SSciPost Physics Submission

A short story of quantum and information thermodynamics

A. Auﬀ`eves Universit´e Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France* alexia.auﬀ[email protected] 2, 2021

Abstract

This Colloquium is a fast journey through the build-up of key thermodynamicalconcepts, i.e. work, heat and irreversibility – and how they relate to information.Born at the time of industrial revolution to optimize the exploitation of thermalresources, these concepts have been adapted to small systems where thermalﬂuctuations are predominant. Extending the framework to quantum ﬂuctuationsis a great challenge of quantum thermodynamics, that opens exciting researchlines e.g. measurement fueled engines or thermodynamics of driven-dissipativesystems. On a more applied side, it provides the tools to optimize the energeticconsumption of future quantum computers.

Contents

References 12 a r X i v : . [ qu a n t - ph ] F e b ciPost Physics Submission Thermodynamics was developed in the XIXth century, providing a uniﬁed framework betweenmechanical sciences and thermometry. At the time, the motivation was very practical, namelyuse temperature to put bodies into motion - as clearly indicated by its name. In otherwords, the goal was to design and optimize thermal engines, i.e. devices that exploit thetransformations of some “working substance” to convert heat into work. Work and heat aretwo ways to exchange energy, and from the ﬁrst law of thermodynamics, there is nothingwrong in converting one into another.However, turning heat into work is like turning lead into gold: It has severe caveats. Themost famous is Kelvin no-go statement: It is not possible to extract work cyclically from asingle hot bath. This no-go statement turned out to become one of the expressions of thesecond law of thermodynamics, which deals with (ir)reversibility. This is how an initiallyapplied area of physics turned out to deliver fundamental concepts like entropy and timearrow.As a matter of fact, the ﬁrst boundary between work and heat was intimately relatedto the (ir)reversible nature of their exchanges. The concept of work comes from mechanicalsciences, and represents a form of energy that can be exchanged reversibly: In principle, thereis no time arrow associated with work exchanges - at least those associated to conservativeforces. Conversely, the heat exchanges between a body and thermal baths are in general notreversible: heat spontaneously circulates from hot to cold bodies. In particular, if a bodycyclically exchanges an amount of heat Q with a hot bath of temperature T h and − Q witha cold bath of temperature T c , the irreversible nature of heat transfers is captured by thephenomenological formula Q (1 /T c − /T h ) ≥

0, with equality if T c = T h .This observation led to deﬁne the entropy change of a body in contact with a bath attemperature T as ∆ S = Q rev /T , where Q rev is the amount of heat exchanged reversibly.More generally, any isothermal heat exchange follows the Clausius inequality ∆ S − Q/T =∆ i S ≥

0. ∆ i S is the so-called entropy production that quantiﬁes the irreversibility of thetransformation. Introducing the system’s internal energy U , and its free energy F = U − T S ,Clausius inequality becomes W − ∆ F = T ∆ i S ≥ . (1)The meaning of Eq.1 is transparent: It is not possible to extract more work than the freeenergy of the system. Reciprocally, to increase the free energy of a system, one has to pay atleast the same amount of work. Since they are natural consequences of the thermodynamicarrow of time, these inequalities are called fundamental bounds. Extending these bounds tothe quantum realm is an important motivation of quantum thermodynamics.Eq.1 provides intuitions on Kelvin no-go statement, as exempliﬁed by the Carnot engine.In this paradigmatic device, work is extracted during the expansion of the gas while it iscoupled to the hot bath - increasing its entropy and thus lowering its free energy by anamount T h ∆ S . ∆ S is ﬁxed by the settings of the engine (number of particles of the gas,minimal and maximal volume of the chamber). Therefore once the maximal volume has beenreached, it is not possible to extract work anymore and one has to “reset” the engine, i.e.bring it back to its initial settings by compressing the gas. If the compression is performed2 ciPost Physics Submission at the same temperature, no net work is extracted from the cycle, hence the need for at leasttwo baths at two diﬀerent temperatures. Latter Maxwell suggested that information could be used to sort out the molecules of the gasand lower its entropy, apparently at no work cost. Such mechanism blatantly violates theSecond Law and Kelvin no go, since no cold bath here would be needed to reset the engine -compression being realized for free, only using information.It took one century to exorcize Maxwell’s demon paradox. With the rise of informationtheory after the Second World War, it became clear that information was not some immaterialconcept that could escape the laws of thermodynamics. This idea is captured by the famous”Information is physical” attributed to Landauer, one of the fathers of information thermody-namics (together with Bennett, Lloyd...). To understand how information and thermodynam-ics are related, Carnot engines are enlightening - however, instead of considering a gas madeof a large number of particles, one can consider a “single-particle gas” rather, that is eitherpositioned on the left or on the right of the chamber. Left or right can be used to encode onebit of information. Denoting by p the probability that the particle is on the left, the Shannonentropy of the probability distribution reads (in bits) H [ p ] = − p log ( p ) − (1 − p ) log (1 − p ).Now imagine that we know the particle is on the left. While it expands to eventually ﬁll thewhole volume, one bit of information is lost, such that the Shannon entropy change reads∆ H = 1 bit. Conversely, from this elementary amount of information, one can extract somework. In agreement with the Second Law, the amount of extractable work is bounded by W = k B T log 2, where T is the temperature of the chamber in which the expansion takesplace. Here k B stands for the Boltzmann constant. This is the basic principle of the so-calledSzilard engine, that evidences the conversion of information into work.This conversion is reversible and its reverse has even stronger practical implications. In-deed, starting from an initial conﬁguration where the particle has equal chances to be on theleft or on the right, and then compressing it, e.g. to the left of the chamber, is what is calledin information theory a RESET operation: Whatever the initial state of the bit, it ends up inthe state “0”. This operation is logically irreversible: When it is performed, the initial statecannot be traced back. However, it is extremely useful since initializing bits is the beginningof any computation. Formula 1 evidences that resetting a bit has a work cost, that cannot belower than W - the bound being reached when the operation is thermodynamically reversible:This is the famous Landauer’s erasure work [1].Of course, the single particle model is a convenient approach to quickly get an intuition ofthe main equations, but it is idealized. First experimental evidences of information to energyconversions (Szilard engines, Landauer’s erasure) have been obtained around 2010-2012 [2, 3]. While introducing information thermodynamics, one has departed from the usual scenery ofmacroscopic thermodynamics that involved large amounts of particles as working substances.Within information thermodynamics, the working substance is now elementary since it solelyinvolves one particle whose phase space reduces to two micro-states, 0 and 1. This new sceneryis the one of “stochastic thermodynamics”, that deals with small enough working substances3 ciPost Physics Submission such that ﬂuctuations become predominant [4].In this new realm, the dynamics of the system results from the action of some externaloperator that drives the system to implement some protocol. The system’s evolution is per-turbed by a thermal bath that induces random, “stochastic” ﬂuctuations. Thus, the dynamicsof the system is described by Markovian, stochastic trajectories is its phase space - one tra-jectory consisting in continuous sequences where the drive controls the system, intertwinedby stochastic jumps imposed by the bath.This new realm sheds new light on the First Law. Work now corresponds to the part ofenergy exchanged with the controller during the continuous sequences. On the other hand,heat is deﬁned as the part of energy stochastically exchanged during the jumps induced by thebath. From an energetic point of view, it appears that the heat/work boundary now reﬂectsthe boundary between noise and control. From that perspective, an engine is a device madeto extract energy from noise, by rectifying the ﬂuctuations it induces.The framework of stochastic thermodynamics also invites to reconsider the meaning ofthe Second Law. As a matter of fact, the laws of physics at the level of single particlesare expected to be reversible, so where does irreversibility come from? There is a simple,operational answer to this question. Let us suppose that the system is initially prepared in awell-deﬁned micro-state. The controller now implements a protocol aimed to bring the systeminto another micro-state. In the absence of a bath, the trajectory is perfectly deterministic.The controller is thus able to reverse the protocol, to bring back the system to its initialmicro-state. However, if a bath remains coupled to the system during the protocol, therandom perturbations it induces prevent the controller from perfectly reversing the trajectoryfollowed by the system, making the protocol irreversible.

Interestingly, stochastic thermodynamics allows us to quantify the amount of irreversibility per trajectory γ . This is captured by the so-called stochastic entropy production ∆ i S [ γ ], thatis deﬁned as ∆ i S [ γ ] = log (cid:18) P F [ γ ] P B [ γ ∗ ] (cid:19) . (2)We have introduced the time-reversed trajectory γ ∗ , and the probability P F [ γ ] (resp. P B [ γ ])of the trajectory γ (resp. γ ∗ ) while the protocol is run forward (resp. backward). With thisdeﬁnition, ∆ i S has no dimension. The meaning of Eq.2 is obvious, entropy production beingpositive if γ is more probable forward than backward.Some particular trajectories may lead to a negative entropy production. However, thisis not contradictory with the Second Law which deals with average values. Namely, thequantity that should remain positive is the average value of the entropy production givenby (cid:104) ∆ i S [ γ ] (cid:105) γ = Σ γ P F [ γ ]∆ i S [ γ ]. This condition is automatically fulﬁlled. This is obvious bynoticing that (cid:104) exp( − ∆ i S [ γ ]) (cid:105) γ = Σ γ P B [ γ ]. In most situations (at the noticeable exception ofso-called absolute irreversibility [5]), this boils down to (cid:104) exp( − ∆ i S [ γ ]) (cid:105) γ = 1 , (3)which is called the Integral Fluctuation Theorem (IFT). From the convexity of the exponential,one easily gets that (cid:104) ∆ i S [ γ ] (cid:105) γ ≥

0, in agreement with the Second Law. The bound is saturated,4 ciPost Physics Submission if and only if ∆ i S [ γ ] = 0 for all γ . This strong condition deﬁnes the equilibrium distribution.As we show below, it is fully equivalent to deﬁne the equilibrium distribution by imposing themicro-reversibility condition.From Eq.2, Eq.3 is a tautology - but it is also the seed of so-called “ﬂuctuation theorems”,Jarzynski Equality (JE) probably being the most famous [6]. To recover it, one considersa system initially prepared at thermal equilibrium, then driven out of equilibrium by someexternal operator. It can be shown that the expression of the stochastic entropy productionexactly matches the classical one, namely ∆ i S [ γ ] = ( W [ γ ] − ∆ F ) /k B T (See below for a simpleapproach). The IFT reads (cid:104) exp( − W [ γ ] /k B T ) (cid:105) γ = exp( − ∆ F/k B T ). This is JE, which hasbeen experimentally veriﬁed on many diﬀerent platforms. While simple, two-points trajectories are interesting since they allow us to build many usefulintuitions. Let us consider a system with micro-states denoted by σ i , i being an integer.The stochastic behavior of the system is fully captured by the probability of jump from thestate j to the state i , P [ σ i | σ j ]. Here we take this probability as a constant for the sake ofsimplicity. Denoting as γ an elementary trajectory γ = ( σ i , σ j ), its forward (resp. backward)probability reads P F [ γ ] = p ( σ i ) P [ σ j | σ i ] (resp. P B [ γ ∗ ] = p ( σ j ) P [ σ i | σ j ]). We have introduced p ( σ ) (resp. p ( σ )) the probability distribution before (resp. after) the jump. The entropyproduced by the trajectory γ reads∆ i S [ γ ] = log (cid:18) p ( σ i ) p ( σ j ) (cid:19) + log (cid:18) P [ σ j | σ i ] P [ σ i | σ j ] (cid:19) . (4)The term on the left is called the boundary term, the term on the right the conditional term.Let us ﬁrst characterize the equilibrium distribution p ∞ ( σ ). According to the deﬁnitionabove, it is characterized by ∆ i S [ γ ] = 0 for all γ , such that p ∞ ( σ j ) P [ σ i | σ j ] = p ∞ ( σ i ) P [ σ j | σ i ]for all ( i, j ). This evidences that the equilibrium distribution fulﬁlls the micro-reversibilitycondition. Reciprocally, any distribution fulﬁlling this condition is an equilibrium distribution.It is now possible to rewrite Eq.4,∆ i S [ γ ] = log (cid:18) p ( σ i ) p ∞ ( σ i ) (cid:19) − log (cid:18) p ( σ j ) p ∞ ( σ j ) (cid:19) . (5)Averaging Eq.5 over all forward trajectories yields (cid:104) ∆ i S [ γ ] (cid:105) γ = − ∆ D ∞ . D ∞ is the distanceto equilibrium, deﬁned as D ∞ = Σ i p ( σ i )(log p ( σ i ) − log p ∞ ( σ i )). This result evidences thateach application of a stochastic map brings the system nearer its equilibrium distribution,which characterizes a relaxation.At this point it is interesting to consider the textbook case where the stochastic behavioris induced by a thermal bath of temperature T . Introducing E ( σ ) the internal energy of thesystem in the micro-state σ , the probability distribution characterizing thermal equilibrium issimply the Boltzmann distribution p ( σ ) = Z − exp( − E ( σ ) /k B T ). Z is the partition function,that allows us to deﬁne the system free energy F = − log( Z ). Let us introduce ∆ S [ γ ] = − log( p ( σ i )) + log( p ( σ j )). ∆ S [ γ ] can be called the system’s “stochastic entropy change”:Once averaged over all forward trajectories, it gives back the standard expression for thesystem’s entropy change. We get Eq.4 becomes ∆ i S [ γ ] = ∆ S [ γ ] − Q [ γ ] /k B T , whose averagevalue is in agreement with the classical deﬁnition (Eq. 1). If the system’s initial states ofthe forward and the backward protocol correspond to the thermal equilibrium (Jarzynski’s5 ciPost Physics Submission protocol), one gets ∆ S [ γ ] = ∆ E [ γ ] − ∆ F . Introducing the stochastic work W [ γ ] = ∆ E [ γ ] − Q [ γ ] yields the stochastic entropy production leading to Jarzynski’s equality. A very important achievement of stochastic thermodynamics has been to incorporate infor-mation in the expression of a ﬂuctuation theorem, giving rise to the so-called ‘GeneralizedIntegral Fluctuation Theorem” [7]. A simple intuition can be grasped, again by consider-ing the case of a two-point trajectory that now involves a system and a demon’s memory.The system has been read by the demon beforehand, such that the system and the memorystate are correlated. Denoting as x and m the system and memory micro-states, p ( x, m )(resp. p ( x ), p ( m )) the joint (resp. marginal) probabilities, the correlation is quantiﬁed bythe stochastic mutual information I ( x, m ) = log( p ( x ) p ( m )) − log( p ( x, m )). Averaged over thedistribution, one recover the usual expression of the mutual information between the systemand the demon, I ( S : M ) = H S + H M − H SM .One focuses on the feedback operation. Namely, the demon exploits its knowledge onthe system to perform some operation on it. Supposing that the memory state is not al-tered by the feedback, the forward (resp. the backward) trajectory reads γ = ( x, m, y )and γ ∗ = ( y, m, x ), and their respective probabilities read P F [ γ ] = p ( x, m ) P [ y | x, m ] and P B [ γ ∗ ] = p ( y, m ) P [ x | y, m ]. An ideal feedback is perfectly deﬁned by the memory state,yielding P [ y | x, m ] = P [ x | y, m ]. We get eventually∆ i S [ γ ] = ∆ S [ γ ] − ∆ I [ γ ] , (6)where ∆ S [ γ ] = log( p ( x )) − log( p ( y )) is the stochastic entropy change of the system. ∆ I [ γ ] = I ( y, m ) − I ( x, m ) is the change of the stochastic mutual information between the systemand the memory, where I k characterizes the joint probability distribution p k .Importantly, this expression puts information and entropy production on an equal footing,allowing to quantitatively address the work value of information. The same kind of argumentas developed for the IFT can indeed be used to demonstrate that ∆ S ≥ ∆ I ( S : M ), opening arigorous path to exorcize Maxwell’s demon. It basically states that information as quantiﬁedby I ( S : M ) is a resource that can be consumed (∆ I ( S : M ) ≤

0) to lower the entropy of asystem at no work cost. Equivalent expressions can be derived when work is extracted fromthe protocol, leading to generalized fundamental bounds W ≥ ∆ F + k B T ∆ I . This can beused to deﬁne eﬃciencies of Maxwell’s demons, e.g. η = W/ (∆ F + k B T ∆ I ). Just like usualengines, maximal eﬃciency is reached when the bound is saturated, i.e. when the process isrun reversibly. Macroscopic thermodynamics has given rise to the concept of engine, as a device that con-verts heat into work. Maximal eﬃciency is reached when the device is operated reversibly,connecting the notion of energetic performance to the thermodynamic arrow of time.These concepts have been extended at the level of single particles by stochastic thermo-dynamics. Here noise plays a key role to deﬁne heat, work, and time arrow. For historicalreasons, thermal noise due to the action of thermal baths was ﬁrst considered. Thus, thermalengines were the ﬁrst to be designed and experimentally implemented, and they still remainthe most studied kind of nano-engines. In the same way, the concepts of entropy production6 ciPost Physics Submission and equilibrium are still widely understood with respect to a thermal bath playing the roleof a reference.However, the framework brought by stochastic thermodynamics is suﬃciently general andﬂexible, such that other kinds of noise can be used as seeds to build “other” thermodynamicalframeworks and explore new physics. We shall adopt this strategy below (See Section 3.2).

Quantum thermodynamics is the converging point of many areas of research, making it a veryexciting ﬁeld where new and transversal concepts are built. Many current lines of research areexposed in reviews and books (See e.g. [8]) and it is not my purpose to summarize them here.I shall rather put them in perspective with respect to the legacy of classical thermodynamicspresented above, and focus on original research topics developed in my group in the past fewyears.Firstly, quantum thermodynamics is the natural follow-up of stochastic thermodynamics,where systems switch from nano to quantum. What are irreversibility, work and heat inthe quantum realm? is one of the most important questions. More technically, one aimsto evidence new and genuinely quantum components in ﬂuctuation theorems, that could berelated to quantum coherence or entanglement. On the more applied side, one importantresearch line investigates if quantum coherence and correlations can be a resource for nano-engines, that would lead them to outperform their classical counterparts. Reciprocally, whatis the energetic cost of ﬁghting against quantum noise?To answer these questions, a natural scenery is provided by quantum open systems -namely, driven quantum systems interacting with one or several baths. In the equationsdescribing the system’s dynamics, the action of the drive is usually modeled by some time-dependent Hamiltonian. Hence, the drive can exchange energy with the system, withoutchanging its von Neumann entropy: This is consistent with the classical deﬁnition of work.Conversely, the action of the bath(s) is non-unitary, such that the system’s entropy is notnecessarily conserved by the interaction: This is reminiscent of a heat exchange.However, there is still no consensus on the deﬁnitions of heat and work in the quantumrealm. One important reason is that the baths interacting with the system are not neces-sarily at thermal equilibrium. Therefore new thermodynamical concepts must be built, inthe absence of temperature. Another reason is genuinely quantum: Quantum measurementperturbs. This is well known, but the energetic consequences of this eﬀect had not been drawnuntil recently.

As mentioned above, a consistent thermodynamical framework can be built for any drivensystem subjected to noise (See Fig.1). For historical reasons, the thermal noise was ﬁrstconsidered - but there are many other kinds of noise in the quantum world. A fundamental oneis the noise induced by projective quantum measurement. The scenery in this case is as simpleas it can be: A quantum system evolving under some time-dependent Hamiltonian on the onehand, and projectively measured at discrete times. Knowing the outcomes of the measurement7 ciPost Physics Submission

Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: September 21, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ~ (1) R (2) U ( t ) = Tr q [ ⇢ q ( t ) H q ( t )] (3)˙ q = Tr q [ ˙ ⇢ q ( t ) H q ( t )] (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) a) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 10, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw W (1)ˆ L µ (2) (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 10, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw Q (1)ˆ L µ (2) (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 16, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ~ (1) L [ ⇢ s ] (2) k b (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 16, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ~ (1) L [ ⇢ s ] (2) k b (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 16, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ~ (1) L [ ⇢ s ] (2) k b (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: September 21, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ~ (1) S (2) U ( t ) = Tr q [ ⇢ q ( t ) H q ( t )] (3)˙ q = Tr q [ ˙ ⇢ q ( t ) H q ( t )] (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 10, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw O (1)ˆ L µ (2) (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) b) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: November 10, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw O (1)ˆ L µ (2) (3) g / (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) P u r ee m i tt e r d e ph a s i n g : a r e s o u r c e f o r a d v a n c e d s o li d - s t a t e s i n g l e ph o t o n s o u r c e s A l e x i a A u ↵ ` e v e s , J e a n - M i c h e l G ´ e r a r d , a nd J e a n - P h ili pp e P o i z a t C E A / C N R S / U J F J o i n tt e a m ” N a n oph y s i c s a n d s e m i c o n d u c t o r s ” , I n s t i t u t N ´ ee l - C N R S , B P , , r u e d e s M a r t y r s , G r e n o b l e C e d e x , F r a n ce a n d C E A / C N R S / U J F J o i n tt e a m ” N a n oph y s i c s a n d s e m i c o n d u c t o r s ” , C E A / I NA C / S P M , r u e d e s M a r t y r s , G r e n o b l e , F r a n ce ⇤ ( D a t e d : N o v e m b e r , ) P A C Snu m b e r s : . . C t ; . . G y ; . . P q ; . . H w Q ( ) H s ( t ) ( ) ( ) g m / ( ) ˙ w = T r q [ ⇢ q ( t ) ˙ H q ( t ) ] ( ) P e ( t ) = T r q ⇢ q ( t ) ✓ z + ◆ ( ) H q ( t ) = T r m [ ⇢ m ( t )( H + V ) ] = h ( ⌫ + ( t )) [ z + ] ( ) ( t ) = T r m [ ⇢ m ( t ) g m ( b + b † ) ] ( ) H m ( t ) = T r q [ ⇢ q ( t )( H m + V ) ] = h ⌦ b † b + h g m P e ( t )( b + b † ) ( ) E m ( t ) = h ⌦T r m [ ⇢ m ( t ) b † b ] ( ) ˙ U = h ˙ ( t ) = ˙ w = ˙ E m ( ) ˙ w = ˙ E m ( ) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: September 21, 2015) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw ~ (1) S (2) U ( t ) = Tr q [ ⇢ q ( t ) H q ( t )] (3)˙ q = Tr q [ ˙ ⇢ q ( t ) H q ( t )] (4)˙ w = Tr q [ ⇢ q ( t ) ˙ H q ( t )] (5) P e ( t ) = Tr q ⇢ q ( t ) ✓ z + 12 ◆ (6) H q ( t ) = Tr m [ ⇢ m ( t )( H + V )] = h ( ⌫ + ( t ))2 [ z + 1] (7) ( t ) = Tr m [ ⇢ m ( t ) g m ( b + b † )] (8) H m ( t ) = Tr q [ ⇢ q ( t )( H m + V )] = h ⌦ b † b + hg m P e ( t )( b + b † ) (9) E m ( t ) = h ⌦Tr m [ ⇢ m ( t ) b † b ] (10)˙ U = h ˙ ( t )2 = ˙ w = ˙ E m (11)˙ w = ˙ E m (12) Pure emitter dephasing : a resource for advanced solid-state single photon sources

Alexia Au↵`eves , Jean-Michel G´erard , and Jean-Philippe Poizat CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors ”,Institut N´eel-CNRS,BP 166, 25, rue des Martyrs, 38042 Grenoble Cedex 9, France and CEA/CNRS/UJF Joint team ” Nanophysics and semiconductors”,CEA/INAC/SP2M, 17 rue des Martyrs, 38054 Grenoble, France ⇤ (Dated: January 18, 2016) PACS numbers: 42.50.Ct; 42.50.Gy; 42.50.Pq ; 42.65.Hw B J ( ✓ ) (1) M (2) ⇤ Electronic address: [email protected]

Figure 1: Classical versus quantum thermodynamics. A system S exchanges work W withan external controller, and heat Q with a stochastic entity. a) Historical framework. Thestochastic entity is a thermal reservoir whose action is symbolized by the dice k B . b) Rebuild-ing quantum thermodynamics on quantum measurement. The stochastic entity is a measuringdevice whose action is symbolized by the dice ¯ h .and the applied Hamiltonian, it is possible to reconstruct at any time the trajectory of purequantum states followed by the system, that consists in continuous sequences intertwined bythe measurement-induced, stochastic quantum jumps. These quantum trajectories are thequantum counterpart of the stochastic trajectories introduced above, that provide the breadand butter of stochastic thermodynamics. However in the present situation, the stochasticityis genuinely quantum, since it is due to measurement back-action.For a given quantum trajectory, the system’s internal energy along time is identiﬁed withthe expectation value of the Hamiltonian along the trajectory. In the spirit of stochasticthermodynamics, work can be deﬁned as the system’s energy change during the unitary se-quences - “heat”, on the other hand, being identiﬁed with the sudden energy changes duringthe quantum jumps. This heat has no classical equivalent, since it comes from the ﬂuctuationsinduced by measurement back-action. Such ﬂuctuations can only take place, if the measuredstate has coherences in the basis of the measured observable. It is thus a quantum eﬀect, dueto quantum coherences. For this reason, my coworkers and I dubbed it “quantum heat” [9].Let us now focus on time arrow. For the sake of simplicity, we focus on a protocoldeﬁned by some initial eigenstate | m (cid:105) of the observable ˆ M , a unitary evolution deﬁned bythe operator ˆ U , and a ﬁnal measurement of ˆ M with stochastic result m k γ . This elementaryquantum trajectory γ is perfectly deﬁned by the two points γ = ( m , m k γ ). Its probabilityreads P F [ γ ] = P [ m k γ | m ], i.e. P F [ γ ] = (cid:10) m k γ (cid:12)(cid:12) ˆ U | m (cid:105) . Averaged over all trajectories, the ﬁnalsystem state is a mixture ρ of pure states | m k (cid:105) with probability p k = (cid:104) m k | ˆ U | m (cid:105) .Reciprocally, the backward protocol consists in measuring the observable ˆ M while the8 ciPost Physics Submission InputOutput > Input |e>|g>Work Quantum heat |+ x > or |- x >? Feedback

Figure 2: Top. Population of the driven qubit excited state P e ( t ) (Rabi oscillation) The state | + x (cid:105) (resp. the state |− x (cid:105) ) gives rise to maximal work extraction in the ﬁeld (resp. from theﬁeld). Bottom. Scheme of a quantum Maxwell?s demon experiment. The qubit exchangeswork with the driving ﬁeld, while it is periodically measured by the demon in the {| + x (cid:105) ; |− x (cid:105)} basis. If the demon measures |− x (cid:105) , it performs a feedback on the qubit to bring it in the state | + x (cid:105) , which is favorable for work extraction.system is in ρ , evolving the system backward with the reversed evolution operator ˆ U † , andperforming a ﬁnal measurement. The probability of the backward trajectory γ ∗ = ( m k γ , m )reads P B [ γ ∗ ] = p k γ P [ m | m k γ ]. Applying Formula 2 immediately gives ∆ i S [ γ ] = − log( p k γ ).Averaged over all trajectories, the entropy production thus reads (cid:104) ∆ i S [ γ ] (cid:105) = ∆ S VN (7)where ∆ S VN is the increase of the Von Neumann entropy of the system along the forwardprotocol. This expression can easily be extended to multi-points trajectories, or to the casewhere the protocol does not start with a pure state. Eq.7 is extremely important since itconnects the Von Neumann entropy - widely used in quantum physics - to entropy production,which is a purely thermodynamical concept. It provides a rigorous demonstration of the well-known “irreversibility of quantum measurement” that fully exploits the relevant framework ofstochastic thermodynamics. In particular, it now allows measuring “how much irreversible”a measurement is. It should now be clear that projective measurement, just like any stochastic process, can beseen as a source of irreversibility and energy - playing a role quite similar to the good oldthermal bath. In particular, a non-zero amount of quantum heat can be exchanged on average ciPost Physics Submission between the system and the measurement channel, as soon as the measured observable doesnot commute with the system’s Hamiltonian. Building on this analogy, my coworkers and Ihave suggested to use quantum measurement as a new kind of energetic resource that could fuelquantum engines (See [ ? ] and Fig.2). The experiment that we have suggested involves a qubitof energy eigenstates denoted | (cid:105) and | (cid:105) , of transition frequency ω as a working substance,that exchanges work with some resonant driving ﬁeld. The mechanism is a classical Rabioscillation, where the qubit’s state evolves as | ψ ( t ) (cid:105) = cos(Ω t/ | (cid:105) + sin(Ω t/ | (cid:105) with Ω theclassical Rabi frequency. Work extraction takes place during stimulated emission, when thequbit provides energy to the ﬁeld. Maximal power extraction is reached when the qubit is inthe coherent superposition | + (cid:105) = ( | (cid:105) + | (cid:105) ) / √ | + (cid:105) . One cycle consists of the fourfollowing steps: (i) Work extraction: after the qubit is initialized in the state | + (cid:105) , it evolves inthe state | φ ( τ ) (cid:105) = cos(Ω τ / | + (cid:105) + sin(Ω τ / |−(cid:105) while an amount of work W = ¯ hω sin(Ω τ / {| + (cid:105) ; |−(cid:105)} basis where |−(cid:105) =( − | (cid:105) + | (cid:105) ) / √

2. During this step, an amount of quantum heat Q q = W is provided by themeasurement channel to the qubit (iii) Feedback to prepare the state | + (cid:105) . This feedback stepcosts no energy since the states | + (cid:105) and |−(cid:105) have the same energy (iv) Erasure of the classicalmemory used to control the feedback loop. The energetic cost of this process is lower boundedby Landauer’s erasure work W L = k B T H [ p ] where T is the temperature of the memory and H [ p ] its Shannon’s entropy.Actually, this machine is nothing but a new kind of Maxwell’s demon engine. The featurethat makes it really quantum is that it does not extract energy from a hot thermal bath,but from the measurement process itself. Thus the two facets of quantum measurement areexploited in this device: Measurement not only allows to extract information, but it alsoprovides energy since it back-acts on the system’s state. Stated in fancy words, with such anengine you can put a body in motion,“ just by looking at it”.For an engine, an important ﬁgure of merit is its yield. It is computed, by comparing thenet extracted work W − W L to the consumed resource Q q i.e. η = 1 − W L /Q q . Interestingly, η → τ (cid:28)

1. This corresponds to the Zeno regime, where measurements are per-formed at such a fast rate that the qubit is “frozen” in the | + (cid:105) state. In this situation, themeasurement outcome is certain and the memory’s entropy vanishes, such that no erasure isneeded. Reaching such a yield means that the quantum heat provided by the measurementchannel is fully converted into work, the engine behaving as a transducer. The power is an-other relevant ﬁgure of merit. It turns out that maximal power is also reached in the Zenoregime, as a simple consequence of the fact that | + (cid:105) is the best state for power extraction- as mentioned above. Unlike classical engines where one has to chose between maximal ef-ﬁciency and maximal power, the present device allows to operate at maximal eﬃciency andpower simultaneously. This is typical of the fact that we have now departed from standardthermodynamics and that new intuitions must be built. Discussion

The engine presented above was the ﬁrst to explicitly exploit “quantum heat”, i.e. measure-ment induced back-action, for the sake of work extraction. Obviously, the main value of theproposal is not its practical interest. It is a proof of concept, that evidences the reality of10 ciPost Physics Submission energy exchanges with a measurement channel. Since then, the concept of quantum heat hasbloomed to give rise to new proposals for measurement driven engines [10, 11], to cool downqubits [12], or to track entanglement generation [13]. Since decoherence is nothing but anunwanted measurement performed by some uncontrolled environment, quantum heat is alsoexpected to be a relevant concept to estimate the energetic costs related to feedback-basedstabilization [9].The quantum heat is the energetic counterpart of the measurement postulate and assuch, it can be perceived diﬀerently by diﬀerent users of quantum theory. It can be seenas a practical, eﬀective quantity allowing to take quantitatively into account the eﬀect ofa measurement on thermodynamical quantities. An interesting line of research now willconsist in “opening the black box”, i.e. modeling the measurement process itself and trackthe energetic and entropic ﬂuxes within the measurement channel. Just like in classicalthermodynamics, where heat and irreversibility are expected to vanish when one reaches acomplete, ultimate description of the system, one could thus expect to ﬁnd that quantumheat is not a fundamental concept.On the other hand, one can be of the opinion that quantum mechanics entirely relies onsome act of measurement, and that whichever the degree of precision in the described mech-anism, its very meaning is built on the irreversible records of measurement outcomes. In thisview, the quantum heat rather appears as a fundamental concept that will always be part ofthe thermodynamical description. Current debates about heat and work in quantum thermo-dynamics reﬂect in some sense the still ongoing debates about the status of the measurementpostulate - quantum thermodynamics providing a new playground, and why not? Maybe newideas and new experiments to explore interpretations of quantum mechanics [11, 14].

As a ﬁnal promising ﬁeld of investigations for quantum thermodynamics, it is worth mention-ing the thermodynamical study of quantum computing in the context of the present school.As a matter of fact, the “quantum advantage” usually put forward to motivate quantum ver-sus classical computing is its reduction in complexity. However, another advantage is thatquantum computing is in principle reversible. Being more speciﬁc, an ideal quantum algo-rithm like the Deutsch problem consists in initializing the data register in the state | (cid:105) of thecomputational basis, a unitary operation, and a ﬁnal measurement in the computational basis.In the absence of noise, the algorithm can be seen as a quantum interference: The quantumstate of the register before the ﬁnal measurement pertains to the computational basis andthe outcome provides a noise-free answer to the asked question. In particular, there is noback-action associated to this ﬁnal measurement, i.e. the act of measurement has no eﬀecton the system’s state. Therefore, it is conceivable that the agent performing the computationrecords the outcome, and then reverses the whole protocol to bring back the register to itsinitial state | (cid:105) , such that there is no heat dissipation associated to the reset.Conversely, activating the gates of the quantum circuit is not energetically free. Simplyconsidering a single qubit gate, the minimal energetic cost to run it is to let the qubit interactresonantly with a coherent ﬁeld during a well-deﬁned time. The energetic cost associated tothis operation can be quantiﬁed as the minimal number of photons put in the coherent ﬁeld.To reduce the cost, it is tempting to work with a small number of photons. However, smallﬁelds get entangled with the qubit, leading to some fundamental noise aﬀecting the process11 ciPost Physics Submission as soon as the ﬁeld is traced out [15, 16]. Preliminary studies show that at least 1000 photonsin the ﬁeld to keep a suﬃciently good ﬁdelity on the gate. For microwave photons interactingwith superconducting qubits, the energetic bill to activate a single gate is typically 10 − J.This is the same order of magnitude as the ultimate heat dissipated by the erasure of a singlebit. Large scale quantum computers will involve a large number of gates, in particular forerror correction. As a consequence, energy will quite certainly play a key role to benchmarkfuture quantum computing architectures.

Acknowledgements

It is a pleasure to thank Ioan Pop, Benjamin Huard and Michel Devoret for their kind invi-tation to provide this manuscript, as well as Patrice Camati for his careful reading.

Funding information

This work was supported by the Foundational Questions InstituteFund (Grant number FQXi-IAF19-05 and FQXi-IAF19-01), the Templeton World CharityFoundation, Inc (Grant No. TWCF0338) and the ANR Research Collaborative Project “Qu-DICE” (ANR-PRC-CES47).

References [1] R. Landauer,

Irreversibility and heat generation in the computing process , IBM Journalof Research and Development pp. 183–191 (1961), doi:10.1147/rd.53.0183.[2] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki and M. Sano

Experimental demonstrationof information-to-energy conversion and validation of the generalized Jarzynski equality. ,Nature Physics , 988 (2010), doi:10.1038/nphys1821.[3] A. B´erut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider and E. Lutz Ex-perimental veriﬁcation of Landauer’s principle linking information and thermodynamics ,Nature , 187 (2012), doi:10.1038/nature10872.[4] U. Seifert,

Stochastic thermodynamics: Principles and perspectives. , Eur. Phys. J. B , 423 (2008).[5] K. Funo, Y. Murashita and M. Ueda, Quantum nonequilibrium equalities with abso-lute irreversibility , New Journal of Physics (7), 075005 (2015), doi:10.1088/1367-2630/17/7/075005.[6] C. Jarzynski, Nonequilibrium equality for free energy diﬀerences , Phys. Rev. Lett. ,2690 (1997), doi:10.1103/PhysRevLett.78.2690.[7] T. Sagawa and M. Ueda, Fluctuation theorem with information exchange: Role ofcorrelations in stochastic thermodynamics , Phys. Rev. Lett. , 180602 (2012),doi:10.1103/PhysRevLett.109.180602.[8] F. Binder, L. A. Correa, C. Gogolin, J. Anders and G. Adesso, eds.,

Thermodynamics inthe Quantum Regime: Fundamental Aspects and New Directions , Fundamental Theoriesof Physics. Springer, Cham, ISBN 978-3-319-99045-3 (2018).12 ciPost Physics Submission [9] C. Elouard, D. A. Herrera-Mart´ı, M. Clusel and A. Auﬀ`eves,

The role of quantummeasurement in stochastic thermodynamics , npj Quantum Information (1) (2017),doi:10.1038/s41534-017-0008-4.[10] C. Elouard and A. N. Jordan, Eﬃcient quantum measurement engines , Physical ReviewLetters (26) (2018), doi:10.1103/physrevlett.120.260601.[11] A. Jordan, C. Elouard and A. Auﬀ`eves,

Quantum measurement engines and theirrelevance for quantum interpretations , Quantum Stud.: Math. Found. (2019),doi:10.1007/s40509-019-00217-2.[12] L. Buﬀoni, A. Solfanelli, P. Verrucchi, A. Cuccoli and M. Campisi,

Quantum measurementcooling , Phys. Rev. Lett. , 070603 (2019), doi:10.1103/PhysRevLett.122.070603.[13] C. Elouard, A. Auﬀ`eves and G. Haack,

Single-shot energetic-based estimator for entan-glement in a half-parity measurement setup , Quantum , 166 (2019), doi:10.22331/q-2019-07-15-166.[14] P. Grangier and A. Auﬀ`eves, What is quantum in quantum randomness? , PhilosophicalTransactions of the Royal Society A: Mathematical, Physical and Engineering Sciences(2018), doi:10.1098/rsta.2017.0322.[15] J. Ikonen, J. Salmilehto, M. M¨ott¨onen

Energy-eﬃcient quantum computing , npj Quan-tum Inf , 17 (2017), doi:10.1038/s41534-017-0015-5.[16] J. Gea-Banacloche, Minimum energy requirements for quantum computation , Phys. Rev.Lett.89