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Thethirdlawasasingleinequality Henrik Wilming1 and Rodrigo Gallego1 1DahlemCenterforComplexQuantumSystems, FreieUniversita¨tBerlin, 14195Berlin, Germany Thethirdlawintheformoftheunattainabilityprinciplestatesthatexactground-statecoolingrequiresinfinite resources. Hereweinvestigatetheamountofnon-equilibriumresourcesneededforapproximatecooling. We considerasresourceanysystemoutofequilibrium,allowingforresourcesbeyondthei.i.d.assumptionandin- cludingtheinputofworkasaparticularcase.Weestablishsufficientconditionsforcoolinginfullgeneralityand showthatforavastclassofnon-equilibriumresourcessufficientandnecessaryconditionsforlow-temperature coolingcanbeexpressedintermsofasinglefunction. Thisfunctionplaysasimilarroleforthethirdlawto the one of the free energy for the second law. From a technical point of view we provide new results about concavity/convexityofcertainRenyi-divergences,whichmightbeofindependentinterest. 7 I. INTRODUCTION fullrankbecauseotherwisetheproblemtrivializes,sinceone 1 0 can, for example, simply swap with a ground-state [35]. We 2 furthermoreassumethatthetargetsystemisinitiallyinther- Pure quantum states are indispensable resources for any malequilibriumwithsomeenvironment. Thenthetransition, n task in quantum information processing. However, the third i.e. the cooling protocol, can be performed by using a ther- a law of thermodynamics (more precisely, the unattainability J mal bath at a fixed inverse temperature β and performing a principle)statesthatcoolingasystemexactlytozerotemper- 7 aturerequiresaninfiniteamountofresources, beingitinthe global unitary that commutes with the total Hamiltonian, so 2 thatenergyisproperlyaccountedfor. Thiskindoftransitions formoftime,space,workorsomeotherresource[1–5]. Sim- have been extensively studied and they can be characterized ilarly, no-go theorems have been put forward for the task of ] byfamiliesoffunctionsMα,theso-calledmonotones,sothat h biterasure–whichiscloselyrelatedtoground-statecooling– p showing that no unitary process on a system and a finite di- atransitionispossibleifandonlyif[32,36,37] - t mensional reservoir can bring the system from a mixed to Mα(ρ ,H )≥Mα(ρ ,H ) ∀α. (1) n a pure state [6–8]. However, these no-go results do not say R R S S a muchabouttheamountofresourcesneededforapproximate Hence, the problem at hand is in principle hard to charac- u q cooling. Indeed, in recent times a sizable number of studies terize since one needs to verify an infinite number of condi- [ dealwithdifferentprotocolstocoolasmallquantumsystem tionstoconcludethatagiventransitionispossible. Themain byunitarilyactingonheatbathandacertainnumberofsys- contribution of the present work is to show that in the limit 2 tems out of equilibrium to be “used up” (known under the whereT issufficientlyclosetozero–i.e. theregimewhere v S 8 name of algorithmic or dynamical cooling) [9–13] or study- the(un)attainabilityproblemisformulated–theinfinitesetof 7 ingparticularmodelsofrefrigeratingsmallquantumsystems monotones appearing in (9) can be essentially reduced to a 4 [14–20], including ones that seem to challenge the unattain- singlemonotone. Wecallthismonotonethevacancyanditis 7 abilityprincipleintermsofrequiredtime[21–26]. definedas 0 Inthisworkwewillfocusinquantifyinginfullgenerality 1. the expenditure of arbitrary systems out of equilibrium that Vβ(ρ,H):=S(ωβ(H)(cid:107)ρ), (2) 0 are needed for approximate cooling. Our scenario is similar 7 totheoneconsideredinalgorithmiccooling,butherewetreat where ωβ(H) is the Gibbs state of at inverse temperature β 1 andS istherelativeentropy. the full thermodynamics of the problem by allowing for re- : Wefindthatsufficientandnecessaryconditionsforcooling, v sourceswithnon-trivialHamiltoniansandaccountingforthe respectively,aregivenby i energy conservation of the total process. We will do this in X the resource theoretic framework of quantum thermodynam- r Vβ(ρR,HR)+K(ρR,HR,ρS,HS,β)≥Vβ(ρS,HS), (3) a ics [27–31], which has proven useful to answer a variety of V (ρ ,H )≥V (ρ ,H ), (4) fundamental questions in quantum thermodynamics, such as β R R β S S establishinganinfinitefamilyofsecondlaws[32],providing whereK(ρ ,H ,ρ ,H ,β) → 0asT → 0. Henceinthe R R S S S fundamentalboundstosingle-shotthermodynamics[28,32– limitofverylowtemperaturecoolingV (ρ ,H )isthekey β S S 34]Recently,therehavealsobeenstudiesfromthisviewpoint quantitythatdeterminesthefundamentallimitations. Impor- ontheproblemofcooling[3–5],howevermostlyfocusingon tantly, V (ρ ,H )divergesasT → 0. Thenecessarycon- β S S S providingnecessaryconditionsintermsofresourcessuchas dition(4)thereforeshowsthataninfiniteamountofresources time,spaceorHilbert-spacedimension. (asmeasuredbyV )isnecessaryforexactground-statecool- β Thetaskofcoolingthatweareconsideringcanbephrased ing. Furthermore we show that for a vast class of resource as finding a cooling protocol between an arbitrary resource systems, forexamplethermalstatesofcoupledharmonicos- described by the state and Hamiltonian ρR and HR, respec- cillators,thefunctionK(ρR,HR,ρS,HS,β)vanishesidenti- tively, and a target system described by ρS and HS so that cally. Hence ρ isathermalstatewithtemperatureT aslowaspossible. S S We will assume that the density matrix of the resource has V (ρ ,H )≥V (ρ ,H ) (5) β R R β S S 2 becomes both a sufficient and necessary condition. That V Moreformally,wesaythatexistthatacoolingprotocolto β playsanimportantroleforthethirdlawhadbeenpreviously ρ ifandonlyifthereexistHamiltoniansH andH andan S B C found in the setting of i.i.d. resources and qubits as target auxiliarystateσ suchthat C systemsintheseminalworkofRef.[27].Here,weextendthe significanceofthequantityVβ toarbitraryscenarios. ρ(cid:48)RS ⊗σC =TrB(cid:0)UρR⊗ωβ(HS)⊗ωβ(HB)⊗σC U†(cid:1) Usually,theunattainabilityprincipleisformulatedwithre- (7) specttotime,argueingthataninfiniteamountoftime(orin- withTr (ρ(cid:48) )=ρ . TheonlyconstraintontheunitaryU is R RS S finitely many cycles of a periodically working machine) are thatitconservestheglobalenergy,i.e., neededtocoolasystemexactlytozerotemperature. Ourre- sultsshow,forexample,thatifthenon-equilibriumresources [U,H +H +H +H ]=0. (8) R S B C aresimplyhotthermalsystems(asintheexampleofathermal machinethatoperatesbetweentwoheatbaths),thesystemto Notethatthisformulationofthecoolingprocesscontainsasa becooledandthecoolingmachinehavetoeffectivelyinteract particularcasepartialcoolinginwhichawedonotstartwith withinfinitelymanysuchresourcesystems(orallpartsofone the target in a Gibbs state. In this case, the initial system of infinitelylargesystem). Thisimpliesthataninfiniteamoutof S,ifitispartiallycooledbeforestartingtheprotocol,canbe timeisneeded,sinceeachsuchinteractiontakesafinitetime simplyincorporatedasapartoftheresourceR. (see[3]forathoroughdiscussionofthispoint). The problem of finding conditions for the existence of a Ourfindingsnotonlyservetoposelimitationstoprotocols transitions of the form (7) has been studied in Ref. [32] for ofalgorithmiccooling,butalsosuggestasurprisingsymmetry diagonalstates,thatis,with[ρ ,H ] = 0and[ρ ,H ] = 0. between the second and third law of thermodynamics. The R R S S Throughout this manuscript we will restrict to such diagonal second law –in its averaged version or in the version of the states, butweemphasizethatthethenecessarycondition(4) Jarzynskiequality[38]–canbeexpressedintermsofthefree- alsoholdsfornon-diagonalstatesaswewillseelater. energydifferencedefinedas Undertheassumptionthatρ andρ arediagonal,onecan R S 1 showthatcoolingtoastateρ ispossibleifandonlyif ∆F (ρ,H)= S(ρ(cid:107)ω (H)). (6) S β β β S (ρ ||ω (H ))≥S (ρ ||ω (H )) ∀α≥0, (9) Inanalogy,weshowthatthethirdlawcanbeexpressedsimi- α R β R α S β S larlyintermsofV (ρ,H)whichsimplyinvertsthearguments β whereS areso-calledRenyi-divergences,whichwillbede- of the relative entropy in Eq. (6) and drops the pre-factor. α fined later. The proof of this statement relies simply in the Thissymmetrybetweenthesecondandthirdlawisquitesur- resultsofRef. [32]togetherwiththeadditivityoftheRenyi- prisingandhintsatthefactthatthesecondandthirdlawcan divergencesundertensor-products. berelatedtotheerrorsoffirstandsecondkindinhypothesis testing[39]. Weleavetheinvestigationofthisdeeperrelation AnimportanttoolthatappearsinEq. (9)istheconceptof betweenthetwoforfuturework. amonotoneof(catalytic)thermaloperations[27,28]. Thisis From a technical point of view, our results rely on certain anyfunctionf whichcanonlydecreaseunder(catalytic)ther- convexity-propertiesofthefunctionα(cid:55)→Sα(ρ||σ),whereSα maloperations. ThefunctionsSα appearingin(9)aremono- are classical Renyi-divergences [39]. We believe that these tones under catalytic thermal operations and more generally resultsmightbeofindependentinterest. under any channel that has the Gibbs state as a fixed point. Importantly,anymonotonef,possiblydifferentfromS ,al- α lows one to construct necessary conditions for a given tran- II. SET-UP sition. We will now show that V is also a monotone under β catalyticthermaloperations. In the following we will use the set-up of catalytic thermaloperations[27,28,32]appliedtothetaskofcooling. In Lemma1(Monotonicity). Thevacancyisanadditivemono- thisset-upweimaginetopossessaresourcegivenbythepair toneundercatalyticthermaloperations. of state and Hamiltonian (ρ ,H ). We can then use an ar- R R bitrary thermal bath at inverse temperature β, that is, a system in a Gibbs state ω (H ) of a Hamiltonian H , and fi- Proof. Firstnotethatthevacancyisonlydefinedonfull-rank β B B nallyanancillarysystem,theso-calledcatalystwitharbitrary states.Fromtheadditivityoftherelativeentropyundertensor- state and Hamiltonian (σ ,H ) in such a way the latter is products it follows that the vacancy is additive over uncor- C C returnedinthesameconfigurationanduncorrelatedfromthe related and non-interacting subsystems. Now let ρ be any restofthesystemsafterimplementingtheprotocol. Thetar- full-rank state with associated Hamiltonian H and consider S get system to be cooled is initially assumed to be in thermal a thermal operation involving a catalyst σ that maps ρ to C equilibriumwiththethermalbathandthereforedescribedby ρ(cid:48). Let the thermal operation be represented as the quantum a Gibbs state (ω (H ),H ). The total compound RSBC channelGonthesystemandcatalystandassumefornowthat β S S is transformed by a cooling protocol, which consists simply thecatalysthasfullrankaswell. SinceG mapsGibbs-states ofaunitarytransformationU whichcommuteswiththetotal at the environment temperature to Gibbs-states at the envi- Hamiltonian. ronmenttemperature,weobtainfromadditivityandthedata- 3 processinginequality III. GENERALSUFFICIENTCONDITIONSFOR COOLING V (ρ(cid:48),H )+V (σ ,H )=V (ρ(cid:48)⊗σ ,H +H ) β S β C C β C S C =S(G(ω (H )⊗ω (H ))(cid:107)G(ρ⊗σ )) Theprocessofcoolinglaidoutintheprevioussectioncan β S β C C in principle be applied to any final state ρ . We will now ≤S(ω (H )⊗ω (H )(cid:107)ρ⊗σ ) S β S β C c assumeforsimplicitythatthefinalstate,asitcorrespondsto =V (ρ,H )+V (σ ,H ). β S β C C acoolingprocess,isoftheformρ =ω (H )withβ very S βS S S large. We can then derive the following completely general Canceling the terms Vβ(σC,HC) we obtain Vβ(ρ(cid:48),HS) ≤ sufficientconditionforcooling. V (ρ,H ). The above argument does not work in case that β S Theorem2(Generalsufficientconditionforcooling). Forev- σ doesnothavefullrank,sinceinthiscaseV (σ ,H ) = C β C C erychoiceofβandH thereisacriticalβ >0suchthatfor ∞. We show in appendix A, however, that catalysts can al- S cr anyβ >β andfull-rankresource(ρ ,H )thecondition waysbechosentohavefullrank. Thisfinishestheproof. S cr R R V (ρ ,H )+K(β ,β,ρ ,H ,H )≥V (ω (H ),H ) β R R S R R S β βS S S (11) Knowing that the vacancy is a monotone under catalytic thermaloperations,wecannoweasilyderiveanecessarycon- is sufficient for cooling. The function K has the prop- ditionforcoolingbyapplyingthevacancytotheresourceand erty K(β ,β,ρ ,H ) → 0 as β → ∞ for any fixed S R R S system: β,H ,ρ >0andH . R R S TheproofofthetheoremisgiveninSec. VI.Nonetheless V (ρ ⊗ω (H ),H +H ) β R β S R S weprovideasketchofthemainideasinvolvedinsuchaproof ≥V (ρ(cid:48) ,H +H ) β RS R S attheendofthissection. ThefunctionK isgivenby ≥V (ω (H )⊗ρ ,H +H ). β β R S R S K(β ,β,ρ ,H ,H )= S R R S Here, the last inequality follows from the fact that one can (cid:26) (cid:18) ∂2 (cid:19) (cid:27) min 0, min S (ρ (cid:107)ω (H )) δ(β ) , always replace the state on any system by an uncorrelated α≤δ(βS)∂α2 α R β R S thermal state at the heat bath’s temperature using a ther- where mal operation. Using additivity of vacancy and the fact that cVoβo(lωinβg(His)g,iHve)nb=y0, we find that a necessary condition for δ(βS):=log(Zβ)/Vβ(ωβS(HS),HS), Zβ =Tr(e−βHS). The bound (11) applies for any possible (diagonal) resource state,howeverfindingK(β ,β,ρ ,H ,H )involvesamin- V (ρ ,H )≥V (ρ ,H ). (10) S R R S β R R β S S imization that although is feasible for low dimensional systems, might be an obstacle for practical purposes when Notethatthisconditionisderivedinfullgeneralityanditap- dealing with large systems and for values of β so that pliestoanyfull-rankρ andρ , possiblynotdiagonal. The S R S K(β ,β,ρ ,H ,H ) cannot be neglected. That said, we monotoneV wasfirstintroducedinRef. [27]. Itsrelevance S R R S β will investigate kinds of resource systems ρ ,H for which for the unattainability principle is clear since if ρ does not R R S K(β ,β,ρ ,H ,H )=0. Inthosecases,thegeneralsuffi- havefullsupport,thenther.h.s. of(10)diverges. Hence,ex- S R R S cientconditiongivenby(11)takentogetherwiththenecessary actcoolingisimpossibleunlesstheresourceρ doesnothave R full support either. In the particular case of ρ = (cid:78)n (cid:37)i condition (5) will imply that that a necessary and sufficient where (cid:37)i has full support, the condition (10) aRlready tie=ll1us conditionisgivensimplyby thatweneedinfiniteresources–infiniteninthiscase–forex- V (ρ ,H )≥V (ρ ,H ), (12) β R R β S S act cooling. Hence, such a simple analysis already suggests thatthequantityV playsacrucialroleforthelimitationson In particular, we will see in the next section, that this holds β cooling. true for large classes of thermal resources. Let us also note that K(β ,β,ρ ,H ,H ), just as the vacancy, is additive To summarize, we have seen, building upon previous lit- S R R S overnon-interactinganduncorrelatedresources. Wewilluse erature, thatthevacancyV establishesnecessaryconditions β this property in section V to investigate the setting of i.i.d- forcooling. However,fornecessaryandsufficientconditions resources. oneshouldinprincipleverifyaninfinitenumberofinequali- Intheresultgivenabove, wehavefocusedonthermaltar- tiesgivenby(9). Ourcontributionwillbetoshowthatthese get states. This is in fact not necessary. We show in the ap- infinitenumberofinequalitiescanbereducedtoasingleone, pendixFthatacompletelyanalogousresultholdsforstatesof whichcanalsobeexpressedintermsofthevacancyforasuf- theform ficiently cold final state ρ . Furthermore, we will show that S foralargefamilyofresourcesystemsthesinglesufficientcon- ρ =(1−(cid:15))|0(cid:105)(cid:104)0|+(cid:15)ρ⊥, (cid:15)(cid:28)1. (13) (cid:15) dition the we find coincides with the necessary (10). Hence, the limits on cooling are entirely ruled by the function V . where ρ⊥ is any density matrix which has full rank on the β This holds for large classes of finite systems, with possibly subspace orthogonal to the ground-state |0(cid:105) and commutes correlatedandinteractingsubsystems. withH . S 4 Sα Sα peratureβ . Inthefollowingwewillderiveasimpleexpres- R 4 4 sionthatallowsustocheckwhether 3 3 K(β ,β,ρ ,H ,H )≥0. ∀ρ ,H (16) S R R S S S 2 2 1 1 andhence(10)becomesanecessaryandsufficientcondition. Thereasoningisbasedonshowingthat α α 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 S (ρ =ω (H )||ω (H )) (17) α R βR R β R FIG. 1. The figure shows the behavior of S (ρ = α S ω (H )(cid:107)ω (H )) (orange) and S (ρ (cid:107)ω (H )) (blue). The is convex for a given range of α < 1, which implies (16). βS S β S α R β R leftplotshowsatargetstatethatisnotverycoldtogetherwithan The convexity of (17) can be determined by looking at the insufficient resource. The transition is not possible since the blue convexity of the average energy as a function of the inverse lineisbelowtheorangelineforα (cid:46) 1.25. Therightplotshows temperatureoftheresource the behavior when ρ becomes very cold. The function becomes S moresimilartoastepfunction.ThefactthatSα(ρR(cid:107)ωβ(HR))(blue x(cid:55)→ExR :=Tr(ωx(HR)HR). (18) curve)islargerthantheorangecurve,whichimpliesthatthetransi- tionispossible, isdeterminedbythebehavioratverysmallvalues In particular, we will see that whenever βR < β and the ofα,anduptoasmallerror,bythefactthatthederivateislargerat function x (cid:55)→ ExR is convex for x ∈ [βR,β] the function α=0. S (ω (H )||ω (H ))isconvexforallα<1. α βR R β R Theorem 3. For resources of the form (ω (H ),H ) that βR R R A. SketchoftheproofofThm.2 are hotter than the bath, that is with βR ≤ β, if ExR = Tr(ω (H )H )isconvexintherangex∈[β ,β],then(10) x R R R is a sufficient and necessary condition for low temperature Aswehaveseenintheprevioussection,asetofsufficient cooling. conditionsforatransitionwithcatalyticthermaloperationsis givenbytheinfinitesetofinequalitiesof(9). Themainidea ThisTheoremsimplifiesconsiderablythetaskofformulat- behindtheproofisthatwhenthetargetsystemissufficiently ingboundsonthethirdlaw,sincetheaverageenergyforther- cold(βS >βcr)itsufficestochecktheconditions(9)forvery malstatesiseasytocomputeformanysystems. InSec. IVA small α. This follows from the fact that for βS > βcr the wediscussseveralclassesofphysicallymotivatedconditions r.h.s. of (9), given by Sα(ρS(cid:107)ωβ(HS)), rapidly saturates to thatimplythatExR =Tr(ωx(HR)HR)isconvex. its maximum value as we increase α and it is concave (see Lastly,letusmentionthatcondition(16)isfulfilledif,fora Fig. 1). Giventhatoneonlyneedstoconsidersmallvaluesof fixedβ ,thebath’sinversetemperatureβissufficientlylarge, R αitispossibletomakeaTaylorexpansionaroundα = 0of without any extra assumption on the convexity of E . This x Sαoftheform implies, that for sufficiently cold baths, (10) is also a suffi- (cid:12) cientandnecessarycondition. ThisisshowninAppendixC, Sα(ρR(cid:107)ωβ(H))≈S0(ρ(cid:107)ωβ(H))+ ∂Sα(ρ∂(cid:107)ωαβ(H))(cid:12)(cid:12)(cid:12) α together with several properties of the Renyi divergences for α=0 thermal states that might be of independent interest and also +k(βS,β,ρR,HR,HS)α2. (14) includeaproofofThm. 3. This reduces the infinite inequalities of (1) to a single one that depends on the derivate of S and an error term k, that α A. Systemsforwhichtheenergyisconvex is related to the error term appearing in Thm. 2 denoted by K. This expansion can be further simplified noting that AsimpliedbyCor. 3,asufficientandnecessarycondition S (ρ(cid:107)ω (H)) = 0. Thevacancycomesintoplaybecause α=0 β for cooling is given by eq. (10) if the resource is a system oftheidentity that is hotter than the heat bath and whose average thermal ∂Sα∂(αρ(cid:107)σ)(cid:12)(cid:12)(cid:12)(cid:12) =S1(σ(cid:107)ρ), (15) ecnoenrvgeyxiitsycoofntvheexeninertghyeiisnfvuelrfisleletdembypearawtuidree.raWngeewoifllphseyesitchael α=0 models. ThisisduetothefollowingLemma,whichweprove which inverts the arguments of the second term on the r.h.s. intheappendix. of (14). Taking all these elements into account and account- Lemma 4 (Equidistant levels). Consider any Hamiltonian ing properly for the precision of the Taylor approximations with equidistant and non-degenerate energy levels. Then the we arrive at an inequality involving only the vacancy and a functionβ (cid:55)→E isconvex. vanishingerrortermasdeterminedbyThm. 2. β Immediate examples of Hamiltonians with equidistant en- ergylevelsaretwo-levelsystemsorharmonicoscillators. But IV. THERMALRESOURCES in fact, the lemma covers a much wider class of models sincethevacancyisunitarilyinvariantandadditiveovernon- We now consider as resource state ρ a (possibly multi- interactingsubsystems. Itfollowsthatalsoanyharmonicsys- R partite)thermalstateofsomeHamiltonianH atinversetem- temandanysystemdescribedbyfreefermionshasaconvex R 5 energyfunction,sincefreebosonicandfermionicsystemscan Wecomputethepartitionfunctioncanbeupperboundedas alwaysbemadenon-interactingbyanormal-modedecompo- d/2 sreitsipoonn.dIntosuacchoallnecotrimonalo-fmnoodne-idnetceoramctpionsgithioarnmthoenyicsiomscpillylactoorrs- Zβ(HRw)= (cid:88) e−βEdx (21) or two-level systems, respectively. Nevertheless, these sys- x=−d/2 tems include highly-correlated (even entangled) systems and ≥eβE/2+(d−1)e−βE/2 (22) nothermodynamiclimitneedstobetaken. Aparticularlyin- (cid:18) (cid:19) 1 d−1 terestingresourcethatisincludedbytheseresultsisthatofhot =eβE/2d + e−βE . (23) d d thermallight,whichhasbeenconsideredbeforeasavaluable resourceforcooling[18]. Hence,wefindthat Itcanfurthermorebecheckedthatforlargebutfinitemany- (cid:18) (cid:19) body systems whose density of states in the bulk is well ap- 1 d−1 V (ρw,Hw)≤−log + e−βE . (24) proximatedbyµ((cid:15))(cid:39)eγ(cid:15)−α(cid:15)2 theaverageenergyE iscon- β R R d d β vexinβ [40]. Finally, for every finite system there is a critical β such Thiscombinedby(10)impliesthatanecessaryconditionfor c thatEβ isconvexforallβ > βc. Thusassoonasβ > βR > coolingtoastateρS isgivenby β ,thesufficientcondition(10)holdsforsmallenoughtarget c (cid:18) (cid:19) 1 d−1 temperatures. V (ρ ,H )≤−log + e−βE . (25) β S S d d Most importantly, note that in order to obtain a state ρ that B. Asourceofwork S isclosetoaGibbsstateatzerotemperaturether.h.s. of(25). ForthisbothE anddhavetodiverge. Since Ourformalismcanalsoincorporateasourceofworkaspar- ticularcaseofaresourceforcooling. Thelimitationsoncool- lim V (ρw,Hw)≤βE, (26) β R R ingasafunctionoftheinputofworkhavebeenstudiedinRef. d→∞ [3].Thereitisshownthatthefluctuationsofwork,ratherthan lim Vβ(ρwR,HRw)≤log(d). (27) E→∞ itsaveragevalue,havetodivergewhenthetargetstatereaches vanishingfinaltemperature. Wewillherederivearesultsim- Thisimplies theunattainabilityprinciple inthesense thatan ilar in spirit using Thm. 2, although we employ a different infinitedenseandwithunboundedenergyisneededforcool- modelfortheworksource. ingtoabsolutezero. LetusmodeltheworksourcebyasystemRwithHamilto- nian V. INDEPENDENTANDIDENTICALLY-DISTRIBUTED d/2 (cid:12) (cid:29)(cid:28) (cid:12) Hw = (cid:88) Ex (cid:12)(cid:12)Ex Ex(cid:12)(cid:12). (19) RESOURCES R d (cid:12) d d (cid:12) x=−d/2 Wewillnowinvestigatethescalingofthelimitsforcooling OnecanseethisHamiltonianasd-dimensionalharmonicos- whentheresourcestateisgivenbyρ = (cid:37)⊗n andHamilto- cillatorswithenergiesboundedbetweenE/2and−E/2. We nian H = (cid:80) Hi where Hi = IR⊗···R⊗I ⊗h ⊗ R i R R 1 i−1 R are interested in the limit of d → ∞. In this case R is sim- I ⊗ ··· ⊗ I . Let us know consider the following task: i+1 n ilar to the model put forward in Ref. [3, 16], with the solely Given fixed (cid:37) ,H ,β,H , find the minimum n so that it is R R S differencethatweconsiderweconsiderhereafinitevalueof possible to cool down the target state to inverse temperature E. Weenforcetheconditionthatthebatteryactsasaenergy β . Alreadythenecessaryconditionforcooling(10)together S reservoirandnotasanentropysink(thatwouldmakethetask with additivity of V poses lower bounds on the necessary β ofcoolingtrivial)byassumingtheworksourcetobeinstate numberofcopiesnnec(β )as S ρw =I/d.Theseassumptionsontheworksourcearejustified R by the fact that it fulfills the second law of thermodynamics. nnec(β )= Vβ(ωβS(HS),HS). (28) ThisisshowninAppendixE. S V ((cid:37) ,h ) β R R Now we will show that the third law can be obtained, in the sense that both E and d have to diverge in order to be ThecontributionofThm. 2inthei.i.d. settingistobeableto abletouseRtocooldownasystemtozerofinaltemperature. giving us a sufficient number of copies nsuff. The condition LetusfirstrecallthatbyLemma4asufficientandnecessary (11)takestheform conditionforcoolingforsucharesourceisgivenby(10).Fur- n[V ((cid:37) ,h )+K(β ,β,(cid:37) ,h ,H )]≥V (ω (H ),H ). thermore,thevacancyoftheworksourceisgivenby β R R S R R S β βS S S Vβ(ρwR,HRw)=S(ωβ(HRw)(cid:107)I/d) Sincethisconditionissufficient,butnotalwaysnecessary,we =Tr(ω (Hw)(log(ω (Hw)−log(I/d))) obtain β R β R =−βTr(ωβ(HRw)HRw)+log(d)−log(Zβ(HRw)) nsuff(β )≤ Vβ(ωβS(HS),HS) . (29) ≤−βE/2+log(d)−log(Zβ(HRw)). (20) S Vβ((cid:37)R,hR)+K(βS,β,(cid:37)R,hR,HS) 6 Since the correction K goes to zero as β → ∞ (the target and S temperaturegoingtozero),weseethat S (ω (cid:107)ω )≤logZ . (36) ∞ βS β β nsuff(β ) lim S =1. (30) Here,thecriticalvalueδ(βS)isgivenby βS→∞ nnec(βS) log(Z ) Itistheninterestingtoknowhownnec scaleswiththetarget δ(βS)= V (ω (Hβ),H ). (37) temperature. We will see in the next section that nnec and β βS S S nsuff scaleasβ forlargeβ . Proof. SeeappendixB. S S Using this result, we can now upper bound the Renyi- divergence on the target by its linear approximation at the A. Vacancyasfreeenergiesandthescalingofthevacancy origin in this parameter regime. Since S(cid:48)(ρ||ω (H)) = 0 β S(ω (H)||ρ)=V (ρ,H),weget Inthespecialcaseofthermalresourceandtargetstates,we β β canreformulatethevacancy,andhencethenecessaryandsuf- S (ω (H )||ω (H ))≤V (ω (H ),H )α, α≤α . α βS S β S β βS S S c ficient condition, also in terms of non-equilibrium free ener- (38) gies. Thisallowstousethelargebodyofresultsinstatistical mechanics to check the condition. Indeed, the vacancy of a Secondly, forsmallenoughtargettemperatureswealsohave thermalstateattemperatureβS simplytakestheform S∞(ωβS(HS)||ωβ(HS)) ≤ Vβ(ωβS(HS),HS)αc. Since α (cid:55)→ S ismonotonouslyincreasing,thesecondlawseq.(9) α V (ω (H),H)=β ∆F (ω (H ),H ), (31) arehencealsosatisfiedif β βS S βS β S S with∆Fβ(ρ,H)=(cid:104)H(cid:105)ρ−(cid:104)H(cid:105)β −(S(ρ)−S(ωβ))/β. Sα(ρR||ωβ(HR))>Vβ(ωβS(HS),HS)α, α≤αc. (39) Inthiscasethecondition(10)reads: Indeed, for small temperatures, we can further restrict the β range of α to the interval [0,δ(βS)), where δ(βS) is given ∆FβR(ωβ(HR),HR)≥ βS∆FβS(ωβ(HS),HS). (32) by: R S (ω (H )||ω (H )) From(31)weseethatforlargeβ wehave(assumingvan- δ(β )= ∞ βS S β S . (40) S S V (ω (H ),H ) ishingground-stateenergy) β βS S S The final step now is given by arguing that the Renyi- V (ω (H),H)=β E , as β →∞. (33) β βS S β S divergenceoftheresourceSα(ρR||ωβ(HR))isapproximately convex in α for α < δ(β ). In particular, if we knew that it Assuming a resource system of n non-interacting identical S was convex (such as in the vase where ER is convex), we particles each described by (ω (h ),h ), we hence obtain β βR R R couldlowerbounditbyitslinearapproximationattheorigin thattheminimumachievabletemperaturescalesas andobtainthenecessaryandsufficientcondition(10). Forthegeneralcasewewillusethatweonlyhavetocheck ES nkBTS = Vβ(ωβR(hβR),hR), TS (cid:28)1. (34) asmndalulsveaTluaeysloorf’sαtahnedorseimmptolyoTbatyailnorexpandSα(ρR||ωβ(HR)) ThisresultissimilartotheasymptoticresultofJanzingetal. S(cid:48)(cid:48)(ρ ||ω (H ))≥V (ρ ,H )α−k(β ,β,ρ ,H )α2. α R β R β R R S R R [27]. (41) Thisyieldsasnewsufficientcondition VI. PROOFOFTHM.2 V (ρ ,H )α+k(β ,β,ρ ,H )α2 ≥V (ω (H ),H )α, β R R S R R β βS S S WewillnowproofThm.2. Beforewegotothedetails,let for all 0 < α ≤ δ(β ). Here, the function S usfirstexplainthegenerallogicbehindtheresult. Itisclear k(β ,β,ρ ,H )≤0isgivenby S R R that to obtain a single necessary and sufficient condition for (cid:26) (cid:27) coolingatlowtemperatures,wehavetoshowthattheinfinite k(β ,β,ρ ,H )=min 0, min S(cid:48)(cid:48)(ρ (cid:107)ω (H )) . S R R α R β R set of second laws in eq. (9) collapse to a single condition. α≤δ(βS) ThefirstimportantstepintheproofisthefollowingLemma. We can now divide the sufficient condition by α and, since Lemma 5 (Concavity at low temperatures). Let β > 0 and k(βS,β,ρR,HR) ≤ 0, replace α by δ(βS) to arrive at the aHamiltonianH withground-statedegeneracyg begiven. finalsufficientcondition S 0 Thereexistsacriticalinversetemperatureβ suchthatforall cr V (ρ ,H )+K(β ,β,ρ ,H ,H )≥V (ω (H ),H ), β >β andforall0<α<δ(β )wehave β R R S R R S β βS S S S cr S withK(β ,β,ρ ,H ,H )= k(β ,β,ρ ,H )δ(β ). This α(cid:55)→S(cid:48)(cid:48)(ω (cid:107)ω )≤0. (35) S R R S S R R S α βS β finishestheproof. 7 VII. SUMMARY resourcesthatdivergewhencoolingasystemtoabsolutezero. Moreparticularly,itisaninterestingquestionforfuturework toobtaintheoptimalsufficientscalingofthesizeoftheheat In this work we have investigated the limits on low tem- bath and the potential “catalyst” τ that is needed to cool the perature cooling when arbitrary systems out of equilibrium systemtothefinallowtemperature[41]. are used as a resource. We provide sufficient and necessary Lastly,theresultsofSec. IVsuggeststhatforalargeclass conditionsthatestablishnovelupperandlowerboundsonthe ofphysicallyrelevantsystemsthethirdlawcanbeexpressed amountofresourcesthatareneededtocoolasystemcloseto simplyasthemonotonicityofthevacancy. Itwouldbeofin- itsgroundstate. Wefoundthatthelimitationsareruledbya terest to specify more general assumptions on a many-body singlequantity,namelythevacancy. Thisisremarkable,since system so that this is the case. On the other hand, there ex- at higher temperatures there is an infinite family of “second ist systems for which the vacancy is not sufficient condition. laws” that need to be checked to determine whether a non- This open the possibility to have families of resources that, equilibriumstatetransitionispossible. although are out of equilibrium, are useless for cooling. We Wehaveonlyfocusedontheamountofnon-equilibriumre- havesomeevidencethatthisfamiliesofstatesmightexist,but sources, asweassumeaccesstoaninfiniteheatbathandwe weleaveitasanopenquestionforfuturework. leaveconsiderationsaboutthetimeandcomplexityofcooling Acknowledgments. We thank Lluis Masanes, Joseph M. protocol aside. These other kind of resources have been ex- Renes and Jens Eisert for interesting discussions and valu- ploredinothercomplementaryworksonthethirdlaw[3–6]. able feedback. This work has been supported by the ERC Itwouldbeinterestingtoseeifthethevacancyplaysaroleto (TAQ), the DFG (GA 2184/2-1) and the Studienstiftung des expressthelimitationsinthesizeoftheheatbathoranyother DeutschenVolkes. [1] W.Nernst,Nachr.Kgl.Ges.Wiss.Gtt.,1(1906). (2015),10.1103/physreve.92.012136. [2] W.Nernst,Ber.Kgl.Pr.Akad.Wiss.,933(1906). [20] P.P.Hofer,M.Perarnau-Llobet,J.B.Brask,R.Silva,M.Huber, [3] L. Masanes and J. Oppenheim, “A derivation (and quan- andN.Brunner, (2016),1607.05218. tification) of the third law of thermodynamics,” (2014), [21] B.Cleuren,B.Rutten, andC.VandenBroeck,PhysicalReview arXiv:1412.3828. Letters108(2012),10.1103/physrevlett.108.120603. [4] J.ScharlauandM.P.Mueller, (2016),1605.06092. [22] A. E. Allahverdyan, K. V. Hovhannisyan, and G. Mahler, [5] R.Silva,G.Manzano,P.Skrzypczyk, andN.Brunner, (2016), Physical Review Letters 109 (2012), 10.1103/phys- 1604.04098. revlett.109.248903. [6] C.DiFrancoandM.Paternostro,ScientificReports3(2013), [23] A. Levy, R. Alicki, and R. Kosloff, Physical Review Letters 10.1038/srep01387. 109(2012),10.1103/physrevlett.109.248901. [7] L.-A.Wu,D.Segal, andP.Brumer,ScientificReports3(2013), [24] O. Entin-Wohlman and Y. Imry, Physical Review Letters 112 10.1038/srep01824. (2014),10.1103/physrevlett.112.048901. [8] F. Ticozzi and L. Viola, Scientific Reports 4 (2014), [25] B.Cleuren,B.Rutten, andC.VandenBroeck,PhysicalReview 10.1038/srep05192. Letters109(2012),10.1103/physrevlett.109.248902. [9] L. J. Schulman and U. V. Vazirani, Proceedings of the thirty- [26] M. Kol, D. Gelbwaser-Klimovsky, R. Alicki, and G. Kur- firstannualACMsymposiumonTheoryofcomputing-STOC izki, Physical Review Letters 109 (2012), 10.1103/phys- 99 (1999),10.1145/301250.301332. revlett.109.090601. [10] P.O.Boykin,T.Mor,V.Roychowdhury,F.Vatan, andR.Vri- [27] D.Janzing,P.Wocjan,R.Zeier,R.Geiss, andT.Beth,Int.J. jen,PNAS99,33883393(2002). Th.Phys.39,2717(2000). [11] L. J. Schulman, T. Mor, and Y. Weinstein, Physical Review [28] M. Horodecki and J. Oppenheim, Nature Comm. 4, 2059 Letters94(2005),10.1103/physrevlett.94.120501. (2013). [12] A. E. Allahverdyan, K. V. Hovhannisyan, D. Janzing, [29] G.Gour,M.P.Mller,V.Narasimhachar,R.W.Spekkens, and and G. Mahler, Phys. Rev. E 84 (2011), 10.1103/phys- N.YungerHalpern,PhysicsReports583,158(2015). reve.84.041109. [30] N.Y.Halpern, (2014),1409.7845. [13] S.RaeisiandM.Mosca,PhysicalReviewLetters114(2015), [31] N. Yunger Halpern and J. M. Renes, Phys. Rev. E 93 (2016), 10.1103/physrevlett.114.100404. 10.1103/physreve.93.022126. [14] A.E.Allahverdyan, K.Hovhannisyan, andG.Mahler,Phys. [32] F.G.S.L.Brandao,M.Horodecki,N.H.Y.Ng,J.Oppenheim, Rev.E81(2010),10.1103/physreve.81.051129. andS.Wehner,PNAS112,3275(2015). [15] N. Linden, S. Popescu, and P. Skrzypczyk, Phys. Rev. Lett. [33] P. Faist, F. Dupuis, J. Oppenheim, and R. Renner, Nature 105,130401(2010). Comm.6,7669(2015). [16] P.Skrzypczyk,N.Brunner,N.Linden, andS.Popescu,J.Phys. [34] N.Y.Halpern,A.J.P.Garner,O.C.O.Dahlsten, andV.Vedral, A44,492002(2011). NewJ.Phys.17,095003(2015). [17] A. Levy, R. Alicki, and R. Kosloff, Phys. Rev. E 85 (2012), [35] Similar arguments can be made for other states without full 10.1103/physreve.85.061126. rank. For example two copies of a rank-2 state in a 4- [18] A. Mari and J. Eisert, Physical Review Letters 108 (2012), dimensional system can be written as a pure state in tensor 10.1103/physrevlett.108.120602. productwithafull-rankstate.Sucharesourcethereforealready [19] R. Silva, P. Skrzypczyk, and N. Brunner, Phys. Rev. E 92 containsapurestatewhichcanthenbemappedtotheground- 8 state. [36] F. Buscemi and G. Gour, “Quantum relative lorenz curves,” (2016),arXiv:1607.05735. [37] G. Gour, “Quantum resource theories in the single-shot regime,” (2016),arXiv:1610.04247. [38] C.Jarzynski,Phys.Rev.Lett.78,2690(1997). [39] M. Tomamichel, SpringerBriefs in Mathematical Physics (2016),10.1007/978-3-319-21891-5. [40] Suchadensityofstatesistypicalformany-bodysystemsinthe bulkoftheenergy,butdeviationstypicallydoappearinthetails ofthedistribution. [41] N.H.Y.Ng,L.Maninska,C.Cirstoiu,J.Eisert, andS.Wehner, NewJ.Phys.17,085004(2015). 9 AppendixA:Catalystscanalwaysbechosenwithfullrank In this section, we show that in our setting catalysts can always be chosen to have full rank. I.e., whenever cooling can be achievedwithacatalystwithoutfull-rankthereexistsacatalystwithfullrankandadifferentHamiltonianthatalsoachievesit. Toseethis,considerabi-partitesystemwithnon-interactingHamiltonianH +H .Thenconsidertheinitialstateρ ⊗σ and 1 2 SB C applyanenergy-preservingunitaryoperationU thatresultsinthestateρ(cid:48) withρ(cid:48) = σ . Here,weimaginethatρ also SBC C C SB includes the state of the heat-bath and thus there can be a built-up of correlations between the catalyst and ρ . Furthermore SB assumethatρ hasfull-rankandthatσ issupportedonlyonasubspaceP ⊂H withcomplementQ=1−P (weidentify SB C 2 (cid:80) thevector-spaceandtheprojectoronthespace). ThusP = |j(cid:105)(cid:104)j|, wherethesumisovertheeigen-statesofσ . Letthe j C spectrumofρ andσ be{p }and{q },respectively. Thenthefinalstate SB C α j (cid:88) ρ(cid:48) = p q U |α(cid:105)(cid:104)α|⊗ |j(cid:105)(cid:104)j|U† (A1) SBC α j α,j isaconvexsum ofthepositivesemi-definiteoperatorsU |α(cid:105)(cid:104)α|⊗ |j(cid:105)(cid:104)j|U†. The sumhassupportonlywithin 1⊗P since otherwisethereducedstateρ(cid:48) wouldalsobesupportedoutsideofP. Hencealsoeverysummandissupportedwithin1⊗P. (cid:80) C Using1⊗P = |α(cid:105)(cid:104)α|⊗ |j(cid:105)(cid:104)j|wethenobtain α,j (cid:88) (1⊗Q)U(1⊗P)U† =1⊗Q U |α(cid:105)(cid:104)α|⊗ |j(cid:105)(cid:104)j|U† =0. (A2) α,j Inotherwords,wehave(1⊗Q)U(1⊗P) = 0andbyasimilarcalculationalso(1⊗P)U(1⊗Q) = 0. Thus,theunitaryU isblock-diagonal. Inparticular, theoperatorV = (1⊗P)U(1⊗P)consideredasanoperatorontheHilbert-spaceH ⊗P 1 is unitary. Since U is energy-preserving by assumption, we can deduce that P = span{|E (cid:105)} for some subset of energy- j eigenstates |E (cid:105)oftheHamiltonianH ofthecatalyst. j 2 ThenV commuteswiththeHamiltonianH +H | ,whereH | denotestheHamiltonianofthecatalyst,butrestrictedto 1 2 P 2 P thesubspaceP. Wecanthusobtainanequivalentcatalystwithfullrankandacorrespondingthermaloperationbyrestrictingσ andH to C 2 thesubspaceP onwhichσ hasfullrankandusingthethermaloperationdefinedbyV: C Vρ ⊗σ | V† =ρ(cid:48) | . (A3) SB C P SBC 1⊗P In particular note that the above analysis also shows that pure catalysts are useless: If a transition can be done with a pure catalyst,itcanalsobedonewithoutacatalyst. AppendixB:ProofofconcavityofRenyidivergenceforlowtemperatures In this section we proof the Lemma 5 about concavity of the Renyi-divergence at low temperatures. The Lemma holds for anyHamiltonianwithpurepointspectrum, agapabovetheground-stateandthepropertythatthepartitionsumexistsforany positivetemperature. Lemma6(Concavityatlowtemperatures). Letβ >0andaHamiltonianH withground-statedegeneracyg begiven. There S 0 existsacriticalinversetemperatureβ suchthatforallβ >β andforall0<α<δ(β )wehave cr S cr S α(cid:55)→S(cid:48)(cid:48)(ω (cid:107)ω )≤0. (B1) α βS β and S (ω (cid:107)ω )≤logZ . (B2) ∞ βS β β Here,thecriticalvalueδ(β )isgivenby S log(Z ) δ(β )= β <1. (B3) S V (ω (H ),H ) β βS S S Proof. Letusfirstprovethatthemax-Renyi-divergenceisupperboundedbythepartitionfunctionattheenvironmenttempera- ture. Supposethatβ >β andletuswrite S (cid:32) (cid:33) 1 (cid:88) e−βEi S (ω (H )(cid:107)ω (H ))= log g e−α(βS−β)Ei(Z /Z )α (B4) α βS S β S α−1 i β βS Z β i (cid:32) (cid:33) 1 (cid:88) e−βEi = log e−α(βS−β)E0(Z /Z )α g e−α(βS−β)(Ei−E0) , (B5) α−1 β βS i Z β i 10 whereE denotethedifferentenergiesofH ,withdegeneraciesg . Assumingw.l.o.g. E =0,wewritethisas i S i 0 (cid:32) (cid:33) α 1 (cid:88) e−βEi S (ω (H )(cid:107)ω (H ))= log(Z /Z )+ log 1+ e−α(βS−β)Ei . (B6) α βS S β S α−1 β βS α−1 Z β i>0 Itisnowobviousthatinthelimitweobtain S (ω (H )(cid:107)ω (H ))= lim S (ω (H )(cid:107)ω (H ))=log(Z )−log(Z )≤logZ . (B7) ∞ βS S β S α→∞ α βS S β S β βS β Asasecondstepletusfindtheconditionforwhichδ(β )<1. Todothatweexpressthevacancyas S V (ω (H ),H )=β E −S +logZ , (B8) β βS S S S β β βS wherewewriteS :=S(ω (H )). Wethusneedthat β β S β E −S(ω (H ))>logZ −logZ . (B9) S β β S β βS Relaxingtothesufficientcriterionβ E −S >logZ =S −βE wethusobtain S β β β β β 2S −βE β > β β. (B10) S E β Letusnowturntotheconcavity. WewillusetherepresentationofS(cid:48)(cid:48)proveninthenextsection,whichisgivenby α 2 (cid:16) (cid:17) S(cid:48)(cid:48)(ω (cid:107)ω )= logZ −logZ +(β −β˜(α))E −(β −β˜(α))2Var(H) , (B11) α βS β (1−α)3 βS β˜(α) S β˜(α) S β˜(α) whereβ˜(α)=β(1−α)+αβ . Sinceweareonlyinterestedinα<δ(β )<1,wehaveβ ≤β˜(α)<β . Wethereforehaveto S S S showthatthetermsintheparenthesisarenegative. Letususethattheaverageenergyismonotonicwithβ andthatZ > 1 β˜(α) toboundthesetermsas parenthesis ≤logZ +(β −β˜(α))E −(β −β˜(α))2Var(H) βS S β˜(α) S β˜(α) ≤logZ +(β −β)E −(β −β˜(α))2Var(H) βS S β S β˜(α) ≤log(d)+(β −β)E −(β −β˜(α))2 min Var(H) . (B12) S max S x x∈[β,β˜(α)] Nowweboundβ˜(α). Todothatweusethatβ˜(α)≤β˜(δ(β ))=:β˜∗(β ). Itisclearthatifwecanboundβ˜∗(β )byaconstant, S S S thetermsintheparenthesisbecomenegativeforsomeβ sincethesecondorderterminβ dominates. Toseethatβ˜∗(β )is S S S indeedupperboundedbyaconstant,weagainwritethevacancyas V (ω (H ),H )=−S(ω )+β E +logZ (B13) β βS S S β S β βS toobtain β∗ := lim β˜∗(β )= lim β(1−δ(β ))+δ(β )β (B14) S S S S βS→∞ βS→∞ logZ logZ =β+ lim β β =β+ β. (B15) βS→∞βSEβ +logZβS −S(ωβ(HS)) S Eβ This finishes the proof that β as claimed in the Lemma exists. We also note that the function β˜∗(β ) is monotonically cr S decreasing for all β such that β˜∗(β ) < 1. Finally, note that eq. (B12) allows to give upper bounds β once one has lower S S c boundsontheenergyvarianceforinversetemperaturesintheinterval[β,β∗].