Photonic Maxwell's demon
Mihai D. Vidrighin, Oscar Dahlsten, Marco Barbieri, M.S. Kim, Vlatko Vedral, Ian A. Walmsley
PPhotonic Maxwell’s demon
Mihai D. Vidrighin,
1, 2
Oscar Dahlsten,
2, 3
Marco Barbieri,
4, 2
M.S. Kim, Vlatko Vedral,
2, 5 and Ian A. Walmsley QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom London Institute for Mathematical Sciences, 35a South Street, Mayfair, W1K 2XF London Dipartimento di Scienze, Universit`a degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy Centre for Quantum Technologies, National University of Singapore 3 Science Drive 2, 117543 Singapore, Republic of Singapore (Dated: October 9, 2015)We report an experimental realisation of Maxwell’s demon in a photonic setup. We show that a measurementat the single-photon level followed by a feed-forward operation allows the extraction of work from intensethermal light into an electric circuit. The interpretation of the experiment stimulates the derivation of a newequality relating work extraction to information aquired by measurement. We derive a bound using this relationand show that it is in agreement with the experimental results. Our work puts forward photonic systems as aplatform for experiments related to information in thermodynamics.
Maxwell’s demon made its appearence in 1867 as part ofa thought experiment discussing the limitations of the secondlaw of thermodynamics [1]. James Clerk Maxwell imaginedthe demon as a microscopic intelligent being, controlling adoor in the wall separating two boxes that contain a gas inthermal equilibrium. The demon would use the door to filterindividual particles of the gas based on their energy, produc-ing an unbalanced particle distribution. The operation seemedto decrease the entropy of the gas without any work invest-ment, contradicting the second law of thermodynamics. Dis-cussions emerging from the apparent paradox played a funda-mental role in revealing the relation between information andthermodynamics [2, 3]. It has been shown that the amount ofwork that the demon can extract from the imbalance of en-ergy between the two boxes which it can create by the sort-ing operation is limited by the information acquired by itsmeasurement of the individual particles. In turn, erasure ofthis information from the demon’s memory requires at leastas much work as was extracted. Maxwell’s demon has seenmany reinterpretations [4, 5] and has come to denote a systemthat achieves a decrease in entropy, or extraction of work byapplying measurement and feedback to a system in thermalequilibrium [6, 7]. Various physical realizations of such sys-tems have recently been demonstrated experimentally [8–11].Spurred by the advancement of experimental techniques,which are allowing the control of physical systems downto the single particle level, there has also been significantprogress in the theoretical analysis of thermodynamics in mi-croscopic systems, including the description of small thermalengines consisting of only a few energy levels [12–14], fluc-tuation theorems [15–21], the role of quantum coherence inthermodynamics [22–27] and resource theories of thermody-namic transformations [28]. As was the case in the 19th cen-tury, when the steam engine and other technologies demandedthe development of thermodynamics of macroscopic systems,the modern analogues involving few or single atoms have alsoled to the emergence of new ideas in microscopic thermody-namics. Here we aim to explore the benefits of photonics asan experimental platform for the study of the role of informa-tion in thermodynamics. This is a powerful platform, due to the experimental tools that have been developed for photonicquantum information processing and other applications thatuse engineered quantum states of light.We present an experiment analogous to Maxwell’s demon,in which light plays the role of the working medium. Insteadof gas particles on two sides of a wall, thermal states are pre-pared in two spatial light modes. We show that a measurementsimilar to photon subtraction [29–32] on each of the opticalmodes and a simple conditional operation (feed-forward) canlead to a difference in average intensity between the two lightmodes. Following this measurement, we let the remaininglight fall on two linear diodes which allow work to be ex-tracted from the unbalanced intensities in the form of energystored in a capacitor, a genuine, practical work reservoir. Thisallows us to link a microscopic measurement to macroscopicwork. Interpretation of the setup as a Maxwell’s demon re-quires a theoretical model in which the work extraction doesnot take place at thermal equilibrium, but as an arbitrary opensystem evolution. Analysing the setup leads us to derive awork-information equality, inspired by methods in [16]. Thisresult is used to relate experimentally measurable quantitiesby providing a bound on the extracted work distribution interms of measurement information. Unlike previously derivedwork-information inequalities, ours does not set a limit on av-erage work extracted per cycle, but on the ratio between av-erage work extraction and single shot fluctuations, a measurerelated to the strength of work discussed in [13]. We verify ex-perimentally that our protocol approaches the derived bound.
SETUP
Our photonics-based experiment, with schematic shownin Figure 1 follows Maxwell’s demon through the char-acteristic steps: measurement, conditional operation andwork extraction. The role of the gas in thermal equi-librium is played by two pulsed light modes, each pre-pared in a thermal state described by the density matrix (cid:0) − e − β hν (cid:1) (cid:80) ∞ n =0 e − β hν n | n (cid:105)(cid:104) n | in the photon number ba-sis where hν is the single photon energy and β the inverse of a r X i v : . [ qu a n t - ph ] O c t FIG. 1. Experimental setup. An optical signal with intensity fluc-tuations that obey thermal statistics is obtained by collecting lightfrom a time-varying laser speckle pattern. This is produced by fo-cusing laser pulses onto a spinning glass diffuser (Arecchi’s wheel).Two modes showing thermal statistics are selected using aperturespositioned in the far field. A measurement on these modes is imple-mented using beam splitters (BS) with high transmittance and single-photon sensitive click detectors (avalanche photodiodes – APD). Anelectromotive source made up of linear photodiodes (PD) charges acapacitor (C). We show that a non-zero average voltage across C,required for charging a battery, can be obtained using informationacquired through the APD measurement. Feed-forward can be im-plemented by swapping the polarity of the capacitor with an effectequivalent to swapping the two light modes after the APD measure-ments. In the experiment, we perform polarity swapping in post pro-cessing. the thermal energy. Light pulses in the two modes have unde-fined phase and energy distributed according to the Boltzmanndistribution. We prepare these states by collecting light from avariable laser speckle pattern, which is produced using a spin-ning glass diffuser as depicted in Figure 1. This type of sourceis known to produce light with thermal fluctuations [33–36]offering the possibility to obtain much higher intensities thatthose achievable by selecting the emission into a single modeof a thermal lamp.The demon’s measurement is similar to photon subtraction[29–32]. Each thermal light mode propagates through a hightransmittance beam splitter and the reflected light is coupledto a single-photon detector. The state inferred form observ-ing a detection event has a different average number of pho-tons than the incident light mode, increased or decreased de-pending on whether the incident light has super-poissonianor sub-poissonian statistics. When this measurement is ap-plied to single-mode thermal light, if the probability of thephoton detection is made very small, the average number ofphotons is doubled in the cases when a photon detection hap-pened [32]. Consider the extreme case in which an incidentlight pulse is either vacuum or populated by a large number ofphotons. Detection of even a single photon from this incidentpulse would distinguish with certainty the two possibilities.In a similar manner, our measurement determines, althoughwith remaining uncertainty, the energy fluctuations of a ther- mal light mode. And if the detection outcomes are ignored,the effect of the measurement amounts to a negligibly smallloss introduced by the high transmittance beam splitter. In oursetup we use on/off detectors which distinguish between vac-uum and one or more photons, giving a simple binary intensitymeasurement. We do not restrict the photon detection rate (thenumber of times that the demon’s detector fires per pulse) tosmall values as is commonly the case in quantum optics ex-periments. Thus the energy fluctuation resolving power of ourmeasurement can be tuned by fixing the amount of light senttowards the single-photon detectors, which sets the photon de-tection rate (see Appendix A). Feed-forward can be imple-mented by swapping the two thermal light modes conditionedon the output of the demon’s measurement, so that on averagethere is more energy on one side. In this way an asymmetricenergy distribution can be created from two equally populatedthermal light modes.For work extraction we propose the detection of the twothermal light modes on two photodiodes, connected with op-posing polarities such that on average they produce zero volt-age. This photodiode circuit includes a capacitor which ischarged according to the fluctuating energy difference be-tween the two detected modes. If an unbalance in the energydistribution of the two modes is produced by measurementand feed-forward, the capacitor will have a non-zero averagecharge which can be used to charge a battery, extracting singlecycle work. We choose this setup for its conceptual simplicity,not aiming to realise an optimal work extraction strategy.In practice, we make some simplifications to the measure-ment and controlled operations, aiming to provide a proof ofprinciple implementation. Firstly we note that photon subtrac-tion can be implemented with imperfect photon detectors andin Appendix A we show that the effect of the demon’s mea-surement does not depend on the detection efficiency. There-fore, we are not required to implement a beam splitter with re-flectance on the order of the inverse average photon number inthe thermal light modes. Instead, we implement a reflectanceof · − , small enough for the average effect on the trans-mitted light to be negligible compared to the thermal fluctu-ations. The detection rates of the demon’s measurement arethen regulated using variable absorbers. Secondly, swappingof the thermal light modes can be implemented on-line usinga variable beam splitter triggered by the single-photon detec-tors. However, as long as the two thermal light modes are wellbalanced, the symmetry of the setup is such that the swappingof the two light modes is indistinguishable from switching thepolarity of the capacitor. Thus, we replace feed-forward by alogical operation on the experimental data, switching the signof the measured capacitor voltage as a function of the single-photon detector outputs. A NON-EQUILIBRIUM WORK-INFORMATION EQUALITYINSPIRED BY THE EXPERIMENT
Non-equilibrium work relations such as the celebratedJarzynski equality [15] link non-equilibrium processes toequilibrium quantities like the thermal free energy. One equal-ity derived by Sagawa and Ueda incorporates the effect ofmeasurement and conditional operations [16, 17], providinga way to derive work-information bounds in Maxwell’s de-mon type scenarios. The effectiveness of a demon’s mea-surement is included in this expression through the mutualinformation quantifying correlations between measurementoutcomes and the measured system. When the initial en-ergy state of the measured system is i and the measuredoutcome is m the point-wise mutual information is I =log ( p ( m | i )) − log( p ( m )) . Here, by p ( m ) we denote theprobability of outcome m and by p ( m | i ) the conditional prob-ability of m given i . The theorem by Sagawa and Ueda reads (cid:104) e β ( W − ∆ F ) − I (cid:105) = 1 where W is single cycle work extrac-tion and ∆ F is the free energy difference between the finaland initial states of the working system. Jensen’s inequalitycan be used to derive from this a bound on extracted work: βW ≤ β ∆ F + I showing that information extracted by mea-surement allows for work extraction, even without free energyconsumption. We note that the entropy of the measurementregister, which can be readily estimated from a set of measure-ment outcomes, provides an upper bound to the mutual infor-mation I . The theorem by Sagawa and Ueda holds for workextraction scenarios where local detailed balance applies.Given the physical complexity of the work extraction setuppresented here, this model does not apply to our experiment.To find a similar relation between information gain and ex-tracted work we require a theoretical model allowing full gen-erality of the work extraction operation. We seek inspirationin our experiment, noting that work extraction can only be per-formed by acting efficient feed-forward. This observation iswith respect to the situation when the demon’s measurementis simply ignored, when no work is extracted. We aim to in-clude this scenario in our theoretical model. This leads us tothe following equality, the first main theoretical result of ourwork: (cid:104) e βW − I (cid:105) f = (cid:104) e βW (cid:105) (1)Here, the left hand term is an average (denoted ’f’) corre-sponding to the situation with feed-forward, controlled by theoutput of the measurement whose efficiency is quantified bythe mutual information I . The right hand side (denoted ’0’)is an average corresponding to the same system, but wherethe measurement and feed-forward steps are missing. Thismeans that the measurement outcomes are simply ignoredwhen the measurement is on average non-disturbing, such asin our setup (the effect of the high transmittance beam splitteris negligible). As we show in the following section, Equation1 can lead to useful results. The model and assumptions un-der which this equality is derived are detailed in the Proofs section. We note that this is valid when the feedback is an en-ergy conserving unitary operation acting only on the thermalsystem measured by the demon and when a non-disturbancecondition applies to the measurement, which we show is thecase for our setup. A BOUND RELATING WORK EXTRACTION TOMEASUREMENT INFORMATION
We now use Equation 1 to derive a bound applicable tomeasurable quantities in the experiment. Let U denote thevoltage created across the capacitor C . The external workreservoir, a battery, can be charged by connecting it to thecapacitor. The energy transfer from capacitor to battery de-pends on the voltage of the battery U , which must be differ-ent from zero for work to be extracted. This energy transferis W = C ( U − U ) U where C ( U − U ) is the charge trans-ported across the battery against the potential difference U after the capacitor and battery are connected. Inserting thisinto Equation 1 we get (cid:104) e βCU U − I (cid:105) f = (cid:104) e βCU U (cid:105) and us-ing Jensen’s inequality, βCU (cid:104) U (cid:105) f − (cid:104) I (cid:105) ≤ log( (cid:104) e βCU U (cid:105) ) .We define (cid:15) = βCU . Since the inequality is valid for anyvalue of U , we can find the tightest bound on (cid:104) U (cid:105) f by opti-mizing with respect to (cid:15) . While this can be done for any dis-tribution of U , a particularly relevant case is that when, ignor-ing the demon’s measurement outcomes, U is normally dis-tributed with mean (cid:104) U (cid:105) and standard deviation σ ( U ) . Then log( (cid:104) e (cid:15)U (cid:105) ) = (cid:15) (cid:104) U (cid:105) + (cid:15) σ ( U ) and optimizing over (cid:15) yieldsthe bound |(cid:104) U (cid:105) f −(cid:104) U (cid:105) | σ ( U ) < (cid:112) (cid:104) I (cid:105) . Using the relation betweenwork extracted and voltage on the capacitor, we can rewritethis equation in terms of work: |(cid:104) W (cid:105) f − (cid:104) W (cid:105) | σ ( W ) < (cid:112) (cid:104) I (cid:105) . (2)A special feature of this bound is that it does not contain β as a scaling factor, with work fluctuations defining the scaleinstead. The fact that the mutual information appears inside asquare root is not surprising since as we repeat the same pro-tocol, average extracted work and mutual information scalelinearly with the number of repetitions while the standard de-viation of the work distribution scales with the square root ofthe number of repetitions. Since for many repetitions of thesame protocol the total work distribution will tend to normal-ity by the law of large numbers, we can use the bound given byEquation 2 for any well behaved single shot work distribution. EXPERIMENTAL RESULTS
We first measure the distribution of pulse energy in the ther-mal light modes, by sampling 4000 consecutive pulses. Thecreation of thermal states is certified by estimating the inten-sity autocorrelation of the light pulses at zero delay, g (2) (0) [37]. For an ideal single mode thermal state this should yield g (2) (0) = 2 . Using the measured pulse energy values, we re-peatably obtained . < g (2) (0) < and no cross correlationin the pulse energy of the two modes. The intensity of the twooptical modes was balanced: the deviation from zero of theaverage voltage produced by the photodiode source in the ab-sence of feed-forward was less than . of the standard de-viation of this voltage, corresponding to thermal fluctuations.In our experiment, we record oscilloscope traces of the volt-age across capacitor C , at the same time recording outcomesof the APD detection, as depicted in Figure 1. In Figure 2we illustrate how the voltage depends on the APD signals andhow, while the average voltage is close to zero when we ignorethe demon’s measurement, it becomes significantly differentfrom zero (relative to fluctuations) when we apply a sign flipconditioned on the APD measurement outcomes. This condi-tional operation emulates unitary feedback on the two thermaloptical modes.The demon’s measurement can be tuned by varying theamount of light sent towards the single-photon detectors. Thisallows us to change the photon detection rates from zero toone (detections per pulse), and as we show in Appendix A,the imbalance that can be created in the two optical modesdepends only on these rates. One might expect that the imbal-ance that can be obtained between the two modes be high-est when the detection probabilities are around / , whichcorresponds to the highest entropy (information content) ofthe measurement register. However, we show in AppendixA that the probabilities that maximise the average unbalanceare actually / and / for the two arms respectively. Forour experiment, we set one of the arms to a detection rate of . ± . and scan the rate p corresponding to the otherarm. There are two different ways in which feed-forwardcould be applied, corresponding to a change of the sign of thevoltage across the capacitor corresponding to either one of thetwo asymmetric outputs of the demon’s measurement. Whichof these two strategies is optimal depends on the choice of p .In Figure 3 we show the average voltage that can beobtained with different measurement and feedback settingsand compare this to the thermodynamic bound introduced inEquation 2. If no information were acquired by the demon’smeasurement, the bound would demand that no work shouldbe extracted. We thus find that the work extraction is causedby the acquisition of information in our setup and that thestrategy that we apply for using the information yields resultsthat are close to a thermodynamic limit. DISCUSSION
The experiment presented in this work links very differentregimes: the single photon regime, a measurement yieldingsingle bit outcomes, intense light fields with thermal occupa-tion number corresponding to very high temperatures and theroom temperature system composed of detectors, electronicsand the environment. We are able to show that the setup is like eventsno click no click/no click/click click click/click/no click a.b.
U(V)U(V)
FIG. 2. Measured voltage on capacitor C . Oscilloscope traces show-ing the voltage on C created by the linear photodiode electromotivesource. The traces are filtered by measurement outcomes. (a) 4000traces, sorted according to binary signals from the two APDs imple-menting the demon’s measurement (click, corresponding to photondetection or no click, corresponding to vacuum). The black dashedline indicates the time at which the maximum voltage was sampled.(b) Histogram depicting the distribution of the maximum voltage onC. Gray – the APD outputs are ignored; blue – a logical operationconditioned by APD outputs is implemented: the sign of the traceis flipped when the two APD signals are click and no click, respec-tively. The dashed vertical lines are showing the averages of the twodistributions. There is a clear displacement in the average voltagewhen a conditional operation is applied, showing that feed-forwardcan produce a non-zero average voltage on the capacitor. The fir-ing rates for the two click detectors were p = 0 . ± . and p = 0 . ± . respectively. Maxwell’s demon, in the sense that work extraction is limitedby information acquired through measurement. However, thequantity bounded by information is not the average work ex-tracted per cycle, as is usually the case [2, 3, 5, 16] but the ratiobetween the average work extraction and the standard devia-tion of the single cycle work distribution. Rather than definingthe energy efficiency of the demon’s control, this quantity de-scribes the purity of the work produced, being related to theconcept of strength of work introduced in [13]. This quan-tity is of relevance in scenarios in which low fluctuations areimportant, such as cooling experiments.Our work demonstrates how photonics can provide a valid |〈 U 〉| σ (U) d n u o b c i m a n y d o m r e h t FIG. 3. Extracted work linked to information. Experimental pointsshowing the absolute average voltage U produced on capacitor C,which is directly proportional to the work that can be extracted bydischarging the capacitor into a battery, weighted by the standard de-viation of this voltage σ ( U ) , as a function of measurement settings.The demon’s measurement is tuned by changing the APD detectionrates. We choose a detection rate of . ± . for the secondAPD and tune the rate p of the first detector between zero (no clicks)and one (a click for every pulse). This setting was chosen becausethe maximum average voltage is obtained when the two detectorshave probabilities of firing / and / , respectively. Blue and or-ange points correspond to two types of feedback: flipping the voltagewhen the measurement yields click/no click (blue) and flipping thevoltage when the measurement yields no click/click (orange). Thedashed lines are simple models based on the average number dif-ference in two multimode thermal states corresponding to a secondorder autocorrelation function g (2) (0) = 1 . , as measured in the ex-periment. Error bars are estimated by binning the experimental dataand computing the variance of the values shown in the figure. Theblack line gives the bound established in Equation 2 in terms of mu-tual information, the computation of which is detailed in AppendixB. experimental platform for thermodynamic scenarios. This of-fers the perspective of moving into the quantum domain, tofurther explore the interface between quantum informationand thermodynamics thanks to the capability to engineer thewave-function of multi-photon states. In addition, the tech-niques presented can be extended to opto-mechanical oscilla-tors [38] and spin-ensembles [39], where single-particle oper-ations can be used to study the link between information andthermodynamics in stationary matter systems. EXPERIMENTAL METHODS
Optical detection using standard silicon linear photodiodesrequires that our source of thermal light have a high averagephoton number per pulse: the source that we used, describedin Figure 1 yields a number of photons per pulse on the or- der of . We achieved this by scattering 4mJ pulses froman amplified Ti:Sapphire laser to produce a laser speckle pat-tern and multimode optical fibers with core size of µ m tocollect the light. A laser speckle pattern with correlation arealarger than the collection aperture is required in order to ob-serve single mode thermal statistics. A fine grit glass diffuserand a relatively tight focusing of the laser, using a cm lensyielded an appropriate speckle pattern. The fiber apertureswere positioned cm away from the glass diffuser in thespeckle pattern. We strongly chirped the laser pulses in orderto avoid nonlinear damage of the glass diffuser which tendsto smooth the diffusing surface. The intensity autocorrelation g (2) (0) of the produced light was repeatedly measured to be . < g (2) (0) < . The g (2) (0) is smaller than for multi-mode thermal light. The mode number in this case is givenby / ( g (2) (0) − [37]. Multimode states are discussed inAppendix A.We used a capacitor with capacitance C = 2 pF and mea-sured the voltage created across it by each laser pulse usingan oscilloscope. The transmittance of the measurement beamsplitters was T = 99 . . We lowered the reflected power us-ing variable neutral density absorbers before coupling the sig-nal into fibers leading to the APDs, allowing the tuning of theAPD photon detection rate between zero and one detectionsper pulse. As we show in Appendix A, for high values of T ,uniform losses in the reflected light are equivalent to a lowerreflectance, in terms of the effect that the measurement has onthe measured light. In terms of overall efficiency, the . loss due to the beam splitter is a negligible effect, making oursetup operationally equivalent to what we would obtain witha much higher transmittance. PROOFS
To define work extraction in our model of the experimen-tal setup, we divide the model system into three parts. Thefirst part starts in thermal equilibrium, with inverse temper-ature β , having no correlations with the rest of the system.It corresponds to the two thermal light modes. The secondpart is the battery, or work reservoir and the third part is de-fined as everything that is neither the battery nor the ther-mal system, which corresponds in our setup to the photodi-odes, capacitor and environment. We define work extractedas the energy increase of the battery. Measurement and feed-forward operate only on the first part, the thermal system (twolight modes). Feed-forward is described by a unitary oper-ation with no energy cost, conditioned on the measurementoutcome. This is an appropriate description of a mode swap-ping operation, which can be in principle implemented by avariable-reflectivity beam splitter that switches without en-ergy consumption. Finally, we impose a condition on the mea-surement, as is detailed below. This is related to the notion ofnon-disturbance and applies to the measurement implementedin our experiment.Let us denote the initial energy eigenstate of the thermalsystem (on which measurement and feedback are performed) | i (cid:105)(cid:104) i | , with energy E i . For the work extraction system (secondand third parts as described above), we use | j (cid:105)(cid:104) j | to denote theinitial state and E j to denote the corresponding energy, ex-cluding the energy of the work reservoir. For the final state ofthe whole system we use | f (cid:105)(cid:104) f | to denote the state and denote E f the corresponding energy, again excluding the work reser-voir. The extracted work is defined W = E i + E j − E f . Theinitial probability distribution of the system’s state in the basisdefined above is p ( i, j ) = Z e − βE i p ( j ) . Let the demon’s mea-surement outcome be denoted m and the effect of the mea-surement be defined by the non-linear map M m yielding nor-malized states: M m ( | i (cid:105)(cid:104) i | ) = (cid:80) k M ( m ) k | i (cid:105)(cid:104) i | M ( m ) † k /p ( m | i ) where M ( m ) k are the measurement operators correspondingto output m . These are positive operators normalized suchthat (cid:80) m,k M ( m ) † k M ( m ) k = . The feedback is representedby unitary operators U m . The work extraction that takesplace after the feedback is modeled as an energy conserv-ing evolution of the whole system, according to a unitaryoperator V . Pointwise mutual information is defined I =log( p ( m | i )) − log( p ( m )) . We denote x ≡ { f, m, i, i (cid:48) } . Usingthese definitions, the left hand side of Equation 1 is (cid:88) x p ( f, m, i, j ) e W − I = (cid:88) x p ( f | m, i, j ) p ( m | i ) p ( i, j ) e βW − log( p ( m | i ))+log( p ( m ) = (cid:88) x p ( f | m, i, j ) p ( m ) 1 Z e − βE i p ( j ) e β ( E i + E j − E f ) = (cid:88) x p ( f | m, i, j ) p ( m ) p ( j ) 1 Z e βE j − βE f (3)where in the second line we used Bayes’ rule.We can write out the probability p ( f | m, i, j ) = (cid:104) f | V (cid:0) U m M m ( | i (cid:105)(cid:104) i | ) U † m ⊗| j (cid:105)(cid:104) j | (cid:1) V † | f (cid:105) . As we showbelow, for the measurement implemented in our setup, wehave (cid:88) i M m ( | i (cid:105)(cid:104) i | ) = . (4)We note that this condition holds for any non-disturbingmeasurement, when M m ( | i (cid:105)(cid:104) i | ) = | i (cid:105)(cid:104) i | . Using this and U m U † m = and (cid:80) m p ( m ) = 1 we get the following ex-pression for the left hand term of Equation 1: (cid:104) e βW − I (cid:105) f == 1 Z (cid:88) m p ( m ) (cid:88) j,f (cid:104) f | V ( ⊗ p ( j ) | j (cid:105)(cid:104) j | ) V † | f (cid:105) e E j − E f = 1 Z (cid:88) j,f (cid:104) f | V ( ⊗ p ( j ) | j (cid:105)(cid:104) j | ) V † | f (cid:105) e E j − E f . (5)Thus using the assumptions of our model, we have obtained anexpression independent of the details of the measurement and feedback operations. Similarly, the right hand side of equation1 is (cid:104) e βW (cid:105) = (cid:88) i,j,f p ( i, j, f ) e βW = (cid:88) i,j,f (cid:104) f | V (cid:18) Z e − βE i | i (cid:105)(cid:104) i | ⊗ p ( j ) | j (cid:105)(cid:104) j | (cid:19) V † | f (cid:105) e β ( E i + E j − E f ) =1 Z (cid:88) j,f (cid:104) f | V ( ⊗ p ( j ) | j (cid:105)(cid:104) j | ) V † | f (cid:105) e E j − E f . (6)This is the same expression as for the left hand term. There-fore Equation 1 holds for our model.We now show that Equation 4 holds for our measurementsetup, described in the main text and depicted schematically inFigure 1. Here we derive this for the idealized, lossless mea-surement and in Appendix A we show that the experimentalimplementation is equivalent to this model. For our setup,the state with energy E i , according to the notation describedabove, is the state with i photons.Let us start with the measurement outcome corresponding tono photons detected ( m = 0 ). When this outcome is recordedgiven the input state | i (cid:105)(cid:104) i | the input is left unchanged: ifno photons were detected in the reflected arm of the beamsplitter, all photons must have been transmitted. Thereforewe have (cid:80) i M ( | i (cid:105)(cid:104) i | ) = . In the case of the outcomecorresponding to a photon detection ( m = 1 ), the situa-tion is not as simple. The measurement operators describingthe detection of k photons are, in the photon number basis, M k = (cid:80) i (cid:113)(cid:0) ik (cid:1) T i − k (1 − T ) k | i − k (cid:105)(cid:104) i | with T the beamsplitter transmittance. The probability of outcome m = 1 when the initial state is | i (cid:105)(cid:104) i | is − T i (with probability T i all photons are transmitted and the measurement outcome is m = 0 ). We thus have (cid:88) i M ( | i (cid:105)(cid:104) i | ) = (cid:88) i,k ≥ M k | i (cid:105)(cid:104) i | M † k / (1 − T i )= (cid:88) i,k ≥ (cid:18) ik (cid:19) T i − k (1 − T ) k − T i | i − k (cid:105)(cid:104) i − k | = (cid:88) i,k ≥ (cid:18) i + kk (cid:19) T i (1 − T ) k − T i + k | i (cid:105)(cid:104) i | . (7)We can bound the coefficients of | i (cid:105)(cid:104) i | from above using − T i + k ≤ − T i +1 ⇒ (cid:88) k ≥ (cid:18) i + kk (cid:19) T i (1 − T ) k − T i + k ≤ (cid:88) k ≥ (cid:18) i + kk (cid:19) T i (1 − T ) k − T i +1 = 1 /T (8)and from below, using the first term of the sum (all terms arepositive): (cid:88) k ≥ (cid:18) i + kk (cid:19) T i (1 − T ) k − T i + k > (cid:18) i + 11 (cid:19) T i (1 − T )1 − T i +1 . (9)As T → , the limit of both the upper and the lower bound is . This shows that lim T → (cid:80) i M ( | i (cid:105)(cid:104) i | ) = , so for hightransmittance beam splitters, the condition given by Equation4 applies approximatively up to an error of the order | − T | . ACKNOWLEDGMENTS
This work was supported by the UK Engineering andPhysical Sciences Research Council (EPSRC EP/K034480/1).M.V. is supported by the Controlled Quantum DynamicsDTC. O.D. is supported by EPSRC, the John Templeton Foun-dation, the Leverhulme Trust, the Oxford Martin School, theNRF (Singapore), the MoE (Singapore) and EU CollaborativeProject TherMiQ (Grant Agreement 618074). M.B. is sup-ported by a Rita Levi-Montalcini fellowship of MIUR. M.S.K.is supported by EPSRC and the Royal Society. We thank B.Smith, A. Gardener, D. Jennings, M. Mitchison, M. Vanner,S. Kolthammer and R. Tetean for useful discussions. [1] J.C. Maxwell, Theory of Heat (Longman, London, 1871).[2] R. Landauer, IBM J. Res. Dev. , 183, (1961).[3] C.H. Bennett, Int. J. Theor. Phys. , 905, (1982).[4] S. Lloyd, Phys. Rev. A, , 3374 (1997).[5] K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys., , 1(2009).[6] L. Szilard, Z. Phys. , 840, (1929).[7] M. Plesch, O. Dahlsten, J. Goold and Vlatko Vedral, Sci. Rep., , 6995 (2014).[8] G.N. Price, S.T Bannerman, K. Viering, E. Narevicius, M. G.Raizen, Phys. Rev. Lett., , 093004 (2008).[9] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano,Nature Phys., , 988 (2010).[10] A. B´erut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillen-schneider, and E. Lutz, Nature, , 187 (2011).[11] J.V. Koski, V.F. Maisi,T. Sagawa, and J.P. Pekola, Phys. Rev.Lett., , 030601 (2014).[12] R. Kosloff, L. Amikam, Annu. Rev. Phys. Chem., , 365(2014).[13] N. Brunner, N. Linden, S. Popescu and P. Skrzypczyk, Phys.Rev. E, , 051117 (2012).[14] M. Horodecki and J. Oppenheim, Nat. Comm., , 2059 (2013).[15] C. Jarzynski, Eur. Phys. J. B, , 331340 (2008).[16] T. Sagawa and M. Ueda, Phys. Rev. Lett., , 090602 (2010).[17] T. Sagawa, and M. Ueda, Phys. Rev. Lett. , 180602 (2012).[18] R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold, and V.Vedral, Phys. Rev. Lett., , 230601 (2013).[19] L. Mazzola, G. De Chiara, and M. Paternostro, Phys. Rev. Lett., , 230602 (2013).[20] D.Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco Jr andC. Bustamante, Nature, , 231 (2005).[21] T.B. Batalh˜ao, A.M. Souza, L. Mazzola, R. Auccaise, R.S.Sarthour, I.S. Oliveira, J. Goold, G. De Chiara, M. Paternos-tro, and R.M. Serra, Phys. Rev. Lett., , 140601 (2014).[22] M. Lostaglio, D. Jennings, T. Rudolph, Nat. Comm., , 6383(2014).[23] L. del Rio, J. ˚Aberg, R. Renner, O. Dahlsten, and V. Vedral,Nature, , 61 (2011). [24] S.W. Kim, T. Sagawa, S. De Liberato, and M.Ueda, Phys. Rev.Lett., , 070401 (2011).[25] R. Kosloff, Entropy, , 2100 (2013).[26] A. Levy and R. Kosloff, Europhys. Lett., , 20004 (2014).[27] R. Uzdin, A. Levy and R. Kosloff, Phys. Rev. X, , 031044(2015)[28] F. G.S. L. Brand˜ao, M. Horodecki, J. Oppenheim, J. M. Renesand R. W. Spekkens, Phys. Rev. Lett., , 250404 (2013).[29] J. Wenger, R. Tuall-Brouri and P. Grangier, Phys. Rev. Lett., ,153601 (2004).[30] A. Oujoumtsev, J. Laurat, R. Tualle-Brouri and P. Grangier, Sci-ence , 83 (2006).[31] V. Parigi, A. Zavatta, M.S. Kim and M. Bellini, Science, ,1890 (2007).[32] A. Zavatta, V. Parigi, M. S. Kim, M. Bellini, New J. Phys., ,123006 (2008).[33] F.T. Arecchi, Phys. Rev. Lett. , 912 (1965).[34] A. Valencia, G. Scarcelli, M. D’Angelo, and Y.H. Shih, Phys.Rev. Lett., , 063601 (2005).[35] V. Parigi, A. Zavatta, and M. Bellini, J. Phys. B, , 114005(2009).[36] A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini,Phys. Rev. Lett., , 140406 (2009).[37] R. Chrapkiewicz, J. Opt. Soc. Am. B, , B8 (2014).[38] M. R. Vanner, M. Aspelmeyer and M. S. Kim, Phys. Rev. Lett., , 010504 (2013).[39] R. McConnell, H. Zhang, J. Hu, S. Cuk and V. Vuletic, Nature, , 439 (2015). APPENDIX A
In this appendix we calculate the effect that the single pho-ton level measurement described in Figure 1 has on a mode ina thermal state. These calculations are used for the theoreticalpredictions depicted in Figure 3.Let T be the measurement beam splitter’s transmittance, R = 1 − T the beam splitter’s reflectance and η the effi-ciency of the single photon detection. A single mode thermalstate can be written (1 − λ ) (cid:80) i λ i | i (cid:105)(cid:104) i | in the photon numberbasis. The POVM element corresponding to detection of kphotons in our setup is M k = (cid:80) i (cid:113)(cid:0) ik (cid:1) T i − k ( Rη ) k | i − k (cid:105)(cid:104) i | with k = 0 corresponding to no detected photons and k ≥ corresponding to at least one photon being detected. Ap-plying these operators to the initial state, we obtain thatthe probability for any of the k ≥ outcomes to occuris p k ≥ = λRη − (1 − Rη ) λ and the mean number of photonsin the corresponding post measurement state is (cid:104) n (cid:105) k ≥ =2 T − (1 − Rη/ λ − (1 − Rη ) zλ (cid:104) n (cid:105) t where (cid:104) n (cid:105) t is the average photon num-ber in the initial thermal state. We thus have that (cid:104) n (cid:105) k ≥ (cid:104) n (cid:105) t =(2 − p k ≥ ) T . The number of photons in the state created whenoutcome k = 0 is observed is (cid:104) n (cid:105) k =0 = (1 − p k ≥ ) (cid:104) n (cid:105) t T .For high transmittance beam splitters, T can be approximatedwith . We thus see that the average number of photons is asimple function of the detection probabilities and that theseprobabilities can be tuned by changing the efficiency η , givenhigh T .We can now calculate the expected average photon numberdifference when the two light modes are swapped according tothe demon’s measurement outcomes. We denote p and p theprobabilities for photon detection to occur in the two modesrespectively. The average difference in the two arms has thefollowing contributions: for no detector firing, the differenceis ( p − p ) (cid:104) n (cid:105) with probability (1 − p )(1 − p ) ; for only thefirst detector firing, ( p − p + 1) (cid:104) n (cid:105) with probability p (1 − p ) ; for only the second detector firing, ( p − p − (cid:104) n (cid:105) with probability (1 − p ) p and for both firing, ( p − p ) (cid:104) n (cid:105) with probability p p . The maximum average photon numberdifference is obtained if the modes are switched in the third ofthese cases, yielding (cid:104) n (cid:105) − (cid:104) n (cid:105) = 16 / (cid:104) n (cid:105) for detectionprobabilities p = 1 / and p = 2 / .The statistics of the light produced in our setup is slightlymulti-mode, which is indicated by the second order autocorre-lation function measured, . < g (2) (0) < . When the ther-mal light is not single mode, the average number of photonsprepared by measurement is different from the single modecase treated above. Let the input state be a mixture of k ther-mal modes. We denote the probability for the single photondetector to fire if only one of the k thermal modes was probed q and the probability for the detector to fire when all modesare probed p . The probability for the detector not to fire whenall k modes are measured is − p = (1 − q ) k . Using the resultabove, we know that the average number of photons createdby this outcome is (1 − q ) (cid:104) n (cid:105) t = (1 − p ) k (cid:104) n (cid:105) t . Using energyconservation, we find the average number of photons corre-sponding to a photon detection outcome: − (1 − p ) k +1 k p . APPENDIX B
Here we calculate the average mutual information charac-terizing the measurement described in Figure 1. We show thatfor high transmittance T ≈ of the measurement beam split-ter and high average photon population of the initial states,this quantity depends only on the single photon detectionprobabilities. The calculation presented here is used to depictin Figure 3 the bound given by Equation 2.The average mutual information is I = (cid:88) m,i p ( m, i ) (log( p ( m | i )) − log( p ( m ))) (10) where we use the same notation as in the main text. Theprobability distribution of the initial state is p ( i ) = (1 − e − β (cid:126) ω ) e − β (cid:126) ω i where (cid:126) ω is the single photon energy. Forhigh average occupation number, the distribution over i issmooth and we can convert the sum to an integral. We in-troduce the variable x = β (cid:126) ω i . m = 0 corresponds to the no-photons-detected outcomeand m = 1 corresponds to the photons-detected outcome.Here we denote the probability of the m = 1 outcome by q .We have for the joint probability of the initial state and mea-surement outcome p (0 , i ) = (1 − e − β (cid:126) ω )(1 − Rη ) i e − β (cid:126) ω i and p (1 , i ) = (1 − e − β (cid:126) ω )(1 − (1 − Rη ) i ) e − β (cid:126) ω i . Convert-ing this to a probability density in terms of the new variable x , we get P (0 , x ) = β (cid:126) ω (1 − e − β (cid:126) ω )(1 − Rη ) xβ (cid:126) ω e − x and P (1 , x ) = β (cid:126) ω (1 − e − β (cid:126) ω )(1 − (1 − Rη ) xβ (cid:126) ω ) e − x . For β (cid:28) , β (cid:126) ω (1 − e − β (cid:126) ω ) ≈ so we can write P (0 , x ) = e c x and since − q = (cid:82) ∞ P (0 , x ) dx = (cid:82) ∞ e c x dx = c , we get c = − q . Finally, we have P (0 , x ) = e − − q x P (1 , x ) = (cid:16) − e − q − q x (cid:17) e − x (11)and we see that these probability densities only depend on q . We can now compute the mutual information in the limitof high occupation numbers and high beam splitter transmit-tance: I = (cid:90) e − − q x ( − q/ (1 − q ) − log(1 − q )) dx + (cid:90) (cid:16) − e − q − q x (cid:17) e − x (cid:16) log(1 − e − q − q x ) − log( q ) (cid:17) dx.dx.