Information gain and loss for a quantum Maxwell's demon
IInformation gain and loss for a quantum Maxwell’s demon
M. Naghiloo, J. J. Alonso, A. Romito, E. Lutz, and K. W. Murch
1, 4 Department of Physics, Washington University, St. Louis, Missouri 63130 Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Institute for Materials Science and Engineering, St. Louis, Missouri 63130
We use continuous weak measurements of a driven superconducting qubit to experimentally studythe information dynamics of a quantum Maxwell’s demon. We show how information gained by ademon who can track single quantum trajectories of the qubit can be converted into work usingquantum coherent feedback. We verify the validity of a quantum fluctuation theorem with feedbackby utilizing information obtained along single trajectories. We demonstrate, in particular, thatquantum backaction can lead to a loss of information in imperfect measurements. We furthermoreprobe the transition between information gain and loss by varying the initial purity of the qubit.
The thought experiment of Maxwell’s demon revealsthe profound connection between information and energyin thermodynamics [1–5]. By knowing the positions andvelocities of each molecule in a gas, the demon can sorthot and cold particles without performing any work, inapparent violation of the second law. Thermodynam-ics must therefore be generalized to incorporate infor-mation in a consistent manner. Classical Maxwell’s de-mon experiments have been realized with cold atoms [6],a molecular ratchet [7], colloidal particles [8, 9], singleelectrons [10, 11] and photons [12]. Recent advances infabrication and control of small systems where quantumfluctuations are dominant over thermal fluctuations allowfor novel studies of quantum thermodynamics [13–19]. Inparticular, Maxwell’s demon has been realized in severalsystems using feedback control to study the role of infor-mation in the quantum regime [20–23]. While these ex-periments probe information and energy dynamics in theregime of single energy quanta, the dynamics either doesnot include quantum coherence or the demon destroysthese coherences through projective measurements [20–23]. Therefore, in either case, the action of the demon canbe understood using entirely classical information. How-ever, in quantum systems the information exchanged in ameasurement may present strikingly nonclassical featuresowing to the measurement backaction [24–27].In this work, we use continuous weak measurementsfollowed by feedback control of a superconducting qubit[28–31] to realize Maxwell’s demon in a truly quantumsituation, where quantum backaction and quantum co-herence contribute to the dynamics. This approach en-ables us to experimentally verify a quantum fluctuationtheorem with feedback [27, 32, 33] at the level of sin-gle quantum trajectories. This fluctuation theorem is anonequilibrium extension of the second law that accountsfor both quantum fluctuations and the information col-lected by the demon. At the same time, this methodallows us to study the role of quantum backaction andquantum coherence in the acquired information. In par-ticular, we show that the average information exchangedwith the detector can be negative due to measurement (c) I Q I Q ω q σ z ω c a a χσ z a a i Ω R σ y P P (a)(b) QubitCavityDrive P P ρ= P P cc* ρ= Detection01 /2/2 M ahd i , H i da j Figure 1: Classical demon vs quantum demon. (a) In theclassical situation the dynamics can be seen as an evolutionof the populations in the definite eigenstates, yet in the quan-tum case (b), the dynamics includes coherences and can nolonger be understood as a classical mixture. (c) The experi-mental configuration consists of a quantum two-level systemcoupled to a cavity mode via a dispersive interaction whichallows for both weak and strong measurements of the qubitstate populations. A resonant drive at the qubit transitionfrequency turns these populations into coherences and viceversa leading to coherent quantum evolution. backaction. Here the loss of information associated withthe perturbing effect of the detector dominates the mea-surement process. By preparing the qubit at differenttemperatures according to a Gibbs distribution, we ex-perimentally map out the full transition between regimesof information gain and information loss [27].
Superconducting circuit —In order to study the infor-mation exchanged with the detector in a genuinely quan-tum situation, we employ quantum measurement tech-niques in a superconducting circuit to realize a quan-tum Maxwell’s demon (Fig. 1a,b). Our setup consistsof a transmon qubit dispersively coupled to a 3D alu-minum cavity by which we readout the state of the qubitwith dispersive measurement using Josephson parametricamplifier operating in phase sensitive mode with a totalmeasurement quantum efficiency of 30% (Fig. 1c). Thecorresponding effective Hamiltonian in the presence of acoherent drive is, H = − ω q σ z − i Ω R σ y cos( ω q t ) − χa † aσ z + ω c a † a, (1) a r X i v : . [ qu a n t - ph ] F e b where σ z , σ y are Pauli matrices, ω q the resonance fre-quency of the qubit, ω c is the cavity frequency, a † , a are creation and annihilation operators, and Ω R is theRabi drive frequency [34]. The quantity χ is the dis-persive coupling rate between the cavity mode and thequbit state. In this measurement architecture [28–31],the qubit-state-dependent phase shift of a weak cavityprobe tone is continuously monitored, resulting in a mea-surement record, r , that is proportional to (cid:104) σ z (cid:105) . Usingthe record r we obtain the qubit conditional state evolu-tion ρ t | r , which depends on the record from time to t ,using the stochastic master equation (SME) [35, 36], ˙ ρ t | r = 1 i (cid:126) [ H R , ρ t | r ] + k ( σ z ρ t | r σ z − ρ t | r )+ 2 ηk ( σ z ρ t | r + ρ t | r σ z − σ z ρ t | r ) ρ t | r ) r ( t ) , (2)where k is the strength of the measurement, η is the ef-ficiency of the detector, and H R = − i Ω R / σ y describesthe qubit drive in the rotating frame [34]. The first twoterms correspond to the standard Lindblad master equa-tion that accounts for unitary evolution and dephasingof the qubit by the dispersive measurement. The thirdterm describes the state update due to the measurementrecord which includes the stochastic measurement signal r ( t ) ∝ (cid:104) σ z (cid:105) ( t ) + dW t , with dW t a zero-mean Gaussiandistributed Wiener increment [37]. Owing to the weakcoupling to the measuring device, information about thestate of the qubit may be gathered without projecting itinto energy eigenstates, thus preserving coherent super-positions. The demon’s information —The SME (2) tracks thestate of knowledge about the qubit obtained by the quan-tum demon. The amount of information exchanged withthe detector depends on both the measurement outcomeand the state of the system. It may be quantified as [27], I ( ρ t | r , r ) = ln P z (cid:48) ( ρ t | r ) − ln P z ( ρ ) , (3)where P z (cid:48) represents the probability of getting the result z (cid:48) = 0 , in the z (cid:48) -basis where the system is diagonal.The stochastic evolution of the information (3) along aquantum trajectory follows as, ˜ I r = (cid:88) z,z (cid:48) = ± [ P z (cid:48) ( ρ t | r ) ln P z (cid:48) ( ρ t | r ) − P z ( ρ ) ln P z ( ρ )]= S ( ρ ) − S ( ρ t | r ) , (4)with the von Neumann entropy S ( ρ ) = − Tr[ ρ ln ρ ] . InEq. (4) the conditional probabilities P z (cid:48) ( ρ t | r ) come fromthe SME corresponding to a single run of the experiment,that is, an individual quantum trajectory. The averagedvalue of the exchanged information is obtained by aver-aging over many trajectories, (cid:104) I (cid:105) = (cid:88) r p ( r ) ˜ I r = S ( ρ ) − (cid:88) r p ( r ) S ( ρ t | r ) , (5) where p ( r ) is the probability density of the measurementrecord r . Equation (5) is the information about the stateof the system gathered by the quantum demon [24, 26,38]. Remarkably, it may positive or negative.For classical measurements, e.g. when the measure-ment operator commutes with the state, Eq. (5) reducesto the classical mutual information which is always pos-itive [39]. By contrast, quantum measurements perturbthe state in addition to acquiring information. Due tothis unavoidable quantum backaction, the uncertainty inthe detector state can be transferred to the system andincrease its entropy. While Eq. (5) is positive for effi-cient measurements, it may become negative for ineffi-cient measurements [24, 26, 38]. Quite generally, mod-elling the detector uncertainty as an average over inac-cessible degrees of freedom, parametrized by a stochasticvariable a , the exchanged information (5) may be writtenas a sum of information gain and information loss, (cid:104) I (cid:105) = I gain − I loss , with I gain = S ( ρ ) − (cid:80) a p ( a, r ) S ( ρ t | r,a ) (cid:62) and I loss = (cid:80) r S ( ρ t | r ) − (cid:80) a p ( a, r ) S ( ρ t | r,a ) (cid:62) [27]. Ex-pression (5) is hence negative whenever the informationloss induced by the quantum backaction is larger thanthe information acquired through the measurement. Feedback protocol —We now turn to the experimentalprocedure and our feedback protocol. The experimentconsists of five steps as depicted in Figure 2. In Step 1,the qubit is initialized in a given thermal state charac-terized by an inverse temperature β . Experimentally, wehave control over β by applying a short excitation pulseto the qubit at the start of the experimental sequence.In Step 2, we perform a first projective measurement,which, when combined with a second projective mea-surement during the final step, will quantify the energychange during the whole protocol [40]. In Step 3, we em-ploy a continuous resonant drive at the qubit transitionfrequency to induce Rabi oscillations of the qubit state.In conjunction with the drive, we continuously probe thecavity with a measurement rate k/ π = 51 kHz generat-ing a measurement record r for the demon to track theevolution according to the SME (2). The axis of the reso-nant drive and measurement basis constrict the evolutionof the qubit to the X – Z plane of the Bloch sphere. Atypical qubit evolution is depicted by solid lines in theStep 3-inset of Fig. 2. The quantum trajectory is val-idated with quantum state tomography as indicated bythe dashed lines [30, 31]. At the final time τ , the de-mon uses the knowledge about the state of the systemto perform a rotation in Step 4 to bring the qubit backto the ground state, and extract work. To implementthe feedback, we perform a random rotation pulse in therange of [0 , π ] and select the correct rotations (withinthe error of ± π/ ) in a post-processing step. This ap-proach avoids long loop delays that occur for realtimefeedback. We eventually finish the experiment with thesecond projective measurement in Step 5. We note thatthe measurement basis ( σ z ) is the same in Steps 2 and 5. ρ = Σ e - β E i i i=0,1 Projectivemeasurement “X” Weak measurement Feedbackrotation Projectivemeasurement “Z”
Step 1: Step 2: Step 3: Step 4: Step 5: -1.0-0.50.00.51.0 2.01.51.00.50.0 Time ( µ s) z tom zx tom x Thermal equilibrium
I W
01 01 01 01 Z i Figure 2: Experimental sequence. Step 1: The qubit is initialized in a thermal state at inverse temperature β . Step 2: As thefirst step in determining the energy change through projective measurements, we perform a projective measurement (labeled“X”) in the energy basis. Step 3: We drive the qubit with a coherent drive characterized by Ω R / π = 0 . MHz while thequantum demon monitors the qubit evolution with a near-quantum-limited detector. The demon’s knowledge about the statecan be expressed in terms of the expectation values x ≡ (cid:104) σ x (cid:105) and z ≡ (cid:104) σ z (cid:105) (solid lines) with the corresponding tomographicvalidation showing that the demon’s expectation values are verified with quantum state tomography [30, 31]. Step 4: Thequantum demon uses the acquired information from the previous step to apply a rotation to bring the qubit to the groundstate. Step 5: We perform a projective measurement labeled “Z” as the second step in a two-point energy measurement. We evaluate averaged quantities by repeating the exper-iment many times.
Experimental results —We begin by experimentallyverifying a quantum fluctuation theorem with feedbackin the form of a generalized quantum Jarzynski equal-ity, (cid:104) exp [ − β ( W − ∆ F ) − I ] (cid:105) = 1 , where W is the workdone on the system by the external driving, ∆ F the equi-librium free energy difference between final and initialstates, and I the information (3). This fluctuation the-orem generalizes the second law to account for quantumfluctuations and information exchange. It has been de-rived for classical systems in Ref. [32] and experimentallyinvestigated in Refs. [8, 11]. It has later been extendedto quantum systems in Refs. [27, 33] and recently experi-mentally studied with a two-level system whose dynamicsis that of a classical (incoherent) mixture [23].In order to test the quantum fluctuation theorem forthe considered two-level system, we write it explicitly as, (cid:104) e − βW − I (cid:105) = P (0) P ( τ ) e − I + P (0) P ( τ ) e − I + P (0) P ( τ ) e + β − I + P (0) P ( τ ) e − β − I = 1 , (6)with ∆ F = 0 since the initial and final Hamiltoniansare here the same. The initial occupation probabilitiesfor ground and excited states are respectively given by P (0) = 1 / (1 + e − β ) and P (0) = e − β / (1 + e − β ) , corre-sponding to the initial thermal distribution. Note thatwe work in units where (cid:126) ω q = 1 . We determine the tran-sition probabilities P ij ( τ ) , i, j = { , } , from the resultsof the projective measurements performed in Steps 2 and5, following the two-point measurement scheme [40], asillustrated in Fig. 3a. We further evaluate the informa-tion term I ij = ln P i ( ρ τ | r ) − ln P j ( ρ ) from the recordedquantum trajectory according to Eq. (3) (Fig. 3b). InFig. 3c (round markers), we show the experimental resultfor Eq. (6) for β = 4 for five different protocol durations τ . We observe that the generalized quantum Jarzynskiequality with feedback is satisfied. However, the fluctu-ation theorem is violated (square markers), as expected,when the information exchange is not taken into account.Every measured trajectory contains a complete set ofinformation by which the expectation value of any (rele-vant) operator can be calculated. In particular, the tran-sition probabilities P ij ( τ ) may be determined directlyfrom the weak measurement data, instead of the out-comes of the two projective measurements. To establishthe consistency between the two approaches, we rewritethe quantum fluctuation theorems with feedback (6) as, (cid:104) e − βW − I (cid:105) = P (0) P ( τ ) e − I + P (0) P ( τ ) e − I + P (0) P ( τ ) e + β − I + P (0) P ( τ ) e − β − I = 1 , (7)with the respective final ground and excited states pop-ulations P ( τ ) = P ( ρ τ | r ) = (1 + z ( τ )) / and P ( τ ) = P ( ρ τ | r ) = (1 − z ( τ )) / , along single quantum trajecto-ries. All these quantities are obtained from the quantumstate tracking in Step 3 and the consequent feedback ro-tation in Step 4. Figure 3b (triangular markers) showsthat Eq. (7) is verified in our experiment and that thetwo approaches are thus indeed consistent.Figure 3d shows the evolution of the information ˜ I r along single quantum trajectories calculated fromEq. (4). The probabilities in Eq. (4) are evaluated ateach time step in the diagonal basis z (cid:48) , as illustrated inFig. 3b [27]. Figure 3e further exhibits the last point of400 trajectories before (after) feedback rotation in red(blue). The red (green) circles indicate the expectation (cid:104) z (cid:105) before (after) the feedback rotation calculated usingweak measurements. On the other hand, the black crossrepresents an independent evaluation of (cid:104) z (cid:105) from the sec-ond projective measurement data. The good agreementbetween the green circle and the black cross validates XZXZ z’ (c)(d) -0.6-0.4-0.20.0 2.01.51.00.50.0 I r Time ( μ s) (a)(b) P P P P P P (e) e - β W e - β W-I I Time ( μ s) ~ Figure 3: Experimental test of the quantum fluctuation the-orem. (a) Transition probabilities are calculated using thetwo projective measurements in Steps 2 and 5. (b) By track-ing a single quantum trajectory we calculate the informationchange (4). The diagonal basis z (cid:48) at time t is indicated. (c)Combining the transition probabilities and the informationchange we verify the quantum fluctuation theorem (6) (roundmarkers), however if the information is ignored ( I = 0 ) thefluctuation theorem is not valid (square markers). The fluc-tuation theorem is also verified at the level of single quantumtrajectories where the projective measurements are not usedto determine transition probabilities (triangle markers). (d)Information change, I , Eq. (3), along single quantum trajec-tories; the dotted curve shows a typical information changeobtained by Eq. (4). The background color shows the distri-bution of information change for many trajectories. The solidcurve is the average information change, (cid:104) I (cid:105) , Eq. (5). Twodashed line shows the Shannon entropy of initial state H (0) (coarse dash) and H (0) − ln(2) (fine dash) which indicate themaximum and minimum limit for information change for agiven initial state. (e) Red (Blue) dots show the qubit statedistribution obtained by trajectories at t = 2 µ s right before(after) the feedback rotation and the average of this distribu-tion indicated by the red (green) circle. The cross indicatesthe reconstruction of the state of the qubit after the feedbackusing projective measurements. that feedback rotations are properly executed.We next study the information dynamics of the meaninformation (cid:104) I (cid:105) , Eq. (5), averaged over many trajectories,and the transition from information gain to informationloss. In Fig. 3d (solid curve), we observe that the meaninformation (cid:104) I (cid:105) for β = 4 averaged over 400 trajectoriesis negative. This negativity of (cid:104) I (cid:105) is a consequence of (a) (b) z I QI Q I XZ XZ I classical I quantum Figure 4: Transition from information gain to informationloss. (a) Classical and quantum noise corresponding to the un-known detector configuration randomly shifts the probe tone(shown here as a coherent state in the quadrature ( I – Q ) spaceof the electromagnetic field) and degrades the total efficiencyof the measurement. The inefficient unraveling of the SME isthe statistical average of possible unravelings correspondingto different detector configurations. (b) Transition from in-formation gain to information loss by changing the purity ofthe initial state of the system controlled by its temperaturevia z = tanh( β/ . The dashed line indicates a linear de-pendence of the information exchange on initial state purityand the solid line indicates the Shannon entropy of the initialstate which is obtainable only with projective measurementscorresponding to the Lanford-Robinson bound [41]. both the quantumness of the dynamics, which generatesstates with coherent superpositions of the eigenstates ofthe measured observable, and of the quantum backactionof the measurement [26]. For a classical measurementfor which the density matrix commutes with the mea-surement operator, (cid:104) I (cid:105) reduces to the (positive) mutualinformation between the measurement result and the en-semble made up of the eigenstates of the density matrix.On the other hand, the quantum backaction of the inef-ficient measurement disturbs the state of the system andreduces our knowledge about it. When this informationloss is larger than the information gained through themeasurement, the total information exchanged is nega-tive, as seen in the experiment.In our setup, the finite efficiency of the measurement,as shown in Fig. 4a, follows from the fact that the de-tector signal is affected by classical and quantum noisewhich induces random shifts of the readout value. Theresulting uncertainty about the state of the detector de-termines the information loss [27]. Meanwhile, the infor-mation gain may be controlled by the purity of the ini-tial state, that is, by its temperature. If we parametrizethe initial thermal state as ρ = ( + z σ z ) /2, with z = (cid:104) z (cid:105)| t =0 = tanh β/ , the information gain, and inturn the averaged information (cid:104) I (cid:105) , is a monotonically de-creasing function of z , as seen in Fig. 4b. The transitionto (cid:104) I (cid:105) < happens for sufficiently pure initial states, thatis, for sufficiently large z , when the initial entropy of thesystem is low enough so that the I loss induces by the mea-surement can overcome I gain . In the limit β → ∞ , theinitial state would reduce to a pure state, correspondingto I gain = 0 . Conclusion —We have experimentally realized a quan-tum Maxwell’s demon using a continuously monitoreddriven superconducting qubit. By determining the in-formation gathered by the demon by tracking individualquantum trajectories of the qubit, we have first verifiedthe validity of a quantum fluctuation theorem with feed-back by using both a weak-measurement approach andthe two-projective measurement scheme. In doing so,we have established the consistency of the two methods.We have further investigated the dynamics of the aver-aged information exchanged with the demon and demon-strated that it may become negative, in stark contrast tothe classical mutual information which is always a posi-tive quantity. Because of the combined effect of the quan-tum coherent dynamics and of the quantum backaction ofthe imperfect measurement, the description of the demonthus requires quantum information.We acknowledge P.M. Harrington, J.T. Monroe, andD. Tan for discussions and sample fabrication. We ac-knowledge research support from the NSF (Grant PHY-1607156) the ONR (Grant 12114811), the John Temple-ton Foundation, and the EPSRC (Grant EP/P030815/1).This research used facilities at the Institute of MaterialsScience and Engineering at Washington University. [1] L. Brillouin,
Science and Information Theory (AcademicPress, 1956).[2] K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. , 1 (2009).[3] H. S. Leff and A. F. Rex, Maxwell’s demon: entropy,information, computing (Princeton University Press,2014).[4] J. M. Parrondo, J. M. Horowitz, and T. Sagawa, Naturephysics , 131 (2015).[5] E. Lutz and S. Ciliberto, Physics Today , 30 (2015).[6] M. G. Raizen, Science , 1403 (2009).[7] V. Serreli, C.-F. Lee, E. R. Kay, and D. A. Leigh, Nature , 523 (2007).[8] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, andM. Sano, Nature Physics , 988 (2010).[9] E. Roldán, I. A. Martínez, J. M. R. Parrondo, andD. Petrov, Nature Physics , 457 (2014), article.[10] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin,Proceedings of the National Academy of Sciences ,13786 (2014).[11] J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola,Phys. Rev. Lett. , 030601 (2014).[12] M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim,V. Vedral, and I. A. Walmsley, Phys. Rev. Lett. ,050401 (2016).[13] T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise,R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara,M. Paternostro, and R. M. Serra, Phys. Rev. Lett. ,140601 (2014). [14] T. B. Batalhão, A. M. Souza, R. S. Sarthour, I. S.Oliveira, M. Paternostro, E. Lutz, and R. M. Serra, Phys.Rev. Lett. , 190601 (2015).[15] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. Quan, and K. Kim, Nature Physics , 193(2015).[16] M. Naghiloo, D. Tan, P. Harrington, J. Alonso,E. Lutz, A. Romito, and K. Murch, arXiv preprintarXiv:1703.05885 (2017).[17] F. Cerisola, Y. Margalit, S. Machluf, A. J. Roncaglia,J. P. Paz, and R. Folman, Nature Communications ,1241 (2017).[18] A. Smith, Y. Lu, S. An, X. Zhang, J.-N. Zhang, Z. Gong,H. T. Quan, C. Jarzynski, and K. Kim, New Journal ofPhysics , 013008 (2018).[19] T. P. Xiong, L. L. Yan, F. Zhou, K. Rehan, D. F. Liang,L. Chen, W. L. Yang, Z. H. Ma, M. Feng, and V. Vedral,Phys. Rev. Lett. , 010601 (2018).[20] P. A. Camati, J. P. S. Peterson, T. B. Batalhão, K. Mi-cadei, A. M. Souza, R. S. Sarthour, I. S. Oliveira, andR. M. Serra, Phys. Rev. Lett. , 240502 (2016).[21] M. A. Ciampini, L. Mancino, A. Orieux, C. Vigliar,P. Mataloni, M. Paternostro, and M. Barbieri, npj Quan-tum Information , 10 (2017).[22] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq, Q. Ficheux, J. Anders, A. Auffèves, R. Azouit,P. Rouchon, and B. Huard, Proceedings of the NationalAcademy of Sciences , 7561 (2017).[23] Y. Masuyama, K. Funo, Y. Murashita, A. Noguchi,S. Kono, Y. Tabuchi, R. Yamazaki, M. Ueda, andY. Nakamura, arXiv preprint arXiv:1709.00548 (2017).[24] H. G. Groenewoeld, International Journal of TheoreticalPhysics , 30 (1971).[25] K. Jacobs, J. Math. Phys. , 012102 (2006).[26] K. Jacobs, Quantum Measurement Theory (Cambridge,2014).[27] K. Funo, Y. Watanabe, and M. Ueda, Phys. Rev. E ,052121 (2013).[28] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, Nature , 162 (2004).[29] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert,K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frun-zio, S. M. Girvin, et al., Science , 178 (2013).[30] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi,Nature , 211 (2013).[31] S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan,K. W. Murch, , and I. Siddiqi, Nature , 570 (2014).[32] T. Sagawa and M. Ueda, Phys. Rev. Lett. , 080403(2008).[33] Y. Morikuni and H. Tasaki, Journal of Statistical Physics , 1 (2011), ISSN 1572-9613.[34] The experimental parameters are characterized by thequbit frequency ω q / π = 5 . GHz, cavity frequency ω c / π = 6 . GHz, Rabi frequency Ω R / π = 0 . MHz,dispersive coupling rate χ/ π = − . MHz, cavity band-width κ/ π = 5 . MHz, measurment strength k/ π = 51 kHz, quantum efficiency η = 0 . , and qubit coherenceproperties T = 5 µ s, T ∗ = 5 µ s.[35] D. Tan, S. J. Weber, I. Siddiqi, K. Mølmer, and K. W.Murch, Phys. Rev. Lett. , 090403 (2015).[36] N. Foroozani, M. Naghiloo, D. Tan, K. Mølmer, andK. W. Murch, Phys. Rev. Lett. , 110401 (2016).[37] K. Jacobs and D. A. Steck, Contemp. Phys. , 279 (2006).[38] M. Ozawa, J. Math. Phys. , 759 (1986).[39] T. Sagawa, Progress of Theoretical Physics , 1 (2012).[40] P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E , 050102 (2007).[41] O. E. Lanford and D. Robinson, J. Math. Phys.9