A two-qubit engine fueled by entangling operations and local measurements
Léa Bresque, Patrice A. Camati, Spencer Rogers, Kater Murch, Andrew N. Jordan, Alexia Auffèves
AA two-qubit engine fueled by entangling operations and local measurements
L´ea Bresque, Patrice A. Camati, Spencer Rogers, Kater Murch, Andrew N. Jordan,
2, 4 and Alexia Auff`eves Universit´e Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Department of Physics, Washington University, St. Louis, Missouri 63130 Institute for Quantum Studies, Chapman University, Orange, CA, 92866, USA
We introduce a two-qubit engine that is powered by entangling operations and projective local quantummeasurements. Energy is extracted from the detuned qubits coherently exchanging a single excitation. Thisengine, which uses the information and back-action of the measurement, is generalized to an N -qubit chain.We show that by gradually increasing the energy splitting along the chain, the initial low energy of the firstqubit can be up-converted deterministically to an arbitrarily high energy at the last qubit by successive neighborswap operations and local measurements. Modeling the local measurement as the entanglement of a qubit witha meter, we identify the measurement fuel as the energetic cost to erase correlations between the qubits. Understanding quantum measurements from a thermody-namic standpoint is one of the grand challenges of quantumthermodynamics, with strong fundamental and practical im-plications in various fields ranging from quantum foundationsto quantum computing. Quantum measurement has a doublestatus: on one hand, it is the process that allows the extractionof information from a quantum system. In the spirit of clas-sical information thermodynamics, its “work cost” was thusquantitatively analyzed as the energetic toll to create correla-tions between the system and a memory [1–3]. On the otheras stochastic processes, quantum measurements also lead towavefunction collapse. Measurements can thus behave asa source of entropy and energy, playing a role similar to abath. The energetic fluctuations generated by the measure-ment backaction have recently been exploited as a new kindof fuel in so-called measurement-driven engines [4–9], andquantum fridges [10, 11].Another core concept, quantum entanglement [12], wasidentified by Schr¨odinger as the characteristic trait of quan-tum physics. This feature of quantum mechanics was origi-nally identified by Einstein, Podolsky, and Rosen [13] in theirattempt to show quantum mechanics was incomplete and laterderided by Einstein as “spooky action at a distance”. It hascome to be viewed as an essential resource in various quan-tum technologies. The spooky action is the consequence ofwavefunction collapse, which happens because the measurednon-local state is not an eigenstate of the local measured ob-servable. In this Letter, we propose to exploit this feature todesign a new generation of quantum measurement poweredengines. Local measurements are performed on still interact-ing entangled systems, allowing to harvest the interaction en-ergy. This contrasts with former entanglement engines pow-ered with thermal resources [14–17]. The measurement-basedfueling mechanism we shall focus on also departs from theoriginal EPR proposal, where the systems sharing the entan-gled state are space-like separated, and not interacting whenthe local measurements take place.We first propose a bipartite engine made of two detunedqubits that become entangled through the coherent exchangeof a quantum of excitation. When the red-detuned qubit A is initially excited, the excitation is partially transferred to (𝑎) + + ⟨𝐻 ./0 ⟩ 𝐸 ⟨𝐻 ⟩⟨𝑉⟩ 𝜔 ; 𝜔 + |1⟩|0⟩ | ⟩ | ⟩ 𝑔 (𝑖)(𝑖𝑖)(𝑖𝑖𝑖) (𝑖𝑖𝑖) ℏ𝜔 ; | ⟩ | ⟩ (𝑖𝑖) (𝑖𝑣)(𝑖𝑣) (𝑏) 𝜃 C |01⟩|10⟩𝜃 D 𝜓 𝑡 G(𝑖)(𝑖𝑖)(𝑖𝑖𝑖) 𝜌 𝑡 G (𝑐) 𝑡 FIG. 1. A two-qubit engine. (a) Scheme of the engine cycle. (i)
Starting from | (cid:105) , the qubits get entangled by coherently exchang-ing an excitation. (ii) A demon performs an energy measurementon qubit B at t = π/ Ω . (iii) Feedback. If B is found in the ex-cited state, a π pulse is applied to each qubit. The energy of B isextracted and A is re-excited. If not, nothing is done. At the end ofthis step, the qubits are back to their initial state. (iv) Reset of thedemon’s memory. (b) Representation of the qubits’ quantum state inthe Bloch sphere spanned by {| (cid:105) , | (cid:105)} . The eigenstates of H are denoted by | θ + (cid:105) and | θ − (cid:105) . At the end of (i) the qubit’s state is | ψ ( t ) (cid:105) . After an unselective measurement, the state is ρ ( t ) . (c)Evolution of (cid:104) H (cid:105) (dotted brown), (cid:104) H loc (cid:105) (dashed blue), and (cid:104) V (cid:105) (solid magenta) as a function of time (See text). the blue-detuned qubit B . Local energy measurements canthen project the excitation into B with a finite probability,resulting in some net energy gain [18]. We provide evidencethat this energy comes from the measurement channel, andcorresponds to the cost of erasing the quantum correlationsbetween the qubits. By exploiting the information carriedby the measurement, one may extract this energy as work, a r X i v : . [ qu a n t - ph ] J u l in a cycle similar to the classical Szilard engine [19] or itsquantum generalization [20]. We demonstrate that in thelimit of small detunings and large values of the couplingsbetween the qubits, work extraction is nearly deterministic.Based on this mechanism of entanglement followed by alocal measurement, we propose a protocol for frequencyup-conversion over an N -qubit chain. Finally, we investigatethe dynamics of the pre-measurement step, where one of thequbits is coupled with a quantum meter. Our analysis revealsa transfer of energy from the qubit-qubit correlations intothe qubits-meter correlations, providing new insights into thephysics of measurement-based engine fueling. An entangled-qubits engine —The basic mechanism of ourengine is illustrated in Figs. 1(a) and 1(b). It involves twoqubits A and B of respective transition frequencies ω A and ω B , whose evolution is ruled by the Hamiltonian H = (cid:88) i = A,B (cid:126) ω i σ † i σ i + (cid:126) g ( t )2 ( σ † A σ B + σ † B σ A ) . (1)We have introduced the lowering operator σ i = | i (cid:105) (cid:104) i | forthe qubit i ∈ { A, B } . The first term of H is the free Hamil-tonian of the qubits. It thus features “local” one-body termsthat we shall denote as H loc . The second term, which we de-note by V , couples the qubits, giving rise to entangled states.The coupling channel can be switched on and off, which ismodeled by the time-dependent coupling strength g ( t ) . In therest of the paper we consider a positive detuning δ = ω B − ω A .For simplicity, we denote the product states | x A (cid:105) ⊗ | y B (cid:105) as | xy (cid:105) , where x, y ∈ { , } .The engine cycle encompasses four steps: (i) Entanglingevolution . At time t = 0 , the qubits are prepared in the state | ψ (cid:105) = | (cid:105) of mean energy (cid:104) H (cid:105) = (cid:104) ψ | H | ψ (cid:105) = (cid:126) ω A .The coupling term is switched on with a strength g . Since | ψ (cid:105) is a product state, its mean energy does not change during thisswitching process, which is thus performed at no cost. Thequbits’ state then evolves into an entangled state | ψ ( t ) (cid:105) wherethe initial excitation gets periodically exchanged between thetwo qubits, with | ψ ( t ) (cid:105) =( c θ e i Ω t/ + s θ e − i Ω t/ ) | (cid:105)− c θ s θ ( e i Ω t/ − e − i Ω t/ ) | (cid:105) . (2)We have defined c θ = cos( θ/ , s θ = sin( θ/ , θ as tan( θ ) = g/δ , and Ω = (cid:112) g + δ the generalized Rabi fre-quency that characterizes the periodic energy exchange. (cid:104) H loc (cid:105) ( t ) and (cid:104) V (cid:105) ( t ) are plotted on Fig. 1(c). As expectedfrom a unitary evolution, their sum remains constant and equalto its initial value (cid:126) ω A . The periodic exchange of the singleexcitation between A and B gives rise to oscillations of thelocal energy component. This evolution is compensated bythe opposite oscillations of the coupling energy (cid:104) V (cid:105) ( t ) ≤ .This term appears here as a binding energy of purely quantumorigin, whose presence ensures that the total energy and thenumber of excitations are both conserved. (ii) Measurement . (cid:104) H loc (cid:105) and |(cid:104) V (cid:105) ( t ) | reach a maximumwhen t = π/ Ω where | ψ ( t ) (cid:105) = i [cos( θ ) | (cid:105) − sin( θ ) | (cid:105) ] .At this time, a local projective energy measurement is per-formed on qubit B , and its outcome is encoded in a classicalmemory M . Here we consider an instantaneous process, per-formed with a classical measuring device. A more elaboratemodel of the measurement will be presented in the last part ofthe paper. In turn, the average qubits’ state becomes a statisti-cal mixture ρ ( θ ) = cos ( θ ) | (cid:105) (cid:104) | + sin ( θ ) | (cid:105) (cid:104) | , eras-ing the quantum correlations between them and thus bringingthe binding energy (cid:104) V ( t ) (cid:105) to zero. The average energy inputby the measurement channel is E meas = −(cid:104) V ( t ) (cid:105) = ∆ (cid:104) H (cid:105) = (cid:126) δ sin ( θ ) ≥ , (3)where ∆ (cid:104)·(cid:105) features the change of mean energy. Conversely,the von Neumann entropy of the qubit pair increases by anamount S meas = − Tr [ ρ ( θ ) log ( ρ ( θ ))] , that reads S meas = − cos ( θ ) log [cos ( θ )] − sin ( θ ) log [sin ( θ )] . (4)We use log , such that all entropies are expressed in bits. Theratio T meas = E meas /S meas characterizes the measurementprocess from a thermodynamic standpoint. For a fixed detun-ing δ , it diverges for large coupling where θ → π/ . It typi-cally scales like T meas ∼ − (cid:126) δ/ [2( π/ − θ ) log ( π/ − θ )] .In this limit of large coupling and small detuning, quantummeasurement can input a finite amount of energy with vanish-ing entropy. This contrasts with isothermal processes, whereenergy and entropy inputs are related by the bath temperature.From an informational standpoint, the measurement cre-ates classical correlations between the memory and the qubitsstates in the basis | (cid:105) , | (cid:105) . If the measurement is ideal, thesecorrelations are perfect, such that the entropies of the qubitsand the memory at the end of the process are equal. Theyare also equal to the mutual information they share, furtherdenoted I meas ( S : M ) .(iii) Feedback . The information stored in the memory isnow processed to extract the energy input by the measure-ment. To do so, the coupling term is switched off at time t . Since the correlations between the qubits have beenerased by the measurement, the switching-off can be imple-mented at no energetic cost. If the excitation is measured in B , which happens with probability P succ ( θ ) = sin ( θ ) , both A and B undergo a resonant π pulse, such that B emits a pho-ton while A absorbs one. The work W = (cid:126) δ is extractedand the qubits are reset to their initial state | (cid:105) . Converselyif the excitation is measured in A , no pulse is implementedand the cycle restarts. Eventually, the mean work extracted is W = E meas . At the end of this feedback step, the qubits’ en-tropy vanishes, and a maximal amount of mutual information | ∆ I ( S : M ) | = I meas ( S : M ) is consumed.(iv) Erasure.
Immediately after the feedback, the memory’sentropy still equals S meas = I meas ( S : M ) . The memory isfinally erased in a cold bath, the minimal work cost of thisoperation being proportional to S meas [21]. 𝐸 " % ( ℏ 𝜔 * ) * 𝑔 = 0.1𝜔 * 𝑔 = 0.05𝜔 * (𝑎) (𝑏) 𝑆 " % ( b i t ) 𝑔 = 0.2𝜔 * 𝑔 = 0.1𝜔 * | Δ 𝐼 𝑆 : 𝑀 | ( b i t ) ℏ𝛿, 𝑔 ≫ 𝛿 𝛿/𝜔 * 𝛿/𝑔 (𝑐) T " % = 𝐸 " % / 𝑆 " %
20 0 0.150.10.05 0.2 0.25𝛿 = 10 GH 𝜔 * 𝛿 = 10 GI 𝜔 * 𝛿 = 10 GJ 𝜔 * 𝛿/𝜔 * 𝜂 = W/E " (𝑑) 𝑔/𝜔 * FIG. 2. Measurement energy vs information as fuel. (a) Energy E meas and (b) entropy S meas inputs as a function of the detuning δ , for var-ious coupling strengths g . (c) Work extraction ratio η = W/E meas (color scale) as a function of δ/g and consumed mutual information | ∆ I ( S : M ) | . The black region corresponds to η = 0 . d) Yield ofinformation to work conversion T meas as a function of g for various δ . Since the whole cycle conserves the number of excitations,the states | (cid:105) and | (cid:105) of the two qubits feature an effectivetwo-level system. This property allows us to picture the qubitsdynamics in the Bloch sphere representation (Fig. 1(b)) wherethe cyclic nature of the evolution becomes evident.The quantum engine described above extends the conceptof measurement-fueled engines, originally proposed forsingle parties as working substances [4–8], to entangledsystems. In those proposals the engine is fueled by quantummeasurement back-action, which can only take place whenthe measured system state bears coherences in the basis ofthe measured observable. Both quantum measurement andcoherence thus contribute to the fueling process. Similarly,in the present bipartite engine, both local measurements andentanglement are necessary for work extraction. Measurement energy vs information as fuel —The engineproposed above exploits two complementary features of quan-tum measurements: on the one hand, they bring energy andentropy, on the other hand, they extract information that canbe further used to convert the energy input into work. Nowfocusing on the measurement and feedback steps, we analyzethese energetic and informational resources, and how they re-spectively impact the performance of the bipartite engine.The mean energy E meas and entropy S meas input by the mea-surement process are plotted in Figs. 2(a) and 2(b) as a func-tion of the detuning δ , for various coupling strengths g . Asindicated in the figure, they are both maximized for δ = g .This also corresponds to a maximal occupation of the mem-ory and mutual information after the measurement step.Converting the measurement energy into work requires the processing of this information during the feedback step. Theconversion is optimal ( W = E meas ) when all information isconsumed, which corresponds to the ideal cycle presentedabove. Non-optimal work extraction results from an incom-plete consumption of information, | ∆ I ( S : M ) | < I meas ( S : M ) , yielding a conversion ratio η = W/E meas < . Wehave modeled such an imperfect feedback in the Suppl. [22].Figure 2(c) features η as a function of δ/g and ∆ I ( S : M ) ,clearly showing the work value of information—the larger theconsumed information, the larger the conversion ratio. Inter-estingly, the figure reveals that work can be extracted even if ∆ I ( S : M ) = 0 . This is the case when P succ ( θ ) > / ,which happens when δ/g < . Then the π -pulses can beblindly applied, still leading to a net work extraction W = (cid:126) δ [sin ( θ ) − cos ( θ )] . This mechanism solely exploits theenergy input by the measurement, but not the extracted infor-mation; it is at play, e.g. in single temperature engines [6, 7].By contrast, information processing is necessary when δ ≥ g .Note that in all non-ideal cases where information is not fullyconsumed, an additional step must be included in the cycle, toreset the qubits’ state.From now on we suppose the feedback to be perfect,such that the information available in the memory is fullyconsumed and all the energy input by the measurementchannel is converted into work. In this situation, the net workextracted is W = E meas . It is thus related to the size of thememory used S meas by the effective parameter T meas definedabove. Interestingly, now T meas is a measure of efficiency ofinformation-to-work conversion. Such efficiency is usuallybounded by the bath temperature in Maxwell’s demons, thatare fueled by a thermal bath [1, 23]. T meas is plotted onFig. 2(d) as a function of g for various values of the detuning δ . As it appears on the figure, it is not bounded and increasesas a function of g . This reveals that in the limit g (cid:29) δ ,a finite amount of work can be extracted by processing avanishingly small amount of information. This effect issimilar to the Zeno regime identified in Ref. [4], where workextraction relies on measurements whose outcomes are nearlydeterministic. Up-conversion –We now propose to exploit this mechanismto implement energy up-conversion. The protocol is basedon the efficient transfer of a single excitation through achain of N qubits of increasing frequency as depictedin Fig. 3(a). We denote the frequency of the qubit i by ω i = ω A + ( i − δ/ ( N − , with i ∈ { , , ..., N } . Asabove, δ = ω B − ω A , such that the frequency of qubit N is ω B and ω = ω A . At time t = 0 , the qubit 1 is excitedand the coupling g between qubit 1 and qubit 2 is switchedon, its Rabi frequency being Ω N = (cid:112) g + ( δ/ ( N − .At time t N = π/ Ω N , the energy of qubit 2 is mea-sured. The process stops if it is found in the groundstate, which happens with probability cos ( θ N ) , where tan( θ N ) = ( N − g/δ = ( N −
1) tan( θ ) . If the excitationis successfully transferred to qubit 2, the coupling between and is switched off and the coupling between and is P $ % && ’ N=4 qubitsN=10 qubitsN=100 qubitsN=2 qubits (𝑎) ℏ𝝎 𝑩 (𝑏) P $ % && ’ 𝑔/𝛿 𝑁 𝑔/𝛿=0.1𝑔/𝛿=0.2𝑔/𝛿=0.3𝑔/𝛿=1 (𝑐) 𝜔 E = 𝜔 F 𝜔 G 𝑔 𝜔 ’HF 𝜔 ’ = 𝜔 I G ’HF ’HF ℏ𝝎 𝑨 G FIG. 3. Entanglement and measurement based up-conversion mech-anism. (a) Scheme of the frequency up-converter (See text). Proba-bility of transfer P N succ as a function of g/δ for various N (b) and asa function of N for various g/δ (c). The grey lines indicate constantvalues as guides to the eye. switched on. The same process is repeated between qubits k and k + 1 until the excitation gets detected in qubit N ,which happens with probability P N succ = sin N − ( θ N ) . P N succ is plotted in Fig. 3(b) and 3(c) as a function of g/δ and N . For fixed values of g and δ , it is clearly advantageous toincrease the number of intermediate qubits. The mechanismat play is reminiscent of the quantum Zeno effect. An analyticdemonstration is presented in the Suppl. [22]. Origin of the measurement fuel —We finally investigate themeasurement-based fueling mechanism, based on the mod-eling of the “pre-measurement process” by which the qubitsare entangled with a quantum meter while still coupled. It iswell-known that such entanglement accounts for the entropyincrease of the measured system. Below we show that it alsoexplains the measurement energy input.The measurement process takes place between t = t and t m , and is depicted in Fig. 4(a). The meter is chosen to be athird qubit m with degenerate energy levels | m (cid:105) and | m (cid:105) . Itis coupled to the qubit B through the Hamiltonian: V m = (cid:126) χ ( t ) σ † B σ B ⊗ σ mx . (5) χ ( t ) is the measurement strength, with χ ( t ) = χ for t =[ t , t m ] and otherwise. We choose χ (cid:29) g , to ensure thereadout takes place on small time-scales with respect to theRabi period. This defines the parameter (cid:15) = g/χ , which issmall but finite since the measurement is implemented on still-interacting qubits.At t − , the meter m is prepared in | m (cid:105) , while A and B are in the entangled state | ψ ( t ) (cid:105) , such that their joint statereads | Ψ( t ) (cid:105) = i (cos( θ ) | m (cid:105) − sin( θ ) | m (cid:105) ) . Since (cid:104) V m ( t ) (cid:105) = 0 , the measurement channel is switched on atno energy cost. The joint qubits-meter system then evolves under the total Hamiltonian H = H (0) + H (1) , where H (0) = H loc + V m (resp. H (1) = V ) rules the evolutionat zeroth order (resp. at first order) in the small parame-ter (cid:15) . The evolution equations are solved at first order inthe Suppl. [22], yielding (cid:12)(cid:12) Ψ (1) ( t ) (cid:11) = (cid:12)(cid:12) Ψ (0) ( t ) (cid:11) + | δ Ψ( t ) (cid:105) where | δ Ψ( t ) (cid:105) is of order (cid:15) . The populations up to first orderare plotted on Fig. 4(b). To lowest order in (cid:15) , the measure-ment is quantum non-demolition, resulting in state (cid:12)(cid:12) Ψ (0) ( t ) (cid:11) [24, 25]. The readout is complete at time t m = t + π/χ where (cid:12)(cid:12) Ψ (0) ( t m ) (cid:11) = i [cos( θ ) | m (cid:105) − sin( θ ) | m (cid:105) ] . Conversely,the first order correction | δ Ψ( t ) (cid:105) accounts for the remainingcoupling between the qubits during the measurement. 𝜒(𝑡) |0 ’ ⟩|1 ’ ⟩ 𝑔(𝑡) 𝜔 , 𝜔 - (𝑎) - ℏ𝜔 , t ⟨𝑉⟩⟨𝑉 ⟩⟨𝐻 ⟩⟨𝐻 ⟩ t ; t (𝑐)(𝑏) D;;’ (𝑡) t t ; 𝑝 ;D;’ (𝑡)𝑝 ;DD’ (𝑡) 𝑝 D;D’ (𝑡)
FIG. 4. Dynamics of measurement-induced energy transfer. (a)Local quantum measurement of qubit B allows for the creationof correlations between the meter m and the AB system and de-stroys correlations between the qubits. (b) Full state decomposi-tion in the {| m (cid:105) , | m (cid:105) , | m (cid:105) , | m (cid:105)} basis during the pre-measurement step. (c) Expectation values of (cid:104) H (cid:105) , (cid:104) H loc (cid:105) , (cid:104) V m (cid:105) ,and (cid:104) V (cid:105) as a function of the pre-measurement time t ∈ [ t , t m ] . Thecurves in the figure are calculated for χ = 10Ω and g = δ . The greylines indicate constant values as guides to the eye. We now focus on energy flow and study the evolution of (cid:104) H loc (cid:105) , (cid:104) V (cid:105) and (cid:104) V m (cid:105) , see Fig. 4(c). Since the process isunitary, these three components sum up to (cid:126) ω A . Perturbativecalculations show that (cid:104) H loc (cid:105) (resp. (cid:104) V (cid:105) and (cid:104) V m (cid:105) ) remainconstant up to first order in (cid:15) (resp. at zeroth order) [22].The first order contribution of the binding energy between A and B reads (cid:104) V (1) (cid:105) = (cid:10) Ψ (0) ( t ) (cid:12)(cid:12) V (cid:12)(cid:12) Ψ (0) ( t ) (cid:11) , and thusscales like the coherences of the AB density matrix in the | (cid:105) , | (cid:105) basis. Its absolute value decreases together withthe quantum correlations between A and B , and vanisheswhen the readout is complete. This evolution is compensatedby an equivalent decrease of (cid:104) V (1) m (cid:105) ( t ) , yielding at time t m : (cid:104) V (1) m ( t m ) (cid:105) = − E meas . Importantly, since V m scales as χ , (cid:104) V (1) m ( t m ) (cid:105) remains finite and of order g even if g/χ (cid:28) .This calculation reveals the direction of the energy flowduring the measurement process: The binding energy initiallylocalized between the qubits is transferred between the qubitsand the meter. This energy flow follows the same dynamicsas the decoherence in the local energy basis, and can beseen as its energetic counterpart. Finally, when the readoutis complete, the qubits-meter coupling must be switchedoff before any further operation can be done on the qubits.This switching off increases the qubits-meter energy by anamount (cid:104)− V m ( t m ) (cid:105) = E meas . In the present non-autonomousscheme, this corresponds to the work cost paid to operate themeasurement channel. Outlook —Our findings advance quantum measurement en-gines to encompass quantum entanglement and energy corre-lations, showing how entanglement engines may be poweredby quantum measurement. From a conceptual standpoint, theyshed new light on the measurement-based fueling process, andprovide a unified view on former analyses based on analogieswith work and heat exchanges. It should be recalled how-ever that the concepts of work and heat were historically de-fined with respect to thermal noise and resources. Our re-sults, on the other hand, are solely based on a stochasticityof quantum nature [26]. They contribute to the emergence ofa new framework—“Quantum energetics”—where thermody-namic concepts will be relevant in the presence of any kindof noise, especially at zero temperature where most quantumtechnology tasks are envisioned [27].In the future, it will be interesting to study the autonomousregimes of our engine where measurement and dissipationbecome time-independent processes, leading to the design ofengines exploiting decoherence as a resource. This wouldbridge the gap with the field of dissipation engineering[28, 29], where dissipation is harnessed to produce nontrivialquantum states and desirable quantum dynamics. Suchreservoir engineering has been recently employed in thecircuit-QED architecture [30–33]—the same experimentalplatform on which we expect to realize our proposed engine.We warmly thank M. Richard and C. Branciard for enlight-ening discussions. AA acknowledges the Agence Nationalede la Recherche under the Research Collaborative Project“Qu-DICE” (ANR-PRC-CES47). P.A.C. acknowledges Tem-pleton World Charity Foundation, Inc. which supported thiswork through the grant TWCF0338. K.M. acknowledges sup-port from NSF No. PHY-1752844 (CAREER) and the Re-search Corporation for Science Advancement. [1] J. Parrondo, J. Horowitz, and T. Sagawa, Nature Phys , 131(2015).[2] T. Sagawa and M. Ueda, Phys. Rev. Lett. , 250602 (2009).[3] K. Jacobs, Phys. Rev. E , 040106(R) (2012).[4] C. Elouard, D. Herrera-Mart´ı, B. Huard, and A. Auff`eves,Phys. Rev. Lett , 260603 (2017).[5] C. Elouard and A. N. Jordan, Phys. Rev. Lett. , 260601(2018).[6] J. Yi, P. Talkner, and Y. W. Kim, Phys. Rev. E , 022108 (2017).[7] X. Ding, J. Yi, Y. W. Kim, and P. Talkner, Phys. Rev. E ,042122 (2018).[8] A. Jordan, C. Elouard, and A. Auff`eves, Quantum Stud.: Math.Found. , 203 (2020).[9] M. H. Mohammady and J. Anders, New J. Phys. , 113026(2017).[10] M. Campisi, J. Pekola, and R. Fazio, New J. Phys. , 053027(2017).[11] L. Buffoni, A. Solfanelli, P. Verrucchi, A. Cuccoli, andM. Campisi, Phys. Rev. Lett , 070603 (2019).[12] E. Schr¨odinger, Proceedings of the Cambridge PhilosophicalSociety , 555?563 (1935).[13] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. , 777(1935).[14] A. Tavakoli, G. Haack, N. Brunner, and J. B. Brask, Phys. Rev.A , 012315 (2020).[15] M. Josefsson and M. Leijnse, Phys. Rev. B , 081408 (2020).[16] X. L. Huang, H. Xu, X. Y. Niu, and Y. D. Fu, Phys. Scr. ,065008 (2013).[17] A. Hewgill, A. Ferraro, and G. De Chiara, Phys. Rev. A ,042102 (2018).[18] A. N. Jordan and M. B¨uttiker, Phys. Rev. Lett. , 247901(2004).[19] L. Szilard, Zeitschrift f¨ur Physik , 840 (1929), [Behav Sci. ,301 (1964)].[20] S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda, Phys.Rev. Lett. , 070401 (2011).[21] R. Landauer, IBM J. Res. Dev. , 183 (1961).[22] See Supplemental Material at [URL will be inserted by pub-lisher].[23] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi, S. Kono,Y. Tabuchi, R. Yamazaki, M. Ueda, and Y. Nakamura, Nat.Comm. , 1291 (2018).[24] P. Grangier, J. A. Levenson, and J.-P. Poizat, Nature , 537(1998).[25] Y. Guryanova, N. Friis, and M. Huber, Quantum , 222 (2020).[26] P. Grangier and A. Auff`eves, Phil. Trans. R. Soc. A ,20170322 (2018).[27] A. M. Timpanaro, J. P. Santos, and G. T. Landi, Phys. Rev. Lett. , 240601 (2020).[28] J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. , 4728(1996).[29] E. Kapit, Quantum Science and Technology , 033002 (2017).[30] Y. Liu, S. Shankar, N. Ofek, M. Hatridge, A. Narla, K. M.Sliwa, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Phys.Rev. X , 011022 (2016).[31] Y. Lu, S. Chakram, N. Leung, N. Earnest, R. K. Naik, Z. Huang,P. Groszkowski, E. Kapit, J. Koch, and D. I. Schuster, Phys.Rev. Lett. , 150502 (2017).[32] S. Touzard, A. Grimm, Z. Leghtas, S. O. Mundhada, P. Rein-hold, C. Axline, M. Reagor, K. Chou, J. Blumoff, K. M. Sliwa,S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, andM. H. Devoret, Phys. Rev. X , 021005 (2018).[33] R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Simon, andD. I. Schuster, Nature566