We give a simple and intuitive proof that the only states which are
completely passive, i.e. those states from which work cannot be extracted even
with infinitely many copies, are Gibbs states at positive temperatures. The
proof makes use of the idea of virtual temperatures, i.e. the association of
temperatures to pairs of energy levels (transitions). We show that (i) passive
states are those where every transition is at a positive temperature, and (ii)
completely passive states are those where every transition is at the same
positive temperature.Comment: 3 pages, no figures. v2: Published versio

Brunner, Nicolas

Silva, Ralph

Skrzypczyk, Paul

English

arXiv

We give a simple and intuitive proof that the only states which are completely passive, i.e., those states from which work cannot be extracted even with infinitely many copies, are Gibbs states at positive temperatures. The proof makes use of the idea of virtual temperatures, i.e., the association of temperatures to pairs of energy levels (transitions). We show that (1) passive states are those where every transition is at a positive temperature and (2) completely passive states are those where every transition is at the same positive temperature.</p

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Passivity, complete passivity, and virtual temperatures

Paul Skrzypczyk

Ralph Silva

Nicolas Brunner

Crossref

Physical Review E

We give a simple and intuitive proof that the only states which are completely passive, i.e., those states from which work cannot be extracted even with infinitely many copies, are Gibbs states at positive temperatures. The proof makes use of the idea of virtual temperatures, i.e., the association of temperatures to pairs of energy levels (transitions). We show that (1) passive states are those where every transition is at a positive temperature and (2) completely passive states are those where every transition is at the same positive temperature.Peer Reviewe

UPCommons

We give a simple and intuitive proof that the only states which are completely passive, i.e., those states from which work cannot be extracted even with infinitely many copies, are Gibbs states at positive temperatures. The proof makes use of the idea of virtual temperatures, i.e., the association of temperatures to pairs of energy levels (transitions). We show that (1) passive states are those where every transition is at a positive temperature and (2) completely passive states are those where every transition is at the same positive temperature.Peer ReviewedPreprin

UPCommons (Universitat Politècnica de Catalunya)

arXiv:1412.5485v1  [quant-ph]  17 Dec 2014A short note on passivity, complete passivity and virtual temperaturesPaul Skrzypczyk,1 Ralph Silva,2 and Nicolas Brunner31ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain2H. H. Wills Physics Laboratory, University of Bristol,Tyndall Avenue, Bristol, BS8 1TL, United Kingdom3Département de Physique Théorique, Université de Genève, 1211 Genève, SwitzerlandWe give a simple and intuitive proof that the only states which are completely passive, i.e. thosestates from which work cannot be extracted even with infinitely many copies, are Gibbs states atpositive temperatures. The proof makes use of the idea of virtual temperatures, i.e. the associationof temperatures to transitions. We show that (i) passive states are those where every transition isat a positive temperature, and (ii) completely passive states are those where every transition is atthe same positive temperature.INTRODUCTIONThe notion of a passive state was introduced in theseminal work of Pusz and Woronowicz [1] as a character-isation of quantum states which cannot be processed toextract work. That is, given a state ρ with HamiltonianH , we ask whether the average energy can be lowered bya unitary transformation on the system, which is other-wise isolated (which is equivalent to a cyclic Hamiltonianprocess1). The change in average energy is denoted byW and given byW = maxUtr(H(ρ− UρU †)), (1)States for which W = 0, i.e. states from which no workcan be extracted, are referred to as passive states. Onthe other hand, states for which W > 0 are termed activestates, and contain extractable work.It is possible to show that passivity can be re-expressedsolely as a property of the state. Namely, a state ρ is pas-sive if and only if it satisfies the following two properties:• [ρ,H ] = 0, i.e. the state is (block) diagonal in theenergy eigenbasis of the Hamiltonian.• Ei > Ej implies λi < λj , where Ei, Ej are energyeigenvalues and λi and λj are the associated pop-ulations of the state ρ. That is, the population oflevels is strictly decreasing as the energy increases.In general one is not only interested in a single copyof a system, but in multiple copies that can be processedjointly. In particular, one may ask how much work canbe extracted from n copies of a state. Crucially, thecomposition of passive systems may not remain passive,hence exhibiting a form of activation. That is, there ex-ist situations where a unitary U acting on n copies of asystem is able to lower the average energy of the total1 i.e. we switch on a time-dependent interaction Hamiltonian V (t)only during 0 ≤ t ≤ τ .system, whilst if one had access to only n − 1 copies ofthe system no such unitary exists. This then naturallyleads to the question of what is the class of states whichremain passive under composition, i.e. from which nowork can be extracted even from an infinite number ofcopies. Such states are termed completely passive. Thecelebrated result of [1] is to show that the set of com-pletely passive states is exactly equivalent to the set ofthermal (or Gibbs) statesρ =1Zexp(−βH), (2)where Z = tr(exp(−βH)) is the partition function andβ = 1/kBT is the inverse temperature. This result putson firm grounds the notion of a heat bath in the form ofan infinitely large Gibbs state, from which no work can beextracted. The importance of these results in the contextof quantum thermodynamics has been highlighted, seefor instance [3–7].This result was originally proven in the context ofC∗ algebras and shortly afterwards translated into theframework of standard quantum mechanics by Lenard[2]. In both cases although the end result is intuitive, theproof of the result does not convey too much intuition.In particular, it is not explicit why it is that passive butnot thermal states can be activated for work extraction.In this short note we provide a simple and intuitiveproof based upon the idea of associating temperatures totransitions. This is the idea behind the concept of virtualtemperatures and virtual qubits introduced in [8], andfurther discussed in [9, 10]. Furthermore, our techniqueillustrates explicitly how work can be extracted from asufficient number of copies of a passive but non-thermalstate, hence giving an upper bound on the number ofcopies needed.VIRTUAL TEMPERATURESWe start by discussing in more detail the idea of as-signing a temperature to a transition. Consider a sys-tem ρ with Hamiltonian H =∑k Ek|k〉〈k| comprised2of d energy eigenstates |k〉, with energy eigenvalues Ek,and assume ρ is diagonal in this basis. In total thereare d(d − 1)/2 transitions between energy levels. Con-sider the transition between the energy states |i〉 and |j〉,and let us assume without loss of generality that the gapEi − Ej > 02. Given populations λi and λj respectively,we associate a virtual inverse temperature βv to the tran-sition, given byβv =logλj − logλiEi − Ej, (3)which arises by identifying the ratio of the populationswith the Boltzman factor, λi/λj = e−βv(Ei−Ej). First wenote that in general the virtual inverse temperature de-fined this way need not be positive. In particular, when-ever λi > λj , i.e. the transition has a population inver-sion, then the temperature will be negative. Second, thereason for calling this a virtual temperature is because bycoupling an external system to this transition, one canprepare physical systems at the virtual temperature, i.e.it behaves in this respect like a real temperature [8].It will also be important to us to understand how vir-tual tempertures transform under composition. To thatend, consider now two systems. For the first system weconsider the transition between states |i〉1 and |j〉1, withenergy gap ∆1 = Ei − Ej , populations λi and λj andvirtual inverse temperature β1v . Likewise for the secondsystem we consider transitions between the states |i′〉2and |j′〉2, with gap ∆2 = Ei′ − Ej′ , populations λi′ , λj′and virtual temperature β2v .Now, the joint system features 2 non-trivial transitions(i) between the pair of levels |i〉1|j′〉2 and |j〉1|i′〉2, and(ii) between |i〉1|i′〉2 and |j〉1|j′〉2. For case (i), the popu-lation of the first level is λiλj′ , the second λjλi′ , and thegap is ∆̃ = ∆2−∆1 (where we have assumed without lossof generality that ∆2 > ∆1). From (3) the inverse virtualtemperature β̃v of the composed transition is given byβ̃v =log(λiλj′ )− log(λjλi′ )∆2 −∆1=(logλj′ − logλi′)− (logλj − logλi)∆2 −∆1=β2v∆2 − β1v∆1∆2 −∆1(4)where we have used equation (3) in the final step. Notethat we obtain the same expression in the case ∆1 > ∆2(i.e. the formula is insensitive to the sign of the gap). Forcase (ii) the populations are now λiλi′ and λjλj′ , gap is∆̃′ = ∆2 + ∆1, and a similar analysis shows that the2 We do not define virtual temperatures for degenerate transitions.Since no work can ever be extracted from such transitions we willnever need such a concept.inverse virtual temperature β̃′v of the transition is givenbyβ̃′v =β2v∆2 + β1v∆1∆2 +∆1(5)Hence we see that the inverse virtual temperatures com-pose linearly in both cases. Finally, we note that β̃′v isalways in between the composed temperatures β1v andβ2v , while β̃v is in fact always larger than the biggest orsmaller than the smallest temperature.PASSIVITYThe notions introduced above will now allow us to re-express the notion of passivity in simple terms. Specif-ically the second requirement for a state to be passive,i.e. that Ei > Ej implies λi < λj , is rephrased in thelanguage of virtual temperatures as saying that the vir-tual temperature of every transition is positive. On thecontrary, if the state has one or more negative virtualtemperatures, work can be extracted by exploiting theassociated population inversion.COMPLETE PASSIVITYWe have seen above that passive states are those whereevery transition is at a positive temperature. We arenow going to show that completely passive states arethose where every transition is at the same positive tem-perature. The proof works by showing that whenevera system has two (or more) transitions at different vir-tual temperatures, then by composing sufficiently manycopies, the combined system always has a transition at anegative temperature, and is therefore not passive.Consider again a system ρ with d levels and considerfirst a transition between states |i〉 and |j〉, with gap∆1 = Ei − Ej and virtual inverse temperature β1v . Letus consider n copies of ρ, and the same transition in eachsystem. Now, by applying the composition rule (5) n− 1times, it is straightforward to see that the joint systemhas a transition between the states |i〉⊗n and |j〉⊗n withgap n∆1 and the same virtual inverse temperature β1v .Similarly, consider another transition of ρ between thestates |i′〉 and |j′〉, with gap ∆2 = Ei′ − Ej′ and virtualinverse temperature β2v > β1v , without loss of generality.Now consider another k copies of ρ. Exactly as above,the joint system has a transition between the states |i′〉⊗kand |j′〉⊗k with gap k∆2 and the virtual inverse temper-ature β2v .Finally, for the n + k copies of ρ together, considerthe transition between the states |i〉⊗n ⊗ |j′〉⊗k and|j〉⊗n ⊗ |i′〉⊗k. This transition has an energy gap ofn∆1 − k∆2, and from the composition rule (4), it has3the inverse virtual temperatureβv =β1vn∆1 − β2vk∆2n∆1 − k∆2(6)Now, since β2v > β1v , i.e. the two transitions are at dif-ferent virtual temperatures, then it is always possible tofind a negative βv, by choosing an appropriate numberof copies n and k such that the numerator of (6) is neg-ative, whilst the denominator is positive. In particular,choosing∆2∆1<nk<∆2∆1β2vβ1v(7)ensures that βv < 0. Thus, any passive state with twovirtual temperatures is not completely passive. The onlypossibility for a completely passive state is thus one con-taining only a single virtual temperature, which is pre-cisely the defining property of a thermal state.Finally, it is worth noting that (7) provides a sufficientcondition for finitely many copies of a state to becomenon-passive: One has to find the smallest number n + ksuch that (7) is satified for any possible pair of virtualtemperatures in the system.CONCLUSIONSIn summary, we have presented what we believe is asimply and insightful alternative proof that the only com-pletely passive states are thermal states. The only no-tion that our proof relies upon is the association of vir-tual temperatures to transitions in a system via Gibbsweights, and can be simply stated as the only completelypassive states are those which contain a single virtualtemperature. To show this we proved that every passivebut not completely passive state has the property thatupon composition one can find a transition at a nega-tive virtual temperature, from which work can then beextracted.This statement is intuitive from the perspective of ther-mal machines, where having access to two baths at dif-fering temperatures is all that is required to build a workextracting machine.Acknowledgements. We acknowledge financial sup-port from the Swiss National Science Foundation (grantPP00P2 138917) and an EU Marie Curie COFUND ac-tion (ICFOnest).[1] W. Pusz and S. L. Woronowicz, Passive States and KMSStates for General Quantum Systems, Comm. Math.Phys. 58, 273 (1978).[2] A. Lenard, Thermodynamical Proof of the Gibbs Formulafor Elementary Quantum Systems, J. Stat. Phys. 19, 575(1978).[3] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki,Thermodynamics of Quantum Information Systems -Hamiltonian Description, Open Syst. Inf. Dyn. 11, 205(2004).[4] A.E. Allahverdyan, R. Balian and Th.M. Nieuwenhuizen,Maximal work extraction from finite quantum systems,Europhys. Lett. 67 565 (2004).[5] R. Alicki, M. Fannes, Entanglement boost for extractablework from ensembles of quantum batteries, Phys. Rev. E87, 042123 (2013).[6] K.V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, andA. Acin, Entanglement Generation is Not Necessary forOptimal Work Extraction, Phys. Rev. Lett. 111, 240401(2013).[7] F.G.S.L. Brandao, M. Horodecki, N.H.Y. Ng, J. Oppen-heim, S. Wehner, The second laws of quantum thermody-namics, arXiv:1305.5278.[8] N. Brunner, N. Linden, S. Popescu, P. Skrzypczyk, Vir-tual qubits, virtual temperatures, and the foundations ofthermodynamics, Phys. Rev. E 85, 051117 (2012).[9] L.A. Correa, J.P. Palao, G. Adesso, D. Alonso, Perfor-mance bound for quantum absorption refrigerators, Phys.Rev. E 87, 042131 (2013).[10] P. Skrzypczyk, A.J. Short, S. Popescu, Extracting workfrom quantum systems, arXiv:1302.2811.

We give a simple and intuitive proof that the only states which are completely passive, i.e., those states from which work cannot be extracted even with infinitely many copies, are Gibbs states at positive temperatures. The proof makes use of the idea of virtual temperatures, i.e., the association of temperatures to pairs of energy levels (transitions). We show that (1) passive states are those where every transition is at a positive temperature and (2) completely passive states are those where every transition is at the same positive temperature

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arXiv:1412.5485v1  [quant-ph]  17 Dec 2014A short note on passivity, complete passivity and virtual temperaturesPaul Skrzypczyk,1 Ralph Silva,2 and Nicolas Brunner31ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain2H. H. Wills Physics Laboratory, University of Bristol,Tyndall Avenue, Bristol, BS8 1TL, United Kingdom3De´partement de Physique The´orique, Universite´ de Gene`ve, 1211 Gene`ve, SwitzerlandWe give a simple and intuitive proof that the only states which are completely passive, i.e. thosestates from which work cannot be extracted even with infinitely many copies, are Gibbs states atpositive temperatures. The proof makes use of the idea of virtual temperatures, i.e. the associationof temperatures to transitions. We show that (i) passive states are those where every transition isat a positive temperature, and (ii) completely passive states are those where every transition is atthe same positive temperature.INTRODUCTIONThe notion of a passive state was introduced in theseminal work of Pusz and Woronowicz [1] as a character-isation of quantum states which cannot be processed toextract work. That is, given a state ρ with HamiltonianH , we ask whether the average energy can be lowered bya unitary transformation on the system, which is other-wise isolated (which is equivalent to a cyclic Hamiltonianprocess1). The change in average energy is denoted byW and given byW = maxUtr(H(ρ− UρU †)), (1)States for which W = 0, i.e. states from which no workcan be extracted, are referred to as passive states. Onthe other hand, states for whichW > 0 are termed activestates, and contain extractable work.It is possible to show that passivity can be re-expressedsolely as a property of the state. Namely, a state ρ is pas-sive if and only if it satisfies the following two properties:• [ρ,H ] = 0, i.e. the state is (block) diagonal in theenergy eigenbasis of the Hamiltonian.• Ei > Ej implies λi < λj , where Ei, Ej are energyeigenvalues and λi and λj are the associated pop-ulations of the state ρ. That is, the population oflevels is strictly decreasing as the energy increases.In general one is not only interested in a single copyof a system, but in multiple copies that can be processedjointly. In particular, one may ask how much work canbe extracted from n copies of a state. Crucially, thecomposition of passive systems may not remain passive,hence exhibiting a form of activation. That is, there ex-ist situations where a unitary U acting on n copies of asystem is able to lower the average energy of the total1 i.e. we switch on a time-dependent interaction Hamiltonian V (t)only during 0 ≤ t ≤ τ .system, whilst if one had access to only n − 1 copies ofthe system no such unitary exists. This then naturallyleads to the question of what is the class of states whichremain passive under composition, i.e. from which nowork can be extracted even from an infinite number ofcopies. Such states are termed completely passive. Thecelebrated result of [1] is to show that the set of com-pletely passive states is exactly equivalent to the set ofthermal (or Gibbs) statesρ =1Zexp(−βH), (2)where Z = tr(exp(−βH)) is the partition function andβ = 1/kBT is the inverse temperature. This result putson firm grounds the notion of a heat bath in the form ofan infinitely large Gibbs state, from which no work can beextracted. The importance of these results in the contextof quantum thermodynamics has been highlighted, seefor instance [3–7].This result was originally proven in the context ofC∗ algebras and shortly afterwards translated into theframework of standard quantum mechanics by Lenard[2]. In both cases although the end result is intuitive, theproof of the result does not convey too much intuition.In particular, it is not explicit why it is that passive butnot thermal states can be activated for work extraction.In this short note we provide a simple and intuitiveproof based upon the idea of associating temperatures totransitions. This is the idea behind the concept of virtualtemperatures and virtual qubits introduced in [8], andfurther discussed in [9, 10]. Furthermore, our techniqueillustrates explicitly how work can be extracted from asufficient number of copies of a passive but non-thermalstate, hence giving an upper bound on the number ofcopies needed.VIRTUAL TEMPERATURESWe start by discussing in more detail the idea of as-signing a temperature to a transition. Consider a sys-tem ρ with Hamiltonian H =∑k Ek|k〉〈k| comprised2of d energy eigenstates |k〉, with energy eigenvalues Ek,and assume ρ is diagonal in this basis. In total thereare d(d − 1)/2 transitions between energy levels. Con-sider the transition between the energy states |i〉 and |j〉,and let us assume without loss of generality that the gapEi − Ej > 02. Given populations λi and λj respectively,we associate a virtual inverse temperature βv to the tran-sition, given byβv =logλj − logλiEi − Ej, (3)which arises by identifying the ratio of the populationswith the Boltzman factor, λi/λj = e−βv(Ei−Ej). First wenote that in general the virtual inverse temperature de-fined this way need not be positive. In particular, when-ever λi > λj , i.e. the transition has a population inver-sion, then the temperature will be negative. Second, thereason for calling this a virtual temperature is because bycoupling an external system to this transition, one canprepare physical systems at the virtual temperature, i.e.it behaves in this respect like a real temperature [8].It will also be important to us to understand how vir-tual tempertures transform under composition. To thatend, consider now two systems. For the first system weconsider the transition between states |i〉1 and |j〉1, withenergy gap ∆1 = Ei − Ej , populations λi and λj andvirtual inverse temperature β1v . Likewise for the secondsystem we consider transitions between the states |i′〉2and |j′〉2, with gap ∆2 = Ei′ − Ej′ , populations λi′ , λj′and virtual temperature β2v .Now, the joint system features 2 non-trivial transitions(i) between the pair of levels |i〉1|j′〉2 and |j〉1|i′〉2, and(ii) between |i〉1|i′〉2 and |j〉1|j′〉2. For case (i), the popu-lation of the first level is λiλj′ , the second λjλi′ , and thegap is ∆˜ = ∆2−∆1 (where we have assumed without lossof generality that ∆2 > ∆1). From (3) the inverse virtualtemperature β˜v of the composed transition is given byβ˜v =log(λiλj′ )− log(λjλi′ )∆2 −∆1=(logλj′ − logλi′)− (logλj − logλi)∆2 −∆1=β2v∆2 − β1v∆1∆2 −∆1(4)where we have used equation (3) in the final step. Notethat we obtain the same expression in the case ∆1 > ∆2(i.e. the formula is insensitive to the sign of the gap). Forcase (ii) the populations are now λiλi′ and λjλj′ , gap is∆˜′ = ∆2 + ∆1, and a similar analysis shows that the2 We do not define virtual temperatures for degenerate transitions.Since no work can ever be extracted from such transitions we willnever need such a concept.inverse virtual temperature β˜′v of the transition is givenbyβ˜′v =β2v∆2 + β1v∆1∆2 +∆1(5)Hence we see that the inverse virtual temperatures com-pose linearly in both cases. Finally, we note that β˜′v isalways in between the composed temperatures β1v andβ2v , while β˜v is in fact always larger than the biggest orsmaller than the smallest temperature.PASSIVITYThe notions introduced above will now allow us to re-express the notion of passivity in simple terms. Specif-ically the second requirement for a state to be passive,i.e. that Ei > Ej implies λi < λj , is rephrased in thelanguage of virtual temperatures as saying that the vir-tual temperature of every transition is positive. On thecontrary, if the state has one or more negative virtualtemperatures, work can be extracted by exploiting theassociated population inversion.COMPLETE PASSIVITYWe have seen above that passive states are those whereevery transition is at a positive temperature. We arenow going to show that completely passive states arethose where every transition is at the same positive tem-perature. The proof works by showing that whenevera system has two (or more) transitions at different vir-tual temperatures, then by composing sufficiently manycopies, the combined system always has a transition at anegative temperature, and is therefore not passive.Consider again a system ρ with d levels and considerfirst a transition between states |i〉 and |j〉, with gap∆1 = Ei − Ej and virtual inverse temperature β1v . Letus consider n copies of ρ, and the same transition in eachsystem. Now, by applying the composition rule (5) n− 1times, it is straightforward to see that the joint systemhas a transition between the states |i〉⊗n and |j〉⊗n withgap n∆1 and the same virtual inverse temperature β1v .Similarly, consider another transition of ρ between thestates |i′〉 and |j′〉, with gap ∆2 = Ei′ − Ej′ and virtualinverse temperature β2v > β1v , without loss of generality.Now consider another k copies of ρ. Exactly as above,the joint system has a transition between the states |i′〉⊗kand |j′〉⊗k with gap k∆2 and the virtual inverse temper-ature β2v .Finally, for the n + k copies of ρ together, considerthe transition between the states |i〉⊗n ⊗ |j′〉⊗k and|j〉⊗n ⊗ |i′〉⊗k. This transition has an energy gap ofn∆1 − k∆2, and from the composition rule (4), it has3the inverse virtual temperatureβv =β1vn∆1 − β2vk∆2n∆1 − k∆2(6)Now, since β2v > β1v , i.e. the two transitions are at dif-ferent virtual temperatures, then it is always possible tofind a negative βv, by choosing an appropriate numberof copies n and k such that the numerator of (6) is neg-ative, whilst the denominator is positive. In particular,choosing∆2∆1<nk<∆2∆1β2vβ1v(7)ensures that βv < 0. Thus, any passive state with twovirtual temperatures is not completely passive. The onlypossibility for a completely passive state is thus one con-taining only a single virtual temperature, which is pre-cisely the defining property of a thermal state.Finally, it is worth noting that (7) provides a sufficientcondition for finitely many copies of a state to becomenon-passive: One has to find the smallest number n + ksuch that (7) is satified for any possible pair of virtualtemperatures in the system.CONCLUSIONSIn summary, we have presented what we believe is asimply and insightful alternative proof that the only com-pletely passive states are thermal states. The only no-tion that our proof relies upon is the association of vir-tual temperatures to transitions in a system via Gibbsweights, and can be simply stated as the only completelypassive states are those which contain a single virtualtemperature. To show this we proved that every passivebut not completely passive state has the property thatupon composition one can find a transition at a nega-tive virtual temperature, from which work can then beextracted.This statement is intuitive from the perspective of ther-mal machines, where having access to two baths at dif-fering temperatures is all that is required to build a workextracting machine.Acknowledgements. We acknowledge financial sup-port from the Swiss National Science Foundation (grantPP00P2 138917) and an EU Marie Curie COFUND ac-tion (ICFOnest).[1] W. Pusz and S. L. Woronowicz, Passive States and KMSStates for General Quantum Systems, Comm. Math.Phys. 58, 273 (1978).[2] A. Lenard, Thermodynamical Proof of the Gibbs Formulafor Elementary Quantum Systems, J. Stat. Phys. 19, 575(1978).[3] R. Alicki, M. Horodecki, P. Horodecki, R. Horodecki,Thermodynamics of Quantum Information Systems -Hamiltonian Description, Open Syst. Inf. Dyn. 11, 205(2004).[4] A.E. Allahverdyan, R. Balian and Th.M. Nieuwenhuizen,Maximal work extraction from finite quantum systems,Europhys. Lett. 67 565 (2004).[5] R. Alicki, M. Fannes, Entanglement boost for extractablework from ensembles of quantum batteries, Phys. Rev. E87, 042123 (2013).[6] K.V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, andA. Acin, Entanglement Generation is Not Necessary forOptimal Work Extraction, Phys. Rev. Lett. 111, 240401(2013).[7] F.G.S.L. Brandao, M. Horodecki, N.H.Y. Ng, J. Oppen-heim, S. Wehner, The second laws of quantum thermody-namics, arXiv:1305.5278.[8] N. Brunner, N. Linden, S. Popescu, P. Skrzypczyk, Vir-tual qubits, virtual temperatures, and the foundations ofthermodynamics, Phys. Rev. E 85, 051117 (2012).[9] L.A. Correa, J.P. Palao, G. Adesso, D. Alonso, Perfor-mance bound for quantum absorption refrigerators, Phys.Rev. E 87, 042131 (2013).[10] P. Skrzypczyk, A.J. Short, S. Popescu, Extracting workfrom quantum systems, arXiv:1302.2811.

A short note on passivity, complete passivity and virtual temperatures

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