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Campisi, Michele; Pekola, J.P.; Fazio, Rosario

Feedback-controlled heat transport in quantum devices

Published in:New Journal of Physics

DOI:10.1088/1367-2630/aa6acb

Published: 01/05/2017

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Please cite the original version:Campisi, M., Pekola, J., & Fazio, R. (2017). Feedback-controlled heat transport in quantum devices: Theory andsolid-state experimental proposal. New Journal of Physics, 19(5), 1-12. [053027]. https://doi.org/10.1088/1367-2630/aa6acb

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

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New J. Phys. 19 (2017) 053027 https://doi.org/10.1088/1367-2630/aa6acb

PAPER

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

Michele Campisi1,4, Jukka Pekola2 andRosario Fazio1,3

1 NEST, ScuolaNormale Superiore & IstitutoNanoscienze-CNR, I-56126 Pisa, Italy2 LowTemperature Laboratory, Department of Applied Physics, AaltoUniversity School of Science, FI-00076Aalto, Finland3 ICTP, StradaCostiera 11, I-34151Trieste, Italy4 Author towhomany correspondence should be addressed

E-mail:michele.campisi@sns.it, pekola@ltl.tkk.fi and rosario.fazio@sns.it

Keywords: quantum thermodynamics,Maxwell demon, superconducting devices

AbstractA theory of feedback-controlled heat transport in quantum systems is presented. It is based onmodelling heat engines as drivenmultipartite systems subject to projective quantummeasurementsandmeasurement-conditioned unitary evolutions. The theory unifies various results presentedpreviously in the literature. Feedback control breaks time reversal invariance. This in turn results inthefluctuation relation not being obeyed. Its restoration occurs through appropriate accounting ofthe gain and use of information viameasurements and feedback.We further illustrate an experimentalproposal for the realisation of aMaxwell demon using superconducting circuits and single-photonon-chip calorimetry. A two-level qubit acts as a trap-door, which, conditioned on its state, is coupledto either a hot resistor or a cold one. The feedbackmechanism alters the temperatures felt by the qubitand can result in an effective inversion of temperature gradient, where heatflows from cold to hotthanks to the gain and use of information.

1. Introduction

In a famous thought experimentMaxwell envisioned amethod for apparently defying the second law ofthermodynamics bymeans of a feedback controlmechanism [1].Maxwell’s idea is based on amalicious demon,an intelligent being that is able to observe themicroscopic dynamics of a system, and acts on it so as to steer ittoward defying the second law. In one ofMaxwell’s original concepts, the system is a container with twochambers, containing respectively a hot gas and a cold gas. The two chambers are separated by awall containinga trap-door, which the demon can open and close at will. The demon observes the erraticmotion of the gasparticles, andwhen he sees a particle in the cold chamber approach the trap-doorwith sufficiently high velocity,he swiftly opens the door so as to let the particle pass through and closes it immediately afterwards. In this way,particle after particle, heatflows from the cold chamber to the hot chamber in contradictionwith the second law.

Advances in nanotechnology have created the possibility of bringingMaxwell demons and similar devicesfrom the realmof thought experiments to the realmof real experiments [2–5]. Theoretical and experimentalstudies so far have both focusedmainly on situations where feedback control is operated as ameasurement-conditioned driving on someworking substance (classical or quantum) coupled to a single temperature, so as towithdraw energy from the latter in contradictionwith the second law as formulated byKelvin. Interestingrealistic proposals have appeared in [6, 7]. Situations where heatflowbetween reservoirs at differenttemperatures is controlled, however, have not been addressed so far, either theoretically or experimentally. Themainmotivation of the present work is that offilling that gap. In the followingwe shall present the general theoryof feedback-controlled heat transport in quantumdevices, and describe a possible experimental realisationthereof.

The theory presented here builds on previousworks concerning fluctuation relations in the presence ofmeasurements without feedback [8, 9] andwith feedback [10], combinedwith an inclusive approachwhere

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quantumheat engines are seen asmechanically drivenmultipartite systems starting in amulti-temperatureinitial state [11–14]. Reference [10] reported on the theory of a one-measurement-based feedback control on aquantumworking substance prepared by contact with a single bath. That formalism is here extended to the caseofmany heat baths, and also repeatedmeasurements, to allow for the study of continuous feedback control ofheatflow in amulti-reservoir scenario. Previous work concerning repeatedmeasurements appeared in [15] forclassical systems in contact with a single bath. Fluctuation relations need to bemodified by amutual informationterm,whichwe shall provide explicitly.

Our experimental proposal is based on the fast developing advancements in experimental solid-statelow-temperature techniques: in particular the calorimetricmeasurement scheme that has beenput forwardbyoneofus and co-workers [16, 17]. As provenby some recent theoretical proposals [13, 18], themethodopens up anewavenue for thepracticalmanagement of heat andworkon a chip bymeans of superconducting devices, particularlysuperconducting qubits.Herewe illustrate the possible implementationof very simple feedback-controlledheattransportwhere the trap-door is realised by a superconducting qubitwhose couplingwith two resistors at differenttemperatures is controlled based on theoutcomes of continuous calorimetricmonitoring of the resistorsthemselves.

2. Theory

Following [14]wemodel a generic heat transport/heat engine scenario as a drivenmultipartite system starting inthe factorised state (see figure 1):

r =b-

⨂ ( )Z

e1

l

H

l0

l l

whereHi is theHamiltonian of each partition including a heat bath and possibly a portion of theworkingsubstance, andZi is the corresponding partition function [14]. Let the totalHamiltonian be

å= +( ) ( ) ( )H t H V t 2l

l

whereV(t) is an interaction term that is switched on for the time interval tÎ [ ]t 0, over which the system ismonitored.We assume that at times < <t t t... K1 2 some observableA ismeasured, thus causing thewavefunction describing the compound to collapse onto the subspace spanned by the eigenvectors belonging tothemeasured eigenvalue aj. Following [10], we shall assume that there can be ameasurement errorwhere theeigenvalue ak is recorded instead of the actual eigenvalue aj. This is assumed to happenwith probability e [ ∣ ]k j .The choice of the interactionV(t) in the interval +( )t t,i i 1 is dictated by the sequence of recorded eigenvalues, ormore simply the recorded sequence ¼ ={ }k k k k, , i i1 2 , that is =( ) ( )V t V tk i

for < < +t t ti i 1. Thecorresponding unitary operator describing the evolution in the time span < < +t t ti i 1 is

ò= ¬¾ +⎡⎣⎢

⎤⎦⎥( )U s H sexp d

t

tk ki

i

i

i

1where¬¾exp denotes time-ordered exponential and = å +( ) ( )H t H V tl lk ki i

.We

Figure 1. Feedback-controlled heat transport. A bipartite system starting in a two-temperature Gibbs state is observed by a demon,whomeasures an observableA. Depending on the outcome aj of themeasurement, the demon applies a quantumgateUj to thebipartite systemwith the aimof beating the second law. Each partition is composed of a heat reservoir and possibly one part of aworking substance. Thewhole system evolves with unitaries interrupted by projections.

2

New J. Phys. 19 (2017) 053027 MCampisi et al

shall denote the unconditioned evolution operator from time t=0 to the time of thefirstmeasurement =t t1 asU0. Note that the sequence of recorded labels k j generally differs from the sequence of labels ¼ ={ }j j j j, , i i1 2specifying inwhich subspace the system state was actually projected at themeasurement times t t t, ,... i1 2 . As iscustomary in the context of the fluctuation theoremwe shall assume that, besides the intermediatemeasurements ofA, allHlʼs aremeasured at times t= =t t0, , giving the eigenvaluesEn

l ,Eml respectively.

The quantity of primary interest is the probability ( )p m nk j, , , that n is obtained in the first energymeasurement, the sequence j is realised, the sequence k is recorded andm is obtained in the final energymeasurement. Herewe have introduced the simplified notations =j jK , =k kK . The explicit expression for

( )p m nk j, , , is

=( ) ( )† †p m n P A U P U A P pk j, , , Tr 3m n m nk j k j, 0 0 ,0

p e= P¬ ( )[ ∣ ] ( )A U k j 4i j i ik j k, i i

where = P b- /p Zen lE

l0 l n

ldenotes the probability of obtaining the eigenvalue = åE En l n

l in thefirstmeasurement; Pn denotes the corresponding projector; pj denotes the projector onto the subspace belonging to

the eigenvalueajofA; the symbol P¬

i denotes i-orderedproduct, that is, p e p eP¬

= ( [ ∣ ]) [ ∣ ]U k j U j ki j i i j K Kk ki i K K

p e p e[ ∣ ] [ ∣ ]U j k U j kj jk k2 2 1 12 2 1 1.

LetD = -E E El ml

nl be the change in energy in the partition l observed in a single realisation of the feedback-

driven protocol. Using the cyclic property of the trace and completeness å =Pn , we obtain the following:

åg rá ñ = =å b- D ( )†A Ae Tr 5E

j kk j k j

,, 0 ,l l l

The proof is reported in appendix A. This relation extends the result presented in [10] to the case of amultipartitesystemwith an initialmulti-temperature state, and to repeatedmeasurements5. The quantity r†A ATr k j k j, 0 ,

represents the probability that the sequences = { }† j j jj ,... ,K 2 1 and = { }† k k kk ,... ,K 2 1 are realised under thebackward evolution specified by the adjoint Kraus operators †Ak j, . The total probability γ does not generally add

to one. The reason for this is that the ith evolution †Uk ioccurs before the ith eigenvalue ji is realised in the

backwardmap. The feedback loop is evidently not symmetric under time reversal, and such a lack of reversibilitybreaks thefluctuation theorem á ñ =bå De 1El l l , which in fact is amanifestation of time-reversal symmetry [19].This is reflected by the fact that the quantum channel specified by theKraus operators Ak j, is generally notunital6. The adjoint of a non-unital quantum channel is not trace-preserving. In the case of feedback control thequantum channel rå †A Aj k k j k j, , 0 , is generally not unital; as a consequence its adjoint is generally not trace-preserving, hencewe have generally g ¹ 1. Lack of unitality generally reflects a lack of time-reversal symmetry.Examples are thermalisationmaps, namelymaps that have a thermal state (not the identity) as afixed point.Physically these are realised bymeans of weak contact of a systemwith a thermal bath, leading to irreversibledynamics. Likewise feedback control breaks the symmetry. This observation reveals some analogy betweenfeedback control and dissipative dynamics.

Before proceeding, let us comment briefly on the origin of the lack of unitality in feedback-controlledsystems, in order to gain insight into the issue. For simplicity let us consider the case of a singlemeasurementK=1. Let us begin by noticing that the quantum channel specified by the Ak j, is trace-preserving.We have

r e p rp e p rp p rp rå = å = å = å =[ ∣ ] [ ∣ ]† †A A k j U U k jTr Tr Tr Tr Trk j k j k j k j k j j k k j j j j j j, , , , , , wherewe have used

the cyclic property of the trace, unitarity =†U Uk k , idempotence p p p=j j j, normalisation eå =[ ∣ ]k j 1k and

completeness på =j . Let us now turn to unitality.We have e på = å [ ∣ ]† †A A k j U Uk j k j k j k j k j k, , , , . If the

evolutionUkwas not dependent on k, that is = ¯U Uk was chosen regardless of the recorded value k (e.g., U ispre-specified or is completely random), one could perform the sumover k using eå =[ ∣ ]k j 1k and then use

på =j to conclude that themap is unital. Feedback, implying explicit dependence on k ofUk, breaks unitality.Unitality would occur also in the case when e [ ∣ ]k j does not depend on j, meaning themeasurement outcome k iscompletely randomand has no correlationwith the actual state j. In sum, if the feedback controlmeasurement isoff, either because one decides not to use the information gathered in themeasurement or because themeasurement gathers no information in thefirst place, unitality is recovered, and the fluctuation theorem isrestored. This result is in agreementwith the established fact that projectivemeasurements without feedbackcontrol do not alter the validity of the fluctuation theorem [8, 20, 21]. Herewe have further learned that noise,i.e. choosing theUʼs between themeasurements completely randomly, also does not affect the integralfluctuation relation.

5For simplicity we restricted the argument to the case of cyclicH(t). The extension to the non-cyclic case is straightforward.

6We recall that a quantum channel specified byKraus operatorsMi, r r å †M Mi i i that is trace-preserving, i.e. å =†M Mi i , is unital

when itmaps the identity into itself: å =†M Mi i .

3

New J. Phys. 19 (2017) 053027 MCampisi et al

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

åb gáD ñ - ( )E ln 6l

l l

In the case when themap is unital we have g = 1, and the second law of thermodynamics is recovered [14].When g > 1 the condition bå áD ñ <E 0l l l is not forbidden, and the apparent violation of the second lawbecomes possible. This occurs with a proper ‘demonic’ design of the feedback control.When g < 1 instead, thesecond law ismore strictly enforced bymeans of an ‘angelic’ intervention.

As shown in [10, 22], in the case of a singlemeasurement (in either classical or quantum systems) thefluctuation relation can be restored if an information theoretic term, in the formofmutual information, isadded to the exponent in the exponential average. Reference [15] reports the extension to the case of repeatedmeasurements in the classical scenario. All these results are for a single-temperature initial state. In the presentset-upwe find aswell an information theoretic correction term (see appendix B for a proof):

á ñ =å b- D - ( )e 1 7E Jl l l k j,

where Jk j, is defined by the following set of equations:

=( )

( ) ( )( )J

p

p p

k j

j k kln

,

:8k j,

å=( ) ( ) ( )p p m nk j k j, , , , 9n m,

å å= =( ) ( ) ( ) ( )p p m n pk k j k j, , , , 10n mj j, ,

e=

P( ) ( )

[ ∣ ]( )p

p

k jj k

k j:

,. 11

i i i

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

åe=

P=( ) ( )

[ ∣ ]( )† †p

p

k jP B U P U B P pj k

k j:

,Tr 12

i i i n mm n m nk j k j

,, 0 0 ,

0

pe

= P¬

=P

( )[ ∣ ]

( )B UA

k j. 13i j

i i i

k j kk j

,,

i i

The operators Bk j, differ from the operators Ak j, by the term containing the conditional probability e [ ∣ ]k ji i .Note that the Bayes rule does not apply here, i.e. generally it is ¹( ) ( ) ( )p p pk j j k k, : . The reason is that j and kare concatenatedwith each other. An outcome ji influences the record ki, which in turn influences the nextoutcome +ji 1 and so on. The quantity Jk j, measures the degree of suchmutual influence or correlation betweenthe two sequences j and k7. In the absence of feedback, namelywhen there is no correlation between the twosequences, Jk j, is null and the standard relation is recovered. Note that, given a feedback rule, generally á ñJk j,

would growwith the lengthK of the sequences, i.e. the number ofmeasurements. It is accordingly expected thatá ñ µJ Kk j, in the large-K regime.

With Jensen’s inequality equation (7) implies

åb áD ñ -á ñ ( )E J . 14l

l l k j,

We thus have found two bounds to bå áD ñEl l l .By looking directly at bå áD ñEl l l as in [14]wehave found a third bound, whose interpretation ismost direct

and straightforward. Let

å år r r= =t ( )† † † †P A U P P U A P A U U A 15n m

m n n mj k

k j k jj k

k j k j, , ,

, 0 0 0 ,,

, 0 0 0 ,

be the systemdensitymatrix at time τ. In the second equality we have used completeness å =Pm m and the factthat the initial state has no coherences in the energy eigenbasis r rå =P Pn n n0 0. Simplemanipulations, similarto those employed in [14], lead to the following salient result:

7Equation (7) is reminiscent of a similar relation reported byVedral [23], see equation (8) there. The two relations differ fundamentally in

various respects, notably in themeaning of themutual information term. In our case itmeasures the correlation between outcomes and theirrecords; in the case of [23] itmeasures the correlation between themeasurements themselves.

4

New J. Phys. 19 (2017) 053027 MCampisi et al

å åb r r ráD ñ = + + Dt t[ ∣∣ ] [ ] ( )E D I 16l

l li

l l0

where

r r r r r r= -t t t t[ ∣∣ ] ( )D Tr ln Tr ln 17l l l l l l0 0

år r r r r= - +t t t t t[ ] ( )I Tr ln Tr ln 18l

l l

r r r rD = - +t t ( )Tr ln Tr ln 190 0

denote theKullback Leibler divergence between the final state rt and the initial state r0, equation (17); the totalamount of correlation (mutual information) that builds up among the partitions as a consequence of theirinteraction during the time span t[ ]0, , equation (18); and the total change in vonNeumann entropy of thewhole compound, equation (19). Here r r= ¢Trt

ll t is the reduced state of partition l at time t ( ¢Trl denotes the

trace over all partitions but the lth). Themutual information I among the partitions of the system (measuring allcorrelations, quantal and classical), which develops generally due to their interactionV(t) (and can also occur inthe absence ofmeasurements and feedback [14]), should not be confusedwith the classicalmutual informationJk j, between the realisation sequence j and the record sequence k caused by the feedbackmechanism.

Both theKullback Leibler divergence r rt[ ∣∣ ]D i i0 and themutual information r[ ]I t are non-negative

quantities.We thus arrive at the central inequality:

åb áD ñ D ( )E . 20l

l l

In the standard no-measurement case, rt is linked to r0 via a unitarymap, hence D = 0 and one recovers theresult of [14], namely b r r rå áD ñ = å +t t[ ∣∣ ] [ ]E D Ii i i i

i i0 , and the second law in its standard form.Note that

when there aremeasurements, but no feedback, rt is linked to r0 via a unitalmap, implying g = 0, á ñ =J 0k j, ,and D 0, hence bå áD ñ DE 0l l l , meaning that, as is already known [8, 20, 21], the second law isnot altered by themere application of projectivemeasurements that interrupt an otherwise unitary dynamics.However, equation (20) clearly indicates that there is a dissipation term associatedwith quantum-mechanicalmeasurements, which is not present in the classical case. In sum, through equation (20)we see that there is athermodynamic cost associatedwith quantummeasurements.

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

åb gáD ñ - -á ñ D[ ] ( )E Jmax ln , , . 21l

l l k j,

3. Illustrative example

To exemplify the theory abovewe consider a prototypicalmodel of a quantumheat enginewhoseworkingsubstance ismade of two qubits [13, 14, 24]. TheirHamiltonian reads

ws

ws= + = + ( )H H H

2 222z z1 2

1 2

where szi denote Pauli operators.We assume that the two qubits have the same level spacing w and are initially

in the state

r = Äb b- -

( )Z Z

e e23

H H

01 2

1 1 2 2

withZi their partition functions. At t=0 the szi ʼs aremeasured by collapsing the two qubits in the state

ñ = ¢ñ ñ∣ ∣ ∣k k k , with ¢ = k k, , .We assume classical error in themeasurement of each qubit, + + = - - =[ ∣ ] [ ∣ ] q, - + = + - = -[ ∣ ] [ ∣ ] q1 for some Î [ ]q 0, 1 . Accordingly the eigenvalues= ¢ j j j, are recordedwith probability e = ¢ ¢ [ ∣ ] [ ∣ ] [ ∣ ]k j k j k j . If the states + +ñ - +ñ - -ñ∣ ∣ ∣, , , , , , are

recordedwe do nothing: = = =+ + - + - -U U U ;, , , else, i.e., if = - +ñ∣k , , we apply a swap operation,=- +U U, SWAP, thatmaps - +ñ∣ , into + -ñ∣ , . The system is now in a joint eigenstate ñ = ñ∣ ∣m U jk of the two

qubits withHamiltonianH, hence thefinalmeasurement of szi is irrelevant. At the end of the process each qubit

is allowed to relax to thermal equilibriumwith its respective thermal bath of inverse temperature bi so as to re-establish the initial state r0. Accordingly, the average energy áD ñEi acquired by each qubit during the process isequal to the average heat that it releases in the bath in the thermal relaxation step. As a result of the feedbackmechanism energymay bewithdrawn from the cold bath and released in the hot one.Note that, due to the factthat the two qubits have same level spacing, the SWAPoperation does not alter their total energy. That is, there isno energy injection by the demon: to steer the energyflowhe only uses information. The set-up is illustrated infigure 2(a).

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New J. Phys. 19 (2017) 053027 MCampisi et al

The relevantprobability chain is a bit simpler than in the general casebecause thefirst energymeasurement is itselfhere also thefirst feedbackmeasurement. It reads e= =( ) [ ∣ ]† †p m k j P U P U P p k j P A A P p, , Tr Trm k j k m j m k j k j m j

0, ,

0 with

e= [ ∣ ]A k j U Pk j k j, . Forγwehave g e r= å [ ∣ ] †k j P U U PTrj k j k k j, 0 . Thefinal state is r e r= å [ ∣ ]k j U Pf j k k j, 0†P Uj k . Theprobability ( )p j k: that theoutcome j is realised conditionedon kbeing recorded is simply themarginal

probabilityp( j) that j is realisedbecause the recordk comes chronologically after the realisationof j andhence cannothave any influenceon it. Thequantity Jk j, boils down then to the logarithmof the ratio ( ) ( ) ( )p j k p j p k, [10], henceits expectation is thenon-negativemutual informationbetween j and k: á ñ = å ( )[ ( ) ( ) ( )]J p j k p j k p j p k, ln ,k j j k, , .

Figures 2(b, c) show b gå áD ñ - -á ñE J, ln ,l l k j, and D , for two choices of b2 and the same b1, as functionsof the error probability q. In accordancewith equation (21)we see that bå áD ñEl l is bounded frombelow by

g- -á ñJln , k j, and D . Independent of all other parameters the refrigerator cannot work in the region <q 1 2where j and k are anticorrelated, while itmay onlywork if >q 1 2. This is captured by g-ln being positive inthe region < <q0 1 2 and negative for < <q1 2 1. At =q 1 2 the outcome and recording are fullyuncorrelated, which restores unitality as discussed above and implies g =ln 0. Regarding D , while it tends tobe close to bå áD ñEl l in the operation region ( >q 1 2), it greatly departs from it in the non-operation region,where it can even take negative values. Notably in both panels there is a value of q for which the bound issaturated by D . Regarding-á ñJk j, , we note that it is everywhere non-positive as expected. Furthermore it issymmetric with respect to -q q1 . This reflects the fact that themutual information does not distinguishbetween correlation and anticorrelation. Themaximum-á ñ =J 0k j, is attained at =q 1 2where j and k areuncorrelated, and the standard fluctuation relation is recovered (i.e., g = 1). In both panels we see thatD > -á ñJk j, .Whether this a generic bound is yet to be understood.Wenote that while both g-ln and-á ñJk j,

are null at q= 1/2, D is non-negative, reflecting the fact that in the absence of feedback there is nonetheless anentropic cost associatedwithmeasurements, as discussed above. Such a cost can be counterbalanced in thepresence of feedback (note thatDH may be negative for ¹q 1 2). Confronting now the two panels, we see thatthe higher the thermal gradient b b-2 1, the larger is the point qwhere the engine starts operating, i.e. wherebå áD ñEl l turns frompositive into negative: as intuition suggests, the steeper the gradient, the bettermust your

measurement be. This feature is captured also by D but not by g- -á ñJln , k j, . Also the smaller the gradient,themore the shape of the function bå áD ñEl l resembles that of g-ln , with the shift between the two beingapproximately the value of D at =q 1 2: that is b gå áD ñ - + D = ∣E lnl l q 1 2.

4. Experimental proposal

The general theory developed above allows for a joint information theoretic and thermodynamic analysis offeedback-controlled dynamics in the broad scenario where a demon can influence not only the amount of workbeing provided by the outside as in previousworks [2–4] but also the heatflowbetween the various parts of acompound system, e.g. the heatflowbetween various heat baths.

Progress in solid-state technology, on the other hand, allows one to realise such feedback-controlled heattransportmechanisms in real devices. The example illustrated above can be realised experimentally byintroducing a feedbackmechanism in the schemewith two superconducting qubits illustrated in [13]. Belowweillustrate a design that permitsmore immediate realisation. It is based on a single qubit and it does not involveany qubit operation, but onlymanipulations of qubit–bath couplings. The proposal that we put forward here isbased on two ingredients that enable unique capabilities, allowing for the implementation of aMaxwell demon

Figure 2. (a) Scheme of a two-qubit feedback-controlled refrigerator. Two qubits are prepared, each in thermal equilibriumwith athermal bath.When the demon sees the cold qubit in the excited state and the hot qubit in the ground state, he swaps them.He thenlets them thermalise, eachwith its own bath, and starts over. He thus transfers heat from the cold bath to the hot bathwithoutinvesting energy. (b, c) bå áD ñEl l , g-ln ,-á ñJk j, and D as functions of the error probability q for b w=1 and twodifferent valuesof b2.

6

New J. Phys. 19 (2017) 053027 MCampisi et al

based on a very simple concept. The two ingredients are a two-level system acting as a quantum trap-door andthe calorimetricmeasurement scheme developed in [16, 17].

The one-qubit set-up is illustrated infigure 3. The two-level system (TLS) is embodied by a superconductingqubit of level spacing w. The two chambers are embodied by two resistors kept at different temperatures. Thequbit and resistors can exchange energy (i.e. heat) in the formof photons of energy w associatedwith the TLSabsorbing/emitting one photon from/to one of the two baths. The resistors are embedded in anRLC loop oftunable resonance frequency. This results in a tunable TLS/resistor coupling.When anRLC circuit is fardetuned fromω, the qubit is effectively decoupled from the resistor, whilemaximal coupling occurs when it is intunewith the qubit. The resonance frequency can be tuned by using a SQUID as a nonlinear and tunableinductor, its inductance being governed by a controllable threadingmagnetic flux.

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

The feedback concept is extremely simple: as soon as a jump down is observed, turn on the interactionwiththe cold resistor and turn off the interactionwith the hot resistor. Vice versa for the observation of a jumpup.This results in a netflowof heat from the cold resistor to the hot one. Based on the above general analysis theapparent violation of the second law is understood in terms of lack of time-reversal symmetry of feedbackcontrol, leading to an overall non-unital dynamics of resistors plus TLS. In a practical realisation one isrealistically not able to fully turn off the interactions. Furthermore therewill be some delay time δ between themeasurement being performed and feedback being realised, giving rise effectively to a possible error e [ ∣ ]k ji ibetween themeasured state ki and actual state ji of the qubit.

5.Modelling

In the followingwemodel the dynamics of the proposed experiment.Wemodel the evolution of the two-levelsystem via a standard Lindbladmaster equation

Figure 3. Set-up of the proposed experiment. A superconducting qubit (black rectangle) embodies aMaxwell demon trap-door, andtwo resistors embedded inRLC circuits embody the two chambers at different temperatures. The qubit andRLC circuits areinductively coupled. Calorimetricmonitoring of photons entering and exiting each resistor is applied, allowing one both tomeasureheat exchanged by each resistor and tomonitor the state of the qubit at any time.When the qubit is up, a feedback algorithmdrives theresonance frequency of the coldRLC circuit out of tunewith the qubit frequency, while keeping the hotRLC in tunewith it (andvice versa) so that an overall heat currentflows from cold to hot. The resonance frequencies of theRLC circuits are controlled bytuning their nonlinear inductive elements, i.e., SQUIDs, via application of externalmagneticflux Fi.

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New J. Phys. 19 (2017) 053027 MCampisi et al

r r r r= - + + [ ] ( )Hi , 24S L R

where s s= - D +( )H E qS x z0 is theHamiltonian of the two-level system expressed in terms of the Paulimatrices sa, and l are Lindblad operators

r s r s r= G + G [ ] [ ] ( )†D D 25l l l

expressed in terms of the super-operator r r r r= - -[ ] † † †D O O O O O O O1

2

1

2and the rising and lowering

spin operators s s†, of theHamiltonianHS, defined via s s s s-ñ = + ñ +ñ = + ñ = -ñ - ñ =∣ ∣ ∣ ∣ ∣ ∣† †, 0, , 0,where - + ñ∣ ( ) is the ground (excited) state ofHS. Here =l L R, denotes either the left or the right reservoir. Therates G

l for jump down/up in the lth resistor are given by

wG =

FD+ D

( )( ) ( )E M

qS 26l

lI l

02 2

20

2

2 2 ,

where w w w w w w= + - -( ) ( )[ ( [ ] )]S S R Q1I l V l l l LC l LC l, ,2 2

, ,2 1 is the current noise spectrum expressed in

terms of the voltage noise spectrum w w= - b w- -( ) ( )S R2 1 eV l l,1l , =Q L C Rl l l l is the quality factor and

w = L C1LC l l l, the resonance frequency of resonator l, expressed in terms of its resistance, inductance andcapacitance R L C, ,l l l. By increasing Lj the rates G

l can be quenched, i.e. the interaction between the TLS andthe lth resistor can be turned off. The symbolMl stands for themutual inductance between the qubit and the lthresistor and F0 is the flux quantum.Note that the rates are in detailed balance:

G = Gb w ( )e 27l ll

The study of heat andworkfluctuations requires the study of the dynamics to be performed at the level of singlequantum-jump trajectories [13, 25], resulting from the unravelling of themaster equation. This is achieved herebymeans of theMonte Carlowavefunction (MCWF)method [26, 27]. In the specific case under study of a two-level system subject to dissipation terms leading to full wavefunction collapse in either state -ñ∣ or +ñ∣ , thisresults in a classical dichotomous Poisson process with rates G

l [13].The basis of our numerical experiment is the generation of such dichotomous Poisson random trajectories.

We chose the right reservoir as the cold one and the left as the hot one. TheTLS is assumed to be initially inequilibriumwith the left bath.We produce a large sample of trajectories and build the normalised histogram

( )h NR of the numberNR of photons entering the right reservoir. Since the heatQR entering the right reservoir isgiven as w=Q NR R , the statistics ( )h NR is the heat statistics. In the absence of feedback it satisfies thefluctuation relation

-= b w-D( )

( )( )h N

h Ne , no feedback. 28R

R

NR

The feedback is introduced as follows. At eachmoment in timewe distinguish between the actual state of thesystem = j and the knowledge = k wehave about it. The latter does not necessarily coincide with theformer becausewe allow for some delay time δ between a jump occurring in the TLS and our knowledge of thestate of the qubit being updated accordingly. The delay time thus effectively introduces an error probabilitye [ ∣ ]between the actual state and our knowledge about it, at each time. At each time, conditioned on theknowledge k of the state, we use either set of rates favouring the interactionwith either the cold or the hot bath.More explicitly, let G ∣

l be the rate for a jump down (up) in the lth bath conditioned on theTLS beingmeasuredto be in state±. In accordancewith equation (26)weuse the following rates:

G =

-G = G

b wb w +

- + + - ( )∣ ∣ ∣A

1 ee 29L L L

L

L

G =

-G = G

b wb w +

- + + - ( )∣ ∣ ∣B

1 ee 30R R R

R

R

G =

-G = G

b wb w -

- - - - ( )∣ ∣ ∣B

1 ee 31L L L

L

L

G =

-G = G

b wb w -

- - - - ( )∣ ∣ ∣A

1 ee 32R R R

R

R

whereA andB are determined by the circuitry parameters and can be tuned via external fluxes Fi.With <B A,thismeans that energy exchangewith the right (cold) bath is larger when the TLS is believed to be down, so that itbecomesmore likely that energyflows out of the cold reservoir. Similarly energy exchangewith the left (hot) bathis larger when the TLS is believed to be up, so that it becomesmore likely that energy flows into the hot reservoir.Overall this results in an effect that is opposite to the natural flow fromhot to cold. The largest effect can beachievedwhen turning off the unwanted interaction completely, namelywhenB=0.Having inmind a realisticset-up, herewe keep the ratioA/Bfinite,meaning partial turning-off is considered.

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

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New J. Phys. 19 (2017) 053027 MCampisi et al

b b b b

we ee e

D = - = D ++ + + - ++ - + - -

[ ∣ ] [ ∣ ][ ∣ ] [ ∣ ]

( )A B

A B

2ln . 33L R

eff eff eff

We thus see that by tuning the ratioA/B the effective temperature gradient can bemanipulated, and if the errorassociatedwith themeasurement is not too big, it can even be inverted as compared to the original thermalgradient bD . So the overall effect of the demon is to change the ‘temperatures felt’ by the TLS. Accordingly thefluctuation relation

-= b w-D( )

( )( )h N

h Ne 34R

R

NReff

is obeyed by the histogram ( )h NR . This immediately allows us to interpret the quantity

we ee e

= -+ + + - ++ - + - -

[ ∣ ] [ ∣ ][ ∣ ] [ ∣ ]

( )JQ A B

A B

2ln 35R

exp

via equation (7) as themutual information encoded in a trajectory alongwhichheatQR is exchangedwith theRbath.Note thatwhenA=B, the feedbackhasno effect and accordingly =J 0exp . Likewise if e e+ + = + -[ ∣ ] [ ∣ ] (hencee e- + = - -[ ∣ ] [ ∣ ]),meaningno correlationbetween state andknowledge thereof, feedback control does notworkand again =J 0exp .Most importantly, the experimentalmutual information Jexp is proportional to theheatexchanged.This allows access to afluctuating information theoretic quantity bymeansof a thermodynamicmeasurement in a realistic experimental scenario.

Figure 4 shows typical histograms ( )h NR for realistic parameters.We also plotted the quantity-( ) ( )h N h Nln R R ,finding good agreementwith the theoretical prediction b w-D NReff . The effective

conditional probabilities e [ ∣ ]k j were obtained by recording for each trajectory the total timewhen the statewas jand the knowledgewas k, and averaging their values over thewhole ensemble of trajectories. The observeddeviation is a consequence of the fact that error here is introduced not in the formof an outcome beingmissed(as assumed in deriving equation (34)), but rather in one being reportedwith some delay.With the histogram

( )h NR we computed b w bå áD ñ = D á ñ = -E N 0.0862l l R , g- = - á ñ = -b wDln ln e 0.1205NR and-á ñ = -J 0.2873exp for the chosen parameters. The computed values are in agreementwith the prediction ofequation (21). The proposed experiment does not allowus tomeasure D , whichwould require access to thefull densitymatrix of system+baths.

5.1. Energy spent by the demonWhat is the energy cost incurred by the demon to open/close the trap-door? To roughly estimate thatwemodelthe LCR circuit as a classical harmonic oscillator (LC circuit) in contact with a heat bath (the resistor) attemperatureT. To open/close the door towards one of the two reservoirs, the demon switches the LC frequencyfrom wi to another frequency wf so as to put it in/off resonancewith the qubit. If the operation is carried out in aquasi-staticmanner, thework done is equal to the change in free energy: w w= ( )W k T lnB f i . The operationwould in this case be reversible, and thework lost when opening the doorwould be retrievedwhen opening it.The overall cost of an open/close cycle would be null in this limiting case. The other limiting case is when theswitch is infinitely fast. The overall cost of a single open/close cycle in this case would be non-negative inaccordancewith the second law of thermodynamics, and amounts to w w w w= -( )W k T 2B f i i f

2 . Theoverall work incurred in a repeated feedback operation is proportional to the number of open/close cycles,which in turn is proportional to the net number of energy quanta being transported, namely the total heattransported. Interestingly we note that the faster the open/close operation, themore effective is the feedbackmechanism, and themore energy needs to be invested.

Figure 4. Left: typical histogram ( )h NR . Right: -( ) ( )h N h Nln R R as a function ofNR. The straight dashed line is b wD NR. Thestraight solid line is b w-D NReff . Here = =k T k T1.1, 1B L B R . These thermal energies are expressed in units of w- .ω alsofixes thetime unit. The delay time is d = ´0.5 103 in those time units. = ´ -A 1 10 3 and = ´ -B 0.5 10 3 correspond to the timescale forthe largest rate = ´t 1.2642 103. The simulation time is t20 . The statistics is built on a sample of ´5 106 trajectories.

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New J. Phys. 19 (2017) 053027 MCampisi et al

6. Conclusions

Wehave developed a general quantum theory of repeated feedback control in a scenario withmultiple heatreservoirs. Themain effect of feedback control is that it induces a generally non-unital dynamics of the fullreservoirs+system compound. As a consequence the standard bound set by the second law of thermodynamicson the dissipation quantifier bå áD ñEl l l is shifted andmay become negative.We have illustrated anexperimental proposal where a single superconducting qubit plays the role of a trap-door that is subject tofeedback control. Themethod envisaged for simultaneouslymeasuring the qubit state and the heat exchangedby each reservoir is single-photon calorimetry.

Acknowledgments

This researchwas supported by aMarie Curie Intra European Fellowshipwithin the 7th EuropeanCommunityFramework Programme through the project NeQuFlux grant no 623085 (MC), byUnicredit Bank (MC), by theAcademy of Finland contract no. 272218 (JP) by theCOST actionMP1209 ‘Thermodynamics in the quantumregime’ and by theCentre forQuantumEngineering at AaltoUniversity, CQE.

AppendixA.Derivation of equation (5)

å

å

å

å r

á ñ =

=

=

=

å å å

åå å

å

b b b

bb b

b

- D -

--

-

( )

† †

†

†

p m n

P A U P U A PZ

P A A PZ

A A

k je , , , e e

Tre

e e

Tre

Tr

E

n m

E E

n mm n m

EE E

mm m

E

j k

j kk j k j

j kk j k j

j kk j k j

, , ,

, , ,, 0 0 ,

, ,, ,

,, 0 ,

l l l l ml

l nl

l nl

l ml

l nl

l ml

Equation (3) and r = å b-åP Zen nE

0l n

lhave been used to obtain the second line. Completeness å =Pn n and

unitarity =†U U0 0 led to the third line. The fourth line follows from the cyclical property of the trace,

idempotence =P P Pm m m and r = å b-åP Zem mE

0l m

l.

Appendix B.Derivation of equation (7)

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

e= =

P- ( ) ( )

( )( )[ ∣ ]

( )p p

p

p

k j

j k k

k j

ke

:

,B.1J

i i i

k j,

hence

å

å

å

eá ñ =

P

=P

=P

å å å

åå å

å

b b b

bb b

b

- D - -

--

-

( ) ( )[ ∣ ]

( )

( ) ( )

† †

†

p m np

k j

P B U P U B PZ

p

P B B PZ

p

k jk

k

k

e , , , e e

Tre

e e

Tre

B.2

E J

n m

E E

i i i

n mm n m

E

l l

E E

mm m

E

l l

j k

j kk j k j

j kk j k j

, , ,

, , ,, 0 0 ,

, ,, ,

l l l l ml

l nl

l nl

l ml

l nl

l ml

k j,

å=P

b-å( ) ( )P

Zp kTr

eB.3

mm

E

l lk,

l ml

år= =( ) ( )p kTr 1 B.4k

0

Equation (3), r = å Pb-åP Zen nE

l l0l n

land equation (13) have been used to obtain the second line.

Completeness å =Pn n and unitarity =†U U0 0 led to the third line. The fourth line follows fromå =†B Bj k j k j, , , which follows by expanding the i-ordered products, applying idempotence p p p=j j j,

10

New J. Phys. 19 (2017) 053027 MCampisi et al

completeness på =j j and unitarity =†U Uj j . r = å Pb-åP Zem mE

l l0l m

llead to thefifth line. Thefinal result

is a consequence of normalisation of r0 and of ( )p k .

AppendixC.Derivation of equation (33)

Under the operation of the demon theTLS experiences effective temperatures of the baths that differ from theiractual value. Tofix ideas, let us for themoment assume no delay time andno error in themeasurement. Thequbit is effectively subject to the following effective rates G = G G = G + -∣ ∣,s s s s

,eff ,eff . Accordingly, the detailedbalance temperatures are shifted:

= G G =b w b w ( ) ( )A Be e C.1L L,eff ,eff

L Leff

= G G =b w b w ( ) ( )B Ae e C.2R R,eff ,eff

R Reff

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

b

wb= +( ) ( )A B

1ln C.3L L

eff

b

wb= +( ) ( )B A

1ln . C.4R R

eff

Let us now introduce the errors e [ ∣ ] related to themeasurement. The stochastic process describing thedynamics of the TLS is still Poissonianwith one rate occurring in the case of correctmeasurement and one rateoccurring in the other case. The idea is thatmonitoring is continuous, or better, occurring with a sampling timeinterval dt, whichwe assume to be short compared to all rates G ∣

R L,, . Let us imagine the system is in state = +j .

There is a probability e + +[ ∣ ] that the observation is = +k and a probability e - +[ ∣ ] that it is = -k . Thus theprobability to undergo a jumpdown in the reservoir s in the interval dt is

e e= + + + - +-G -G + -( ) [ ∣ ] [ ∣ ] ( )∣ ∣p td e e C.5t td ds s

e e+ + - G + - + - G + - [ ∣ ]( ) [ ∣ ]( ) ( )∣ ∣t t1 d 1 d C.6s s

e e= - + + G + - + G + -( [ ∣ ] [ ∣ ] ) ( )∣ ∣ t1 d C.7s s

= e e- + + G + - + G + - ( )( [ ∣ ] [ ∣ ] )∣ ∣e C.8tds s

Similarly for the jumpup.Overall the TLS experiences the new rates

e eG = + + G + - + G + -[ ∣ ] [ ∣ ] ( )∣ ∣ C.9s s s,eff

e eG = + - G + - - G + -[ ∣ ] [ ∣ ] ( )∣ ∣ . C.10s s s,eff

Accordingly,

e ee e

=G

G=

+ + G + - + G+ - G + - - G

b w

+ -

+ -

[ ∣ ] [ ∣ ][ ∣ ] [ ∣ ]

( )∣ ∣

∣ ∣e C.11s

s

s s

s s

,eff

,eff

seff

b

we ee e

=+ + G + - + G+ - G + - - G

+ -

+ -

[ ∣ ] [ ∣ ][ ∣ ] [ ∣ ]

( )∣ ∣

∣ ∣1

ln . C.12ss s

s s

eff

Plugging in the explicit expressions we get

b b

we ee e

= ++ + + - ++ - + - -

[ ∣ ] [ ∣ ][ ∣ ] [ ∣ ]

( )A B

A B

1ln C.13L L

eff

b b

we ee e

= ++ - + - -+ + + - +

[ ∣ ] [ ∣ ][ ∣ ] [ ∣ ]

( )A B

A B

1ln . C.14R R

eff

Hence equation (33).

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