Feedback-controlled heat transport in quantum devices: theory and solid-state experimental proposal

In a famous thought experiment Maxwell envisioned a method for apparently defying the second law of thermodynamics by means of a feedback control mechanism [1]. Maxwell's idea is based on a malicious demon, an intelligent being that is able to observe the microscopic dynamics of a system, and acts on it so as to steer it toward defying the second law. In one of Maxwell's original concepts, the system is a container with two chambers, containing respectively a hot gas and a cold gas. The two chambers are separated by a wall containing a trap-door, which the demon can open and close at will. The demon observes the erratic motion of the gas particles, and when he sees a particle in the cold chamber approach the trap-door with sufficiently high velocity, he swiftly opens the door so as to let the particle pass through and closes it immediately afterwards. In this way, particle after particle, heat flows from the cold chamber to the hot chamber in contradiction with the second law.

Advances in nanotechnology have created the possibility of bringing Maxwell demons and similar devices from the realm of thought experiments to the realm of real experiments [2–5]. Theoretical and experimental studies so far have both focused mainly on situations where feedback control is operated as a measurement-conditioned driving on some working substance (classical or quantum) coupled to a single temperature, so as to withdraw energy from the latter in contradiction with the second law as formulated by Kelvin. Interesting realistic proposals have appeared in [6, 7]. Situations where heat flow between reservoirs at different temperatures is controlled, however, have not been addressed so far, either theoretically or experimentally. The main motivation of the present work is that of filling that gap. In the following we shall present the general theory of feedback-controlled heat transport in quantum devices, and describe a possible experimental realisation thereof.

The theory presented here builds on previous works concerning fluctuation relations in the presence of measurements without feedback [8, 9] and with feedback [10], combined with an inclusive approach where quantum heat engines are seen as mechanically driven multipartite systems starting in a multi-temperature initial state [11–14]. Reference [10] reported on the theory of a one-measurement-based feedback control on a quantum working substance prepared by contact with a single bath. That formalism is here extended to the case of many heat baths, and also repeated measurements, to allow for the study of continuous feedback control of heat flow in a multi-reservoir scenario. Previous work concerning repeated measurements appeared in [15] for classical systems in contact with a single bath. Fluctuation relations need to be modified by a mutual information term, which we shall provide explicitly.

Our experimental proposal is based on the fast developing advancements in experimental solid-state low-temperature techniques: in particular the calorimetric measurement scheme that has been put forward by one of us and co-workers [16, 17]. As proven by some recent theoretical proposals [13, 18], the method opens up a new avenue for the practical management of heat and work on a chip by means of superconducting devices, particularly superconducting qubits. Here we illustrate the possible implementation of very simple feedback-controlled heat transport where the trap-door is realised by a superconducting qubit whose coupling with two resistors at different temperatures is controlled based on the outcomes of continuous calorimetric monitoring of the resistors themselves.

Following [14] we model a generic heat transport/heat engine scenario as a driven multipartite system starting in the factorised state (see figure 1):

$\begin{eqnarray}&&{\rho }_{0}=\mathop{\displaystyle \bigotimes }\limits_{l}\displaystyle \frac{{{\rm{e}}}^{-{\beta }_{l}{H}_{l}}}{{Z}_{l}}\end{eqnarray} \tag{ 1 }$

where H_i is the Hamiltonian of each partition including a heat bath and possibly a portion of the working substance, and Z_i is the corresponding partition function [14]. Let the total Hamiltonian be

$\begin{eqnarray}&&H(t)=\displaystyle \sum _{l}\,{H}_{l}+V(t)\end{eqnarray} \tag{ 2 }$

where V(t) is an interaction term that is switched on for the time interval $t\in [0,\tau ]$ over which the system is monitored. We assume that at times ${t}_{1}\lt {t}_{2}\lt ...{t}_{K}$ some observable A is measured, thus causing the wavefunction describing the compound to collapse onto the subspace spanned by the eigenvectors belonging to the measured eigenvalue a_j. Following [10], we shall assume that there can be a measurement error where the eigenvalue a_k is recorded instead of the actual eigenvalue a_j. This is assumed to happen with probability $\varepsilon [k| j]$. The choice of the interaction V(t) in the interval $({t}_{i},{t}_{i+1})$ is dictated by the sequence of recorded eigenvalues, or more simply the recorded sequence $\{{k}_{1},{k}_{2},\ldots {k}_{i}\}={{\bf{k}}}_{i}$, that is $V(t)={V}_{{{\bf{k}}}_{i}}(t)$ for ${t}_{i}\lt t\lt {t}_{i+1}$. The corresponding unitary operator describing the evolution in the time span ${t}_{i}\lt t\lt {t}_{i+1}$ is ${U}_{{{\bf{k}}}_{i}}=\overleftarrow{\exp }\left[{\int }_{{t}_{i}}^{{t}_{i+1}}{{\rm{d}}{s}{H}}_{{{\bf{k}}}_{i}}(s)\right]$ where $\overleftarrow{\exp }$ denotes time-ordered exponential and ${H}_{{{\bf{k}}}_{i}}(t)={\sum }_{l}{H}_{l}+{V}_{{{\bf{k}}}_{i}}(t)$. We shall denote the unconditioned evolution operator from time t = 0 to the time of the first measurement $t={t}_{1}$ as U₀. Note that the sequence of recorded labels ${{\bf{k}}}_{j}$ generally differs from the sequence of labels $\{{j}_{1},{j}_{2},\ldots {j}_{i}\}={{\bf{j}}}_{i}$ specifying in which subspace the system state was actually projected at the measurement times ${t}_{1},{t}_{2}\,,...{t}_{i}$. As is customary in the context of the fluctuation theorem we shall assume that, besides the intermediate measurements of A, all H_l's are measured at times $t=0,t=\tau$, giving the eigenvalues E_n^l, E_m^l respectively.

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The quantity of primary interest is the probability $p(m,{\bf{k}},{\bf{j}},n)$ that n is obtained in the first energy measurement, the sequence ${\bf{j}}$ is realised, the sequence ${\bf{k}}$ is recorded and m is obtained in the final energy measurement. Here we have introduced the simplified notations ${\bf{j}}={{\bf{j}}}_{K}$, ${\bf{k}}={{\bf{k}}}_{K}$. The explicit expression for $p(m,{\bf{k}},{\bf{j}},n)$ is

$\begin{eqnarray}&&p(m,{\bf{k}},{\bf{j}},n)={\rm{Tr}}{P}_{m}{A}_{{\bf{k}},{\bf{j}}}{U}_{0}{P}_{n}{U}_{0}^{\dagger }{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}{p}_{n}^{0}\end{eqnarray} \tag{ 3 }$

$\begin{eqnarray}&&{A}_{{\bf{k}},{\bf{j}}}={\overleftarrow{{\rm{\Pi }}}}_{i}\phantom{\rule{}{2.5ex}}({U}_{{{\bf{k}}}_{i}}{\pi }_{{j}_{i}}\sqrt{\varepsilon [{k}_{i}| {j}_{i}]}\phantom{\rule{}{2.5ex}})\end{eqnarray} \tag{ 4 }$

where ${p}_{n}^{0}={{\rm{\Pi }}}_{l}{{\rm{e}}}^{-{\beta }_{l}{E}_{n}^{l}}/{Z}_{l}$ denotes the probability of obtaining the eigenvalue ${E}_{n}={\sum }_{l}{E}_{n}^{l}$ in the first measurement; P_n denotes the corresponding projector; ${\pi }_{j}$ denotes the projector onto the subspace belonging to the eigenvalue a_j of A; the symbol ${\overleftarrow{{\rm{\Pi }}}}_{i}$ denotes i-ordered product, that is, ${\overleftarrow{{\rm{\Pi }}}}_{i}({U}_{{{\bf{k}}}_{i}}{\pi }_{{j}_{i}}\varepsilon [{k}_{i}| {j}_{i}])={U}_{{{\bf{k}}}_{K}}{\pi }_{{j}_{K}}\sqrt{\varepsilon [{j}_{K}| {k}_{K}]}\,\cdots {U}_{{{\bf{k}}}_{2}}{\pi }_{{j}_{2}}\sqrt{\varepsilon [{j}_{2}| {k}_{2}]}{U}_{{{\bf{k}}}_{1}}{\pi }_{{j}_{1}}\sqrt{\varepsilon [{j}_{1}| {k}_{1}]}$.

Let ${\rm{\Delta }}{E}_{l}={E}_{m}^{l}-{E}_{n}^{l}$ be the change in energy in the partition l observed in a single realisation of the feedback-driven protocol. Using the cyclic property of the trace and completeness $\sum {P}_{n}={\mathbb{1}}$, we obtain the following:

$\begin{eqnarray}&&\langle {{\rm{e}}}^{-{\displaystyle \sum }_{l}{\beta }_{l}{\rm{\Delta }}{E}_{l}}\rangle =\gamma =\displaystyle \sum _{{\bf{j}},{\bf{k}}}{\rm{Tr}}{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{\rho }_{0}{A}_{{\bf{k}},{\bf{j}}}\end{eqnarray} \tag{ 5 }$

The proof is reported in appendix A. This relation extends the result presented in [10] to the case of a multipartite system with an initial multi-temperature state, and to repeated measurements⁵ . The quantity ${\rm{Tr}}{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{\rho }_{0}{A}_{{\bf{k}},{\bf{j}}}$ represents the probability that the sequences ${{\bf{j}}}^{\dagger }=\{{j}_{K}\,,...{j}_{2},{j}_{1}\}$ and ${{\bf{k}}}^{\dagger }=\{{k}_{K}\,,...{k}_{2},{k}_{1}\}$ are realised under the backward evolution specified by the adjoint Kraus operators ${A}_{{\bf{k}},{\bf{j}}}^{\dagger }$. The total probability γ does not generally add to one. The reason for this is that the ith evolution ${U}_{{{\bf{k}}}_{i}}^{\dagger }$ occurs before the ith eigenvalue j_i is realised in the backward map. The feedback loop is evidently not symmetric under time reversal, and such a lack of reversibility breaks the fluctuation theorem $\langle {{\rm{e}}}^{{\sum }_{l}{\beta }_{l}{\rm{\Delta }}{E}_{l}}\rangle =1$, which in fact is a manifestation of time-reversal symmetry [19]. This is reflected by the fact that the quantum channel specified by the Kraus operators ${A}_{{\bf{k}},{\bf{j}}}$ is generally not unital⁶ . The adjoint of a non-unital quantum channel is not trace-preserving. In the case of feedback control the quantum channel ${\sum }_{{\bf{j}},{\bf{k}}}{A}_{{\bf{k}},{\bf{j}}}{\rho }_{0}{A}_{{\bf{k}},{\bf{j}}}^{\dagger }$ is generally not unital; as a consequence its adjoint is generally not trace-preserving, hence we have generally $\gamma \ne 1$. Lack of unitality generally reflects a lack of time-reversal symmetry. Examples are thermalisation maps, namely maps that have a thermal state (not the identity) as a fixed point. Physically these are realised by means of weak contact of a system with a thermal bath, leading to irreversible dynamics. Likewise feedback control breaks the symmetry. This observation reveals some analogy between feedback control and dissipative dynamics.

Before proceeding, let us comment briefly on the origin of the lack of unitality in feedback-controlled systems, in order to gain insight into the issue. For simplicity let us consider the case of a single measurement K = 1. Let us begin by noticing that the quantum channel specified by the ${A}_{k,j}$ is trace-preserving. We have ${\rm{Tr}}{\sum }_{k,j}{A}_{k,j}\rho {A}_{k,j}^{\dagger }={\sum }_{k,j}\varepsilon [k| j]{\rm{Tr}}\,{U}_{k}{\pi }_{j}\rho {\pi }_{j}{U}_{k}^{\dagger }={\sum }_{k,j}\varepsilon [k| j]{\rm{Tr}}{\pi }_{j}\rho {\pi }_{j}={\sum }_{j}{\rm{Tr}}{\pi }_{j}\rho {\pi }_{j}={\rm{Tr}}\,\rho$, where we have used the cyclic property of the trace, unitarity ${U}_{k}^{\dagger }{U}_{k}={\mathbb{1}}$, idempotence ${\pi }_{j}{\pi }_{j}={\pi }_{j}$, normalisation ${\sum }_{k}\varepsilon [k| j]=1$ and completeness $\sum {\pi }_{j}={\mathbb{1}}$. Let us now turn to unitality. We have ${\sum }_{k,j}{A}_{k,j}{A}_{k,j}^{\dagger }={\sum }_{k,j}\varepsilon [k| j]{U}_{k}{\pi }_{j}{U}_{k}^{\dagger }$. If the evolution U_k was not dependent on k, that is ${U}_{k}=\bar{U}$ was chosen regardless of the recorded value k (e.g., $\bar{U}$ is pre-specified or is completely random), one could perform the sum over k using ${\sum }_{k}\varepsilon [k| j]=1$ and then use $\sum {\pi }_{j}={\mathbb{1}}$ to conclude that the map is unital. Feedback, implying explicit dependence on k of U_k, breaks unitality. Unitality would occur also in the case when $\varepsilon [k| j]$ does not depend on j, meaning the measurement outcome k is completely random and has no correlation with the actual state j. In sum, if the feedback control measurement is off, either because one decides not to use the information gathered in the measurement or because the measurement gathers no information in the first place, unitality is recovered, and the fluctuation theorem is restored. This result is in agreement with the established fact that projective measurements without feedback control do not alter the validity of the fluctuation theorem [8, 20, 21]. Here we have further learned that noise, i.e. choosing the U's between the measurements completely randomly, also does not affect the integral fluctuation relation.

Let us now turn to thermodynamics. Using Jensen's inequality, equation (5) implies

$\begin{eqnarray}&&\displaystyle \sum _{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \geqslant -\mathrm{ln}\gamma \end{eqnarray} \tag{ 6 }$

In the case when the map is unital we have $\gamma =1$, and the second law of thermodynamics is recovered [14]. When $\gamma \gt 1$ the condition ${\sum }_{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \lt 0$ is not forbidden, and the apparent violation of the second law becomes possible. This occurs with a proper 'demonic' design of the feedback control. When $\gamma \lt 1$ instead, the second law is more strictly enforced by means of an 'angelic' intervention.

As shown in [10, 22], in the case of a single measurement (in either classical or quantum systems) the fluctuation relation can be restored if an information theoretic term, in the form of mutual information, is added to the exponent in the exponential average. Reference [15] reports the extension to the case of repeated measurements in the classical scenario. All these results are for a single-temperature initial state. In the present set-up we find as well an information theoretic correction term (see appendix B for a proof):

$\begin{eqnarray}&&\langle {{\rm{e}}}^{-{\displaystyle \sum }_{l}{\beta }_{l}{\rm{\Delta }}{E}_{l}-{J}_{{\bf{k}},{\bf{j}}}}\rangle =1\end{eqnarray} \tag{ 7 }$

where ${J}_{{\bf{k}},{\bf{j}}}$ is defined by the following set of equations:

$\begin{eqnarray}&&{J}_{{\bf{k}},{\bf{j}}}=\mathrm{ln}\displaystyle \frac{p({\bf{k}},{\bf{j}})}{p({\bf{j}}\,:{\bf{k}})p({\bf{k}})}\end{eqnarray} \tag{ 8 }$

$\begin{eqnarray}&&p({\bf{k}},{\bf{j}})=\displaystyle \sum _{n,m}p(m,{\bf{k}},{\bf{j}},n)\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray}&&p({\bf{k}})=\displaystyle \sum _{n,{\bf{j}},m}p(m,{\bf{k}},{\bf{j}},n)=\displaystyle \sum _{{\bf{j}}}p({\bf{k}},{\bf{j}})\end{eqnarray} \tag{ 10 }$

$\begin{eqnarray}&&p({\bf{j}}\,:{\bf{k}})=\displaystyle \frac{p({\bf{k}},{\bf{j}})}{{{\rm{\Pi }}}_{i}\varepsilon [{k}_{i}| {j}_{i}]}.\end{eqnarray} \tag{ 11 }$

The symbol $p({\bf{k}},{\bf{j}})$ represents the joint probability that the sequence ${\bf{j}}$ is realised and the sequence ${\bf{k}}$ is recorded, while $p({\bf{k}})$ is the probability that ${\bf{k}}$ is recorded. The symbol $p({\bf{j}}\,:{\bf{k}})$ stands for the probability that the sequence ${\bf{j}}$ is realised, conditioned on ${\bf{k}}$ being the record. More explicitly,

$\begin{eqnarray}&&p({\bf{j}}\,:{\bf{k}})=\displaystyle \frac{p({\bf{k}},{\bf{j}})}{{{\rm{\Pi }}}_{i}\varepsilon [{k}_{i}| {j}_{i}]}=\displaystyle \sum _{n,m}{\rm{Tr}}{P}_{m}{B}_{{\bf{k}},{\bf{j}}}{U}_{0}{P}_{n}{U}_{0}^{\dagger }{B}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}{p}_{n}^{0}\end{eqnarray} \tag{ 12 }$

$\begin{eqnarray}&&{B}_{{\bf{k}},{\bf{j}}}={\overleftarrow{{\rm{\Pi }}}}_{i}({U}_{{{\bf{k}}}_{i}}{\pi }_{{j}_{i}})=\displaystyle \frac{{A}_{{\bf{k}},{\bf{j}}}}{{{\rm{\Pi }}}_{i}\sqrt{\varepsilon [{k}_{i}| {j}_{i}]}}.\end{eqnarray} \tag{ 13 }$

The operators ${B}_{{\bf{k}},{\bf{j}}}$ differ from the operators ${A}_{{\bf{k}},{\bf{j}}}$ by the term containing the conditional probability $\varepsilon [{k}_{i}| {j}_{i}]$. Note that the Bayes rule does not apply here, i.e. generally it is $p({\bf{k}},{\bf{j}})\ne p({\bf{j}}\,:{\bf{k}})p({\bf{k}})$. The reason is that ${\bf{j}}$ and ${\bf{k}}$ are concatenated with each other. An outcome j_i influences the record k_i, which in turn influences the next outcome ${j}_{i+1}$ and so on. The quantity ${J}_{{\bf{k}},{\bf{j}}}$ measures the degree of such mutual influence or correlation between the two sequences ${\bf{j}}$ and ${\bf{k}}$ ⁷ . In the absence of feedback, namely when there is no correlation between the two sequences, ${J}_{{\bf{k}},{\bf{j}}}$ is null and the standard relation is recovered. Note that, given a feedback rule, generally $\langle {J}_{{\bf{k}},{\bf{j}}}\rangle$ would grow with the length K of the sequences, i.e. the number of measurements. It is accordingly expected that $\langle {J}_{{\bf{k}},{\bf{j}}}\rangle \propto K$ in the large-K regime.

With Jensen's inequality equation (7) implies

$\begin{eqnarray}&&\displaystyle \sum _{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \geqslant -\langle {J}_{{\bf{k}},{\bf{j}}}\rangle .\end{eqnarray} \tag{ 14 }$

We thus have found two bounds to ${\sum }_{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$.

By looking directly at ${\sum }_{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$ as in [14] we have found a third bound, whose interpretation is most direct and straightforward. Let

$\begin{eqnarray}&&{\rho }_{\tau }=\displaystyle \sum _{n,{\bf{j}},{\bf{k}},m}{P}_{m}{A}_{{\bf{k}},{\bf{j}}}{U}_{0}{P}_{n}{\rho }_{0}{P}_{n}{U}_{0}^{\dagger }{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}=\displaystyle \sum _{{\bf{j}},{\bf{k}}}{A}_{{\bf{k}},{\bf{j}}}{U}_{0}{\rho }_{0}{U}_{0}^{\dagger }{A}_{{\bf{k}},{\bf{j}}}^{\dagger }\end{eqnarray} \tag{ 15 }$

be the system density matrix at time τ. In the second equality we have used completeness ${\sum }_{m}{P}_{m}={\mathbb{1}}$ and the fact that the initial state has no coherences in the energy eigenbasis ${\sum }_{n}{P}_{n}{\rho }_{0}{P}_{n}={\rho }_{0}$. Simple manipulations, similar to those employed in [14], lead to the following salient result:

$\begin{eqnarray}&&\displaystyle \sum _{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle =\displaystyle \sum _{i}D[{\rho }_{\tau }^{l}| | {\rho }_{0}^{l}]+I[{\rho }_{\tau }]+{\rm{\Delta }}{ \mathcal H }\end{eqnarray} \tag{ 16 }$

where

$\begin{eqnarray}&&D[{\rho }_{\tau }^{l}| | {\rho }_{0}^{l}]={\rm{Tr}}{\rho }_{\tau }^{l}\mathrm{ln}{\rho }_{\tau }^{l}-{\rm{Tr}}{\rho }_{\tau }^{l}\mathrm{ln}{\rho }_{0}^{l}\end{eqnarray} \tag{ 17 }$

$\begin{eqnarray}&&I[{\rho }_{\tau }]=-\displaystyle \sum _{l}{\rm{Tr}}{\rho }_{\tau }^{l}\mathrm{ln}{\rho }_{\tau }^{l}+{\rm{Tr}}{\rho }_{\tau }\mathrm{ln}{\rho }_{\tau }\end{eqnarray} \tag{ 18 }$

$\begin{eqnarray}&&{\rm{\Delta }}{ \mathcal H }=-{\rm{Tr}}{\rho }_{\tau }\mathrm{ln}{\rho }_{\tau }+{\rm{Tr}}{\rho }_{0}\mathrm{ln}{\rho }_{0}\end{eqnarray} \tag{ 19 }$

denote the Kullback Leibler divergence between the final state ${\rho }_{\tau }$ and the initial state ${\rho }_{0}$, equation (17); the total amount of correlation (mutual information) that builds up among the partitions as a consequence of their interaction during the time span $[0,\tau ]$, equation (18); and the total change in von Neumann entropy of the whole compound, equation (19). Here ${\rho }_{t}^{l}={\rm{Tr}}{^{\prime} }_{l}{\rho }_{t}$ is the reduced state of partition l at time t (${{\rm{Tr}}}_{l}^{\prime }$ denotes the trace over all partitions but the lth). The mutual information I among the partitions of the system (measuring all correlations, quantal and classical), which develops generally due to their interaction V(t) (and can also occur in the absence of measurements and feedback [14]), should not be confused with the classical mutual information ${J}_{{\bf{k}},{\bf{j}}}$ between the realisation sequence ${\bf{j}}$ and the record sequence ${\bf{k}}$ caused by the feedback mechanism.

Both the Kullback Leibler divergence $D[{\rho }_{\tau }^{i}| | {\rho }_{0}^{i}]$ and the mutual information $I[{\rho }_{t}]$ are non-negative quantities. We thus arrive at the central inequality:

$\begin{eqnarray}&&\displaystyle \sum _{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \geqslant {\rm{\Delta }}{ \mathcal H }.\end{eqnarray} \tag{ 20 }$

In the standard no-measurement case, ${\rho }_{\tau }$ is linked to ${\rho }_{0}$ via a unitary map, hence ${\rm{\Delta }}{ \mathcal H }=0$ and one recovers the result of [14], namely ${\sum }_{i}{\beta }_{i}\langle {\rm{\Delta }}{E}_{i}\rangle ={\sum }_{i}D[{\rho }_{\tau }^{i}| | {\rho }_{0}^{i}]+I[{\rho }_{\tau }]$, and the second law in its standard form. Note that when there are measurements, but no feedback, ${\rho }_{\tau }$ is linked to ${\rho }_{0}$ via a unital map, implying $\gamma =0$, $\langle {J}_{{\bf{k}},{\bf{j}}}\rangle =0$, and ${\rm{\Delta }}{ \mathcal H }\geqslant 0$, hence ${\sum }_{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \geqslant {\rm{\Delta }}{ \mathcal H }\geqslant 0$, meaning that, as is already known [8, 20, 21], the second law is not altered by the mere application of projective measurements that interrupt an otherwise unitary dynamics. However, equation (20) clearly indicates that there is a dissipation term associated with quantum-mechanical measurements, which is not present in the classical case. In sum, through equation (20) we see that there is a thermodynamic cost associated with quantum measurements.

Combining equations (6), (14) and (20), the second law of thermodynamics in the presence of feedback control takes the form

$\begin{eqnarray}&&\displaystyle \sum _{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \geqslant \max [-\mathrm{ln}\gamma ,-\langle {J}_{{\bf{k}},{\bf{j}}}\rangle ,{\rm{\Delta }}{ \mathcal H }].\end{eqnarray} \tag{ 21 }$

To exemplify the theory above we consider a prototypical model of a quantum heat engine whose working substance is made of two qubits [13, 14, 24]. Their Hamiltonian reads

$\begin{eqnarray}&&H={H}_{1}+{H}_{2}=\displaystyle \frac{{\hslash }\omega }{2}{\sigma }_{z}^{1}+\displaystyle \frac{{\hslash }\omega }{2}{\sigma }_{z}^{2}\end{eqnarray} \tag{ 22 }$

where ${\sigma }_{z}^{i}$ denote Pauli operators. We assume that the two qubits have the same level spacing ${\hslash }\omega$ and are initially in the state

$\begin{eqnarray}&&{\rho }_{0}=\displaystyle \frac{{{\rm{e}}}^{-{\beta }_{1}{H}_{1}}}{{Z}_{1}}\otimes \displaystyle \frac{{{\rm{e}}}^{-{\beta }_{2}{H}_{2}}}{{Z}_{2}}\end{eqnarray} \tag{ 23 }$

with Z_i their partition functions. At t = 0 the ${\sigma }_{z}^{i}$'s are measured by collapsing the two qubits in the state $| k\rangle =| k^{\prime} \rangle | k^{\prime\prime} \rangle$, with $k^{\prime} ,k^{\prime\prime} =\pm ,\,\pm$. We assume classical error in the measurement of each qubit, $\epsilon [+| +]=\epsilon [-| -]=q$, $\epsilon [-| +]=\epsilon [+| -]=1-q$ for some $q\in [0,1]$. Accordingly the eigenvalues $j=j^{\prime} ,j^{\prime\prime}$ are recorded with probability $\varepsilon [k| j]=\epsilon [k^{\prime} | j^{\prime} ]\epsilon [k^{\prime\prime} | j^{\prime\prime} ]$. If the states $| +,\,+\rangle ,| -,\,+\rangle ,| -,\,-\rangle$, are recorded we do nothing: ${U}_{+,+}={U}_{-,+}={U}_{-,-}={\mathbb{1}};$ else, i.e., if $k=| -,\,+\rangle$, we apply a swap operation, ${U}_{-,+}={U}_{{\rm{SWAP}}}$, that maps $| -,\,+\rangle$ into $| +,\,-\rangle$. The system is now in a joint eigenstate $| m\rangle ={U}_{k}| j\rangle$ of the two qubits with Hamiltonian H, hence the final measurement of ${\sigma }_{z}^{i}$ is irrelevant. At the end of the process each qubit is allowed to relax to thermal equilibrium with its respective thermal bath of inverse temperature ${\beta }_{i}$ so as to re-establish the initial state ${\rho }_{0}$. Accordingly, the average energy $\langle {\rm{\Delta }}{E}_{i}\rangle$ acquired by each qubit during the process is equal to the average heat that it releases in the bath in the thermal relaxation step. As a result of the feedback mechanism energy may be withdrawn from the cold bath and released in the hot one. Note that, due to the fact that the two qubits have same level spacing, the SWAP operation does not alter their total energy. That is, there is no energy injection by the demon: to steer the energy flow he only uses information. The set-up is illustrated in figure 2(a).

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The relevant probability chain is a bit simpler than in the general case because the first energy measurement is itself here also the first feedback measurement. It reads $p(m,k,j)={\rm{Tr}}{P}_{m}{U}_{k}{P}_{j}{U}_{k}^{\dagger }{P}_{m}{p}_{j}^{0}\varepsilon [k| j]={\rm{Tr}}{P}_{m}{A}_{k,j}{A}_{k,j}^{\dagger }{P}_{m}{p}_{j}^{0}$ with ${A}_{k,j}=\sqrt{\varepsilon [k| j]}{U}_{k}{P}_{j}$. For γ we have $\gamma ={\sum }_{j,k}\varepsilon [k| j]{\rm{Tr}}{P}_{j}{U}_{k}^{\dagger }{\rho }_{0}{U}_{k}{P}_{j}$. The final state is ${\rho }_{f}={\sum }_{j,k}\varepsilon [k| j]{U}_{k}{P}_{j}{\rho }_{0}{P}_{j}{U}_{k}^{\dagger }$. The probability $p(j\,:k)$ that the outcome j is realised conditioned on k being recorded is simply the marginal probability p(j) that j is realised because the record k comes chronologically after the realisation of j and hence cannot have any influence on it. The quantity ${J}_{k,j}$ boils down then to the logarithm of the ratio $p(j,k)/p(j)p(k)$ [10], hence its expectation is the non-negative mutual information between j and k: $\langle {J}_{k,j}\rangle ={\sum }_{j,k}p(j,k)[\mathrm{ln}p(j,k)/p(j)p(k)]$.

Figures 2(b, c) show $\sum \,{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle ,-\mathrm{ln}\gamma ,-\langle {J}_{k,j}\rangle$ and ${\rm{\Delta }}{ \mathcal H }$, for two choices of ${\beta }_{2}$ and the same ${\beta }_{1}$, as functions of the error probability q. In accordance with equation (21) we see that $\sum {\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$ is bounded from below by $-\mathrm{ln}\gamma ,-\langle {J}_{k,j}\rangle$ and ${\rm{\Delta }}{ \mathcal H }$. Independent of all other parameters the refrigerator cannot work in the region $q\lt 1/2$ where j and k are anticorrelated, while it may only work if $q\gt 1/2$. This is captured by $-\mathrm{ln}\gamma$ being positive in the region $0\lt q\lt 1/2$ and negative for $1/2\lt q\lt 1$. At $q=1/2$ the outcome and recording are fully uncorrelated, which restores unitality as discussed above and implies $\mathrm{ln}\gamma =0$. Regarding ${\rm{\Delta }}{ \mathcal H }$, while it tends to be close to $\sum {\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$ in the operation region ($q\gt 1/2$), it greatly departs from it in the non-operation region, where it can even take negative values. Notably in both panels there is a value of q for which the bound is saturated by ${\rm{\Delta }}{ \mathcal H }$. Regarding $-\langle {J}_{k,j}\rangle$, we note that it is everywhere non-positive as expected. Furthermore it is symmetric with respect to $q\to 1-q$. This reflects the fact that the mutual information does not distinguish between correlation and anticorrelation. The maximum $-\langle {J}_{k,j}\rangle =0$ is attained at $q=1/2$ where j and k are uncorrelated, and the standard fluctuation relation is recovered (i.e., $\gamma =1$). In both panels we see that ${\rm{\Delta }}{ \mathcal H }\gt -\langle {J}_{k,j}\rangle$. Whether this a generic bound is yet to be understood. We note that while both $-\mathrm{ln}\gamma$ and $-\langle {J}_{k,j}\rangle$ are null at q = 1/2, ${\rm{\Delta }}{ \mathcal H }$ is non-negative, reflecting the fact that in the absence of feedback there is nonetheless an entropic cost associated with measurements, as discussed above. Such a cost can be counterbalanced in the presence of feedback (note that ${\rm{\Delta }}H$ may be negative for $q\ne 1/2$). Confronting now the two panels, we see that the higher the thermal gradient ${\beta }_{2}-{\beta }_{1}$, the larger is the point q where the engine starts operating, i.e. where $\sum {\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$ turns from positive into negative: as intuition suggests, the steeper the gradient, the better must your measurement be. This feature is captured also by ${\rm{\Delta }}{ \mathcal H }$ but not by $-\mathrm{ln}\gamma ,-\langle {J}_{k,j}\rangle$. Also the smaller the gradient, the more the shape of the function $\sum {\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$ resembles that of $-\mathrm{ln}\gamma$, with the shift between the two being approximately the value of ${\rm{\Delta }}{ \mathcal H }$ at $q=1/2$: that is $\sum {\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle \simeq -\mathrm{ln}\gamma +{\rm{\Delta }}{ \mathcal H }{| }_{q=1/2}$.

The general theory developed above allows for a joint information theoretic and thermodynamic analysis of feedback-controlled dynamics in the broad scenario where a demon can influence not only the amount of work being provided by the outside as in previous works [2–4] but also the heat flow between the various parts of a compound system, e.g. the heat flow between various heat baths.

Progress in solid-state technology, on the other hand, allows one to realise such feedback-controlled heat transport mechanisms in real devices. The example illustrated above can be realised experimentally by introducing a feedback mechanism in the scheme with two superconducting qubits illustrated in [13]. Below we illustrate a design that permits more immediate realisation. It is based on a single qubit and it does not involve any qubit operation, but only manipulations of qubit–bath couplings. The proposal that we put forward here is based on two ingredients that enable unique capabilities, allowing for the implementation of a Maxwell demon based on a very simple concept. The two ingredients are a two-level system acting as a quantum trap-door and the calorimetric measurement scheme developed in [16, 17].

The one-qubit set-up is illustrated in figure 3. The two-level system (TLS) is embodied by a superconducting qubit of level spacing ${\hslash }\omega$. The two chambers are embodied by two resistors kept at different temperatures. The qubit and resistors can exchange energy (i.e. heat) in the form of photons of energy ${\hslash }\omega$ associated with the TLS absorbing/emitting one photon from/to one of the two baths. The resistors are embedded in an RLC loop of tunable resonance frequency. This results in a tunable TLS/resistor coupling. When an RLC circuit is far detuned from ω, the qubit is effectively decoupled from the resistor, while maximal coupling occurs when it is in tune with the qubit. The resonance frequency can be tuned by using a SQUID as a nonlinear and tunable inductor, its inductance being governed by a controllable threading magnetic flux.

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When a photon enters/exits one of the two resistors, its electronic temperature undergoes a positive/negative jump followed by a fast decay. Two calorimeters [16, 17] continuously monitor the two resistors and count how many photons enter/exit them. This allows for a directional full counting statistics of heat. Most remarkably it also allows one to infer the state of the TLS at each time. If an absorption (in either resistor) is observed, it means the TLS jumped down, hence it was up before the absorption was detected and is down afterwards. This allows one to access the quantum state trajectory of the TLS experimentally.

The feedback concept is extremely simple: as soon as a jump down is observed, turn on the interaction with the cold resistor and turn off the interaction with the hot resistor. Vice versa for the observation of a jump up. This results in a net flow of heat from the cold resistor to the hot one. Based on the above general analysis the apparent violation of the second law is understood in terms of lack of time-reversal symmetry of feedback control, leading to an overall non-unital dynamics of resistors plus TLS. In a practical realisation one is realistically not able to fully turn off the interactions. Furthermore there will be some delay time δ between the measurement being performed and feedback being realised, giving rise effectively to a possible error $\varepsilon [{k}_{i}| {j}_{i}]$ between the measured state k_i and actual state j_i of the qubit.

In the following we model the dynamics of the proposed experiment. We model the evolution of the two-level system via a standard Lindblad master equation

$\begin{eqnarray}&&\dot{\rho }=-{\rm{i}}[{H}_{S},\rho ]+{{ \mathcal L }}_{L}\rho +{{ \mathcal L }}_{R}\rho \end{eqnarray} \tag{ 24 }$

where ${H}_{S}=-{E}_{0}({\rm{\Delta }}{\sigma }_{x}+q{\sigma }_{z})$ is the Hamiltonian of the two-level system expressed in terms of the Pauli matrices ${\sigma }_{\alpha }$, and ${{ \mathcal L }}_{l}$ are Lindblad operators

$\begin{eqnarray}&&{{ \mathcal L }}_{l}\rho ={{\rm{\Gamma }}}_{l}^{\downarrow }D[\sigma ]\rho +{{\rm{\Gamma }}}_{l}^{\uparrow }D[{\sigma }^{\dagger }]\rho \end{eqnarray} \tag{ 25 }$

expressed in terms of the super-operator $D[O]\rho =O\rho {O}^{\dagger }-\tfrac{1}{2}{O}^{\dagger }O\rho -\tfrac{1}{2}\rho {O}^{\dagger }O$ and the rising and lowering spin operators ${\sigma }^{\dagger },\sigma$ of the Hamiltonian H_S, defined via ${\sigma }^{\dagger }| -\rangle =| +\rangle ,{\sigma }^{\dagger }| +\rangle =0,\sigma | +\rangle =| -\rangle ,\sigma | -\rangle =0$, where $| -(+)\rangle$ is the ground (excited) state of H_S. Here $l={L},{R}$ denotes either the left or the right reservoir. The rates ${{\rm{\Gamma }}}_{l}^{\downarrow \uparrow }$ for jump down/up in the lth resistor are given by

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{l}^{\downarrow \uparrow }=\displaystyle \frac{{E}_{0}^{2}{M}_{l}^{2}}{{{\hslash }}^{2}{{\rm{\Phi }}}_{0}}\displaystyle \frac{{{\rm{\Delta }}}^{2}}{({q}^{2}+{{\rm{\Delta }}}^{2})}{S}_{I,l}(\pm \omega )\end{eqnarray} \tag{ 26 }$

where ${S}_{I,l}(\omega )={S}_{V,l}(\omega ){[{R}_{l}^{2}(1+{Q}_{l}^{2}{[\omega /{\omega }_{{LC},l}-{\omega }_{{LC},l}/\omega ]}^{2})]}^{-1}$ is the current noise spectrum expressed in terms of the voltage noise spectrum ${S}_{V,l}{(\omega )=2{R}_{l}{\hslash }\omega (1-{{\rm{e}}}^{-{\beta }_{l}{\hslash }\omega })}^{-1}$, ${Q}_{l}=\sqrt{{L}_{l}/{C}_{l}}/{R}_{l}$ is the quality factor and ${\omega }_{{LC},l}=1/\sqrt{{L}_{l}{C}_{l}}$ the resonance frequency of resonator l, expressed in terms of its resistance, inductance and capacitance ${R}_{l},{L}_{l},{C}_{l}$. By increasing L_j the rates ${{\rm{\Gamma }}}_{l}^{\downarrow \uparrow }$ can be quenched, i.e. the interaction between the TLS and the lth resistor can be turned off. The symbol M_l stands for the mutual inductance between the qubit and the lth resistor and ${{\rm{\Phi }}}_{0}$ is the flux quantum. Note that the rates are in detailed balance:

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{l}^{\downarrow }={{\rm{e}}}^{{\beta }_{l}{\hslash }\omega }{{\rm{\Gamma }}}_{l}^{\uparrow }\end{eqnarray} \tag{ 27 }$

The study of heat and work fluctuations requires the study of the dynamics to be performed at the level of single quantum-jump trajectories [13, 25], resulting from the unravelling of the master equation. This is achieved here by means of the Monte Carlo wavefunction (MCWF) method [26, 27]. In the specific case under study of a two-level system subject to dissipation terms leading to full wavefunction collapse in either state $| -\rangle$ or $| +\rangle$, this results in a classical dichotomous Poisson process with rates ${{\rm{\Gamma }}}_{l}^{\downarrow \uparrow }$ [13].

The basis of our numerical experiment is the generation of such dichotomous Poisson random trajectories. We chose the right reservoir as the cold one and the left as the hot one. The TLS is assumed to be initially in equilibrium with the left bath. We produce a large sample of trajectories and build the normalised histogram $h({N}_{R})$ of the number N_R of photons entering the right reservoir. Since the heat Q_R entering the right reservoir is given as ${Q}_{R}={\hslash }{N}_{R}\omega$, the statistics $h({N}_{R})$ is the heat statistics. In the absence of feedback it satisfies the fluctuation relation

$\begin{eqnarray}&&\displaystyle \frac{h({N}_{R})}{h(-{N}_{R})}={{\rm{e}}}^{-{\rm{\Delta }}\beta {\hslash }\omega {N}_{R}},\qquad \mathrm{no}\ \mathrm{feedback}.\end{eqnarray} \tag{ 28 }$

The feedback is introduced as follows. At each moment in time we distinguish between the actual state of the system $j=\pm$ and the knowledge $k=\pm$ we have about it. The latter does not necessarily coincide with the former because we allow for some delay time δ between a jump occurring in the TLS and our knowledge of the state of the qubit being updated accordingly. The delay time thus effectively introduces an error probability $\varepsilon [\pm | \pm ]$ between the actual state and our knowledge about it, at each time. At each time, conditioned on the knowledge k of the state, we use either set of rates favouring the interaction with either the cold or the hot bath. More explicitly, let ${{\rm{\Gamma }}}_{l}^{\downarrow \uparrow | \pm }$ be the rate for a jump down (up) in the lth bath conditioned on the TLS being measured to be in state ±. In accordance with equation (26) we use the following rates:

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{L}^{\downarrow | +}=\displaystyle \frac{A}{1-{{\rm{e}}}^{-{\beta }_{L}{\hslash }\omega }}\qquad {{\rm{\Gamma }}}_{L}^{\uparrow | +}={{\rm{\Gamma }}}_{L}^{\downarrow | +}{{\rm{e}}}^{-{\beta }_{L}{\hslash }\omega }\end{eqnarray} \tag{ 29 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{R}^{\downarrow | +}=\displaystyle \frac{B}{1-{{\rm{e}}}^{-{\beta }_{R}{\hslash }\omega }}\qquad {{\rm{\Gamma }}}_{R}^{\uparrow | +}={{\rm{\Gamma }}}_{R}^{\downarrow | +}{{\rm{e}}}^{-{\beta }_{R}{\hslash }\omega }\end{eqnarray} \tag{ 30 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{L}^{\downarrow | -}=\displaystyle \frac{B}{1-{{\rm{e}}}^{-{\beta }_{L}{\hslash }\omega }}\qquad {{\rm{\Gamma }}}_{L}^{\uparrow | -}={{\rm{\Gamma }}}_{L}^{\downarrow | -}{{\rm{e}}}^{-{\beta }_{L}{\hslash }\omega }\end{eqnarray} \tag{ 31 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{R}^{\downarrow | -}=\displaystyle \frac{A}{1-{{\rm{e}}}^{-{\beta }_{R}{\hslash }\omega }}\qquad {{\rm{\Gamma }}}_{R}^{\uparrow | -}={{\rm{\Gamma }}}_{R}^{\downarrow | -}{{\rm{e}}}^{-{\beta }_{R}{\hslash }\omega }\end{eqnarray} \tag{ 32 }$

where A and B are determined by the circuitry parameters and can be tuned via external fluxes ${{\rm{\Phi }}}_{i}$. With $B\lt A$, this means that energy exchange with the right (cold) bath is larger when the TLS is believed to be down, so that it becomes more likely that energy flows out of the cold reservoir. Similarly energy exchange with the left (hot) bath is larger when the TLS is believed to be up, so that it becomes more likely that energy flows into the hot reservoir. Overall this results in an effect that is opposite to the natural flow from hot to cold. The largest effect can be achieved when turning off the unwanted interaction completely, namely when B = 0. Having in mind a realistic set-up, here we keep the ratio A/B finite, meaning partial turning-off is considered.

Because of the feedback the fluctuation relation (28) is not obeyed. However, it can be proved (see appendix C) that, due to the feedback mechanism, the TLS feels the effective temperature gradient

$\begin{eqnarray}&&{\rm{\Delta }}{\beta }^{\mathrm{eff}}={\beta }_{L}^{\mathrm{eff}}-{\beta }_{R}^{\mathrm{eff}}={\rm{\Delta }}\beta +\displaystyle \frac{2}{{\hslash }\omega }\mathrm{ln}\displaystyle \frac{\varepsilon [+| +]A+\varepsilon [-| +]B}{\varepsilon [+| -]A+\varepsilon [-| -]B}.\end{eqnarray} \tag{ 33 }$

We thus see that by tuning the ratio A/B the effective temperature gradient can be manipulated, and if the error associated with the measurement is not too big, it can even be inverted as compared to the original thermal gradient ${\rm{\Delta }}\beta$. So the overall effect of the demon is to change the 'temperatures felt' by the TLS. Accordingly the fluctuation relation

$\begin{eqnarray}&&\displaystyle \frac{h({N}_{R})}{h(-{N}_{R})}={{\rm{e}}}^{-{\rm{\Delta }}{\beta }_{\mathrm{eff}}{\hslash }\omega {N}_{R}}\end{eqnarray} \tag{ 34 }$

is obeyed by the histogram $h({N}_{R})$. This immediately allows us to interpret the quantity

$\begin{eqnarray}&&{J}_{\exp }=-\displaystyle \frac{2{Q}_{R}}{{\hslash }\omega }\mathrm{ln}\displaystyle \frac{\varepsilon [+| +]A+\varepsilon [-| +]B}{\varepsilon [+| -]A+\varepsilon [-| -]B}\end{eqnarray} \tag{ 35 }$

via equation (7) as the mutual information encoded in a trajectory along which heat Q_R is exchanged with the R bath. Note that when A = B, the feedback has no effect and accordingly ${J}_{\exp }=0$. Likewise if $\varepsilon [+| +]=\varepsilon [+| -]$ (hence $\varepsilon [-| +]=\varepsilon [-| -]$), meaning no correlation between state and knowledge thereof, feedback control does not work and again ${J}_{\exp }=0$. Most importantly, the experimental mutual information ${J}_{\exp }$ is proportional to the heat exchanged. This allows access to a fluctuating information theoretic quantity by means of a thermodynamic measurement in a realistic experimental scenario.

Figure 4 shows typical histograms $h({N}_{R})$ for realistic parameters. We also plotted the quantity $\mathrm{ln}h({N}_{R})/h(-{N}_{R})$, finding good agreement with the theoretical prediction $-{\rm{\Delta }}{\beta }_{\mathrm{eff}}{\hslash }\omega {N}_{R}$. The effective conditional probabilities $\varepsilon [k| j]$ were obtained by recording for each trajectory the total time when the state was j and the knowledge was k, and averaging their values over the whole ensemble of trajectories. The observed deviation is a consequence of the fact that error here is introduced not in the form of an outcome being missed (as assumed in deriving equation (34)), but rather in one being reported with some delay. With the histogram $h({N}_{R})$ we computed $\sum {\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle ={\hslash }\omega {\rm{\Delta }}\beta \langle {N}_{R}\rangle =-0.0862$, $-\mathrm{ln}\gamma =-\mathrm{ln}\langle {{\rm{e}}}^{{\rm{\Delta }}\beta {\hslash }\omega {N}_{R}}\rangle =-0.1205$ and $-\langle {J}_{\exp }\rangle =-0.2873$ for the chosen parameters. The computed values are in agreement with the prediction of equation (21). The proposed experiment does not allow us to measure ${\rm{\Delta }}{ \mathcal H }$, which would require access to the full density matrix of system+baths.

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5.1. Energy spent by the demon

We have developed a general quantum theory of repeated feedback control in a scenario with multiple heat reservoirs. The main effect of feedback control is that it induces a generally non-unital dynamics of the full reservoirs+system compound. As a consequence the standard bound set by the second law of thermodynamics on the dissipation quantifier ${\sum }_{l}{\beta }_{l}\langle {\rm{\Delta }}{E}_{l}\rangle$ is shifted and may become negative. We have illustrated an experimental proposal where a single superconducting qubit plays the role of a trap-door that is subject to feedback control. The method envisaged for simultaneously measuring the qubit state and the heat exchanged by each reservoir is single-photon calorimetry.

This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme through the project NeQuFlux grant no 623085 (MC), by Unicredit Bank (MC), by the Academy of Finland contract no. 272218 (JP) by the COST action MP1209 'Thermodynamics in the quantum regime' and by the Centre for Quantum Engineering at Aalto University, CQE.

$\begin{eqnarray*}\langle {{\rm{e}}}^{-{\displaystyle \sum }_{l}{\beta }_{l}{\rm{\Delta }}{E}_{l}}\rangle & = & \displaystyle \sum _{n,{\bf{j}},{\bf{k}},m}p(m,{\bf{k}},{\bf{j}},n){{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{m}^{l}}{{\rm{e}}}^{\displaystyle \sum {\beta }_{l}{E}_{n}^{l}}\\ & = & \displaystyle \sum _{n,{\bf{j}},{\bf{k}},m}{\rm{Tr}}{P}_{m}{A}_{{\bf{k}},{\bf{j}}}{U}_{0}{P}_{n}{U}_{0}^{\dagger }{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}\displaystyle \frac{{{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{n}^{l}}}{Z}{{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{m}^{l}}{{\rm{e}}}^{\displaystyle \sum {\beta }_{l}{E}_{n}^{l}}\\ & = & \displaystyle \sum _{{\bf{j}},{\bf{k}},m}{\rm{Tr}}{P}_{m}{A}_{{\bf{k}},{\bf{j}}}{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}\displaystyle \frac{{{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{m}^{l}}}{Z}\\ & = & \displaystyle \sum _{{\bf{j}},{\bf{k}}}{\rm{Tr}}{A}_{{\bf{k}},{\bf{j}}}^{\dagger }{\rho }_{0}{A}_{{\bf{k}},{\bf{j}}}\end{eqnarray*}$

Equation (3) and ${\rho }_{0}={\sum }_{n}{P}_{n}{{\rm{e}}}^{-\sum {\beta }_{l}{E}_{n}^{l}}/Z$ have been used to obtain the second line. Completeness ${\sum }_{n}{P}_{n}={\mathbb{1}}$ and unitarity ${U}_{0}{U}_{0}^{\dagger }={\mathbb{1}}$ led to the third line. The fourth line follows from the cyclical property of the trace, idempotence ${P}_{m}{P}_{m}={P}_{m}$ and ${\rho }_{0}={\sum }_{m}{P}_{m}{{\rm{e}}}^{-\sum {\beta }_{l}{E}_{m}^{l}}/Z$.

Using equation (11), the exponentiated fluctuating mutual information can be conveniently expressed as

$\begin{eqnarray}&&{{\rm{e}}}^{-{J}_{{\bf{k}},{\bf{j}}}}=\displaystyle \frac{p({\bf{j}}\,:{\bf{k}})p({\bf{k}})}{p({\bf{k}},{\bf{j}})}=\displaystyle \frac{p({\bf{k}})}{{{\rm{\Pi }}}_{i}\varepsilon [{k}_{i}| {j}_{i}]}\end{eqnarray} \tag{ B.1 }$

hence

$\begin{eqnarray}\langle {{\rm{e}}}^{-{\displaystyle \sum }_{l}{\beta }_{l}{\rm{\Delta }}{E}_{l}-{J}_{{\bf{k}},{\bf{j}}}}\rangle & = & \displaystyle \sum _{n,{\bf{j}},{\bf{k}},m}p(m,{\bf{k}},{\bf{j}},n){{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{m}^{l}}{{\rm{e}}}^{\displaystyle \sum {\beta }_{l}{E}_{n}^{l}}\displaystyle \frac{p({\bf{k}})}{{{\rm{\Pi }}}_{i}\varepsilon [{k}_{i}| {j}_{i}]}\\ & = & \displaystyle \sum _{n,{\bf{j}},{\bf{k}},m}{\rm{Tr}}{P}_{m}{B}_{{\bf{k}},{\bf{j}}}{U}_{0}{P}_{n}{U}_{0}^{\dagger }{B}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}\displaystyle \frac{{{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{n}^{l}}}{{{\rm{\Pi }}}_{l}{Z}_{l}}{{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{m}^{l}}{{\rm{e}}}^{\displaystyle \sum {\beta }_{l}{E}_{n}^{l}}p({\bf{k}})\\ & = & \displaystyle \sum _{{\bf{j}},{\bf{k}},m}{\rm{Tr}}{P}_{m}{B}_{{\bf{k}},{\bf{j}}}{B}_{{\bf{k}},{\bf{j}}}^{\dagger }{P}_{m}\displaystyle \frac{{{\rm{e}}}^{-\displaystyle \sum {\beta }_{l}{E}_{m}^{l}}}{{{\rm{\Pi }}}_{l}{Z}_{l}}p({\bf{k}})\end{eqnarray} \tag{ B.2 }$

$\begin{eqnarray}&&=\,\displaystyle \sum _{{\bf{k}},m}{\rm{Tr}}{P}_{m}\displaystyle \frac{{{\rm{e}}}^{-\sum {\beta }_{l}{E}_{m}^{l}}}{{{\rm{\Pi }}}_{l}{Z}_{l}}p({\bf{k}})\end{eqnarray} \tag{ B.3 }$

$\begin{eqnarray}&&=\,{\rm{Tr}}{\rho }_{0}\displaystyle \sum _{{\bf{k}}}p({\bf{k}})=1\end{eqnarray} \tag{ B.4 }$

Equation (3), ${\rho }_{0}={\sum }_{n}{P}_{n}{{\rm{e}}}^{-\sum {\beta }_{l}{E}_{n}^{l}}/{{\rm{\Pi }}}_{l}{Z}_{l}$ and equation (13) have been used to obtain the second line. Completeness ${\sum }_{n}{P}_{n}={\mathbb{1}}$ and unitarity ${U}_{0}{U}_{0}^{\dagger }={\mathbb{1}}$ led to the third line. The fourth line follows from ${\sum }_{{\bf{j}}}{B}_{{\bf{k}},{\bf{j}}}{B}_{{\bf{k}},{\bf{j}}}^{\dagger }={\mathbb{1}}$, which follows by expanding the i-ordered products, applying idempotence ${\pi }_{j}{\pi }_{j}={\pi }_{j}$, completeness ${\sum }_{j}{\pi }_{j}={\mathbb{1}}$ and unitarity ${U}_{j}{U}_{j}^{\dagger }={\mathbb{1}}$. ${\rho }_{0}={\sum }_{m}{P}_{m}{{\rm{e}}}^{-\sum {\beta }_{l}{E}_{m}^{l}}/{{\rm{\Pi }}}_{l}{Z}_{l}$ lead to the fifth line. The final result is a consequence of normalisation of ${\rho }_{0}$ and of $p({\bf{k}})$.

Under the operation of the demon the TLS experiences effective temperatures of the baths that differ from their actual value. To fix ideas, let us for the moment assume no delay time and no error in the measurement. The qubit is effectively subject to the following effective rates ${{\rm{\Gamma }}}_{s}^{\downarrow ,\mathrm{eff}}={{\rm{\Gamma }}}_{s}^{\downarrow | +},{{\rm{\Gamma }}}_{s}^{\uparrow ,\mathrm{eff}}={{\rm{\Gamma }}}_{s}^{\uparrow | -}$. Accordingly, the detailed balance temperatures are shifted:

$\begin{eqnarray}&&{{\rm{e}}}^{{\beta }_{L}^{\mathrm{eff}}{\hslash }\omega }={{\rm{\Gamma }}}_{L}^{\downarrow ,\mathrm{eff}}/{{\rm{\Gamma }}}_{L}^{\uparrow ,\mathrm{eff}}=(A/B){{\rm{e}}}^{{\beta }_{L}{\hslash }\omega }\end{eqnarray} \tag{ C.1 }$

$\begin{eqnarray}&&{{\rm{e}}}^{{\beta }_{R}^{\mathrm{eff}}{\hslash }\omega }={{\rm{\Gamma }}}_{R}^{\downarrow ,\mathrm{eff}}/{{\rm{\Gamma }}}_{R}^{\uparrow ,\mathrm{eff}}=(B/A){{\rm{e}}}^{{\beta }_{R}{\hslash }\omega }\end{eqnarray} \tag{ C.2 }$

where we used the explicit expressions of equation (26). This implies the effective temperatures

$\begin{eqnarray}&&{\beta }_{L}^{\mathrm{eff}}=\displaystyle \frac{1}{{\hslash }\omega }\mathrm{ln}(A/B)+{\beta }_{L}\end{eqnarray} \tag{ C.3 }$

$\begin{eqnarray}&&{\beta }_{R}^{\mathrm{eff}}=\displaystyle \frac{1}{{\hslash }\omega }\mathrm{ln}(B/A)+{\beta }_{R}.\end{eqnarray} \tag{ C.4 }$

Let us now introduce the errors $\varepsilon [\pm | \pm ]$ related to the measurement. The stochastic process describing the dynamics of the TLS is still Poissonian with one rate occurring in the case of correct measurement and one rate occurring in the other case. The idea is that monitoring is continuous, or better, occurring with a sampling time interval dt, which we assume to be short compared to all rates ${{\rm{\Gamma }}}_{R,L}^{\uparrow ,\downarrow | \pm }$. Let us imagine the system is in state $j=+$. There is a probability $\varepsilon [+| +]$ that the observation is $k=+$ and a probability $\varepsilon [-| +]$ that it is $k=-$. Thus the probability to undergo a jump down in the reservoir s in the interval dt is

$\begin{eqnarray}&&p({\rm{d}}{t})=\varepsilon [+| +]{{\rm{e}}}^{-{{\rm{\Gamma }}}_{s}^{\downarrow | +}{\rm{d}}{t}}+\varepsilon [-| +]{{\rm{e}}}^{-{{\rm{\Gamma }}}_{s}^{\downarrow | -}{\rm{d}}{t}}\end{eqnarray} \tag{ C.5 }$

$\begin{eqnarray}&&\simeq \,\varepsilon [+| +](1-{{\rm{\Gamma }}}_{s}^{\downarrow | +}{\rm{d}}{t})+\varepsilon [-| +](1-{{\rm{\Gamma }}}_{s}^{\downarrow | -}{\rm{d}}{t})\end{eqnarray} \tag{ C.6 }$

$\begin{eqnarray}&&=\,1-(\varepsilon [+| +]{{\rm{\Gamma }}}_{s}^{\downarrow | +}+\varepsilon [-| +]{{\rm{\Gamma }}}_{s}^{\downarrow | -}){\rm{d}}{t}\end{eqnarray} \tag{ C.7 }$

$\begin{eqnarray}&&=\,{{\rm{e}}}^{-(\varepsilon [+| +]{{\rm{\Gamma }}}_{s}^{\downarrow | +}+\varepsilon [-| +]{{\rm{\Gamma }}}_{s}^{\downarrow | -}){\rm{d}}{t}}\end{eqnarray} \tag{ C.8 }$

Similarly for the jump up. Overall the TLS experiences the new rates

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{s,\mathrm{eff}}^{\downarrow }=\varepsilon [+| +]{{\rm{\Gamma }}}_{s}^{\downarrow | +}+\varepsilon [-| +]{{\rm{\Gamma }}}_{s}^{\downarrow | -}\end{eqnarray} \tag{ C.9 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{s,\mathrm{eff}}^{\uparrow }=\varepsilon [+| -]{{\rm{\Gamma }}}_{s}^{\uparrow | +}+\varepsilon [-| -]{{\rm{\Gamma }}}_{s}^{\uparrow | -}.\end{eqnarray} \tag{ C.10 }$

Accordingly,

$\begin{eqnarray}&&{e}^{{\beta }_{s}^{\mathrm{eff}}{\hslash }\omega }=\displaystyle \frac{{{\rm{\Gamma }}}_{s,\mathrm{eff}}^{\downarrow }}{{{\rm{\Gamma }}}_{s,\mathrm{eff}}^{\uparrow }}=\displaystyle \frac{\varepsilon [+| +]{{\rm{\Gamma }}}_{s}^{\downarrow | +}+\varepsilon [-| +]{{\rm{\Gamma }}}_{s}^{\downarrow | -}}{\varepsilon [+| -]{{\rm{\Gamma }}}_{s}^{\uparrow | +}+\varepsilon [-| -]{{\rm{\Gamma }}}_{s}^{\uparrow | -}}\end{eqnarray} \tag{ C.11 }$

$\begin{eqnarray}&&{\beta }_{s}^{\mathrm{eff}}=\displaystyle \frac{1}{{\hslash }\omega }\mathrm{ln}\displaystyle \frac{\varepsilon [+| +]{{\rm{\Gamma }}}_{s}^{\downarrow | +}+\varepsilon [-| +]{{\rm{\Gamma }}}_{s}^{\downarrow | -}}{\varepsilon [+| -]{{\rm{\Gamma }}}_{s}^{\uparrow | +}+\varepsilon [-| -]{{\rm{\Gamma }}}_{s}^{\uparrow | -}}.\end{eqnarray} \tag{ C.12 }$

Plugging in the explicit expressions we get

$\begin{eqnarray}&&{\beta }_{L}^{\mathrm{eff}}={\beta }_{L}+\displaystyle \frac{1}{{\hslash }\omega }\mathrm{ln}\displaystyle \frac{\varepsilon [+| +]A+\varepsilon [-| +]B}{\varepsilon [+| -]A+\varepsilon [-| -]B}\end{eqnarray} \tag{ C.13 }$

$\begin{eqnarray}&&{\beta }_{R}^{\mathrm{eff}}={\beta }_{R}+\displaystyle \frac{1}{{\hslash }\omega }\mathrm{ln}\displaystyle \frac{\varepsilon [+| -]A+\varepsilon [-| -]B}{\varepsilon [+| +]A+\varepsilon [-| +]B}.\end{eqnarray} \tag{ C.14 }$

Hence equation (33).