}={\mu }_{0}+2L$, if ${y}_{m}\lt {\mu }_{0}+L$ we do nothing. We then replace ${\mu }_{0}\to {\mu }_{0}+2L$ and start over again after some transient relaxation time. By repeatingly performing this feedback protocol, we can actually move the average position of the particle to the right, ideally without performing work. Here, ${\rm{\Delta }}F=0$ holds throughout the whole process. Furthermore, one can also extract work from the system by this feedback control if the particle is transported against a potential gradient as, e.g., in the experiment [35]. For a single step of the ratchet, where we put ${\mu }_{0}=0$ for simplicity, the measured efficacy with feedback control is given by

$$\begin{eqnarray}&&{\gamma }_{m}=\displaystyle \frac{1}{2}\left({\rm{erfc}}\left[-\displaystyle \frac{L}{\sqrt{\tfrac{1}{\beta f}+2{\sigma }_{m}^{2}}}\right]+{{\rm{e}}}^{8{f}^{2}{L}^{2}{\beta }^{2}{\sigma }_{m}^{2}}{\rm{erfc}}\left[-L\displaystyle \frac{1+4f\beta {\sigma }_{m}^{2}}{\sqrt{\tfrac{1}{\beta f}+2{\sigma }_{m}^{2}}}\right]\right).\end{eqnarray} \tag{ 42 }$$

The derivation follows the same steps as in appendix A.5 but the integral of ${y}_{m}^{* }$ is splitted at L instead of 0. Equation (42) differs from the efficacy parameter γ of the original information ratchet [25],

$$\begin{eqnarray}&&\gamma ={\rm{erfc}}\left[-\displaystyle \frac{L}{\sqrt{\tfrac{1}{\beta f}+2{\sigma }_{m}^{2}}}\right].\end{eqnarray} \tag{ 43 }$$

In figure 4 (left) we plot the solutions of the two equations above as function of the variance of the measurement ${\sigma }_{m}$. The two equations coincide for the case of perfect measurement. However, for finite values of ${\sigma }_{m}$ the efficacy γ of the feedback control (dashed line) is lower than for perfect measurement: if the measurement has an error, then the potential will be shifted even though the real position of the particle may not be greater than L. Then we may actually apply work to the system instead of extracting it and the average value of extracted work is lower for noisier measurements.

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If we look at the work we measure using the same apparatus as we have used to measure y_m (line), we see that with increasing measurement error ${\sigma }_{m}$, the measured efficacy ${\gamma }_{m}$ also increases in strong contrast to γ. Since the measured work is given in terms of the measured position y_m of the particle, we always apply the 'correct' feedback scheme from the observer's point of view. Thus, we (the observer) always think that we extract work. This can also be seen in the distribution of measured (purple) and system (blue) work in figure 4 (right), where the probability of measured work is only non-zero for ${W}_{m}\lt 0$. To support this claim even further, we can calculate the average measured and system work by integration of equation (32) over z and y_m, where the integral is non-zero only if ${y}_{m}\gt L$. The difference of them results in

$$\begin{eqnarray}&&\langle W[{\bf{y}}| {y}_{m}]\rangle -\langle W[{\bf{z}}| {y}_{m}]\rangle =-4{fL}{\sigma }_{m}^{2}\sqrt{\displaystyle \frac{f\beta }{\pi (1+\kappa )}}\exp \left\{-\displaystyle \frac{f\beta {L}^{2}}{1+\kappa }\right\}\leqslant 0,\end{eqnarray} \tag{ 44 }$$

where $\kappa =2\beta f{\sigma }_{m}^{2}$. Thus, on average the measured extracted work (note that in our convention work is positive if it is done on the system) from the system will be greater than the true extracted work and even increases with ${\sigma }_{m}$. For a larger value of ${\sigma }_{m}$ the probability distribution ${p}_{m}^{\prime }({y}_{m})$ (see equation (13)) of the measured position y_m is broader (i.e., has a larger variance) than $p(z)$, but still has the same mean value as $p(z)$. Then, measurement outcomes with ${y}_{m}\gt L$ are more frequent and ${\gamma }_{m}$ increases.

Similarly to the derivation of the MJE of the TLS without feedback we find with feedback for a c.p.d. but at this point unspecified driving protocol an approximation for the modification of the original JE (see appendix A.6 for details):

$$\begin{eqnarray}{\gamma }_{m} & \approx & \displaystyle \sum _{{z}_{m}\in \{g,{e}\}}\left[(1-\eta ){p}_{{\lambda }^{\dagger }({z}_{m})}({z}_{m})\exp \left(-\eta \beta \displaystyle \int {\rm{d}}{t}\,{\dot{\omega }}_{{\lambda }^{\dagger }(t,{z}_{m})}({p}_{{e},{\lambda }^{\dagger }({z}_{m})}(t)-{p}_{g,{\lambda }^{\dagger }({z}_{m})}(t))\right)\right.\\ & & \left.-\eta {p}_{{\lambda }^{\dagger }({\bar{z}}_{m})}({z}_{m})\exp \left(-\eta \beta \displaystyle \int {\rm{d}}{t}\,{\dot{\omega }}_{{\lambda }^{\dagger }(t,{\bar{z}}_{m})}({p}_{{e},{\lambda }^{\dagger }({\bar{z}}_{m})}(t)-{p}_{g,{\lambda }^{\dagger }({\bar{z}}_{m})}(t))\right)\right].\end{eqnarray} \tag{ 45 }$$

Here, ${p}_{z,{\lambda }^{\dagger }({y}_{m})}(t)$ is the probability for the system to be in state z (ground or excited) at time t in the backward process with the backward protocol according to the measurement outcome y_m (ground or excited state) in the forward process. We again note that we do not apply feedback in the backward process and that equation (45) is valid under exactly the same conditions as discussed below equation (28). Furthermore, ${\omega }_{{\lambda }^{\dagger }(t,{y}_{m})}$ is the energy gap as defined in section 3.3 with the time-reversed protocol according to the outcome of the forward process. For a c.p.d. protocol with non-differentiable points the integral in equation (45) is again split into parts at the respective points. For most driving protocols with feedback we have considered numerically (not shown here) equation (45) is a very good approximation.

For a driving protocol that is not continuous in time, we find a different expression. Here, we assume as in the case without feedback, that before and after t_m the protocol is constant and that a quench is performed at time t_m. We then find for the MJE (see also appendix A.6)

$$\begin{eqnarray}&&{\gamma }_{m}=\displaystyle \sum _{{z}_{m}\in \{g,e\}}[(1-\eta ){p}_{{z}_{m},{\lambda }^{\dagger }({z}_{m})}({t}_{m})+\eta {p}_{{z}_{m},{\lambda }^{\dagger }({\bar{z}}_{m})}({t}_{m}){{\rm{e}}}^{\beta {\rm{\Delta }}{\omega }_{{\lambda }^{\dagger }({\bar{z}}_{m})}({z}_{m})}],\end{eqnarray} \tag{ 46 }$$

where ${p}_{{z}_{m},{\lambda }^{\dagger }({\bar{z}}_{m})}({t}_{m})$ denotes the probability of the system to be in state z_m at time t_m in the backward process with the backward protocol according to the measurement outcome ${y}_{m}={\bar{z}}_{m}$. Here, we introduced the complementary state ${\bar{z}}_{m}$ to z_m (i.e., if ${z}_{m}=g$ then ${\bar{z}}_{m}=e$ and vice versa). Furthermore, ${\rm{\Delta }}{\omega }_{{\lambda }^{\dagger }({\bar{z}}_{m})}({z}_{m})\equiv {\omega }_{{\lambda }^{\dagger }({t}_{f},{\bar{z}}_{m})}({z}_{m})-{\omega }_{{\lambda }^{\dagger }(0,{\bar{z}}_{m})}({z}_{m})$ and ${\omega }_{{\lambda }^{\dagger }(t,{\bar{z}}_{m})}({z}_{m})={\varepsilon }_{{\lambda }^{\dagger }(t,{\bar{z}}_{m})}({z}_{m})-{\varepsilon }_{{\lambda }^{\dagger }(t,{\bar{z}}_{m})}({\bar{z}}_{m})$.

We will now discuss an example of a protocol with a quench in detail in the next paragraph.

4.3.1. Conditional swap

As a specific example, for which we can extract work from a single heat bath by measuring the state of the TLS at time t_m, we discuss a feedback operation which we calll a conditional swap: if at time t_m the measured state of the TLS y_m is the excited state, we interchange the two energy levels such that we extract work of $\omega ={\varepsilon }_{e}-{\varepsilon }_{g}$ if the system state z_m is the excited one and perform work of $-\omega$ if ${z}_{m}=g$. If ${y}_{m}=g$ we do nothing. We compare our findings (see equation (46)) of this conditional swap to the corresponding expression of the efficacy parameter γ, which is given for this specific example by

$$\begin{eqnarray}&&\gamma =(1-\eta )2{p}_{g,{\lambda }^{\dagger }(g)}({t}_{m})+2\eta {p}_{e,\lambda (g)}({t}_{m}).\end{eqnarray} \tag{ 47 }$$

Note that in the model of the conditional swap ${p}_{g,{\lambda }^{\dagger }(g)}({t}_{m})={p}_{e,{\lambda }^{\dagger }(e)}({t}_{m})$ and ${p}_{e,{\lambda }^{\dagger }(g)}({t}_{m})=\,{p}_{g,{\lambda }^{\dagger }(e)}({t}_{m})$.

We show the difference of γ (dashed) and ${\gamma }_{m}$ (line) for different values of η in figure 5 (left). As one can see, for a perfect measurement they result in the same value. However if η is greater than zero, the two differ. The explanation is very similar to the one of the information ratchet discussed in section 4.2: if the measurement y_m involves errors, the two states are sometimes interchanged even though the system may be in the ground state resulting in work applied to the system instead of extracting work from the system. If we look at the work distribution of the system (see figure 5 right top), one can see that for values $\eta \gt 0$, the extracted work becomes less whereas the probability of applying work to the system increases with measurement error (note that in our convention work is negative if it is done by the system). Then the efficacy parameter is lower than without measurement error. On the other hand, if we look at the measured work (see figure 5 right bottom), which is calculated from the measured state of the system, we only measure positive work extraction from the system by perfoming the conditional swap. Furthermore, the probability of measuring the excited state of the system is always larger than the actual probability of the system to be in the excited state if ${p}_{e}({t}_{m})\lt 1/2$ (as in our case),

$$\begin{eqnarray}&&{p}_{{y}_{m}=e}^{\prime }({t}_{m})=(1-\eta ){p}_{e}({t}_{m})+\eta {p}_{g}({t}_{m})={p}_{e}({t}_{m})+\eta (1-2{p}_{e}({t}_{m}))\geqslant {p}_{e}({t}_{m}).\end{eqnarray} \tag{ 48 }$$

Therefore, the probability of extracting work from the system and therefore ${\gamma }_{m}$ increases with larger values of η.

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In this section we derive the analytic expression of the MJE for an OBP in a harmonic potential in one-dimension, namely equation (25). We assume the external control parameter $\lambda (t)$ to be c.p.d. throughout this section. The discretized work along a trajectory ${\bf{z}}$ given the Hamiltonian in equation (22) becomes

$$\begin{eqnarray}&&W[{\bf{z}}]=\displaystyle \sum _{i}(\delta {f}_{{\lambda }_{i}}{z}_{i-1}^{2}-2\delta {[f\mu ]}_{{\lambda }_{i}}{z}_{i-1}+\delta {[f{\mu }^{2}]}_{{\lambda }_{i}}),\end{eqnarray} \tag{ A.1 }$$

where $\delta {f}_{{\lambda }_{i}}={f}_{{\lambda }_{i}}-{f}_{{\lambda }_{i-1}}$, $\delta {[f\mu ]}_{{\lambda }_{i}}={f}_{{\lambda }_{i}}{\mu }_{{\lambda }_{i}}-{f}_{{\lambda }_{i-1}}{\mu }_{{\lambda }_{i-1}}$ and $\delta {[f{\mu }^{2}]}_{{\lambda }_{i}}={f}_{{\lambda }_{i}}{\mu }_{{\lambda }_{i}}^{2}-{f}_{{\lambda }_{i-1}}{\mu }_{{\lambda }_{i-1}}^{2}$. For the example considered here, it holds that ${z}_{i}^{* }\,=\,{z}_{i}$ and ${y}_{i}^{* }\,=\,{y}_{i}$.

By factorizing ${ \mathcal P }[{\bf{z}},{\bf{y}}]$ (see equation (16)) one can express the right-hand side of the general equation (20) as

$$\begin{eqnarray}{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }} & = & \displaystyle \int { \mathcal D }[{{\bf{z}}}^{\dagger }]\,{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \int {{\rm{d}}{y}}_{0}\ldots \displaystyle \int {{\rm{d}}{y}}_{N}\displaystyle \prod _{i}[{p}_{m}({y}_{i}| {z}_{i})\\ & & \times \exp \{\beta (\delta {f}_{{\lambda }_{i+1}^{\dagger }}{y}_{i}^{2}-2\delta {[f\mu ]}_{{\lambda }_{i+1}^{\dagger }}{y}_{i}-\delta {f}_{{\lambda }_{i+1}^{\dagger }}{z}_{i}^{2}+2\delta {[f\mu ]}_{{\lambda }_{i+1}^{\dagger }}{z}_{i})\}].\end{eqnarray} \tag{ A.2 }$$

Assuming a normal distribution of ${p}_{m}({y}_{i}| {z}_{i})$ (see equation (24)) we find after integration over all y_k:

$$\begin{eqnarray}{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }} & = & \left(\displaystyle \prod _{i}\displaystyle \frac{1}{\sqrt{1-2\beta \delta {f}_{{\lambda }_{i+1}^{\dagger }}{\sigma }_{m}^{2}}}\right)\\ & & \times \displaystyle \int { \mathcal D }[{{\bf{z}}}^{\dagger }]\,{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\exp \left\{-\displaystyle \sum _{i}\displaystyle \frac{2{\beta }^{2}{\sigma }_{m}^{2}}{2\beta \delta {f}_{{\lambda }_{i+1}^{\dagger }}{\sigma }_{m}^{2}-1}{(\delta {[f\mu ]}_{{\lambda }_{i+1}^{\dagger }}-\delta {f}_{{\lambda }_{i+1}^{\dagger }}{z}_{i})}^{2}\right\}.\end{eqnarray} \tag{ A.3 }$$

Note that for the integral over y_k to converge the standard deviation of the measurement must obey

$$\begin{eqnarray}&&{\sigma }_{m}^{2}\lt \displaystyle \frac{1}{2\beta | \delta {f}_{{\lambda }_{k+1}^{\dagger }}| }.\end{eqnarray} \tag{ A.4 }$$

This means that in an experimental setup (or also for simulations), in which the width of the potential is varied between two measurements by a finite value $\delta {f}_{{\lambda }_{k+1}^{\dagger }}$, the deviation of measured and system coordinate cannot be arbitrarily large.

We first look at the integral of equation (A.3): in the limit $N\to \infty$ the time steps ${\rm{d}}{t}={t}_{f}/N$ become infinitesimal and we can write the term in the exponential approximately as

$$\begin{eqnarray}&&\exp \left\{-\displaystyle \sum _{i}\displaystyle \frac{2{\beta }^{2}{\sigma }_{m}^{2}}{2\beta \delta {f}_{{\lambda }_{i+1}^{\dagger }}{\sigma }_{m}^{2}-1}{(\delta {[f\mu ]}_{{\lambda }_{i+1}^{\dagger }}-\delta {f}_{{\lambda }_{i+1}^{\dagger }}{z}_{i})}^{2}\right\}\\ &&\quad \approx \exp \left\{2{\beta }^{2}{\sigma }_{m}^{2}{\rm{d}}{t}{\displaystyle \int }_{0}^{{t}_{f}}{\rm{d}}{t}{({[f\mu ]}_{{\lambda }^{\dagger }(t)}^{\prime }-{f}_{{\lambda }^{\dagger }(t)}^{\prime }z(t))}^{2}\right\}\equiv \star ,\end{eqnarray} \tag{ A.5 }$$

where the prime (e.g, $f^{\prime}$) denotes a derivative with respect to time t. Note that the additional dt in front of the integral is correct. Furthermore, this step is only exact provided that the protocol is differentiable. However, as long as it is continuous and only non-differentiable at a finite number of points $0\lt {t}_{1}\lt \ldots \,\lt \,{t}_{K}\lt {t}_{f}$ this argument can be easily generalized by splitting the integral at the respective places (i.e., ${\int }_{0}^{{t}_{1}}{\rm{d}}{t}+{\int }_{{t}_{1}}^{{t}_{2}}{\rm{d}}{t}\,+\,\cdots \,+\,{\int }_{{t}_{K}}^{{t}_{f}}{\rm{d}}{t}$) and by observing that due to the continuity $\delta {f}_{{\lambda }_{i+1}^{\dagger }}$ and $\delta {[f\mu ]}_{{\lambda }_{i+1}^{\dagger }}$ remain infinitesimal small at all points. Then, by the mean value theorem of integration we know that there exists a $\xi \in [0,{t}_{f}]$ such that

$$\begin{eqnarray}&&\star =\exp \{2{\beta }^{2}{\sigma }_{m}^{2}{\rm{d}}{t}{({[f\mu ]}_{{\lambda }^{\dagger }(\xi )}^{\prime }-{f}_{{\lambda }^{\dagger }(\xi )}^{\prime }z(\xi ))}^{2}\}.\end{eqnarray} \tag{ A.6 }$$

And hence, this term becomes 1 for $N\to \infty$, i.e., ${\rm{d}}{t}\to 0$.

Therefore, equation (A.3) simplifies to

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\displaystyle \prod _{i}\displaystyle \frac{1}{\sqrt{1-2\beta \delta {f}_{{\lambda }_{i+1}^{\dagger }}{\sigma }_{m}^{2}}}\approx \displaystyle \prod _{i}(1+{\sigma }_{m}^{2}\beta \delta {f}_{{\lambda }_{i+1}^{\dagger }}),\end{eqnarray} \tag{ A.7 }$$

which holds for $N\to \infty$. In the last step, we write the product as an exponential and use an approximation of the logarithm up to first order:

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\exp \left(\displaystyle \sum _{i}\mathrm{ln}(1+{\sigma }_{m}^{2}\beta \delta {f}_{{\lambda }_{i+1}^{\dagger }})\right)\approx \exp ({\sigma }_{m}^{2}\beta ({f}_{{\lambda }_{0}}-{f}_{{\lambda }_{N}})).\end{eqnarray} \tag{ A.8 }$$

Taking the limit $N\to \infty$, ${f}_{{\lambda }_{0}}={f}_{\lambda ({t}_{0})}$ and ${f}_{{\lambda }_{N}}={f}_{\lambda ({t}_{N})}$, we arrive at equation (25).

Here, we derive equation (26), where we assume that the stiffness of the harmonic potential as well as the position are instantaneously changed at the same time t_m. Since the driving protocol is constant before and after t_m, it holds that $\delta {f}_{{\lambda }_{k+1}^{\dagger }}=0$ as well as $\delta {[f\mu ]}_{{\lambda }_{k+1}^{\dagger }}=0$ for all $k\ne m$. In this case the right-hand side of equation (20) reads after integration over all y_k and z_k with $k\ne m$:

$$\begin{eqnarray}{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }} & = & \displaystyle \int {{\rm{d}}{z}}_{m}\,{p}^{\dagger }({z}_{m})\displaystyle \int {{\rm{d}}{y}}_{m}\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{m}^{2}}}\exp \left\{-\displaystyle \frac{{({z}_{m}-{y}_{m})}^{2}}{2{\sigma }_{m}^{2}}\right\}\\ & & \times \exp \{\beta \delta {f}_{{\lambda }_{m+1}^{\dagger }}{y}_{m}^{2}-2\delta {[f\mu ]}_{{\lambda }_{m+1}^{\dagger }}{y}_{m}-\delta {f}_{{\lambda }_{m+1}^{\dagger }}{z}_{m}^{2}+2\delta {[f\mu ]}_{{\lambda }_{m+1}^{\dagger }}{z}_{m}\}.\end{eqnarray} \tag{ A.9 }$$

For a quench it holds that $\delta {f}_{{\lambda }_{m+1}^{\dagger }}={f}_{\lambda (0)}-{f}_{\lambda ({t}_{f})}\equiv -{\rm{\Delta }}f$ and equivalently $\delta {[f\mu ]}_{{\lambda }_{m+1}^{\dagger }}={f}_{\lambda (0)}{\mu }_{\lambda (0)}-{f}_{\lambda ({t}_{f})}{\mu }_{\lambda ({t}_{f})}\equiv -{\rm{\Delta }}[f\mu ]$. Then, the integration over y_m yields

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}\\ &&\quad =\displaystyle \int {{\rm{d}}{z}}_{m}{p}^{\dagger }({z}_{m})\displaystyle \frac{1}{\sqrt{1+2{\rm{\Delta }}f\beta {\sigma }_{m}^{2}}}\exp \left\{\displaystyle \frac{2{\beta }^{2}{\sigma }_{m}^{2}}{1+2{\rm{\Delta }}f\beta {\sigma }_{m}^{2}}{(-{\rm{\Delta }}[f\mu ]+{\rm{\Delta }}{{fz}}_{m})}^{2}\right\}.\end{eqnarray} \tag{ A.10 }$$

For the integral over y_m to converge, it must again hold that ${\sigma }_{m}^{2}\lt {(2\beta | {\rm{\Delta }}f)}^{-1}$.

We now use, that for the harmonic potential the probability distribution of the position of the OBP in equilibrium (initial system state) is Gaussian distributed with mean ${\mu }_{\lambda ({t}_{f})}$ and variance ${(2\beta {f}_{\lambda ({t}_{f})})}^{-1/2}$ in the time-reversed protocol. The integration over z_m then finally yields equation (26).

Note that, the integral over z_m only converges if

$$\begin{eqnarray}&&{\sigma }_{m}^{2}\leqslant \displaystyle \frac{1}{2\beta | {\rm{\Delta }}f| }\displaystyle \frac{{f}_{\lambda ({t}_{f})}}{{f}_{\lambda (0)}}\,.\end{eqnarray} \tag{ A.11 }$$

In this section we derive the analytic expression of the MJE for a driven TLS, namely equation (28). We assume that the protocol $\lambda (t)$ changes continuously and is piecewise differentiable as in appendix A.1. For the TLS it also holds that ${z}^{* }\,=\,z$ and ${y}^{* }\,=\,y$. The work along a trajectory ${\bf{z}}$ can be discretized as

$$\begin{eqnarray}&&W[{\bf{z}}]=\displaystyle \sum _{i}({\varepsilon }_{{\lambda }_{i}}({z}_{i-1})-{\varepsilon }_{{\lambda }_{i-1}}({z}_{i-1}))\equiv \displaystyle \sum _{i}\delta {\varepsilon }_{{\lambda }_{i}}({z}_{i-1}).\end{eqnarray} \tag{ A.12 }$$

Equivalently, the measured work is given by ${W}_{m}[{\bf{y}}]={\sum }_{i}\delta {\varepsilon }_{{\lambda }_{i}}({y}_{i-1})$.

Then we can evaluate the right-hand side of equation (20) analytically as follows:

$$\begin{eqnarray}&&\begin{array}{l}{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}\\ \quad =\displaystyle \sum _{{{\bf{z}}}^{\dagger }}{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{i}\left(\displaystyle \sum _{{y}_{i}}[(1-\eta ){\delta }_{{y}_{i},{z}_{i}}+\eta (1-{\delta }_{{y}_{i},{z}_{i}})]{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({y}_{i})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}\right)\\ \quad =\displaystyle \sum _{{{\bf{z}}}^{\dagger }}{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{i}((1-2\eta )+\eta [{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}(g)-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}+{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({e})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}]).\end{array}\end{eqnarray} \tag{ A.13 }$$

Here, $\sum _{{{\bf{z}}}^{\dagger }}={\sum }_{{z}_{N}}\,\ldots \,{\sum }_{{z}_{0}}$ denotes all the sums over z_k and $\delta {\varepsilon }_{{\lambda }_{k}^{\dagger }}({z}_{k-1})$ is defined as in equation (A.12) with the time-reversed protocol ${\lambda }^{\dagger }(t)$. To further simplify equation (A.13) we introduce the complementary state ${\bar{z}}_{k}$ such that ${\bar{z}}_{k}\ne {z}_{k}$ for all k, i.e. if ${z}_{k}=e$ then ${\bar{z}}_{k}=g$ and vice versa. Consequently,

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\displaystyle \sum _{{{\bf{z}}}^{\dagger }}{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{i}((1-2\eta )+\eta [1+{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({\bar{z}}_{i})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}]).\end{eqnarray} \tag{ A.14 }$$

For large N we approximate

$$\begin{eqnarray}&&1+{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({\bar{z}}_{i})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}\approx 2+\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({\bar{z}}_{i})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))\equiv 2+\beta {\delta }_{i},\end{eqnarray} \tag{ A.15 }$$

such that we can write equation (A.14) simply as

$$\begin{eqnarray}&&{\langle {e}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{z}^{\dagger },{y}^{\dagger }}=\sum _{{z}^{\dagger }}{{ \mathcal P }}^{\dagger }[{z}^{\dagger }]\prod _{i}(1+\eta \beta {\delta }_{i}).\end{eqnarray} \tag{ A.16 }$$

Writing the product explicitely yields

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\displaystyle \sum _{{{\bf{z}}}^{\dagger }}{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\displaystyle \sum _{n=0}^{N}\left(\displaystyle \frac{1}{n!}{(\eta \beta )}^{n}\displaystyle \sum _{{i}_{1}\ne \ldots \ne {i}_{n}}{\delta }_{{i}_{1}}\times \ldots \times {\delta }_{{i}_{n}}\right).\end{eqnarray} \tag{ A.17 }$$

We now make the crucial assumption that ${{ \mathcal P }}^{\dagger }[{z}_{{k}_{1}},\ldots ,{z}_{{k}_{n}}]\approx {p}^{\dagger }({z}_{{k}_{1}})\,\ldots {p}^{\dagger }({z}_{{k}_{n}})$. Then,

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\displaystyle \sum _{{{\bf{z}}}^{\dagger }}\displaystyle \sum _{n=0}^{N}\displaystyle \frac{{(\eta \beta )}^{n}}{n!}\displaystyle \sum _{{i}_{1},\ldots ,{i}_{n}}{p}^{\dagger }({z}_{{i}_{1}}){\delta }_{{i}_{1}}\ldots {p}^{\dagger }({z}_{{i}_{n}}){\delta }_{{i}_{n}}-{ \mathcal R }.\end{eqnarray} \tag{ A.18 }$$

To ensure this equality, we introduced a 'rest' term ${ \mathcal R }$ of the form

$$\begin{eqnarray}{ \mathcal R } & = & \displaystyle \frac{{(\eta \beta )}^{2}}{2!}\displaystyle \sum _{i}\displaystyle \sum _{{z}_{i}}{p}^{\dagger }{({z}_{i})}^{2}{\delta }_{i}^{2}\\ & & +\displaystyle \frac{{(\eta \beta )}^{3}}{3!}\displaystyle \sum _{{ij}}\displaystyle \sum _{{z}_{i},{z}_{j}}{p}^{\dagger }{({z}_{i})}^{2}{\delta }_{i}^{2}{p}^{\dagger }({z}_{j}){\delta }_{j}+\displaystyle \frac{{(\eta \beta )}^{3}}{3!}\displaystyle \sum _{i}\displaystyle \sum _{{z}_{i}}{p}^{\dagger }{({z}_{i})}^{3}{\delta }_{i}^{3}+\ldots \,\end{eqnarray} \tag{ A.19 }$$

taking care of the sums where at least two of the indices ${i}_{1},\ldots ,{i}_{n}$ are equal. But then all terms of ${ \mathcal R }$ are at least of the order ${ \mathcal O }\left(\tfrac{1}{N}\right)$ and therefore vanish for $N\to \infty$. Hence, we are left with evaluating

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\displaystyle \sum _{i=0}^{N}\displaystyle \frac{1}{n!}{(\eta \beta )}^{n}\displaystyle \sum _{{i}_{1},..,{i}_{n}}\displaystyle \,\sum \text{}\,_{{z}_{{i}_{1}},\ldots ,{z}_{{i}_{n}}}{p}^{\dagger }({z}_{{i}_{1}}){\delta }_{{i}_{1}}\ldots {p}^{\dagger }({z}_{{i}_{n}}){\delta }_{{i}_{n}}.\end{eqnarray} \tag{ A.20 }$$

Taking the limit $N\to \infty$, we can write

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{\delta t\to 0}\displaystyle \frac{{\delta }_{k}}{\delta t}=\mathop{\mathrm{lim}}\limits_{\delta t\to 0}\displaystyle \frac{\delta {\varepsilon }_{{\lambda }_{k+1}^{\dagger }}({\bar{z}}_{k})-\delta {\varepsilon }_{{\lambda }_{k+1}^{\dagger }}({z}_{k})}{\delta t}={\dot{\varepsilon }}_{{\lambda }^{\dagger }({t}_{k+1})}({\bar{z}}_{k})-{\dot{\varepsilon }}_{{\lambda }^{\dagger }({t}_{k+1})}({z}_{k}),\end{eqnarray} \tag{ A.21 }$$

where we again assumed that the protocol is differentiable (see the remark below for the case of a c.p.d. protocol). Evaluating the sums over ${z}_{{i}_{k}}$ and writing the sums over i_k as integrals (by taking $N\to \infty$), equation (A.20) finally reads

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}\approx \displaystyle \sum _{n=0}^{\infty }\displaystyle \frac{1}{n!}{(-1)}^{n}\left(\eta \beta {\int }_{0}^{{t}_{f}}{\rm{d}}{t}\,{\dot{\omega }}_{{\lambda }^{\dagger }(t)}({p}_{{\rm{e}}}^{\dagger }(t)-{p}_{g}^{\dagger }(t))\right),\end{eqnarray} \tag{ A.22 }$$

where we denote the time derivative of the energy gap of the TLS by ${\dot{\omega }}_{{\lambda }^{\dagger }({t}_{k})}={\dot{\varepsilon }}_{{\lambda }^{\dagger }({t}_{k})}(e)-{\dot{\varepsilon }}_{{\lambda }^{\dagger }({t}_{k})}(g)$ and the probability of the system to be in the ground/exited state at time t_i by ${p}_{g/e}^{\dagger }({t}_{i})$, both in the backward protocol of the driving scheme. Note that equation (A.22) is exact up to first order in η.

Finally, we remark that for a c.p.d. protocol with non-differentiable points at $0\lt {t}_{1}\lt \ldots \,\lt \,{t}_{K}\lt {t}_{f}$ the result above readily generalizes and in equation (A.22) we have to split the integral at the respective points as

$$\begin{eqnarray}&&{\int }_{0}^{{t}_{f}}{\rm{d}}{t}={\int }_{0}^{{t}_{1}}{\rm{d}}{t}+{\int }_{{t}_{1}}^{{t}_{2}}{\rm{d}}{t}\,+\,\cdots \,+\,{\int }_{{t}_{K}}^{{t}_{f}}{\rm{d}}{t}.\end{eqnarray} \tag{ A.23 }$$

In this section we derive equation (29), i.e. an expression for the MJE of a TLS, where the energy levels are changed instantaneously at one moment in time t_m with $0\lt {t}_{m}\lt {t}_{f}$ and are constant before and after. Since the energy levels are constant before and after t_m, it follows that $\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i})=0$ for all $i\ne m$ and also $\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({\bar{z}}_{i})=0$ for all $i\ne m$. Then the right-hand side of equation (20) simplifies to

$$\begin{eqnarray}\begin{array}{rcl}{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }} & = & \displaystyle \sum _{{{\bf{z}}}^{\dagger }}{{ \mathcal P }}^{\dagger }[{{\bf{z}}}^{\dagger }]\,[1-\eta (1-{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{m+1}^{\dagger }}({z}_{m})-\delta {\varepsilon }_{{\lambda }_{m+1}^{\dagger }}({\bar{z}}_{m}))})]\\ & = & \displaystyle \sum _{{z}_{m}\in \{g,{e}\}}{p}_{{z}_{m}}^{\dagger }({t}_{m})[1-\eta (1-{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{m+1}^{\dagger }}({z}_{m})-\delta {\varepsilon }_{{\lambda }_{m+1}^{\dagger }}({\bar{z}}_{m}))})].\end{array}\end{eqnarray} \tag{ A.24 }$$

Summing over z_m, equation (A.24) can be written as

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }-{W}^{\dagger })}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=1-\eta (1-{p}_{g}^{\dagger }({t}_{m}){{\rm{e}}}^{\beta {\rm{\Delta }}{\omega }^{\dagger }}-{p}_{{e}}^{\dagger }({t}_{m}){{\rm{e}}}^{-\beta {\rm{\Delta }}{\omega }^{\dagger }}),\end{eqnarray} \tag{ A.25 }$$

where ${\rm{\Delta }}{\omega }^{\dagger }={\omega }_{{\lambda }^{\dagger }({t}_{f})}-{\omega }_{{\lambda }^{\dagger }(0)}$. Note that this equation is exact for $N\to \infty$ ($\delta t\to 0$).

For the derivation of equation (39), the MJE under feedback, we assume that ${\mu }_{\lambda (0)}=0$ initially and changes instantaneously at t_m to ${\mu }_{\lambda ({t}_{f})}$ if ${y}_{m}\gt 0$. Similarly, the width ${f}_{\lambda (t)}$ changes from ${f}_{\lambda (0)}$ to ${f}_{\lambda ({t}_{f})}$ instantaneously if ${y}_{m}\gt 0$. Since the form and the position of the potential is fixed before and after applying the feedback, it holds $\delta {H}_{{\lambda }_{k+1}({y}_{m})}({y}_{k})=0$ for all $k\ne m$ and the same is true for z_k. Then the measured efficacy parameter reads after integration over all z_k and y_k with $k\ne m$:

$$\begin{eqnarray}&&{\gamma }_{m}=\int {{\rm{d}}{z}}_{m}\int {{\rm{d}}{y}}_{m}{p}_{{\lambda }^{\dagger }({y}_{m})}({z}_{m}){p}_{m}({y}_{m}| {z}_{m}){{\rm{e}}}^{\beta (\delta {H}_{{\lambda }_{m+1}^{\dagger }({y}_{m})}({y}_{m})-\delta {H}_{{\lambda }_{m+1}^{\dagger }({y}_{m})}({z}_{m}))}.\end{eqnarray} \tag{ A.26 }$$

The integral of y_m splits into two parts: one in which we alter the potential (${y}_{m}\gt 0$) and one where we do nothing (${y}_{m}\lt 0$):

$$\begin{eqnarray}&&{\langle {{\rm{e}}}^{\beta ({W}_{m}^{\dagger }[{{\bf{y}}}^{\dagger }| {y}_{m}]-{W}^{\dagger }[{{\bf{z}}}^{\dagger }| {y}_{m}])}\rangle }_{{{\bf{z}}}^{\dagger },{{\bf{y}}}^{\dagger }}=\displaystyle \int {{\rm{d}}{z}}_{m}{\displaystyle \int }_{-\infty }^{0}{{\rm{d}}{y}}_{m}{p}_{{\lambda }^{\dagger }({y}_{m})}({z}_{m}){p}_{m}({y}_{m}| {z}_{m})\\ &&\quad +\displaystyle \int {{\rm{d}}{z}}_{m}{\displaystyle \int }_{0}^{\infty }{{\rm{d}}{y}}_{m}{p}_{{\lambda }^{\dagger }({y}_{m})}({z}_{m}){p}_{m}({y}_{m}| {z}_{m}){{\rm{e}}}^{\beta (\delta {H}_{{\lambda }_{m+1}^{\dagger }({y}_{m})}({y}_{m})-\delta {H}_{{\lambda }_{m+1}^{\dagger }({y}_{m})}({z}_{m}))}.\end{eqnarray} \tag{ A.27 }$$

The conditional probability ${p}_{m}({y}_{m}| {z}_{m})$ is again assumed to be Gaussian with a standard deviation of ${\sigma }_{m}$ (see equation (24)). Moreover, the probability ${p}_{{\lambda }^{\dagger }({y}_{m}\lt 0)}({z}_{m})$ (no feedback) is the canonical distribution of the harmonic potential centered at ${\mu }_{\lambda (0)}$ and width ${f}_{\lambda (0)}$ and the probability ${p}_{{\lambda }^{\dagger }({y}_{m}\geqslant 0)}({z}_{m})$ (feedback) is the canonical distribution centered at ${\mu }_{\lambda ({t}_{f})}$ and width ${f}_{\lambda ({t}_{f})}$, because we are in equilibrium before applying the backwards protocol. Then the first term of equation (A.27) becomes 1/2 after integration of z_m and y_m. If feedback ist applied (${y}_{m}\gt 0$) it holds

$$\begin{eqnarray}&&\delta {H}_{{\lambda }_{m+1}^{\dagger }({y}_{m})}({y}_{m})-\delta {H}_{{\lambda }_{m+1}^{\dagger }}({z}_{m})=({f}_{\lambda (0)}-{f}_{\lambda ({t}_{f})})({y}_{m}^{2}-{z}_{m}^{2})-2{f}_{\lambda ({t}_{f})}{\mu }_{\lambda ({t}_{f})}({z}_{m}-{y}_{m}).\end{eqnarray} \tag{ A.28 }$$

Then after integration over z_m and y_m of the second part of equation (A.27) one arrives at equation (39).

Here, we derive the analytic expression of the MJE for a driven TLS under feedback, equation (45). We again assume that the driving protocol changes continuously and depends on the measurement outcome y_m at time t_m. Then, the measured efficacy parameter of the TLS is given by

$$\begin{eqnarray}&&{\gamma }_{m}=\displaystyle \sum _{{{\bf{z}}}^{\dagger }}\displaystyle \sum _{{{\bf{y}}}^{\dagger }}{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger }]\displaystyle \prod _{k}[(1-\eta ){\delta }_{{y}_{k},{z}_{k}}+\eta (1-{\delta }_{{y}_{k},{z}_{k}})]{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{k+1}({y}_{m})}({y}_{k})-\delta {\varepsilon }_{{\lambda }_{k+1}({y}_{m})}({z}_{k})}.\end{eqnarray} \tag{ A.29 }$$

Since the driving protocol depends on y_m, we can write ${\gamma }_{m}$ as:

$$\begin{eqnarray}{\gamma }_{m} & = & \displaystyle \sum _{{{\bf{z}}}^{\dagger }}\displaystyle \sum _{{y}_{m}}{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger }][(1-\eta ){\delta }_{{y}_{m},{z}_{m}}+\eta (1-{\delta }_{{y}_{m},{z}_{m}})]{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{m+1}({y}_{m})}({z}_{m})-\delta {\varepsilon }_{{\lambda }_{m+1}({y}_{m})}({y}_{m}))}\\ & & \times \displaystyle \prod _{k\ne m}\left[\displaystyle \sum _{{y}_{k}}(1-\eta ){\delta }_{{y}_{k},{z}_{k}}+\eta (1-{\delta }_{{y}_{k},{z}_{k}}){{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{k+1}({y}_{m})}({z}_{k})-\delta {\varepsilon }_{{\lambda }_{k+1}({y}_{m})}({y}_{k}))}\right].\end{eqnarray} \tag{ A.30 }$$

Summing over all y_k results in

$$\begin{eqnarray}{\gamma }_{m} & = & \displaystyle \sum _{{{\bf{z}}}^{\dagger }}\displaystyle \sum _{{y}_{m}}{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger }][(1-\eta ){\delta }_{{y}_{m},{z}_{m}}+\eta (1-{\delta }_{{y}_{m},{z}_{m}})]{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{m+1}({y}_{m})}({z}_{m})-\delta {\varepsilon }_{{\lambda }_{m+1}({y}_{m})}({y}_{m}))}\\ & & \times \displaystyle \prod _{k\ne m}((1-2\eta )+\eta [1+{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({\bar{z}}_{i})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}])\\ & \approx & \,\displaystyle \sum _{{{\bf{z}}}^{\dagger }}\displaystyle \sum _{{y}_{m}}{{ \mathcal P }}_{{\lambda }^{\dagger }({y}_{m})}[{{\bf{z}}}^{\dagger }][(1-\eta ){\delta }_{{y}_{m},{z}_{m}}+\eta (1-{\delta }_{{y}_{m},{z}_{m}})]\\ & & \times \displaystyle \prod _{k\ne m}((1-2\eta )+\eta [1+{{\rm{e}}}^{\beta (\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({\bar{z}}_{i})-\delta {\varepsilon }_{{\lambda }_{i+1}^{\dagger }}({z}_{i}))}]).\end{eqnarray} \tag{ A.31 }$$

For the last step we approximated

$$\begin{eqnarray}&&\exp \{\beta (\delta {\varepsilon }_{{\lambda }_{m+1}({y}_{m})}({y}_{m})-\delta {\varepsilon }_{{\lambda }_{m+1}({y}_{m})}({z}_{m})\}\approx 1\end{eqnarray} \tag{ A.32 }$$

for the single point at k = m. This is justified because the final integral does not depend on the value of a single point as long as we change the protocol continuously. Following the same intermediate steps as in appendix A.3 we arrive at

$$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{m} & \approx & \displaystyle \sum _{{z}_{m},{y}_{m}}[(1-\eta ){\delta }_{{y}_{m},{z}_{m}}+\eta (1-{\delta }_{{y}_{m},{z}_{m}})][{p}_{{\lambda }^{\dagger }({y}_{m})}({z}_{m})\\ & & \left.\times \exp \left\{-\beta \eta \displaystyle \int {\rm{d}}{t}\,{\dot{\omega }}_{\lambda ({y}_{m},t)}^{\dagger }({p}_{e,{\lambda }^{\dagger }({y}_{m})}(t)-{p}_{g,{\lambda }^{\dagger }({y}_{m})}(t))\right\}\right].\end{array}\end{eqnarray} \tag{ A.33 }$$

Finally, by summing over y_m we arrive at equation (45).

For an instantenous change of the driving protocol, where we assume that the Hamiltonian of the TLS is constant before and after the quench at time t_m, $\delta {\varepsilon }_{{\lambda }_{i+1}({y}_{m})}({z}_{m})$ and $\delta {\varepsilon }_{{\lambda }_{i+1}({y}_{m})}({y}_{m})$ are the only terms different from zero. Then, equation (46) follows immediately from evaluating the sum over y_m in equation (A.29).