PRX QUANTUM 1, 010309 (2020)

Quasiprobability Distribution for Heat Fluctuations in the Quantum Regime

Amikam Levy 1,2,3,* and Matteo Lostaglio 4,5,†

1Department of Chemistry, University of California, Berkeley, California 94720, USA

2The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv

University, Tel Aviv 69978, Israel3Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel

4ICFO—Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels

(Barcelona) 08860, Spain5QuTech, Delft University of Technology, P.O. Box 5046, Delft, GA 2600, The Netherlands

(Received 8 October 2019; revised 10 July 2020; accepted 8 September 2020; published 25 September 2020)

The standard approach to deriving fluctuation theorems fails to capture the effect of quantum cor-relation and coherence in the initial state of the system. Here, we overcome this difficulty and derivethe heat-exchange-fluctuation theorem in the full quantum regime by showing that the energy exchangebetween two locally thermal states in the presence of initial quantum correlations is faithfully capturedby a quasiprobability distribution. Its negativities, being associated with proofs of contextuality, are prox-ies of nonclassicality. We discuss the thermodynamic interpretation of negative probabilities and provideheat-flow inequalities that can only be violated in their presence. Remarkably, testing these fully quantuminequalities, at an arbitrary dimension, is no more difficult than testing traditional fluctuation theorems. Wetest these results on data collected in a recent experiment studying the heat transfer between two qubitsand give examples for the capability of witnessing negative probabilities at higher dimensions.

DOI: 10.1103/PRXQuantum.1.010309

I. INTRODUCTION

Consider two systems, C for “cold” and H “hot,” in ther-mal states at temperatures TC < TH . If the overall systemis isolated, energy is conserved, and there are no initial cor-relations between C and H , heat will flow on average fromthe hot to the cold body:

Q ≡ Q(H → C) ≤ 0, (1)

as mandated by the second law of thermodynamics.Equation (1) can be derived for both classical and quantumsystems and understood as an average over heat fluctua-tions that satisfy Jarzynski’s exchange fluctuation theorem(XFT) [1].

The situation is less straightforward if the initial stateis locally thermal but correlated, since correlations allowfor temporary “back flows,” i.e., Q > 0. [2,3]. However,

*amikamlevy@gmail.com†lostaglio@gmail.com

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license. Fur-ther distribution of this work must maintain attribution to theauthor(s) and the published article’s title, journal citation, andDOI.

as long as the system is simply classically correlated, wecan still understand these flows as emerging from averagesof underlying classical fluctuations [4].

This picture breaks down in the presence of quantumcorrelations. In quantum theory, access to the energy of thesystem, which is necessary for defining fluctuations classi-cally, can be acquired only via a measurement process thatdestroys the very quantum properties we wish to investi-gate. A striking example is given by strong heat back flows,i.e., the observation of an amount of heat flowing fromC to H larger than (�β)−1 log d, where �β = βC − βH ,βX = 1/(kTX ), in which k is Boltzmann’s constant and dis the local dimension. Observation of such flows impliesthat the underlying state is entangled [3]. If we attempt tounderstand this flow as originating from underlying (classi-cal) heat fluctuations, the energy measurements required toaccess them destroy the entanglement and so the effect dis-appears. In fact, any stochastic interpretation of heat flowsthat reproduces the observed Q faces severe no-go theo-rems [5,6]. This is the reason why the leading proposalfor defining and measuring heat and work fluctuations,the “two-projective-measurement” (TPM), clearly cannotreproduce quantum effects such as strong heat back flows.

We propose that quantum heat fluctuations should beassociated with a quasiprobability distribution. This arisesnaturally from a correction to the TPM distribution and

2691-3399/20/1(1)/010309(19) 010309-1 Published by the American Physical Society

AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

can be accessed by weak measurements. We show that theassociated negative “probabilities” have a clear thermody-namic interpretation as contributions to the heat flows thatcannot be explained within the framework of stochasticthermodynamics. These quasiprobabilities satisfy an XFTthat incorporates quantum correlations in the initial stateand includes in the relevant classical limits the originalXFT of Ref. [1] as well as its extension to the classi-cally correlated systems of Ref. [4]. More importantly, ourresults provide heat-flow inequalities that can only be vio-lated in the presence of negativities. These violations are aproxy for a strong form of nonclassicality known as con-textuality and, remarkably, can be evaluated by means ofTPM alone.

The challenges posed by the construction of a theoryof heat fluctuations reproducing the observed flows are astrong indication that certain flow configurations are inher-ently quantum mechanical. We propose that quasiprobabil-ities, a tool already successfully applied in fields such asquantum optics and quantum computing, can be importantdiagnostic tools for exploring quantum effects in thermo-dynamics as our ability to measure heat flows improves.

In Sec. II, we present the problem of capturing quan-tum phenomena in fluctuation theorems and introduce theconcept of quasiprobabilities as the solution. In Sec. III,we introduce a quasiprobability distribution for heat fluc-tuation and discuss its thermodynamic interpretation, itsquantum nature, and the classical limit. Section IV focuseson the two-qubit system and presents heat-flow inequalitiesthat witness negativity. We further demonstrate the resulton recent experimental data. In Sec. V, we present the fullquantum heat-exchange-fluctuation theorem (QXFT), andextend the results on heat-flow inequalities to arbitrary-dimension systems. In Sec. VI, we show that witnessingnegativity in higher-dimensional systems is possible usingprojective energy measurement alone and we demonstratethis on a two-qutrit system. Finally, in Sec. VII, wesummarize and discuss future possible inquires.

A. Comparison with previous works

In recent years, several proposals have attempted to gobeyond some of the limitations faced by TPM schemes. InRef. [7], a modified quantum Jarzynski equality is derived,which accounts for the thermodynamic cost of the mea-surement. This in turn has led to approaches based on an(initial) one-time measurement scheme for closed [8] andopen [9] systems, which solve experimental and concep-tual difficulties related to performing the second projectivemeasurement in the TPM scheme. Here, on the other hand,we want to confront the loss of information about quantumeffects caused by the first projective measurement of theTPM scheme.

The definition of fluctuations in the quantum regime isnot unique. There is in fact a zoo of such proposals that

reduce to the TPM scheme in the case of initial statesdiagonal in the energy basis, and the first moments ofwhich coincide with the average energy change (of theundisturbed system) (see the review in Ref. [10]). Theseproposals necessarily fall into one of two categories [5,6]:

(1) Lack of positivity: approaches that give up theidea of assigning probabilities to fluctuations, by allowing“negative probabilities” (Margenau-Hill quasiprobability[11], full counting statistics [12–14], and consistent his-tories [15,16]). These approaches call for an interpretationof such negativities.

(2) Lack of σ -additivity: approaches based on probabil-ities, which give up the linearity in the density operator ofthe initial state (Hamilton-Jacobi framework [17], quan-tum Bayesian network [18], and one-time final energymeasurement [19]). For example, if we consider two prepa-rations ρHeads and ρTails, in these approaches the workprobability for 1

2ρHeads + 12ρTails (tossing a fair coin and

preparing ρHeads or ρTails accordingly) is not the equal mix-ture of the work probability for ρHeads and ρTails. Theseapproaches require an interpretation for the lack of thisproperty, which is normally a consequence of the standardprobabilistic interpretation.

Independently of the preferred alternative, given the pro-liferation of definitions one needs to set clear objectivesfor these investigations. It is not complex to come up withnovel definitions of quantum fluctuations, leading to suit-ably complicated fluctuation theorems that differ from thestandard ones due to the presence of quantum coherence.To make a proposal relevant, however, one should attemptto solve three main problems:

(a) Give an answer to the above-mentioned conceptualquestions, providing a thermodynamic interpretation of thefailure of properties expected in the statistical interpreta-tion and showing how these are recovered in the classicallimit.

(b) Find a definition of fluctuations leading to a notionof “quantum signature” that can be identified with a clear-cut notion of nonclassicality. In other words, one shoulddefine a precise notion of classicality and show that theidentified quantum signature is an operational featurethat cannot be reproduced by any model satisfying thatclassicality property.

(c) Show that such quantum signatures are experimen-tally accessible without requiring a very large amountof control (such as knowledge of the initial state or thedynamics). Ideally, witnessing quantum signatures shouldnecessitate no more control than that required by theoriginal TPM scheme.

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What we set out to do in this work is to show that all theseproblems can be solved simultaneously in the context ofheat fluctuations.

II. THE PROBLEM OF QUANTUM HEATFLUCTUATIONS

The impasse in the definition of heat and work fluctu-ations can be traced back to the information-disturbancetrade-off of quantum mechanics. Imagine the cold and hotsystems, C and H , initially described by a density operatorρCH , evolve under closed dynamics described by a unitaryU, ρCH (τ ) = UρCH U†. The average heat flow reads

Q := Tr [ρCH HC] − Tr [ρCH (τ )HC] . (2)

In order to study fluctuations, classically we want to seethis quantity as emerging from a microscopic process inwhich energy q is exchanged with probability p(q) suchthat Q = ∫

dqp(q)q. In this case, the system C startsat some phase-space point z0 with probability p(z0) andevolves deterministically to zτ , in which case q = E(z0)−E(zτ ), with E(z) the energy at point z. Hence

p(q) =∫

dz0p(z0)δ(q − (E(z0)− E(zτ ))). (3)

Quantumly, however, the initial energy is generally notwell defined. Noncommutativity forbids the constructionof a joint probability (linear in ρCH ) that correctly repro-duces the energy distributions of the initial and final states[5].

This long-standing problem dates back many decadesand the standard way to overcome it has been to definefluctuations by a two-point-measurement (TPM) scheme[20,21]. The TPM scheme involves (1) projectively mea-suring the energy of C at the start, (2) letting the sys-tem dynamics unfold, and (3) projectively measuring theenergy of C again at the end. pTPM(q) is then defined asthe probability of recording an energy difference betweenthe outcomes of the two projective measurements equal toq. However, in this scheme QTPM := ∫

dqpTPM(q)q �= Q,since effects due to noncommutativity are expunged bythe initial invasive measurement. The widely used TPMmethod has the advantage that, by construction, classicalfluctuation theorems hold for pTPM(q). The drawback isthat we are effectively describing the heat fluctuations ofa state disturbed (decohered) by the initial energy mea-surement, rather than those of the original state ρCH .Quantum effects such as strong heat back flows provablynever appear. Hence this solution cannot be satisfactoryin the fully quantum regime, when ρCH includes energeticcoherence and quantum correlations.

A. A different solution: quasiprobabilities

We argue that the root cause of the present difficulties isthe attempt to give a classical stochastic account of quan-tum heat fluctuations. Since quantum mechanics does notallow for the construction of joint probabilities of noncom-muting quantities, a natural framework is to describe themby means of quasiprobabilities [22], i.e., real-valued func-tions p(q) that sum to 1 but are allowed to take valuesoutside [0, 1]. While relatively uncommon in the ther-modynamic setting (but see, e.g., Refs. [6,11,12,14,23]),the use of quasiprobabilities to probe quantum effects iswidespread in other fields of quantum sciences, includingquantum transport [24–26], quantum optics [27,28], quan-tum computing [29,30], and condensed matter [31,32].Quasiprobabilities originated with the attempt to give aphase-space description to quantum mechanics despitenoncommutativity of position and momentum. The mostfamous example is the Wigner function [28], which asso-ciates a quantum state with a quasiprobability distributionover positions and momenta the marginals of which cor-rectly reproduce the statistics collected in the correspond-ing quantum measurements [33].

The analogy to our scenario is that, in order to recoverEq. (2), we wish to reproduce the correct quantum statisticsof energy measurements performed on the initial ρCH andfinal ρCH (τ ) states. Formally, if pW

iCiH →fCfH denotes thequasiprobability of starting with energies (EC

iC , EHiH ) before

the process U and ending up in the final energies (ECfC

, EHfH)

afterward, then require that

∑fC,fH

pWiCiH →fCfH = Tr

[ρCH�

iCiH]

:= piCiH ,

∑iC,iH

pWiCiH →fCfH = Tr

[ρCH (τ )�

fCfH]

:= pfCfH (τ ), (4)

where �iCiH ≡ �iC ⊗�iH and �iX are energy-projectionoperators such that HX = ∑

iX EXiX�

iX (X = C, H ). Notealso that piCiH (piCiH (τ )) denotes the statistics of energymeasurements carried out at the beginning (end) of theexperiment. The condition in Eq. (4) ensures that the fluc-tuations pW

iCiH →fCfH reproduce the average heat flows ofEq. (2):

∑iC,iH ,fC,fH

pWiCiH →fCfH (EiC − EfC) = Q. (5)

III. A QUASIPROBABILITY FOR QUANTUMHEAT FLUCTUATIONS

How do we choose a quasiprobability distributionamong the many possible candidates? The analogy withquantum optics suggests that different choices will be rel-evant in different scenarios. Nevertheless, one candidate

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

stands out. Recall that the TPM distribution reads

pTPMiCiH →fCfH := Tr

[�fCfH (τ )�iCiHDρCH

], (6)

where �fCfH (τ ) := U†�fCfH U and D denotes dephasing,the operation that removes all off-diagonal elements in theenergy basis. Moreover, the TPM characteristic function isassociated with a two-time correlation function computedon a decohered version DρCH of the initial state ρCH [20].This naturally suggests the reintroduction of coherence andquantum correlations by removing such dephasing:

pWiCiH →fCfH := Re Tr

[�fCfH (τ )�iCiHρCH

], (7)

where we focus here on the real part (the imaginary partmay be of independent interest [34]). This is our proposalfor a quasiprobability distribution for heat fluctuations inthe quantum regime.

Since every ρCH can be additively decomposed as a termDρCH (block-)diagonal in the energy basis plus energeticcoherences χCH , i.e., ρCH = DρCH + χCH , corrections tothe standard scheme emerge only if χCH �= 0:

pWiCiH →fCfH =pTPM

iCiH →fCfH +Re Tr[�fCfH (τ )�iCiHχCH

].

The quantum correction to pTPMiCiH →fCfH

is bounded by‖χCH‖, which is a coherence measure [35,36]. pW satisfiesthe marginal properties of Eq. (4) and hence it repro-duces the quantum heat flow of Eq. (5). As expected,when χCH �= 0, in general pW does not describe a stochas-tic process, since it can be negative: in fact, pW

iCiH →fCfH ∈[−1/8, 1] [37]. It is worth noting that pW recovers thedefinition of a quasiprobability distribution for two non-commuting observables given by Margenau and Hill in1961 [38] and known since the late 1930s [39]. It alsotakes the same form as the work quasiprobability proposedby Allahverdyan in Ref. [11], the negativities of whichwere, however, dismissed as an unphysical feature. Here,we present a different picture.

First, while pW can show negativities, it can beexperimentally accessed in a probing scheme that nat-urally extends the TPM scheme (for more details, seeAppendix A). To estimate pW, one has to replace the firstof the two projective measurements of the TPM schemewith a weak measurement, which is achieved by veryweakly coupling the system to either a continuous-variablepointer or a qubit probe, which is then projectively mea-sured. Appealingly, pW can hence be reconstructed in a“minimally disturbing” version of the TPM scheme.

Second, negativity is a precise signature of nonclassical-ity. In fact, pW

iCiH →fCfHis proportional to a quantity inferred

from weak measurements and known as the (general-ized) weak value [40], first introduced by Aharonov et al.[41,42] and later generalized to mixed states [43,44]. It

then follows from the extensions [6,34] of Pusey’s theorem[45] that the negativities of pW are witnesses of a strongform of nonclassicality known as generalized contextuality[46]. Specifically, the statistics collected by the weak-measurement scheme probing pW cannot be explained byany noncontextual hidden-variable model, a claim valideven in the presence of noise [34]. This is directly related tothe absence of any classical stochastic process explainingthe relevant statistics [47].

Thus far, we showed that pW is a natural extension ofthe TPM scheme, both formally and experimentally; that itreproduces the heat flows of Q, including strong heat backflows due to initial entanglement; and that its negativity is aprecise signature of strong nonclassicality. However, whatis the thermodynamic significance of negativity and howdoes it impact the observed heat flows? We now answerthese questions.

A. Thermodynamic role of negativities in the heatfluctuations

To single out the meaning of negativities, let us split thequasiprobability into positive and negative components:pW

iCiH →fCfH = p+iCiH →fCfH + p−

iCiH →fCfH , where p±iCiH →fCfH =

pWiCiH →fCfH if pW

iCiH →fCfH is positive (negative) and zero oth-erwise. Recall that, with our sign convention, Q > 0 meansback flow. Then, the net flow Q can be decomposed intodirect (H → C) and back (C → H ) flows:

Q = Qback − Qdirect, (8)

where the “back” term includes all contributions to theflow C → H and the “direct” term includes all contribu-tions H → C:

Qback =∑

EiC>EfC

(p+iCiH �fCfH

− p−fCfH �iCiH

)�EiCfC , (9)

Qdirect =∑

EiC>EfC

(p+fCfH �iCiH

− p−iCiH �fCfH

)�EiCfC . (10)

These equations give a very suggestive interpretation ofthe role of negative probabilities in heat flows (see Fig. 1).Since EiC > EfC , Qback has two positive contributions: (1)from the transition iC → fC, which removes energy from C(as expected); and (2) from the transition fC → iC whenp−

fCfH �iCiH< 0. Classically, the transition fC → iC would

add energy to C, yet it contributes to the back flow fromC to H when the correspondent quasiprobability turnsnegative.

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

(a)

Classical Contributions

(b)

Quantum Contributions

ÆÆ

ÆÆ

ÆÆ

ÆÆ

FIG. 1. A schematic illustration of the classical and quantum contributions to the overall heat flow Q between a cold (C) and ahot (H ) system. Q is an average of direct (H → C) and back flows (C → H ). (a) The classical contributions (positive-probabilityevents) correspond to standard contributions, where the loss of energy from H contributes to the flow toward C and vice versa. (b) Thequantum contributions (negative-probability events) effectively act as contributions in which the loss of energy from H contributes tothe flow (C → H ).

Symmetrically, a transition taking energy away fromC can contribute to the direct flow, when p−

iCiH →fCfH< 0.

These counterintuitive heat-flow contributions signal thebreakdown of a stochastic description and the onset ofquantum effects, as depicted in Fig. 1.

In particular, we can understand strong flows as theresult of negative probabilities, which are destroyed bythe measurement stage of the TPM scheme (a projectiveenergy measurement on C and H forces pW > 0). Further-more, Eq. (10) suggests that negativities may lead to strongdirect flows as well.

It is important to bear in mind that negativity will notalways result in strong heat flows. Since only Q = Qback −Qdirect is observed, negativities from Qback and Qdirect mustnot cancel each other if we are going to observe an overalleffect in the heat flow.

B. Classical limit

When discussing quantum effects in heat or work fluctu-ations, it is important to distinguish between two differentforms:

1. Quantumness due to correlations or entanglement inthe initial state, such as that responsible for strong heatback flows.

2. Quantumness due to the superpositions created by Uduring the dynamics.

The first form of nonclassicality is the one that prevents astraightforward description of quantum heat exchanges bymeans of a classical stochastic process. Classicality in thefirst sense emerges in the expected regimes. The typicalmechanism is decoherence. If C and H are not protectedfrom the external environment, superpositions of differentenergy states are quickly suppressed (χCH → 0) and so

pW → pTPM ≥ 0, i.e., negativity disappears. Decoherencemay be also due to the measurement process, which is themechanism that introduces classicality in the TPM scheme.

Another mechanism relevant at mesoscopic scales iscoarse graining. Suppose that one can only access the heatexchanges between two clusters, each containing N par-ticles, but one cannot keep track of the process at thelevel of the individual constituents. We idealize the sit-uation via an average Hamiltonian HCH ≈ ∑N

i=1(HCi +HHi)/N and dynamics U ≈ ⊗N

i=1UCiHi . Then, whateverthe initial state ρCH is, pW ≈ pTPM + O(1/N ) [48]. Theratio between fluctuations and average flows, on theother hand, is suppressed as O(1/

√N ). There is hence

an intermediate “mesoscopic” regime in which quantumeffects are irrelevant but fluctuations are not yet negligi-ble.

The second form of nonclassicality has been studiedin Ref. [49]. In this study, it has been shown that for aninitially uncorrelated bipartite system, energy cannot flowwithout generation of quantum correlations in the form ofdiscord during the unitary process. This form of quantum-ness is the one responsible for the differences between theclassical stochastic process predicted by the TPM schemeand the one obtained within a fully classical description(e.g., a classical-mechanics model), whenever the latter iswell defined. Note that since we recover the TPM probabil-ity in the regimes discussed above, we can exploit previousresults on the emergence of this second form of classicalitywhen the superpositions created by U during the dynamicscan be neglected [50].

We conclude that our definition of heat fluctuationsappears to provide a clear path to the emergence of clas-sicality, from negative quasiprobability all the way up to aclassical-mechanical model, passing through the standardTPM scheme that plays the role of a “quasiclassical”description.

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IV. HEAT FLOWS WITNESSING NEGATIVITIESIN A TWO-QUBIT SYSTEM

By attempting to derive a stochastic interpretation ofquantum fluctuations, we arrive at a picture in which quan-tum effects are incorporated as nonclassical features in theheat exchange, such as contributions to the energy flowfrom C to H associated with transitions to higher levels inC. From a practical perspective, however, the next step isto ask if these nonclassical features can be directly relatedto observable effects and if these effects can be accessedin realistic experiments. We answer both questions in thepositive.

We start with the archetypal (and fully solvable) ideal-ized two-qubit scenario. Since U injects no work, nontriv-ial energy exchanges can only happen if the two qubits areresonant.

Inequality 1. Let ρCH be a two-qubit system with ther-mal marginals (βC �= βH ), HC = HH = E|1〉〈1| and U aunitary injecting no work, [U, HC + HH ] = 0. If pW isnon-negative,

|Q| ≤ 2 + eβH E + eβCE

eβCE − eβH E |QTPM|. (11)

See Appendix B 1 for the proof. This “heat-flow inequal-ity” shows that strong enough direct or back flows of heatcan only occur when negativity is at play in pW. Note thatto violate the bound of Eq. (11), we necessarily witness|Q| > |QTPM|, i.e., a direct or inverse flow bigger than thecorresponding flow observed in the TPM scheme. Whileviolations of Eq. (1), reported in Ref. [51], can occur in apurely classical setup, due to initial correlations in the ener-getic degrees of freedom, violations of Inequality 1 implynegativity, which is a proxy for genuinely quantum effects.

A. Experimental verification

Next, we analyze the violations of Inequality 1. Inrecent NMR experimental setups [51,52], the heat backflow between two local-thermal qubits with initial corre-lations has been measured. Here, we focus on the data ofRef. [51] and show that, while other general approaches(e.g., Ref. [3]) fail to witness quantum effects for this setup,Inequality 1 is violated.

In the experiment, the heat-back-flow Q between twonuclear spin 1/2, prepared in an initial locally thermalstate, has been measured. We denote the Pauli matrices byσi and i = x, y, z. The initial Hamiltonian is HHC = HH +HC, in which HC(H) = hν(1 − σC(H)

z ) and h is Planck’sconstant. A locally thermal state is prepared with the form

ρHC = ρH ⊗ ρC + γ |01〉〈10| + γ ∗|10〉〈01|, (12)

where ρH(C) = e−βH(C)HH(C)/Tr[e−βH(C)HH(C)

]. The unitless

rescaled inverse temperatures reported in the experiment

are βH = 0.9618 and βC = 1.13, with the gap ν = 1 kHz.In the correlated scenario, γ = −0.19, whereas in theuncorrelated scenario γ = 0.

During the process, the nuclear spins are coupled via theinteraction Hamiltonian Hint = Jπ�/2(σH

x σCy − σH

y σCx ),

where J = 215.1 Hz. Since [Hint, HHC] = 0, this gener-ates an energy-preserving unitary. A simple yet importantobservation is the following: the measurement of the heatexchange when γ = 0 corresponds to the measurementof QTPM for the case γ = −0.19. This follows from thefact that the initial energy measurement on C and H (thefirst step of the TPM protocol) has the effect of settingγ = 0. Due to this observation, the data collected in theexperiment suffice to test Inequality 1.

In Fig. 2, we plot the heat transfer between qubits C andH for various interaction times. The green line correspondsto the heat flow when the qubits are initially quantum cor-related, whereas the orange line corresponds to initiallyuncorrelated qubits. Equivalently, this corresponds to themeasurements of Q (green) and QTPM (orange). Q > 0corresponds to back flow from C to H , whereas Q < 0corresponds to direct flow from H to C.The shaded areasindicate the interaction times in which negativity in heat-transfer quasiprobability is present. The dark blue shadedarea indicates the interaction times for which Inequal-ity 1 successfully witnesses negativity for the relevantexperimental parameters.

While in the experiment only the back-flow regime hasbeen explored, which is sufficient for detecting negativ-ity, here we theoretically extend the interaction time to theregime in which direct flow is observed. We see that inthis regime also, the measurement of the heat flows wouldallow us to detect the negativity of the quasiprobability,showing that quantum effects are not only present in backflows. We further extend the investigation of Inequality 1beyond the specific parameters of Ref. [51] (see Fig. 3). We

0 1 2 3 4

–0.5

0.0

0.5

Interaction time (ms)

Hea

t (pe

V)

FIG. 2. The heat transfer Q and QTPM (green and orangecurves, respectively) for the experiment in Ref. [51]. The shadedareas indicate the regime of negative probabilities. The dark blueshaded area indicates the violation of Inequality 1.

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

2.5 3.0 3.5

–0.15

–0.10

–0.05

0.00

q

h

FIG. 3. The region of negative probabilities (large trianguloid)and violations of Inequality 1 (darker trianguloid) in the parame-ter space (θ , η). θ characterizes the unitary evolution (the rotationin the |01〉, |10〉 subspace) and η the initial energetic super-position between |01〉, |10〉. In Appendix B, we show that forfixed βC, βH and a simple reparametrization, the parameter spaceis restricted to (θ , η, P00), where P00 is the population in thestate |00〉. In the experiment reported in Ref. [51], βC = 1.13,βH = 0.962, P00 = 0.547, and η = −0.19, which is indicated bythe dashed line at the bottom of the figure.

conclude that negativity can be detected for a wide rangeof parameters in accessible experiments.

Using the experimental parameters of Ref. [51], thesmallest eigenvalue of the partial transpose ρTH

CH of ρCH

is approximately 0.0014; hence ρTHCH ≥ 0. By the Peres-

Horodecki criterion, ρCH is separable. This shows thatthe heat fluctuations can have negativities, as detected byInequality 1, even when ρCH is not entangled.

We mention in passing a nontrivial result that holdsspecifically for two-qubit systems: the existence of quan-tum correlations in the energy degrees of freedom is nec-essary for witnessing a back flow for appropriate energy-preserving dynamics (see Appendix B). This is the reasonwhy the orange curve in Fig. 2 is all below the x axis.

B. Nonideal heat exchange

While it is commonplace to assume that U injects anamount of work into the system that can be neglected incalculations, strictly speaking, Inequality 1 holds only inan ideal case. Next, we extend Inequality 1 to realistic heat-exchange scenarios, in which (1) the energy levels are notresonant and (2) the dynamics inject some work.

Inequality 2. Let ρCH be a two-qubit system with thermalmarginals (βC �= βH ), HC = EC|1〉〈1|, HH = EH |1〉〈1|and let U denote unitary dynamics. Suppose that

1. U is close to an energy-preserving unitary U :‖U − U‖ ≤ ε.

2. The gap is not too wide: � := |EC − EH |/2E <(1 − R)(1 + R), where E := (EC + EH )/2 is the averageenergy and R = (1 + eβH EH )/(1 + eβCEC).

3. The energy flow is large enough that we can distin-guish a heat contribution: |Q| > 2εE.

If pW is non-negative,

|Q| ≤ 1 + R +�(1 + R)1 − R −�(1 − R)

|QTPM| + 4εE2 +�(1 + R)

1 − R −�(1 − R).

(13)

When ε → 0 and � → 0, we recover Inequality 1. Theabove inequality is suitable to be applied in realistic exper-imental scenarios. Note that the condition of unitarity is notcrucial and that Inequality 2 can be extended to quantumchannels. The proof of this inequality, stronger bounds, anda case study are presented in Appendix B 2. This showsthat analog heat-flow inequalities can be derived beyondthe ideal regime.

V. A QUANTUM-EXCHANGE-FLUCTUATIONTHEOREM

The above heat-flow inequality is limited to two-qubitsystems. We now derive a heat-flow inequality validfor arbitrary finite-dimensional systems. This is obtainedthrough the derivation of a QXFT in the spirit of Ref. [11,53].

Denote by �I := IfCfH − IiCiH , with

IiCiH = logTr[�iCiHρCH

]Tr[�iCρC

]Tr[�iHρH

] , (14)

the elements of the classical mutual information.Assuming that EiC − EfC ≈ EfH − EiH for all nonzeropW

iCiH →fCfH , we obtain the XFT in the full quantum regime

⟨e�I+�β�EiCfC

⟩W

= 1 + χ . (15)

The right-hand side of Eq. (15) consists of 1, the contribu-tion from the populations, and χ , involving contributionsfrom the coherence terms in the energy eigenbasis

χ =∑l,k �=m

ρll

ρkkRe{ρkm〈l|U|k〉〈m|U†|l〉}, (16)

which are absent in the TPM scheme. Note the following:

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

1. If ρCH is classically correlated in the energy basis,χ = 0, recovering the main result of Ref. [4].

2. If, furthermore, ρCH is uncorrelated, �I = 0 and werecover Jarzynski’s original result [1].

Equation (15) is thus the generalization of the XFT to thefull quantum regime. The proof of the quantum XFT isprovided in Appendix C.

A. Witnessing negativity in an arbitrary dimension

Using the quantum XFT, we derive a general inequalitywitnessing negativity in any dimension.

Inequality 3. Let ρCH be an arbitrary finite-dimensionalsystem with thermal marginals (βC �= βH ) and let U bea unitary that injects no work (EiC − EfC = EfH − EiH ). IfpW is non-negative,

Q ≤ −〈�I〉W

�β+ log(1 + χ)

�β. (17)

This result can be derived by applying Jensen’s inequal-ity to Eq. (15). 〈�I〉W in Eq. (17) is a classical mutualinformationlike term satisfying 〈�I〉W = 〈�I〉TPM if theinitial state only has classical correlations in the energybasis and 〈�I〉W = 0 in the absence of initial quantumand classical correlations [54]. Violations of the inequal-ity imply that some quasiprobabilities are necessarilynegative. Note that the inequality becomes trivial if noinitial coherence is present (χCH = 0), as expected. InAppendix C, we provide a second heat-flow inequality thatis valid for general finite-dimensional systems.

For finite-dimensional systems with equal Hamilto-nians and nondegenerate energy gaps, we prove inAppendix C an additional generic property of negativity:if all quasiprobabilities that contribute to direct or backflow are negative, then the flows are necessarily strong,i.e., |Q| > |QTPM|.

B. Nonideal heat exchange

Inequality 3 can be generalized to situations in whichthe injected work cannot be neglected. If, for allnonzero pW

iCiH →fCfH , one has |EiC − EfC − (EfH − EiH )| ≤ε, almost the same derivation allows us to generalizeEq. (17) to

pWnon-negative ⇒ Q ≤ −〈�I〉W

�β+ log(1 + χ)

�β− βHε

�β.

(18)

A symmetric inequality can be derived for the energyflowing into C.

C. Case study: heat exchange between two qutrits

In this section, we investigate the heat flows betweentwo qutrits and show that Inequality 3 can witness neg-ativity in these higher-dimensional systems as well. TheHamiltonian of the qutrits is assumed to take the formHC = HH = ∑2

n=0 En|n〉〈n|, where E0 = 0 and there is nodegeneracy of the energy gaps (the “Bohr spectrum”). Thestate ρCH = ∑8

i,j =0 ρij |i〉〈j | can be expressed by the ele-ments ρij = ρ∗

ji = ηij e−iξij √ρiρj ∀ i �= j , the populationρi ≡ ρii, and the parameters ηij ∈ [0, 1] and ξij ∈ R. Here,we use the natural labeling (00, 01, 02, 10, 11, 12, . . .) ≡(0, 1, 2, 3, 4, 5, . . .).

By imposing the constraint that the marginal statesTrC [ρCH ] and TrH [ρCH ] are thermal, and following thesame reasoning as in the two-qubit scenario, heat flowsbetween two qutrits are determined by: the six param-eters ηij , ξij for (i, j ) = {(01, 10) ≡ (1, 3), (02, 20) ≡(2, 6), (12, 21) ≡ (5, 7)}; and four undetermined popula-tions ρi that must comply with the non-negativity of ρCH .Furthermore, all possible heat flows achievable by energy-preserving unitaries can be realized by U = ⊕3

i=1U(i),where U(i) are real 2d rotations within the three above-mentioned manifolds, fixed by three angles θ01, θ02, andθ12. In Appendix C, we provide the explicit structure ofthe state ρCH and the energy-preserving unitary. In Fig. 4,we plot the regions in parameter space θ01 and θ02 in

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

q01

q 02

FIG. 4. The negativity of the quasiprobabilities (all shadedareas) and the violation of Inequality 4 (yellow shaded area) fordifferent interaction protocols determined by θ01 and θ02. Param-eters: βH = 0.3, βC = 1.3, θ12 = θ02, E1 = 1, E2 = 1.15, ξij =0, ηij = 1 for (i, j ) = {(1, 3), (2, 6), (5, 7)}, ρ0 = 0.3, ρ5 = 0.03,ρ7 = 0.07, and ρ8 = 0.06.

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

which these probabilities turn negative (all shaded areas).The yellow shaded area indicates the regions in whichInequality 3 is being violated. This clearly shows thatour method also allows us to detect negativities at higherdimensions. A more detailed analysis and a straightforwardgeneralization to an arbitrary dimension are presented inAppendix C.

VI. WITNESSING NEGATIVITY WITHPROJECTIVE ENERGY MEASUREMENTS IN AN

ARBITRARY DIMENSION

The testing of violations of Eq. (17) is a way to probethe fully quantum regime in arbitrary finite-dimensionalsystems. However, it has a significant drawback of beingexperimentally very demanding, as χ depends on all theentries in the initial state, as well as the details of thedynamics. Standard fluctuation-theorem experiments, onthe other hand, involve only projective energy measure-ments. The drawback of these is the inability to probequantum effects due to initial quantum correlations andenergy coherence. Can our method be adapted to obtaina scheme satisfying both desiderata, that is, (1) probing thefully quantum regime and (2) using only the experimen-tal capabilities required by the traditional TPM quantumfluctuation theorems? Here, we answer this question in thepositive.

We consider two d-dimensional systems and for sim-plicity we assume that there are no matching gaps in theirenergy spectra (formally, the “nondegenerate Bohr spec-trum”). The most general energy-preserving unitary is thena direct product of two-dimensional rotations. This allowsus to derive the following result (see Appendix C for theproof).

Inequality 4. Consider two arbitrary finite-dimensionalsystems with nondegenerate Bohr spectra undergoing anenergy-preserving unitary. If pW is non-negative, then

Q ∈ [QTPM − 2λ−, QTPM + 2λ+]

, (19)

where

λ− =∑

EiC>EfC

pTPMiCiH →fCfH�EiCfC , (20)

λ+ =∑

EiC<EfC

pTPMiCiH →fCfH�EfCiC . (21)

and �Enm := En − Em.Remarkably, to confirm a violation of Eq. (19) and

witness negativity, it suffices to have the experimentalcapability of performing projective energy measurements.In fact, λ− and λ+ and QTPM can all be inferred from

the data already required to test standard TPM fluctua-tion theorems. To infer Q, one can prepare N samplesof the initial state and use half to estimate the averageenergy before the dynamics and half to estimate the aver-age energy after the dynamics. The difference between thetwo is the heat Q. In other words, if one can protect theinitial state from decoherence, negativity can be witnessedin any of the platforms used to test TPM fluctuation theo-rems. It is worth noting that Inequality 4 does not requirelocal thermality. In other words, it can also be tested mea-suring energy exchange between two quantum systems ina generic state.

To illustrate this result, we consider the two-qutrits casestudy of Sec. V C. Figure 5 presents the negativity ofthe quasiprobability related to heat exchange (all shadedareas) and the violation of the Inequality 4 (green and yel-low areas) for different interaction protocols determinedby the unitary angles θ01 and θ02. The different insets rep-resent different initial states, determined by the parameterη = ηij for (i, j ) = {(1, 3), (2, 6), (5, 7)} that quantifies theamount of initial quantum correlations between the states.The yellow area represents violation of the lower bound

0 1 2 3 4 5 6 701234567

q 02

h = 1

0 1 2 3 4 5 6 701234567

h = 0.6

0 1 2 3 4 5 6 701234567

q01

q 02

h = 0.3

0 1 2 3 4 5 6 701234567

q01

h = 0.05

FIG. 5. The negativity of the quasiprobabilities (all shadedareas) and the violation of Inequality 4 (yellow and green shadedarea) for different interaction protocols determined by θ01 and θ02.The insets shows different initial states, determined by the param-eter η = ηij for (i, j ) = {(1, 3), (2, 6), (5, 7)} Parameters: βH =0.3, βC = 1.3, θ12 = θ02, E1 = 1, E2 = 1.15, ξij = 0, ρ0 = 0.3,ρ5 = 0.03, ρ7 = 0.07, and ρ8 = 0.06.

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

of Inequality 4 and corresponds to negativity in the directflow as illustrated in the quantum contributions of Fig. 1(upper-right process, red arrows). The green area, on theother hand, represents violations of the upper bound ofInequality 4, which correspond to negativity in the backflow as illustrated in the quantum contributions of Fig. 1(lower-right process, blue arrows). The upper-left inset ofFig. (5) with η = 1 corresponds to the parameters used forFig. 4. This indicates that Inequalities 3 and 4 can wit-ness negativity in different regimes. As already remarked,however, the testing of Inequality 4 is much simpler.

VII. OUTLOOK

Recent no-go results [5,6] strongly suggest that, if weare to probe the fully quantum thermodynamic regime, dueto noncommutativity we need to renounce a straightfor-ward statistical interpretation of microscopic fluctuationprocesses. Moving in this direction, here we introduce anatural quasiprobability for heat fluctuations the negativityof which captures nonclassical effects in the strong formof contextuality. This formally implies the impossibilityof describing the weak-measurement scheme probing pW

as the result of any underlying classical stochastic pro-cess. Differently from previous approaches, we show thatnegativities in the heat fluctuations have a thermodynamicinterpretation and that certain heat flows between twolocally thermal states at different temperatures can onlyhappen in their presence. To prove the latter, we developfamilies of heat-flow inequalities that can only be violatedin the presence of negativities. We use these tools to wit-ness nonclassicality by analyzing data collected in a recenttwo-qubit experiment. We also present heat-flow inequal-ities for generic finite-dimensional systems, the testing ofwhich only requires the control already needed to test stan-dard TPM fluctuation theorems, with the advantage thatone is testing the fully quantum regime.

Moving forward, we propose that a quasiprobabilityapproach to fluctuations can advance our understanding ofthe thermodynamic behavior of realistic devices and thestudy of heat flows in more complex networks. Classicallyunachievable heat-flow configurations can also be used asa witness of nonclassicality in thermal machines and couldunlock performance boosts in thermodynamic protocols.We also expect that the same ideas as developed here canbe applied to work fluctuations as well. Building on theseminal paper of Allahverdyan [11], this may provide themost direct experimental test of the framework developedhere.

More broadly, this work also opens up the possibility ofapplying to nonequilibrium thermodynamics the extensivetechnical tools based on quasiprobability representationsdeveloped to identify quantum effects and advantages inquantum optics, quantum foundations, quantum comput-ing, and condensed-matter physics.

ACKNOWLEDGMENTS

We thank David Jennings and Karen Hovhannisyanfor helpful discussions. M.L. thanks Eran Rabani for thekind hospitality during a visit to University of Califor-nia, Berkeley, where part of this work was completed.M.L. acknowledges financial support from the EuropeanUnion’s Marie Sklodowska-Curie individual Fellowships(Grants No. H2020-MSCA-IF-2017 and No. GA794842),the Spanish Ministerio de Econom y Competitividad(Ministry of Economy and Competitiveness, MINECO)(Severo Ochoa Grant No. SEV-2015-0522 and ProjectNo. QIBEQI FIS2016-80773-P), the Fundacio Cellex andthe Generalitat de Catalunya (Centres de Recerca deCatalunya, CERCA Program and SGR 1381). This work ispart of the “Photonics at Thermodynamic Limits” EnergyFrontier Research Center funded by the U.S. Departmentof Energy, Office of Science, Office of Basic EnergySciences under Award No. DE-SC0019140.

A.L. and M.L. contributed equally to this work.

APPENDIX A: ESTIMATING THEMARGENAU-HILL QUASIPROBABILITY

a. Traditional scheme

The scheme for measuring pWiCiH →fCfH consists of a

weak measurement at the start of the protocol and a projec-tive measurement at the end. Specifically, define a familyof measurement schemes where at the start the system pro-jectors �iCiH are coupled for a unit time to the momentumP of a one-dimensional pointer device through the inter-action Hamiltonian �iCiH ⊗ P. The pointer is initially ina pure state (πs2)−1/4

∫dxe−x2/2s2 |x〉. Then the dynamics,

represented by some unitary U, take place on CH . At theend, a final projective energy measurement is performedon CH and outcome (fC, fH ) is observed with probabilityqfCfH .

One can then verify that the expected position of thepointer given that some energies (fC, fH ) are observed inthe final energy measurement, denoted by 〈X 〉|fCfH , can bedirectly related to pW

iCiH →fCfH in the weak-measurementlimit s → ∞ (large initial spread of the pointer):

〈X 〉|fCfH qfCfHs→∞−→ pW

iCiH →fCfH . (A1)

The same expression gives pTPMiCiH →fCfH if s → 0 (very

sharp pointer). For a derivation see, for example, AppendixB of Ref. [6]. In this sense, the protocol probingpW

iCiH →fCfH can be understood as a “minimally invasive”version of the standard TPM scheme.

b. Qubit probe

An alternative scheme to estimate pWiCiH →fCfH uses only

a qubit pointer rather than a continuous system [34,56]

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

and, for completeness, is briefly discussed here. Take aqubit ancilla in a state |ψε〉 = cos ε|0〉 − sin ε|1〉. Cou-ple the system and the ancilla through the unitary V =�iCiH ⊥ ⊗ I +�iCiH ⊗ σz (generated by the HamiltonianHint = �iCiH ⊗ |1〉〈1| by V = e−igHintt, setting t = π/g).The ancilla is then measured in the |±〉 = (|0〉 ± |1〉)/√2basis. Denoting the two outcomes by ±, the correspondingKraus operators on the system are

M± = 〈±|V|ψε〉 = cos ε√2

I ± sin ε√2(�iCiH −�iCiH ⊥),

(A2)

with the correspondent positive-operator-valued measure-ment (POVM) on the system being

E± = M †±M± = (1 ∓ sin 2ε)

I

2± sin 2ε�iCiH . (A3)

Denote the joint probability of observing outcome + on thepointer and outcome (fC, fH ) on the system by qfCfH ,+(ε).Also define, with obvious notation, qfCfH ,−(ε). Then, if�qfCfH (ε) := qfCfH ,+(ε)− qfCfH ,−(ε),

�qfCfH (ε)

= Tr[�fCfH (τ )⊗ |+〉〈+|V(ρCH ⊗ |ψε〉〈ψε |)V†]

− Tr[�fCfH (τ )⊗ |−〉〈−|V(ρCH ⊗ |ψε〉〈ψε |)V†]

= sin 2ε(2pWiCiH →fCfH − pfCfH ),

where�fCfH (τ )= U†�fCfH U and pfCfH = Tr[�fCfH (τ )ρCH

]is the probability of observing energy outcomes (fCfH ) ifno measurement scheme is performed (V = I). Clearly,

pWiCiH →fCfH = �qfCfH (ε)

2 sin 2ε+ pfCfH

2. (A4)

This gives a way of reconstructing pWiCiH →fCfH from the

joint statistics on the pointer and the system in the above-mentioned measurement scheme, together with the proba-bilities pfCfH that can be inferred from a second experimentwhere no measurement scheme is applied. Note that, giventhe knowledge of ε from the initialization of the ancilla,one can reconstruct pW

iCiH →fCfH without taking the limitε → 0.

APPENDIX B: HEAT FLOW BETWEEN TWOQUBITS

a. Reduction of the parameters

The most general two-qubit unitary satisfying energyconservation takes the form

U =

⎛⎜⎜⎝

1 0 0 00 ei(κ+λ) cos θ −ei(κ−φ) sin θ 00 ei(κ+φ) sin θ ei(κ−λ) cos θ 00 0 0 1

⎞⎟⎟⎠ . (B1)

Let D be the global dephasing operation in the energybasis,

D(·) =∑

E∈spec(HC+HH )

�E(·)�E , (B2)

where spec(X ) is the spectrum of X and �E is the pro-jector on the eigenspace of energy E. One can verify that,for all U with [U, HC + HH ] = 0, neither pTPM nor pW areaffected by the application of D on the initial state ρCH . Forthe two-qubit case this implies that, without loss of gener-ality, we can take the state with thermal marginals to havethe form

ρCH =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

P00 0 0 0

01zC

− P00 ηeiξ 0

0 ηe−iξ 1zH

− P00 0

0 0 0zCzH − zC − zH

zCzH+ P00

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,

(B3)

where zC(H) = 1 + e−βC(H) . The two-qubit case must beresonant to have nontrivial dynamics, due to the fact that Uconserves energy (we discuss how to weaken this assump-tion later). We then set HC = HH = |1〉〈1| by renormaliz-ing the temperature. Since ρCH has to be a valid densityoperator, P00 must satisfy

P00 ≤ 1zH

, (B4)

P00 ≥ zC + zH − zCzH

zCzH. (B5)

The MH and the TPM probabilities satisfy

pW01,10 = pTPM

01,10 + η cos θ sin θ cos ξ ,

pW10,01 = pTPM

10,01 − η cos θ sin θ cos ξ , (B6)

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

pTPM01,10 =

(1zC

− P00

)sin2 θ ,

pTPM10,01 =

(1

zH− P00

)sin2 θ . (B7)

The probabilities pW and pTPM are independent of κ;hence we set κ = 0. Furthermore, pTPM are independentof λ, ξ , and φ, while pW only depends upon them bya factor cos(λ+ ξ + φ). A simple reparametrization thenallows one to set λ = φ = 0. One can also set ξ = 0

by a redefinition of η. We can always take η ≥ 0 by areparametrization of θ (if η ≤ 0 map θ �→ −θ ). In themain text, we allow η ∈ R for an easier comparison to theexperimental parameters of Ref. [51]. The final forms are

U =

⎛⎜⎝

1 0 0 00 cos(θ) − sin(θ) 00 sin(θ) cos(θ) 00 0 0 1

⎞⎟⎠ , (B8)

ρCH =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

P00 0 0 0

01zC

− P00 η 0

0 η1

zH− P00 0

0 0 0zCzH − zC − zH

zCzH+ P00

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, (B9)

where zC(H) = 1 + e−βC(H) and η ≥ 0.A direct computation of the heat

Q := Tr [ρCH HC] − Tr[UρCH U†HC

]

returns

Q = −η cos ξ sin 2θ + sin2 θ

(1

1 + eβC− 1

1 + eβH

).

(B10)

Note that Q does not depend on P00 and that ρCH ≥ 0implies |η| ≤ √

(1/zC − P00)(1/zH − P00).Concerning QTPM,

QTPM := Tr [D(ρCH )HC])− Tr[UD(ρCH )U†HC

].

We have

QTPM = sin2 θ

(1

1 + eβC− 1

1 + eβH

). (B11)

Hence QTPM ≤ 0, i.e., for the two-qubit case, no back flowexists in the TPM scheme. Note that, as is well known,Q(η = 0) = QTPM. That is, the average heat flows of theTPM scheme coincide with that of a state from which allinitial coherence has been removed.

b. Back flow and quantum correlations in two-qubitsystems

Next, we show that for a state that is diagonal in theenergy basis (even if it includes classical correlations), no

heat back flow can be observed. This can be checked bysimply replacing η = 0 in Eq. (B10), which gives

Q(η = 0) = QTPM ≤ 0. (B12)

Hence, η �= 0 is necessary for back flow.

1. Proof of Inequality 1

Assume that we observe a direct flow, i.e., Q < 0. FromEqs. (B6) and (B7), this that implies θ �= 0. Since we canthen restrict to θ ∈ (0,π), Eq. (B12) and β1 �= β2 implyQTPM < 0. Hence, we can define α := Q/QTPM. FromEq. (B6), we obtain

2pW10,01 = (α + 1)pTPM

10,01 − (α − 1)pTPM01,10. (B13)

We have α > 0 and, due to Eq. (B4), pTPM01,10 > 0. Then

pW10,01 ≥ 0 implies that

pTPM10,01

pTPM01,10

≥ α − 1α + 1

= Q − QTPM

Q + QTPM . (B14)

We further note that, from Eq. (B7),

pTPM10,01

pTPM01,10

= zC(1 − zH P00)

zH (1 − zCP00)≤ 1 + eβH

1 + eβC, (B15)

where equality is obtained for the minimal value of P00 inEq. (B5). From Eqs. (B14) and (B15), we have

1 + eβH

1 + eβC≥ Q − QTPM

Q + QTPM , (B16)

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

which is the statement of Inequality 1 for Q < 0.In a similar manner, we can repeat the calculations for

Q > 0. In this case,

2pW01,10 = (α + 1)pTPM

01,10 − (α − 1)pTPM10,01. (B17)

Again, we have QTPM < 0, which implies α < 0, andpTPM

10,01 > 0. Hence pW01,10 ≥ 0 implies that

pTPM10,01

pTPM01,10

≥ α + 1α − 1

= Q + QTPM

Q − QTPM . (B18)

From Eqs. (B15) and (B18),

1 + eβH

1 + eβC≤ Q + QTPM

Q − QTPM . (B19)

This is Inequality 1 in the case Q > 0.

2. Nonideal heat exchange: Inequality 2

a. Assumptions and some consequences

We find the tolerance of Inequality 1 to imperfections.The relevant assumptions are as follows:

A1 Rather than assuming that the dynamics U on CHsatisfy [U, HCH ] = 0 (no work injected), we only assumethat there exists some U close to U (‖U − U‖ ≤ ε) thatsatisfies [U, HCH ] = 0 (small work injected).

A2 In the setting where no work is injected, the twolocal thermal states support heat exchanges only if theyare resonant. We drop this assumption and define the localHamiltonians HC(H) = EC(H)|1〉〈1|, where EH − EC = δ ishow much they are off resonance. We assume that this gapis not too large: if the average energy is E = 1

2 (EH + EC),

� ≡ |δ|2E

≤ eβCEC − eβH EH

2 + eβCEC + eβH EH≡ 1 − R

1 + R, (B20)

where we define by R the ratio R = (1 + eβH EH )/(1 +eβCEC).

A3 Since some work is now injected, we will only con-sider energy flows that are far enough from zero that theycannot be simply due to work being injected directly intothe local system by the driving. Our conclusions will hencebe valid only for dynamics U inducing energy flows

|Q(U)| ≥ 2Eε. (B21)

Above, ‖ · ‖ denotes the Shatten infinite norm

‖A‖ := max|v〉 s.t. ‖v‖=1

‖A|v〉‖. (B22)

We will also use the Shatten 1-norm or trace norm

‖A‖1 := Tr[√

A†A]

. (B23)

Let us start with some inequalities that will be useful laterand then deal with the two cases (direct flow and backflow) separately.

Using Hölder’s inequality |Tr[B†A

] | ≤ ‖B‖‖A‖1 andthe submultiplicativity of the norm ‖A‖1 ,

|pWiCiH →fCfH (U)− pW

iCiH →fCfH (U)|=∣∣∣ReTr

[U†(�fCfH )U�iCiHρCH

]

− ReTr[U†(�fCfH )U�iCiHρCH

] ∣∣∣≤ 1

2‖U† − U†‖‖�fCfH U(�iCiHρCH + ρCH�

iCiH )‖1

+ 12‖U − U‖‖�fCfH U†(�iCiHρCH + ρCH�

iCiH )‖1

≤ 2‖U − U‖‖�fCfH ‖1‖U†‖1‖�iCiH ‖1‖ρCH‖1

≤ 2‖U − U‖ ≤ 2ε. (B24)

In a similar manner, one can bound

|pTPMiCiH →fCfH (U)− pTPM

iCiH →fCfH (U)| ≤ 2ε. (B25)

Define pW ≡ pW(U) and pTPM ≡ pTPM(U). Using Eq. (B6),the heat under the energy-preserving unitary U can beexpressed as

Q(U) = pW10,01EH − pW

01,10EC (B26)

= (pW10,01 − pW

01,10)E + 12(pW

10,01 + pW01,10)δ

= (pW10,01 − pW

01,10)E

+ 12(pTPM

10,01 + pTPM01,10)δ,

QTPM(U) = (pTPM10,01 − pTPM

01,10)E

+ 12(pTPM

10,01 + pTPM01,10)δ.

From Eqs. (B24)–(B26),

|Q(U)− Q(U)| ≤ 2Eε,

|QTPM(U)− QTPM(U)| ≤ 2Eε. (B27)

We will now study two cases separately.

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

b. Direct flow

Following the same procedure as in Sec. 1, some algebragives

2pW10,01 = (α + 1)pTPM

10,01 − (α − 1)

×(

pTPM01,10 − δ

2E(pTPM

10,01 + pTPM01,10)

). (B28)

Using assumption (A3) and the fact that Q(U) ≤ 0 (directflow),

Q(U)+ 2Eε ≤ 0. (B29)

This and Eq. (B27) imply Q(U) ≤ 0 and so α :=Q(U)/QTPM(U) ≥ 0. Assuming that pW ≥ 0, Eq. (B24)implies 2pW

10,01 + 4ε ≥ 0. This allows us to derive fromEq. (B28) the inequality

pTPM10,01

pTPM01,10

− (α − 1)α + 1

+ (α − 1)α + 1

δ

2E

(1 + pTPM

10,01

pTPM01,10

)

+ 4ε(α + 1)pTPM

01,10≥ 0. (B30)

Recalling from Eq. (B15) that pTPM10,01/p

TPM01,10 ≤ (1 + eβH EH )/

(1 + eβCEC) ≡ R,

Q(U) ≥ 1 + R −�(1 + R)1 − R −�(1 + R)

QTPM(U)

+ 4ε1 −�(1 + R)

1 − R −�(1 + R)QTPM(U)

pTPM01,10

≥ 1 + R −�(1 + R)1 − R −�(1 + R)

QTPM(U)

− 4Eε(1 −�(1 + R))(1 +�)

1 − R −�(1 + R). (B31)

In the first inequality, we use assumption (A2) and inthe second we use the relation QTPM(U)/pTPM

01,10 ≥ −E(1 +�). Finally, expressing the results for the nonideal heatexchange, we apply Eq. (B27) to the last inequality andobtain for the direct flow

Q(U) ≥ 1 + R −�(1 + R)1 − R −�(1 + R)

QTPM(U)

− 4Eε(1 −�(1 + R))(2 −�)

1 − R −�(1 + R). (B32)

Note also that this weaker inequality holds:

Q(U) ≥ 1 + R −�(1 + R)1 − R −�(1 + R)

QTPM(U)

− 4Eε2 +�(1 + R)

1 − R −�(1 + R). (B33)

c. Back flow

Following a procedure as in Sec. 1, some algebra gives

2pW01,10 = (α + 1)pTPM

01,10 − (α − 1)

×(

pTPM10,01 + δ

2E(pTPM

01,10 + pTPM10,01)

), (B34)

Using assumption (A3) and the fact that Q(U) ≥ 0 (backflow),

Q(U)− 2Eε ≥ 0. (B35)

This and Eq. (B27) imply that Q(U) ≥ 0 and α :=Q(U)/QTPM(U) ≤ 0. Assuming that pW ≥ 0, we have2pW

01,10 + 4ε ≥ 0. This relation gives

pTPM10,01

pTPM01,10

− α + 1α − 1

+ δ

2E

(1 + pTPM

10,01

pTPM01,10

)

+ 4ε(α − 1)pTPM

01,10≥ 0. (B36)

As before, since pTPM10,01/p

TPM01,10 ≤ R and using assumption

(A2), some algebra gives

Q(U) ≤ −1 + R +�(1 + R)1 − R −�(1 + R)

QTPM(U)

− 4ε1

1 − R −�(1 + R)QTPM(U)

pTPM01,10

(B37)

≤ 1 + R +�(1 + R)1 − R −�(1 + R)

QTPM(U)

+ 4Eε1 − R +�(1 + R)1 − R −�(1 + R)

,

where we use the relation −QTPM(U)/pTPM01,10 ≤ E(1 − R +

�(1 + R)). Applying Eq. (B27),

Q(U) ≤ −1 + R +�(1 + R)1 − R −�(1 + R)

QTPM(U)

+ 4Eε(2 − R +�(1 + R))1 − R −�(1 + R)

. (B38)

While the two separate inequalities are stronger, the direct-and back-flow inequalities for nonideal heat exchange inEqs. (B33) and (B38) can also be cast into a compact,symmetric representation:

|Q| ≤ 1 + R +�(1 + R)1 − R −�(1 + R)

|QTPM|

+ 4Eε2 +�(1 + R)

1 − R −�(1 + R). (B39)

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

0 0.002 0.004 0.0060

0.08

0.16

0.24

e

D

FIG. 6. The shaded area indicates the parameter regime inwhich Inequality 2 can be violated, i.e., negativity can bedetected for nonideal heat-exchange processes. Here, ωC = 1.2kHz, J = 220 Hz, TC = 3.48 · 10−8 K, TH = 1.74 · 10−7 K, γ =−0.19, and t = 4 ms.

3. Nonideal heat exchange: example

We study the violation of Inequality 2. Our starting pointis the setup considered in the experiment in Ref. [51] withtwo modifications. First, we add a small perturbation, suchthat the unitary is no longer energy preserving. In particu-lar, we add an interaction Hamiltonian of the form Jxσ

Hx σ

Cx

, where Jx determines the distance between the unitaries ε.Second, the two qubits have different Hamiltonians that arecharacterized by � = (ωH − ωC)/(ωH + ωC). In Fig. 6,we present the tolerance of our inequality to nonidealheat exchange. The shaded area indicates the violation ofInequality 2 as a function of ε and �.

APPENDIX C: EXTENSIONS TO ARBITRARYFINITE-DIMENSIONAL SYSTEMS

1. Proof of the Quantum XFT

Here, we provide a proof of the quantum XFT in themain text. Traditionally, fluctuation theorems are derivedby considering the ratio of the forward- and backward-process probabilities [1,4]. Hence, we consider

pWiCiH →fCfH

pWfCfH →iCiH

= ReTr[U†(�fCfH )ρ iCiH

]ReTr

[U(�iCiH )ρ fCfH] e−�I−�β�EiCfC ,

(C1)

where pWfCfH →iCiH

:= ReTr[U(�iCiH )�fCfHρCH

], ρ iCiH =

�iCiHρCH/Tr[�iCiHρCH

], and �I := IfCfH − IiCiH , with

IiCiH = log Tr[�iCiHρCH

]/Tr

[�iCρC

]Tr[�iHρH

],(C2)

the elements of the classical mutual information. Notethat we assume that EiC − EfC ≈ EfH − EiH for allnonzero pW

iCiH →fCfH . One can then obtain the quan-tum XFT from Eq. (C1) by multiplying both sides bypW

fCfH →iCiH e�I+�β�EiCfC and summing over all indexes.

2. Alternative bound to Inequality 3

Here, we derive an alternative bound (Inequality 4) forthe detection of negativity. However, we note that, for allthe case studies considered, we find Inequality 3 to besuperior to Inequality 4.

A direct calculation of∑

piCiH →fCfHe�β�EiCfC leads to

the following:

〈e�β�EiCfC〉 = 1 + J (C3)

= 1 + ReTr[U†(ρC ⊗ ρH )(c(ρ)+ q(ρ))

],

where

c(ρ) =∑iCiH

�iCiH (ρCH − ρC ⊗ ρH )�iCiH

Tr[�iCρC

]Tr[�iHρH

] , (C4)

q(ρ) =∑iCiH

�iCiHρCH�iCiH ⊥

Tr[�iCρC

]Tr[�iHρH

] . (C5)

Note that c(ρCH ) = 0 if the populations of ρCH coincidewith those of ρC ⊗ ρH , even if ρCH has coherence. Thisis why we use the notation “c” to indicate “classical cor-relations.” On the other hand, q(ρCH ) = 0 whenever thereis no coherence, even in the presence of classical correla-tions; hence the notation “q.” The term 1 in Eq. (C3) isobtained noting that

ReTr

⎡⎣U†(ρC ⊗ ρH )

⎛⎝∑

iCiH

�iCiH (ρC ⊗ ρH )�iCiH

Tr[�iCρC]Tr[�iHρH ]

⎞⎠⎤⎦

= ReTr

⎡⎣U†(ρC ⊗ ρH )

⎛⎝∑

iCiH

�iCiH

⎞⎠⎤⎦ = 1.

Inequality 5. Let ρCH be an arbitrary finite-dimensionalsystem with thermal marginals (βC �= βH ) and let U bean energy-preserving unitary, [U, HC + HH ] = 0. If pW isnon-negative,

Q ≤ (�β)−1 log(1 + J ). (C6)

The result is an application of Jensen’s inequality toEq. (C3). Using norm inequalities, it can be shown that

J ≤ ‖c(ρCH )‖ + ‖q(ρCH )‖, (C7)

so the right-hand side is finite.

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

3. Heat flow between two qudits

To study the heat flow between two qudits, we firstconsider the case of two qutrits, which can then be gen-eralized easily to d-dimensional systems. Setting E0 =0, we take HC = HH = ∑2

n=1 En|n〉〈n| and assume no

degeneracy of the energy gaps (the “Bohr spectrum”)throughout this section (see Fig. 7). By the same reasoningas in the two-qubit scenario, for the purpose of studyingheat flows we can reduce a general two-qutrit state to theform

ρCH =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ρ0 0 0 0 0 0 0 0 00 ρ1 0 ρ13 0 0 0 0 00 0 ρ2 0 0 0 ρ26 0 00 ρ31 0 ρ3 0 0 0 0 00 0 0 0 ρ4 0 0 0 00 0 0 0 0 ρ5 0 ρ57 00 0 ρ62 0 0 0 ρ6 0 00 0 0 0 0 ρ75 0 ρ7 00 0 0 0 0 0 0 0 ρ8

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (C8)

where ρij = ρ∗ji = ηij e−iξij √ρiρj , with ηij ∈ [0, 1], ξij ∈ R, and we use the natural labeling (00, 01, 02, 10, 11, 12, . . .)

≡ (0, 1, 2, 3, 4, 5, . . .). If we impose the constraints Tr [ρCH ] = 1 and TrC(H) [ρCH ] = ρH(C), with

ρX =⎛⎝ 1 0 0

0 e−βX E1 00 0 e−βX E2

⎞⎠Tr [ρX ]−1 , (C9)

we are left with the free parameters ηij , ξij and four undetermined populations ρi that must comply with the non-negativityof ρCH .

The general energy-preserving unitary takes the form

U =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0 0 0 00 ei(κ1+λ01)c01 0 −ei(κ1−φ01)s01 0 0 0 0 00 0 ei(κ2+λ02)c02 0 0 0 −ei(κ2−φ02)s02 0 00 ei(κ1+φ01)s01 0 ei(κ1−λ01)c01 0 0 0 0 00 0 0 0 eiκ3 0 0 0 00 0 0 0 0 ei(κ4+λ12)c12 0 −ei(κ4−φ12)s12 00 0 ei(κ2+φ02)s02 0 0 0 ei(κ2−λ02)c02 0 00 0 0 0 0 ei(κ4−φ12)s12 0 ei(κ4−λ12)c12 00 0 0 0 0 0 0 0 eiκ5

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(C10)

FIG. 7. A schematic of the energy levels of the two qudits andthe couplings between the two.

Here, we use the short-cut notation cnm := cos(θnm) andsnm := sin(θnmt), where θnm = θmn, φnm = φmn and λnm =λmn are parameters characterizing the coupling betweenthe nth level of the first qutrit and the mth levelof the second qutrit. Within the manifolds (01,02,12),κ are global phases that do not change the energy-transfer transition probabilities. One can note the similaritybetween the U for the qutrits with that of the qubits.The heat-flow probabilities can be split into contribu-tions from independent manifolds (01,02,12) that possessthe same structure as seen in Eq. (B1). The general-ization to higher dimensions is now straightforward, as

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QUASIPROBABILITY DISTRIBUTION FOR HEAT... PRX QUANTUM 1, 010309 (2020)

the structure of ρCH and U in Eqs. (C8) and (C10) ispreserved.

We can express the transition probability that is relatedto heat transfer for arbitrary finite dimensions with nonde-generate gaps as

∀En > Em, pWnm �mn

= ρnd+m sin2(θnm)− 12ηnm

√ρnd+mρmd+n sin(2θnm)

× cos(ξnm + φnm + λnm) (C11)

∀En < Em, pWnm �mn

= ρmd+n sin2(θnm)+ 12ηnm

√ρnd+mρmd+n sin(2θnm)

× cos(ξnm + φnm + λnm).

In a similar manner, we can calculate the transition proba-bilities obtained from the TPM scheme:

∀En > Em, pTPMnm �mn = ρnd+m sin2(θnm) (C12)

∀En < Em, pTPMnm �mn = ρmd+n sin2(θnm).

Here, we set E0 = 0 and order the levels such that E0 ≤E1 ≤ E2 ≤ · · · and d is the dimension of the qudit. Next,we show that if all quasiprobabilities that contribute todirect or back flow are negative, then necessarily |Q| >|QTPM|. The difference �Q = Q − QTPM reads

�Q = −∑

En>Em

ηnm√ρnd+mρmd+n sin(2θnm)

× cos(ξnm + φnm + λnm)(En − Em). (C13)

For the direct flow (Q < 0), if ∀En > Em we havepW

nm �mn < 0, then from Eq. (C11) we immediately concludethat �Q < 0. For the back flow (Q > 0), if ∀En < Em wehave pW

nm �mn < 0, then from Eq. (C11) we now have that�Q > 0, which is exactly what we wanted to prove. Notethat Eq. (C13) also suggests a way of maximizing the dif-ference �Q for a given initial state. Choosing a protocolsuch that ξnm + λnm = −φnm and θnm = ±π/4 for all n, m,we obtain

|�Q|max =∑

En>Em

ηnm√ρnd+mρmd+n(En − Em). (C14)

In Fig. 8, we consider the two-qutrit setup of Eq. (C8). Weplot the negativity in the direct flow and compare it to theviolations of Inequalities 3 and 5 for different protocols.The protocols are determined by varying θ01 and θ02.

4. Proof of inequality 4

We define ψnm = ξnm + φnm + λnm and �Enm = En −Em. Using Eqs. (C11) and (C12), the symmetry θnm = θmn,

Inequality 5

Inequality 3

FIG. 8. The direct-flow negativity of the probabilities specifiedin the figure and the violation of Inequalities 3 and 5 for differ-ent interaction protocols determined by θ01 and θ02. Parameters:βH = 0.3, βC = 1.3, θ12 = θ02, E1 = 1, E2 = 1.15, ξ = φ = λ =0, ηij = 1 for (i, j ) = {(1, 3), (2, 6), (5, 7)}, ρ0 = 0.3, ρ5 = 0.03,ρ7 = 0.07, and ρ8 = 0.06.

ψnm = ψmn, and noting that

Q =∑

En>Em

pWnm �mn�Enm +

∑En<Em

pWnm �mn�Enm

=∑

En>Em

pWnm �mn�Enm −

∑En>Em

pWmn �nm�Enm,

using QTPM = Q(η = 0) the heat exchanged can beexpressed as

Q = QTPM

−∑

En>Em

ηnm√ρnd+mρmd+n sin(2θnm) cos(ψnm)�Enm.

(C15)

Using Eqs. (C11) and (C12) for En > Em, non-negativityimplies that

2pTPMnm �mn > ηnm

√ρnd+mρmd+n sin(2θnm) cos(ψnm). (C16)

We conclude that

Q > QTPM − 2∑

En>Em

pTPMnm �mn�Enm. (C17)

Violation of this inequality witnesses negativity in thedirect flow, as illustrated by the upper-right process ofFig. 1 and proves the lower bound of inequality 4. The

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AMIKAM LEVY and MATTEO LOSTAGLIO PRX QUANTUM 1, 010309 (2020)

upper bound on the heat can be obtained in a similarmanner. To see this, note that the heat can be written as

Q = QTPM +∑

En<Em

ηnm√ρnd+mρmd+n sin(2θnm)

× cos(ψnm)�Emn. (C18)

Using the non-negativity of Eq. (C11) for En < Em, wehave

Q < QTPM + 2∑

En<Em

pTPMnm �mn�Emn. (C19)

Violation of this inequality witnesses negativity in the backflow as illustrated by the lower-right process of Fig. 1 andproves the upper bound of Inequality 4.

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