The Resource Theory of Quantum States Out of Thermal Equilibrium
Fernando G. S. L. Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, Robert W. Spekkens
TThe Resource Theory of Quantum States Out of Thermal Equilibrium
Fernando G.S.L. Brand˜ao,
1, 2
Micha(cid:32)l Horodecki,
3, 4
Jonathan Oppenheim, Joseph M. Renes,
6, 7 and Robert W. Spekkens Departamento de F´ısica, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil Centre for Quantum Technologies, National University of Singapore, Singapore Institute for Theoretical Physics and Astrophysics, University of Gda´nsk, Gda´nsk, Poland National Quantum Information Centre of Gda´nsk, Sopot, Poland Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom Institut f¨ur Angewandte Physik, TU Darmstadt, Darmstadt, Germany Institut f¨ur Theoretische Physik, ETH Zurich, Z¨urich, Switzerland Perimeter Institute for Theoretical Physics, Waterloo, Canada (Dated: October 8, 2013)The ideas of thermodynamics have proved fruitful in the setting of quantum information theory, inparticular the notion that when the allowed transformations of a system are restricted, certain statesof the system become useful resources with which one can prepare previously inaccessible states.The theory of entanglement is perhaps the best-known and most well-understood resource theory inthis sense. Here we return to the basic questions of thermodynamics using the formalism of resourcetheories developed in quantum information theory and show that the free energy of thermodynamicsemerges naturally from the resource theory of energy-preserving transformations. Specifically, thefree energy quantifies the amount of useful work which can be extracted from asymptotically-manycopies of a quantum system when using only reversible energy-preserving transformations and athermal bath at fixed temperature. The free energy also quantifies the rate at which resourcestates can be reversibly interconverted asymptotically, provided that a sublinear amount of coherentsuperposition over energy levels is available, a situation analogous to the sublinear amount of classicalcommunication required for entanglement dilution.
Quantum resource theories are specified by a restric-tion on the quantum operations (state preparations,measurements, and transformations) that can be imple-mented by one or more parties. This singles out a setof states which can be prepared under the restricted op-erations. If the parties facing the restriction acquire aquantum state outside the restricted set of states, thenthey can use this state to implement measurements andtransformations that are outside the class of allowed op-erations, consuming the state in the process. Thus, suchstates are useful resources.A few prominent examples serve to illustrate the idea:if two or more parties are restricted to communicatingclassically and implementing local quantum operations,then entangled states become a resource [1]; if a party isrestricted to quantum operations that have a particularsymmetry, then states that break this symmetry becomea resource [2–4]; if a party is restricted to preparing statesthat are completely mixed and performing unitary oper-ations, then any state that is not completely mixed, i.e.any state that has some purity, becomes a resource [5].In this Letter we develop the quantum resource the-ory of states that are athermal (relative to temperature T ). This provides a useful new formulation of nonequi-librium thermodynamics for finite-dimensional quantumsystems, and allows us to apply new mathematical toolsto the subject. The restricted class of operations whichdefines our resource theory are those that can be achievedthrough energy-conserving unitaries and the preparationof any ancillary system in a thermal state at temperature T , as first studied by Janzing et al. [6] in the context of Landauer’s principle. Here the ancillary systems canhave an arbitrary Hilbert space and an arbitrary Hamil-tonian, and may be described as having access to a sin-gle heat bath at temperature T . States that are not inthermal equilibrium at temperature T , that is, which areathermal, are the resource in this approach.Quantum resource theories provide answers to ques-tions such as: How does one measure the quality of dif-ferent resource states? Can one particular resource statebe converted to another deterministically? If not, can itbe done nondeterministically, and if so with what prob-ability? What if one has access to a catalyst? A par-ticularly fundamental problem, addressed in this Letter,is to identify the equivalence classes of states that are reversibly interconvertible in the limit of asymptotically-many copies of the resource and to determine the ratesof interconversion. We show that all athermal states arereversibly interconvertible asymptotically and that theinterconversion rate is governed by the free energies ofthe states involved.The great merit of the resource theory approach is itsgenerality. Rather than considering the behavior of theproperty of interest for some particular system with par-ticular dynamics (as is typical in thermodynamics), oneconsiders instead the fundamental limits that are im-posed by the restriction defining the resource and thelaws of quantum theory . On the practical side, a betterunderstanding of a given resource helps determine howbest to implement the tasks that make use of it, and,more fundamentally, such an understanding may serveto clarify what sorts of resources are even relevant for a r X i v : . [ qu a n t - ph ] O c t a given task. For instance, entanglement is commonlyasserted to be the necessary resource for tasks in whichthe use of quantum systems yields improved performanceover the use of classical systems. But in quantum metrol-ogy it is asymmetry which is relevant, not entanglement.Finally, the resource theory approach provides a frame-work for organizing and consolidating the results in agiven field, thermodynamics being particularly in needof such a framework, as well as synthesizing new re-sults. Indeed, studying the interconvertibility of finite resources leads to useful notions of free energy in thatcase, as shown by two of us in [7], and to a more detailed,quantitative treatment of the second law, by three of usin [8]. Results similar to the former were also reportedby ˚Aberg [9] and Egloff et al. [10], who investigated thework extractable from finite resources. Allowed Operations & Resource States.—
We now de-fine the restricted class of operations and the resourcestates more precisely. Given a quantum system withHilbert space H and Hamiltonian H , the restrictedoperations are the completely-positive trace-preserving(CPTP) maps E : L ( H ) → L ( H ) of the form E ( ρ ) = Tr (cid:16) V ( ρ ⊗ ¯ γ ) V † (cid:17) , (1)where ¯ γ is the thermal (Gibbs) state of an arbitrary ancil-lary system with Hamiltonian ¯ H at inverse temperature β = 1 /k B T , and V is an arbitrary unitary operation onthe joint system which commutes with the total Hamil-tonian: [ V , H ⊗ I + I ⊗ ¯ H ] = 0. Observe that E ( γ ) = γ ,where γ is the Gibbs state associated with H . Any otherstate ρ (cid:54) = γ is a resource state. While we here consider thecase that input and output systems and their Hamiltoni-ans are identical, this framework can be easily extendedto the more general case, as done by Janzing et al. [6].The allowed operations are particularly relevant forthermodynamics because they cannot, on their own, beused to do work. Moreover, it is not too difficult to seethat various different kinds of athermal states can beused, via the restricted class of operations, to do work:for thermal states at a temperature T (cid:48) distinct from T (hence athermal relative to T ), work can be drawn usinga heat engine (such states simulate having a second heatbath at a different temperature); for pure states within adegenerate energy eigenspace, work can be drawn usinga Szilard engine [11]; for pure energy eigenstates, workcan be drawn directly by an energy-conserving unitary.One is led to expect that work can be extracted from any athermal state. We shall show that asymptotically thisis indeed the case.It is important to note the differences between the re-source theory framework and the more usual approachesto thermodynamics. Chiefly, all sources and sinks of en-ergy and entropy must be explicitly accounted for: onlyenergy and entropy-neutral operations on the systemand thermal reservoir are allowed, rather than specificenergy- or entropy-changing operations more commonin an open-system approach. All interactions between the system and reservoir are due to the unitary V andnot an interaction term in the total Hamiltonian. More-over, no attempt is made a priori to restrict the allowedoperations to be physically realistic; indeed we assumethe experimenter has complete control over V . This en-sures that the restrictions we find are truly fundamen-tal, though ultimately the operations needed to estab-lish our main result are mappings between macroscopicobservables and do not require fine-grained, microscopiccontrol. These apparent differences notwithstanding, weshow in the Appendix that a number of different classesof operations for thermodynamics are in fact equivalent. Resource Interconvertibility & Free Energy .—A centralquestion in any resource theory is that of resource in-terconversion: Which resources can be transformed intowhich others, and how easily? Generally there exists apartial order, or quasiorder , of resources: We say A ≥ B if resource A can be transformed into B using the allowedoperations. Functions which respect this quasiorder areknown as resource monotones . For instance, the relativeentropy of entanglement is a well-known resource mono-tone relative to local operations and classical communi-cation (LOCC) [1].Here we are interested in determining the optimal rate R ( A → B ) at which resource A can generate resource B ,in the limit of an infinite supply of A , that is, the largest R such that A ⊗ n ≥ B ⊗ nR for n → ∞ . A simple argu-ment, going back to Carnot [12], implies that if the trans-formation is reversible in the sense that R ( B → A ) (cid:54) = 0,then the rate at which two resources can be reversibly in-terconverted must achieve the optimal rate. Otherwise,it would be possible to generate arbitrary amounts of aresource state from a small number via cyclic transfor-mations to and from another resource state.That reversible interconversion is optimal (when pos-sible) gives a simple means of characterizing the inter-conversion rate by using a “standard” reference resource.Consider a transformation from A to B which proceedsvia the standard resource C : A → C → B . Follow-ing this with B → A must give a combined transforma-tion of unit rate, again to avoid the possibility of spon-taneously generating resources. Composing the rates, wehave R ( A → C ) R ( C → B ) R ( B → A ) = 1, or R ( A → B ) = R ( A → C ) R ( B → C ) , (2)using the fact that R ( A → B ) R ( B → A ) = 1. Withthis framework, we need only define the relative entropy D ( ρ || γ ) = Tr[ ρ (ln ρ − ln γ )] to state the main result ofthis Letter. Theorem 1
Using thermal operations at backgroundtemperature T , asymptotic interconversion at nonvanish-ing rate is possible between all states ρ and σ of a systemwith Hamiltonian H . For γ the Gibbs state of tempera-ture T associated with H , the optimal rate is given by R ( ρ → σ ) = D ( ρ || γ ) D ( σ || γ ) . (3)Simple calculation reveals that D ( ρ || γ ) = βF β ( ρ ) − βF β ( γ ), where F β ( ρ ) ≡ (cid:104) H (cid:105) ρ − k B T S ( ρ ) is the free energyand S ( ρ ) = − Tr[ ρ ln ρ ] the von Neumann entropy. Thus,the free energy directly determines the optimal rate ofresource interconversion in our resource theory.To prove the result we shall employ the connection tofree energy by constructing protocols for both distillation of resource states into a standard state and formation ofresource states from standard states. The standard stateis chosen to have energy but no entropy, so as to representavailable work.Before doing so, it is enlightening to note that, assum-ing reversible interconversion is possible, Eq. (3) followseasily from [13, Theorem 1], [14, Theorem 4]. This re-sult states that any asymptotically-continuous resourcemonotone f determines the interconversion rate via itsregularization f ∞ ( ρ ) = lim n →∞ n f ( ρ ⊗ n ) as R ( ρ → σ ) = f ∞ ( ρ ) /f ∞ ( σ ), provided the latter is nonzero and finite.Here, f ( ρ ) = D ( ρ || γ ) is an athermality monotone (i.e.for all thermal operations E , D ( E ( ρ ) || γ ) ≤ D ( ρ || γ )) bycontractivity of the relative entropy under quantum op-erations and the fact that E ( γ ) = γ . Its regularizationis nonzero and finite since f ∞ ( ρ ) = f ( ρ ), which followsfrom the additivity of the relative entropy and the factthe thermal state of n identical systems is just n copiesof the thermal state of one system. Finally, asymptoticcontinuity follows from extensivity of energy by usingProposition 2 of [15]; we leave the simple derivation ofthis to the Appendix.Extensivity is crucial to the conclusion. For instance,˜ f ( ρ ) = D ( γ || ρ ) (note the reversed order of ρ and γ ) isalso an athermality monotone, but does not lead to theinterconversion rate; the extensivity argument fails and˜ f is not asymptotically continuous. Nonetheless, ˜ f ( ρ )plays an important role in determining the resource re-quirements for creating low-temperature states [6]. Distillation and Formation Protocols.—
In order to es-tablish Theorem 1, let us now turn to the distillationand formation protocols. For purposes of exposition, wespecialize to the case of resources having just two nonde-generate energy levels, call them | (cid:105) and | (cid:105) , i.e. qubits.This nevertheless captures the essential aspects of theproblem. We first consider the distillation and forma-tion of quasiclassical resources ρ , meaning [ ρ, H ] = 0and take up the case of non-stationary resources after-wards. In what follows we sketch the steps required tocomplete the proof and leave the somewhat cumbersomemathematical details to the Appendix.Both distillation and formation protocols must sat-isfy three requirements, up to error terms smaller than O ( n ): (1) energy conservation, (2) unitarity, and (3)equality of input and output dimensions. Without lossof generality, we may take the total Hamiltonian to be H = E (cid:80) i | (cid:105) i (cid:104) | for some energy E , where the sumruns over all the qubits.We begin the distillation protocol with (cid:96) copies of theGibbs state γ of H and n copies of the resource ρ , where ρ = (1 − p ) | (cid:105)(cid:104) | + p | (cid:105)(cid:104) | for arbitrary 0 ≤ p ≤ γ = (1 − q ) | (cid:105)(cid:104) | + q | (cid:105)(cid:104) | for q = e − βE / (1+ e − βE ). Theaim is to effect a transformation of the form γ ⊗ (cid:96) ⊗ ρ ⊗ n → σ ( k ) ⊗ | (cid:105)(cid:104) | ⊗ m by an energy-conserving unitary, suchthat m is as large as possible. The resulting exhaust state σ of k systems is arbitrary, though as an aside we showthat the optimality of the protocol implies that it hasnear-Gibbs form in the Appendix. We denote by R = mn the rate of distillation and (cid:15) = nl the ratio between initialresource states and Gibbs states. The Gibbs states arefree, so we allow (cid:15) → n → ∞ .We now use the fact that for large n , ρ ⊗ n consists ofmixtures of basis states corresponding to length- n binarystrings with roughly np t of 1s in a stringis known as its type , and more concretely we have that,to an error which vanishes as n → ∞ , ρ ⊗ n ≈ (cid:88) t p t P t . (4)Here the t summation runs over strongly typical types,the types for which t = np ± O ( √ n ) [16], and P t de-notes the projector onto the type t . An entirely similarstatement holds for γ ⊗ (cid:96) . For simplicity we shall first pre-tend that both γ ⊗ (cid:96) and ρ ⊗ n consist of a single type andsubsequently show how to extend the argument to allstrongly-typical types.We begin with a single composite type, a concatena-tion of a type coming from the resource state and onefrom the reservoir state. This corresponds to a uniformmixture of strings of length n + (cid:96) , each of which consists oftwo substrings: the first having (cid:96)q
1s and the second np e lh ( q ) × e nh ( p ) such strings, where h ( p ) = − p ln p − (1 − p ) ln(1 − p ) is the binary entropy,expressed in nats.Now consider a transformation which maps thesestrings to new strings having at least m
1s in the right-most positions, (cid:96) (cid:122) (cid:125)(cid:124) (cid:123) . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) (cid:96)q n (cid:122) (cid:125)(cid:124) (cid:123) . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) np → . . .
00 1 . . . (cid:124) (cid:123)(cid:122) (cid:125) rk (cid:124) (cid:123)(cid:122) (cid:125) k . . . (cid:124) (cid:123)(cid:122) (cid:125) m where k = (cid:96) + n − m expresses conservation of dimen-sion , and r and m are to be determined. Conservation ofenergy requires that the number of 1s is conserved, hence (cid:96)q + np = rk + m . Unitarity requires that there are at leastas many strings of length k with rk
1s as the number ofinitial strings: e kh ( r ) ≥ e (cid:96)h ( q )+ nh ( p ) . Roughly speaking,this is conservation of entropy . Using these three condi-tions, we find that the transformation is possible for any R such that h ( q ) + (cid:15)h ( p ) ≤ (1 + (cid:15) − R(cid:15) ) h (cid:18) q + (cid:15)p − (cid:15)R (cid:15) − (cid:15)R (cid:19) . (5)We now expand this with respect to (cid:15) to first order andlet (cid:15) →
0. This means the heat reservoir is much largerthan the resource systems. As a result we obtain thatthe following rate can be achieved R = h ( q ) − h ( p ) + β ( p − q ) h ( q ) + β (1 − q ) = D ( ρ || γ ) D ( | (cid:105)(cid:104) ||| γ ) , (6)establishing one direction necessary for Theorem 1.In the above argument we worked with a single com-posite type, whereas in actuality the initial state is amixture of these. We thus apply the protocol sepa-rately to each composite type, assuming the number m of output excited states to be the same for all inputtypes, with m fitted to the composite type containingthe fewest strings (i.e. the one consisting of strings with (cid:96)q − O ( √ (cid:96) ) + np − O ( √ n ) 1’s). To proceed as above weneed to ensure that any variations from the above condi-tions are small relative to n . Thus, we need to simulta-neously fulfill √ (cid:96) (cid:28) m = Rn , in order for R from (6) tobe achievable, and (cid:96) (cid:29) n , in order for (cid:15) →
0. Choosing (cid:96) = ( Rn ) / therefore ensures that our estimate (6) willbe accurate in the limit n → ∞ . [17]The formation protocol is similar to the distillationprotocol and is again based on considering type trans-formations satisfying the three requirements of energyconservation, unitarity, and dimension conservation. Themajor difference is that whereas the ideal distillation out-put is simply the fixed-type state | (cid:105) ⊗ m , the ideal forma-tion output must recreate a good approximation to theprobabilistic mixture of type classes found in ρ ⊗ n .We construct the formation protocol in three stages.The first two are similar to the distillation protocol. Inthe first, a given type class of the Gibbs state togetherwith the standard resource is transformed into a desiredtype class of the target resource ρ ⊗ n . In the second,the transformation is extended to all the strongly-typicaltypes of the Gibbs state. Finally, in the third step anadditional number of Gibbs states are used to probabilis-tically select which type class of the target should beoutput, in order to recreate the appropriate distributionover types of the target state. In principle this step isirreversible, but since the number of type classes growsonly polynomially with n , the number of extra resourcesrequired for the third step of the formation protocol van-ishes in the n → ∞ limit. The similarity of the first twosteps with the distillation protocol then ensures that theformation protocol achieves the inverse rate.Distillation for arbitrary resource states is related tothat of stationary states, and we can recycle part of theprevious distillation protocol. Suppose the resource statehas the diagonal form ρ = p | φ (cid:105)(cid:104) φ | + (1 − p ) | φ (cid:105)(cid:104) φ | ,for arbitrary orthogonal states | φ k (cid:105) , implying an averageenergy of (cid:104) E (cid:105) = ( p |(cid:104) φ | (cid:105)| + (1 − p ) |(cid:104) φ | (cid:105)| ) E . In n instances of ρ the total energy will overwhelmingly likelybe n (cid:104) E (cid:105) ± O ( √ n ). Now imagine projecting the resourcestate onto the various energy subspaces, destroying anycoherence between them. Just as in (4), ρ ⊗ n is supportedalmost entirely on its typical subspace, whose size notlarger than e nS ( ρ )+ O ( √ n ) . Thus, the state support in ev-ery energy subspace is at most this large. Now we may imagine applying the same scheme asin the previous distillation protocol, creating as manycopies of | (cid:105) as possible. The three conditions now be-come k = (cid:96) + n − m , (cid:96)qE + n (cid:104) E (cid:105) = rkE + mE , and e kh ( r ) ≥ e (cid:96)h ( q )+ nS ( ρ ) . An entirely similar derivation leadsagain to the distillation rate found in (6). Finally, sincethe distillation operations commute with the Hamilto-nian, they commute with the projection onto energy sub-spaces. Thus we may instead imagine that this projectionis performed after the distillation step. Such a projec-tion has no effect on the work systems, while the formof the exhaust state is irrelevant, and therefore we maydispense with the projection step altogether.The formation of arbitrary resource states is more com-plicated than their distillation. Strictly speaking, thedesired transformation is impossible, since the inputs arestates diagonal in the energy basis and the allowed trans-formations cannot change this fact. However, to createthe appropriate coherences between energy subspaces itsuffices to use a small additional resource in the form ofa superposition over energy eigenstates.In particular, a system in a superposition of energylevels acts as a reference system which lifts the superse-lection rule of energy conservation, as in [18, 19], allowingone to create arbitrary coherences over energy levels onthe system. However, since ρ ⊗ n is almost entirely sup-ported on energy levels in the range n (cid:104) E (cid:105) ± O ( √ n ), theformation process requires only a reference system madefrom order √ n qubits. The extra resource of the refer-ence system is thus of a size sublinear in n and does notaffect the rate calculations. This creates an interestingasymmetry between distillation and formation, akin toa similar phenomenon in the resource theory of entan-glementmt, where distillation of entangled states doesnot require any communication but formation requiresan amount sublinear in the number of inputs n . Conclusions .—We have shown that well-known resultsfrom thermodynamics can be derived quite naturallywithin the framework of the resource theory of energy-preserving transformations and auxiliary thermal states.We should emphasize that although the procedures wehave described for the conversion of resource states mayseem quite unnatural from a physical point of view, theiruse is to establish the “in principle” interconversion rategiven in Theorem 1. Any more realistic reversible trans-formation, for instance the Hamiltonian method of [20]or the sequential protocol (for quasiclassical resources)of [21], will necessarily extract the same amount of work.
Acknowledgments .—We thank Jochen Rau and Do-minik Janzing for helpful conversations. JMR acknowl-edges support from the Center for Advanced Security Re-search Darmstadt (CASED). RWS acknowledges supportfrom the Government of Canada through NSERC andthe Province on Ontario through MRI. MH thanks thesupport by Foundation for Polish Science TEAM projectcofinanced by the EU European Regional DevelopmentFund for preparing the final version of this paper. Partof this work was done at National Quantum InformationCentre of Gdansk. The authors thank the hospitalityof Institute Mittag Leffler within the program Quantum Information Science (2010), where part of this work wasdone. [1] R. Horodecki, P. Horodecki, M. Horodecki, andK. Horodecki, Rev. Mod. Phys. , 865 (2009).[2] D. Janzing and T. Beth, IEEE Trans. Inf. Theory ,230 (2003).[3] G. Gour and R. W. Spekkens, New J. Phys. , 033023(2008).[4] I. Marvian and R. W. Spekkens, arXiv:1105.1816 [quant-ph] (2011).[5] M. Horodecki, P. Horodecki, and J. Oppenheim, Phys.Rev. A , 062104 (2003).[6] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth,Int. J. Theor. Phys. , 2717 (2000).[7] M. Horodecki and J. Oppenheim, Nat. Com. (2013).[8] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Op-penheim, and S. Wehner, arXiv:1305.5278 [quant-ph](2013).[9] J. ˚Aberg, Nat. Com. (2013).[10] D. Egloff, O. C. O. Dahlsten, R. Renner, and V. Vedral,arXiv:1207.0434 (2012).[11] J. Oppenheim, M. Horodecki, P. Horodecki, andR. Horodecki, Phys. Rev. Lett. , 180402 (2002). [12] E. Fermi, Thermodynamics (Courier Dover Publications,1956).[13] M. Horodecki, J. Oppenheim, and R. Horodecki, Phys.Rev. Lett. , 240403 (2002).[14] G. Gour, I. Marvian, and R. W. Spekkens, Phys. Rev.A , 012307 (2009).[15] K. Horodecki, M. Horodecki, P. Horodecki, and J. Op-penheim, Phys. Rev. Lett. , 200501 (2005).[16] T. M. Cover and J. A. Thomas, Elements of InformationTheory , 2nd ed. (Wiley-Interscience, 2006).[17] Additionally, the number of strings of a given type np is given by poly( n ) e nh ( p ) , not simply e nhx ( p ) . However,in the limit n → ∞ , the polynomial factor is again notrelevant in our estimates.[18] Y. Aharonov and L. Susskind, Phys. Rev. , 1428(1967).[19] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev.Mod. Phys. , 555 (2007).[20] R. Alicki, M. Horodecki, P. Horodecki, andR. Horodecki, Open Syst. Inf. Dyn. , 205 (2004).[21] P. Skrzypczyk, A. J. Short, and S. Popescu,arXiv:1302.2811 [quant-ph] (2013) APPENDIX
This appendix contains eight sections. The first shows that the relative entropy distance to the Gibbs state is anasymptotically continuous function. The next four sections discuss in detail the state transformation protocols for thecase of two-level systems. Section II presents a distillation protocol for quasi classical states, while section III describesa formation protocol also for quasi classical states. The following two sections extend these protocols to the case ofarbitrary nonstationary two-level resources. Then in Section VI we outline how the results can be easily generalizedto higher dimensions by considering as an example the distillation protocol for quasi classical states. Section VIIdiscusses some characteristics of the exhaust states produced in these protocols. Finally, Section VIII discusses theequivalence of our formulation to other models of thermodynamics and the degree of control one needs to implementour thermal operations.
Appendix A: Extensivity and asymptotic continuity
To show that D ( ρ || γ ) is asymptotically-continuous, we make use of the following, from [1]: Proposition 2
Suppose a function f satisfies (1) “approximate affinity” | pf ( ρ ) + (1 − p ) f ( σ ) − f ( pρ + (1 − p ) σ )) | ≤ c, (A1) for some constant c > and any p such that ≤ p ≤ , and (2) “subextensivity” f ( ρ ) ≤ M log d , where M > isconstant and d = dim( H ) for H the state space on which ρ has support. Then f is asymptotically continuous: | f ( ρ ) − f ( ρ ) | ≤ M (cid:107) ρ − ρ (cid:107) log d + 4 c. (A2)The entropy relative to the Gibbs state, f ( ρ ) := D ( ρ || σ ), satisfies both conditions. To see the first, let τ = (cid:80) k p k τ k for some arbitrary set of density operators { τ k } and probability distribution p k and let ω be another arbitrary densityoperator. Then D ( τ || ω ) = Tr (cid:34)(cid:88) k p k ( − τ k log ω + τ k log τ ) (cid:35) (A3)= Tr (cid:34)(cid:88) k p k ( τ k log τ k − τ k log ω + τ k log τ − τ k log τ k ) (cid:35) (A4)= (cid:88) k p k D ( τ k || ω ) + (cid:88) k p k S ( τ k ) − S ( τ ) , (A5)where in the second line we have added and subtracted Tr( τ log τ k ). Since S ( τ ) ≤ (cid:80) k p k S ( τ k ) + H ( p k ) where H ( p k )denotes the Shannon entropy for the distribution p k , and since the relative entropy is convex, this implies0 ≤ (cid:88) k p k D ( τ k || ω ) − D ( τ || ω ) ≤ H ( p k ) . (A6)Finally, letting ω = γ and { τ k } = { ρ, σ } with distribution ( p, − p ), we find pD ( ρ || γ ) + (1 − p ) D ( σ || γ ) − D ( pρ + (1 − p ) σ || γ ) ≤ h ( p ) ≤ , (A7)where h is the binary entropy function.The fact that the second condition is satisfied, i.e., that the entropy relative to the Gibbs state is subextensive,follows from the fact that the maximum energy of the system is extensive. First, note that D ( ρ || γ ) = βF β ( ρ ) − βF β ( γ ),where F β ( ρ ) = (cid:104) H (cid:105) ρ − β S ( ρ ) is the free energy. Thus, the maximum of D ( ρ || γ ) occurs for ρ = | E max (cid:105)(cid:104) E max | where | E max (cid:105) is the eigenstate of maximum energy. Direct calculation shows D ( | E max (cid:105)(cid:104) E max ||| γ ) = Tr [ | E max (cid:105)(cid:104) E max | (log | E max (cid:105)(cid:104) E max | − log γ )] = −(cid:104) E max | log γ | E max (cid:105) (A8)= βE max + log Z β = βE max + log (cid:88) k e − βE k ≤ βE max + log d. (A9)Here we have assumed that the energy values E j >
0. When the maximum energy is extensive, i.e. E max ≤ K log d for some constant K , we obtain D ( | E max (cid:105)(cid:104) E max ||| γ ) ≤ M log d for M = βK + 1. Appendix B: Distillation of quasiclassical states
For simplicity of presentation, we consider qubit systems with the Hamiltonian given by H = (cid:80) i | (cid:105) i (cid:104) | , where thesum runs over all involved qubits. We start with l copies of the Gibbs state γ and n copies of the resource state ρ ,where ρ = (1 − p ) | (cid:105)(cid:104) | + p | (cid:105)(cid:104) | ; γ = (1 − q ) | (cid:105)(cid:104) | + q | (cid:105)(cid:104) | , (B1)with q = e − β / (1 + e − β ) and β the inverse temperature, which we take as a constant parameter. The aim is toobtain the maximal number of copies possible of qubits in the pure excited state | (cid:105) by implementing a unitary thatcommutes with H and taking the partial trace over some subsystem, γ ⊗ l ⊗ ρ ⊗ n → σ ( k ) ⊗ | (cid:105)(cid:104) | ⊗ m . (B2)We denote R = mn (the rate of distillation) and (cid:15) = nl (the ratio between the number of used Gibbs states and thenumber of resource states). The Gibbs states are free, so we accept that (cid:15) asymptotically vanishes.In the protocol we shall use the fact that up to a small error (vanishing for a large number of qubits) ρ ⊗ n ≈ (cid:88) t p t P t (B3)where t run over strongly typical types, i.e. the types containing strings with the number of 1’s within the interval( np − O ( √ n ) , np + O ( √ n )), and P t denotes the projector onto type t . Similarly γ ⊗ l ≈ (cid:88) t q t Q t (B4)again with q t ≈ qn . The errors in both approximations are smaller than 2 −√ n when quantified by the trace norm.For simplicity we shall first pretend that both γ ⊗ l and ρ ⊗ n consist of a single type. Then further we will show howto extend the argument to a mixture of types.So we start with a tensor product of two types (one from Gibbs, the other from ρ ), i.e. an equal mixture of stringsof length l + n . The string consist of two substrings: the first has ql pn l (cid:122) (cid:125)(cid:124) (cid:123) . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) lq n (cid:122) (cid:125)(cid:124) (cid:123) . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) np . (B5)There are roughly 2 lh ( q ) × nh ( p ) (B6)such strings (with the error being a multiplicative poly( n ) factor).We now apply a unitary transformation to these strings to map them into strings of the same total length whichhave m l (cid:122) (cid:125)(cid:124) (cid:123) . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) lq n (cid:122) (cid:125)(cid:124) (cid:123) . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) np → . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) rk (cid:124) (cid:123)(cid:122) (cid:125) k . . . (cid:124) (cid:123)(cid:122) (cid:125) m (B7)where k = l + n − m (B8)(conservation of dimension), and r and m are about to be determined. First, r is fixed by conservation of energy ,which requires that the number of 1’s is conserved: lq + np = rk + m.. (B9)Then unitarity requires that there are at least as many strings of length k with rk kh ( r ) ≥ lh ( q )+ nh ( p ) . (B10)Roughly speaking this is conservation of entropy . Using (B9) and (B10) we obtain that our transformation is possibleif h ( q ) + (cid:15)h ( p ) ≤ (1 + (cid:15) − R(cid:15) ) h (cid:18) q + (cid:15)p − (cid:15)R (cid:15) − (cid:15)R (cid:19) , (B11)where recall that (cid:15) is the ratio of Gibbs states used, and R the ratio of pure excited states obtained. We now expandthis with respect to (cid:15) to first order and let (cid:15) →
0. This means that we take many more Gibbs states than the resourcestates. In other words, we presume a heat reservoir that is arbitrarily larger than the size of our system. As a result,recalling that q = e − β / (1 + e − β ), we obtain that the following rate can be achieved R = h ( q ) − h ( p ) + β ( p − q ) h ( q ) + β (1 − q ) = S ( ρ || γ ) S ( | (cid:105)(cid:104) ||| γ ) , (B12)i.e., the protocol achieves the upper bound obtained by the monotonicity argument in the main text.So far we have worked with a single type. But our initial state is actually a mixture of products of types Q t ⊗ P t (cid:48) .We thus apply the protocol separately to each type, with the same number m of required output excited states, forall types, with this number being fitted to the product of less numerous types (i.e. the one with the smallest numberof 1’s, namely ( np − O ( √ n ))( lq − O ( √ l )) 1’s). Note, however, that the variations will disappear asymptotically, aswe divide the equations by l . Also the approximation to the number of strings in each type by the exponential of theentropy is correct up to a multiplicative polynomial factor, which is also irrelevant asymptotically. Appendix C: Formation of quasiclassical states
We are going to construct the formation protocol in three stages. The first is to show that for a particular type T q of the l copies of the Gibbs state, we can create a particular type T p of the state we want to form. We then showthat we can do this for all types of the Gibbs state. Finally we show how to correctly get the distribution over typesof the target state.We shall need the following useful lemma: Lemma 3 (Birkhoff primitive) The following operation can be done by means of thermal operations with arbitraryaccuracy: ρ → (cid:88) k p k U k ρU † k (C1) where p k is an arbitrary probability distribution, and the U k commute with the Hamiltonian. In particular, a randompermutation of the systems is a valid operation. Remark.
The accuracy depends on the number of Gibbs states that are used, but in our paradigm they are for free.
Proof.
First, let us note that the following unitary transformation preserves energy: (cid:88) i | i (cid:105)(cid:104) i | ⊗ ˜ U i (C2)provided that | i (cid:105) are eigenvectors of the reservoir Hamiltonian and the ˜ U i commute with the system Hamiltonian.To obtain the required transformation, we take the initial state of the reservoir to be l copies of the Gibbs state.Let q i denote the probability distribution of single strings. We now divide the set of eigenvectors of the reservoirHamiltonian into sets, denoted S k , such that the sum of the probability distribution over i within each set yieldsapproximately the probability p k from (C1), that is, (cid:80) i ∈ S k q i ≈ p k . This can be done with arbitrary accuracy bytaking l large enough, since q i ≤ max { q, − q } l . Then, for every i ∈ S k , we set ˜ U i = U k . (cid:3) We want to form n copies of the state ρ = (1 − p ) | (cid:105)(cid:104) | + p | (cid:105)(cid:104) | from a pure excited state. Let us first show howwe can form any of the typical types of the state. Formation of a maximally mixed state over a fixed type.
Consider a fixed type T p of the state of n systems that we want to create, and let it be one with np l copies of the Gibbs state γ ,and m copies of the excited state | (cid:105) . We consider a final exhaust system consisting of k two-level systems. Considera typical type T q of the Gibbs state; it has lq ≈ lh ( q ) strings up to some 2 √ lh ( q ) factor.We want to map these initial strings onto the N final strings in the type T p as follows. Take { u i } i to be the set of FIG. 1. Mapping of strings in the formation protocol strings in T p and for each string u i consider some set { v ij } j of strings on the exhaust system. We now map each ofthe initial strings to some string u i v ij . This is illustrated in Figure 1.For sets { v ij } j corresponding to different values of i , we can take the number of strings in each set to be the sameor off by 1, simply by assigning the strings in an order determined by fixing j = 1, then incrementing the i registeruntil i = N , then incrementing the j register by 1, reseting i to 1 and again incrementing the i register until i = N and repeating. This is all done to ensure that when we trace out the exhaust system, we get an even mixture overpermutations within the type class.We can now use the analogues of equations (B8), (B9) and (B10) to ensure that we can perform the unitary whichimplements this mapping: m + l = n + k (C3) lq + m = rk + np (C4)2 lh ( q ) ≤ kh ( r )+ nh ( p ) (C5)We now take l ∝ m α with 1 < α < α > (cid:15) = n/l small, and α < √ l is sublinear in m and we can ignore such terms.This maps type T q of the l Gibbs states onto the type T p of ρ ⊗ n . We can map each of the initial Gibbs types onto T p in this manner using a unitary U pq . For each such mapping, the above three equations will change, but only bysome √ l factors which we took to be sublinear in m . We can thus choose m to ensure conservation of energy in theworst case of Eq. (C4), and kh ( r ) is chosen to ensure the inequality Eq. (C5) in the worst case.We now need to implement U pq conditioned on the initial type T q . Since the typical initial types are on orthogonaland diagonal subspaces, we first do a conditional copying of the type class q onto an initialised register. We thenact U pq conditioned on this register. Since the number of types is polynomial in l , the register only needs to be ofsize log l , and thus this resource does not matter as it is sublinear in m . It is an interesting question whether theformation protocol can be made to work without this sublinear supply of pure states.We denote by U p the unitary that creates a particular type T p . To get the distribution over types, we simply usethe Birkoff primitive of Lemma 3 to implement (cid:80) p | p (cid:105)(cid:104) p | ⊗ U p . This is irreversible, but since the number of typicaltypes is polynomial in n , the rate of entropy that is created by this procedure is negligible, i.e. logarithmic in n . It isnot hard to see that Eqs. (C3-C5) give the required rate (i.e. the inverse of the distillation rate). Appendix D: Distillation of arbitrary states
We now extend distillation to the case where our states are not diagonal in the energy eigenbasis.Consider a state ρ = p | φ (cid:105)(cid:104) φ | +(1 − p ) | φ (cid:105)(cid:104) φ | . The average energy of the state is (cid:104) E (cid:105) = p |(cid:104) φ | (cid:105)| +(1 − p ) |(cid:104) φ | (cid:105)| .As before, we consider n copies of ρ and l copies of systems in a Gibbs state γ . Regarding ρ ⊗ n , only the blocks withenergy E ∈ [ n (cid:104) E (cid:105) − √ n, n (cid:104) E (cid:105) + √ n ] will be relevant, i.e. we haveTr (cid:32)(cid:88) E P E ρ ⊗ n (cid:33) ≥ − − O ( n ) (D1)0where the sum runs over E ∈ [ n (cid:104) E (cid:105) − O ( √ n ) , n (cid:104) E (cid:105) + O ( √ n )], with P E the projector onto the energy E eigenspace(this follows from Eq. (E14), proven in Section E).Our protocol has two stages:(i) unitary rotation within energy blocks of a resource system (consisting of n qubits) solely.(ii) drawing work by string permutations on the total system resource (with n qubits) plus heat bath (with l qubits)We write down the resource state in the energy eigenbasis. As said above, only blocks with energy ≈ nE will appear.We will use the fact that the state is, up to exponentially vanishing error, equal to its projection onto the typicalsubspace, having dimension 2 nS ( ρ )+ O ( √ n ) where S ( ρ ) is the von Neumann entropy of ρ . Therefore, within every blockthe rank of the state is not larger than ≈ nS (as a projection cannot increase the rank).Now stage ( i ) is the following: within each energy block, we apply unitary rotation, which diagonalizes the staterestricted to the block in the energy basis. Then there is stage ( ii ), in which we apply the protocol of distillation ofquasiclassical states as in Section F, i.e. we permute strings in such a way that the output strings have m m systems in a pure excited state (note that all coherences initially present in thestate are now left in the garbage). Using the same notation as in Section F, for a product of two single types, e.g.with lq and n (cid:104) E (cid:105) k = m + m − llq + n (cid:104) E (cid:105) = rk + m kh ( r ) ≥ lh ( q )+ nS ( ρ ) As before, taking the limit nl → R is achievable, provided it satisfies R ≤ h ( q ) − S ( ρ ) + β ( (cid:104) E (cid:105) − q ) h ( q ) + β (1 − q ) = S ( ρ || γ ) S ( | (cid:105)(cid:104) ||| γ ) . (D2)Thus also for states that are not quasiclassical, we can reach the upper bound given by the relative entropy distancefrom the Gibbs state. Appendix E: Formation of arbitrary states
We now show that we can achieve reversibility even in the case of states that are not quasiclassical. To do so,however, we must allow the use of a sublinear amount of states that are a superposition over energy eigenstates. Thisis a reasonable assumption since the rate at which such states are consumed vanishes in the asymptotic limit. Thisis very similar to the fact that in entanglement theory, distillation requires no communication but formation requiresa sublinear amount of it. Or, instead, formation requires a state which is a superposition over different amounts ofentanglement. The superposition over different amount of entanglement (known as entanglement spread [3–6]), isanalogous to the superposition over energy eigenstates in the present context. In the athermality context, distillationof work requires no superposition over energy eigenstates, but formation does.Suppose we want to implement some unitary U = (cid:88) ij u ij | E i (cid:105)(cid:104) E j | (E1)that does not conserve energy. We introduce a state that acts as a reference frame for time, | H (cid:105) = (cid:88) f ( h ) | h (cid:105) , (E2)where | h (cid:105) denotes an energy eigenstate, and we implement U inv = (cid:88) ij u ij | E i (cid:105)(cid:104) E j | ⊗ | h − E i + E j (cid:105)(cid:104) h | (E3)on the system and reference frame. If we are interested in implementing U on the state | ψ (cid:105) = (cid:88) i ∈S c i | E i (cid:105) , (E4)1then we do so by implementing U inv on | ψ (cid:105) ⊗ | H (cid:105) . Note that we must ensure that the reference frame system hasenergy levels with gaps of size | E i − E j | for every transition appearing in U . If the energy spread of f ( h ) is largecompared to the largest value of | E i − E j | in an energy transition induced by U , then the state of the reference frameis not disturbed very much in the process.For the problem in which we are interested, this is indeed the case because on the typical subspace, the variation inenergy is sublinear in the number of copies of the state we want to create. To see how this works by way of example,note that if in Eq. (E2), we take f ( h ) to be 1 / √ N for energies h ∈ { , ..., N } and f ( h ) = 0 otherwise, then removinga unit of energy and adding it to another system, does not change the state of Eq. (E2) much. i.e. the inner productbetween | H (cid:105) and (cid:80) h | h − (cid:105)(cid:104) h || H (cid:105) approaches 1, because N (cid:88) h =1 √ N (cid:104) h | N − (cid:88) h =0 √ N | h (cid:105) = 1 − N , (E5)which approaches 1 for large N . States like that of Eq. (E2) therefore allow us to lift the superselection rule forenergy, without being consumed much in the process.This gives some insight into embezzling states [2]. These are resource states that are often used in entanglementtheory in similar situations. For instance, one can use a state similar in form to Eq. (E2) | E (cid:105) = (cid:88) f ( k ) | φ (cid:105) ⊗ kAB | (cid:105) ⊗ ( n − k ) AB (E6)which is a superposition of a different number of entangled EPR pairs | φ (cid:105) AB . These states can be used to implementoperations which need to create superpositions over amounts of entanglement (entanglement spread). Just as removingone unit of energy, doesn’t change the state of Eq. (E2) much, likewise, removing one EPR pair from the state ofEq. (E6) and adding it to another system doesn’t change the embezzling state by much. We can embezzle energy, justas one can embezzle entanglement. We therefore see that a superposition over some resource can create an embezzlingstate for that resource, and will allow us to lift some superselection rule or restriction.With this small superposition over energy states, let us now show that we can create an arbitrary state at a rategiven by the relative entropy distance to the Gibbs state. Let ρ := p | φ (cid:105)(cid:104) φ | + (1 − p ) | φ (cid:105)(cid:104) φ | and ρ ⊗ n = (cid:88) k,g p k | Ψ k,g (cid:105)(cid:104) Ψ k,g | , (E7)with | Ψ k,g (cid:105) := π g | φ (cid:105) ⊗ k ⊗ | φ (cid:105) ⊗ n − k , (E8)for π g a permutation.The idea of the protocol is as follows: we will first create a diagonal state (cid:37) n = (cid:88) p k | t k , s g (cid:105)(cid:104) t k , s g | (E9)which has the same spectrum as ρ ⊗ n and where each eigenstate has the same average energy as an eigenstate in thetypical subspace of ρ ⊗ n . From the result of the previous section it is not hard to see that this can be done at a rategiven by the relative entropy distance of ρ to the Gibbs state, since in the limit of many copies, the regularised relativeentropy distance is the same. We would then like to rotate the diagonal basis to the Ψ k,s -basis. This cannot be doneby unitaries which commute with the Hamiltonian unless we allow for a reference frame | H (cid:105) which is a superpositionover energy states. We then want to show that the reference frame which allows us to break the energy superselectionrule is consumed at a vanishingly small rate. We do so by showing that the reference frame superposition is over asize sublinear in n . This can be understood as coming from the fact that in the typical subspace, the superpositionover different types is sublinear.We consider only typical | Ψ k,g (cid:105) with k ∈ Typ ρ := [ np − √ n, np + √ n ]. Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ ⊗ n − (cid:88) k ∈ Typ ρ ,g p k | Ψ k,g (cid:105)(cid:104) Ψ k,g | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ − Ω( √ n ) (E10)Let also | Ψ k,g (cid:105) := (cid:88) t (cid:48) ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) (E11)2where | t (cid:48) , s (cid:48) (cid:105) is an eigenstate of the Hamiltonian with energy t (cid:48) ( s (cid:48) labels the degeneracy). From Eq. (E8) it followsthat the sum in Eq. (E11) will be peaked around only a few energy values t (cid:48) . Indeed, with | φ (cid:105) := a | (cid:105) + b | (cid:105) , (E12)and | φ (cid:105) := b | (cid:105) − a | (cid:105) , (E13)set Typ := [ nE t − √ n, nE t + √ n ], where E t := (cid:0) ( n − t ) | b | + t | a | (cid:1) /n . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) t (cid:48) / ∈ Typ t ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 2 − Ω( √ n ) (E14)Note that since E t := (cid:0) ( n − t ) | b | + t | a | (cid:1) /n the degeneracy of each energy state | t k , g s (cid:105) is at least as large as thedegeneracy of | Ψ ks (cid:105) .Now we construct the reference frame. Let | w (cid:105) be an energy eigenstate with energy n ( p | b | + (1 − p ) | a | ) − n / . Itis needed to pad the dimension of the reference frame, since although the probability that it happens is vanishinglysmall, the unitary does connect states with large energy difference. After the protocol, we will see that | w (cid:105) will hardlybe changed, and thus is only used as a catalyst. We define the reference system as follows | H (cid:105) := 1 (cid:112) | H | (cid:88) h ∈ H | h (cid:105) (E15)with | h (cid:105) := | h (cid:48) (cid:105) ⊗ | w (cid:105) , where | h (cid:48) (cid:105) is an energy eigenstate of energy h (cid:48) and H := { , ..., n / } .Consider the energy preserving unitary U := (cid:88) h,t,s,t (cid:48) ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105)(cid:104) t k , s g | ⊗ | h + t − t (cid:48) (cid:105)(cid:104) h | . (E16)Then in the sequel we prove that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) U (cid:88) k ∈ Typ ρ ,s p k | t k , s g (cid:105)(cid:104) t k , s g | ⊗ | H (cid:105)(cid:104) H | U † − ρ ⊗ n ⊗ | H (cid:105)(cid:104) H | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ O ( n − / ) . (E17)We first analyze the action of U in | s, t (cid:105) ⊗ | H (cid:105) : U ( | t k , s g (cid:105) ⊗ | H (cid:105) ) = (cid:88) t (cid:48) ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) ⊗ (cid:32) (cid:112) | H | (cid:88) h ∈ H | h + t k − t (cid:48) (cid:105) (cid:33) = | ν (cid:105) + | ν (cid:105) + | ν (cid:105) (E18)where the non-normalized pure states | ν k (cid:105) are given by | ν (cid:105) := (cid:88) t (cid:48) ∈ Typ t ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) ⊗ | H (cid:105) , (E19) | ν (cid:105) := (cid:88) t (cid:48) ∈ Typ t ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) ⊗ | err t (cid:48) (cid:105) (E20)with | err t (cid:48) (cid:105) := 1 (cid:112) | H | (cid:88) h ∈ H | h + t − t (cid:48) (cid:105) − | H (cid:105) , (E21)3and | ν (cid:105) := (cid:88) t (cid:48) / ∈ Typ t ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) ⊗ (cid:32) (cid:112) | H | (cid:88) h ∈ H | h + t − t (cid:48) (cid:105) (cid:33) . (E22)Set t k = E k and let us take s g = g . We can do the latter since as we mentioned, the degeneracy of | t k , s g (cid:105) is largerthan the degeneracy of | Ψ k,g (cid:105) . Then, (cid:107) U ( | t k , s g (cid:105) ⊗ | H (cid:105) ) − | Ψ k,g (cid:105) ⊗ | H (cid:105)(cid:107) ≤ (cid:107)| ν (cid:105) − | Ψ k,g (cid:105) ⊗ | H (cid:105)(cid:107) + (cid:107)| ν (cid:105)(cid:107) + (cid:107)| ν (cid:105)(cid:107) . (E23)We now show that the three terms in the R.H.S. are small. For | ν (cid:105) we first note that for t (cid:48) ∈ Typ t (cid:107)| err t (cid:48) (cid:105)(cid:107) ≤ n − / . (E24)by taking the worst case. Then (cid:107)| ν (cid:105)(cid:107) = (cid:88) t (cid:48) ∈ Typ t ,s (cid:48) | c kgt (cid:48) s (cid:48) | (cid:107)| err t (cid:48) (cid:105)(cid:107) ≤ max t (cid:48) ∈ Typ t (cid:107)| err t (cid:48) (cid:105)(cid:107) ≤ n − / . (E25)For | ν (cid:105) , in turn, we have (cid:107)| ν (cid:105)(cid:107) ≤ (cid:88) t (cid:48) / ∈ Typ ,s (cid:48) | c kgt (cid:48) s (cid:48) | (cid:107)| err t (cid:48) (cid:105)(cid:107) ≤ (cid:88) t (cid:48) / ∈ Typ ,s (cid:48) | c kgt (cid:48) s (cid:48) | ≤ − Ω( √ n ) . (E26)Finally, for | ν (cid:105) , (cid:107)| ν (cid:105) − | Ψ k,g (cid:105) ⊗ | H (cid:105)(cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) t (cid:48) ∈ Typ t ,s (cid:48) c kgt (cid:48) s (cid:48) | t (cid:48) , s (cid:48) (cid:105) − | Ψ k,g (cid:105) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ − Ω( √ n ) . (E27)From Eq. (E23) it thus follows that (cid:107) U ( | t k , s g (cid:105) ⊗ | H (cid:105) ) − | Ψ k,g (cid:105) ⊗ | H (cid:105)(cid:107) ≤ O ( n − / ) . (E28)Since (cid:107)| ψ (cid:105)(cid:104) ψ | − | φ (cid:105)(cid:104) φ |(cid:107) ≤ √ (cid:107)| ψ (cid:105) − | φ (cid:105)(cid:107) for every two states | ψ (cid:105) , | φ (cid:105) , we find (cid:13)(cid:13) U ( | t k , s g (cid:105)(cid:104) t k , s g | ⊗ | H (cid:105)(cid:104) H | ) U † − | Ψ k,g (cid:105)(cid:104) Ψ k,g | ⊗ | H (cid:105)(cid:104) H | (cid:13)(cid:13) ≤ O ( n − / ) . (E29)Eq. (E17) then follows from the triangle inequality for trace-norm and Eq. (E10). Appendix F: Distillation of quasiclassical states in arbitrary dimensions
In this section we present the details of the distillation protocol for quasiclassical states for the general case of d -dimensional systems. This is presented as an example of how the results can be extended to arbitrary dimensionsusing arguments very similar to those used in the two-dimensional case.The input to the protocol consists of n copies of the initial resource ρ and (cid:96) copies of the Gibbs state γ of the sameHamiltonian H . Since the states are quasiclassical, the overall state of the input is fully described by a collection ofstrings s ∈ { , . . . , d − } n + (cid:96) listing the energy level occupied by each system, each string weighted by its probabilityof occurrence. Here d is the total number of energy levels of H , and the first n entries of s correspond to the state of ρ and the remainder correspond to γ . If some of the energy levels are degenerate, we simply work in the eigenbasisof ρ to remove the degeneracy in labeling. Because permutations within each of the two substrings do not change the4overall probability, we can therefore instead work with the collection of occupation frequencies f of the state, whichdescribe the number of systems in the ground state, first excited state, second excited state, and so on (divided bythe total number of systems), again each weighted by an appropriate probability.We would now like to define a protocol which creates as many standard resources in the form of work as possible.In the qubit case the standard resource had a very simple form, namely the excited state of the Hamiltonian. Here,however, the setup is more cumbersome, as there are d − | (cid:105) can betransferred to two instances of the state | (cid:105) ). To handle this issue most simply, we imagine that, in addition to theresource and thermal states, we also have a work system at our convenience. The work system is capable of acceptingarbitrary amounts of energy, i.e. it has energy transitions which precisely correspond to those of H , but it cannotaccept any entropy. Now the goal of the protocol is to change the occupation of the energy levels so as to transfer asmuch energy to the work system as possible.Let us now restrict attention to a fixed occupation frequency f ρ of the resource and a fixed f γ for the Gibbs state.We will later design the protocol so that it works for every such frequency pair which has appreciable probability.Suppose that we now change the occupation numbers by an amount described by the vector − n x (whose prefactor ischosen for later convenience). This results in a new occupation vector ν defined by( n + (cid:96) ) ν ≡ n f ρ + (cid:96) f γ − n x . (F1)For this to be an allowable transformation in our framework, this mapping must satisfy two constraints: energyconservation and unitarity. In contrast to the qubit case, here the input and output dimensions are equal by design.Energy conservation is simply enforced by requiring the work system to take up the change in energy of theinput systems. Using the vector H to describe the energy of each energy level, the initial energy is given by E in = n H · f ρ + (cid:96) H · f γ while the final energy is E out = n H · f ρ + (cid:96) H · f γ − n H · x . Energy conservation is then the statementthat the work extracted is given by W = E in − E out = n H · x .Unitarity is enforced by making sure that the total number of configurations (strings) consistent with each occu-pation vector is conserved by the process. The total number of possible input strings in this case, N in , is just theproduct of the multinomial cofficients using the frequency vectors: N in = M ( n f ρ ) M ( (cid:96) f γ ) = n !( n ( f ρ ) )! · · · ( n ( f ρ ) d − )! (cid:96) !( (cid:96) ( f γ ) )! · · · ( (cid:96) ( f γ ) d − )! (F2)The maximum number of strings N out = M (( n + (cid:96) ) ν ) which can be created in the n + (cid:96) systems given the newoccupation frequency ν is the multinomial coefficient of the new occupation frequency vector, N out = M (( n + (cid:96) ) ν ) = ( n + (cid:96) )!(( n + (cid:96) ) ν )! · · · (( n + (cid:96) ) ν d − )! . (F3)Therefore, a sufficient condition for unitarity is N in ≤ N out , or M ( n f ρ ) M ( (cid:96) f γ ) ≤ M (( n + (cid:96) ) ν ).It can be shown that the multinomial coefficients obey the bounds e ( ne ) d f · · · f d nH ( f ) ≤ n !( nf )! · · · ( nf d )! ≤ ne d − nH ( f ) , (F4)and therefore M ( n f ) ≈ nH ( f ) ± O (log n ) . The unitarity condition then becomes nH ( f ρ ) − O (log n ) + (cid:96)H ( f γ ) − O (log (cid:96) ) ≤ ( n + (cid:96) ) H ( ν ) + O (log( n + (cid:96) )) (F5)Defining (cid:15) = n(cid:96) we may express this as (cid:15)H ( f ρ ) + H ( f γ ) ≤ (1 + (cid:15) ) H ( ν ) + O ( log (cid:96)(cid:96) ) . (F6)Using the expression for ν from (F1) and assuming that (cid:15) (cid:28) H ( ν ) = H (cid:18) n f ρ + (cid:96) f γ − n x n + (cid:96) (cid:19) (F7)= H ( f γ ) + (cid:15) [( f γ + x − f ρ ) · − H ( f γ ) + ( x − f ρ ) · log f γ ] + O ( (cid:15) ) (F8)= H ( f γ ) − (cid:15) ( f ρ − x ) · log f γ − (cid:15)H ( f γ ) + O ( (cid:15) ) . (F9)5Here is the vector of all ones, and we have made use of the fact that f · = 1 for any frequency vector f , which alsoimplies x · = 0. Combining this with (F5) we obtain the relation − x · log f γ ≤ D ( f ρ || f γ ) + O ( log (cid:96)(cid:96) ) . (F10)The next step is to fix the protocol to the worst case among the likely frequency vectors f ρ and f γ . Their probabilitiessharply peaked around the individual distributions ρ and γ , respectively. Specifically, fixing an error parameter δ ,the probability that || f ρ − ρ || ≥ δ is less than a quantity of order e − nδ . The variations of likely f ρ from ρ itself areagain O ( √ n ) as in the argument presented in the main text (there the statement was phrased in terms of the numberof 1’s and not the type class or frequency distribution itself). Thus we may choose (cid:96) = ( Rn / ) to ensure that (F1)and (F5) hold with f ρ replaced with ρ and similarly for γ , at least to terms sublinear in n . We conclude that even inthe worst, but still probable case, we have − x · log γ ≤ D ( ρ || γ ) − O ( √ n ) . (F11)Now log γ = − β H − log Z , so this condition becomes β H · x ≤ D ( ρ || γ ) − O ( √ n ) . (F12)This equation gives the minimum amount of extractable work among all the likely frequencies, which is taken to bethe target amount for the process. As the extraction is unitary for each frequency, and these correspond to disjointquantum states, we can thus find a unitary for the entire input capable of generating β (cid:104) D ( ρ || γ ) − O ( √ n ) (cid:105) units ofuseful work per input resource state, with probability greater than 1 − O ( √ n ). Appendix G: Structure of the exhaust state
When doing a transformation of ρ ⊗ n into σ ⊗ m , at the end of the protocol we actually obtain σ ⊗ m ⊗ π k and wetrace out π k , which lives in k = Ω( n ) copies of the system. Although π n is usually far away, in fidelity, from manycopies of a Gibbs state, we show that its reductions are very close to a Gibbs state. The main observation is thatbecause π n should be useless for extracting more copies of σ at a non-zero rate, we must have S ( π k || ρ ⊗ kβ ) ≤ k − δ , (G1)for δ >
0. But by subadditivity of the entropy we have S ( π k || ρ ⊗ kβ ) = − S ( π k ) − k (cid:88) l =1 Tr( ρ k,l log ρ β ) ≥ − k (cid:88) l =1 S ( π k,l ) − k (cid:88) l =1 Tr( ρ k,l log ρ β )= k (cid:88) l =1 S ( π k,l || ρ β ) , (G2)where π k,l := Tr \ l ( π k ) is the reduced state of π k that is obtained by partial tracing all the systems except the l -thone. Let us assume for simplicity that all the π k,l are identical. Then S ( π k, || ρ β ) ≤ k − δ , (G3)which by Pinsker’s inequality implies (cid:107) π k, − ρ β (cid:107) ≤ Ω( k − δ ) . (G4)More generaly, repeating the same argument for larger blocks we get that (cid:107) Tr L,L +1 ,...,k ( π k ) − ρ ⊗ Lβ (cid:107) ≤ Ω( Lk − δ ) . (G5)6 Appendix H: Equivalence and degree of control for thermal operations
Here, we address two questions. The first is how our paradigm, where we use unitaries V which commute withthe total Hamiltonian H , relates to other approaches. The second is how much control an experimenter needs overthe choice of unitaries V . To answer the first question, consider a common approach to thermodynamics, which is tomanipulate thermodynamical systems using an external apparatus. In this model, the systems are manipulated usinga time-dependent Hamiltonian, H ( t ). Another approach is to add an interaction term H int between various systemswe are trying to manipulate (e.g. the resource, and the heat bath), and then bring these systems into contact withone another. Let us now see that these are equivalent to considering unitaries V which commute with the originalHamiltonian H .First, observe that in the case of a time-dependent Hamiltonian H ( t ), we can simply include the clock as oneof our systems. Letting τ be the coordinate operator of the clock system and Π τ such that [ τ, Π τ ] = − i , define H indep = H ( τ ) + Π τ . The τ observable faithfully records the time t , as can be seen by solving the Heisenbergequations of motion to get τ ( t ) = τ (0) + t . Now consider a joint density matrix for the system plus clock of the form ξ ( t ) = ρ ( t ) ⊗ | t (cid:105)(cid:104) t | , where | t (cid:105) are the eigenstates of τ . The time-independent Hamiltonian H indep acting on the state ξ ( t ) will generate the equation of motions of H ( t ) acting on ρ ( t ), but will also conserve energy. To see this, recall thatthe product rule of derivatives gives dξ ( t ) dt = dρ ( t ) dt ⊗ | t (cid:105)(cid:104) t | + ρ ( t ) ⊗ ddt | t (cid:105)(cid:104) t | , (H1)while the Heisenberg equation of motion gives dξ ( t ) dt = i [ H indep , ξ ( t )]= i [ H ( t ) , ρ ( t )] ⊗ | t (cid:105)(cid:104) t | + ρ ( t ) ⊗ [Π τ , | t (cid:105)(cid:104) t | ] (H2)Comparing the above two equations we have ˙ ρ ( t ) = i [ H ( t ) , ρ ( t )] as claimed. That a system in a pure state stays ina pure state, can be achieved by having the clock have a large coherent superposition over energy levels, thus thechange in it’s state can be made arbitrarily small, as explained in Section E. We thus can go from a picture with achanging Hamiltonian, to one with a fixed one. The model with time-dependent Hamiltonian is therefore equivalentto the one considered here, with fixed Hamiltonian.Likewise, in the case where an interaction term is added, we can take the total Hamiltonian to be H tot = H + H int and assume that initially, ( H + H int ) | ψ (cid:105) ≈ H | ψ (cid:105) i.e. the systems are initially far apart. They can then evolve unitarily,such that the systems interact, and then move far enough apart that the interaction terms are negligible again. Insuch a picture, an eigenstate of the initial Hamiltonian H will evolve into an eigenstate of H with the same energy (byconservation of energy, and the fact that the interaction is negligible at initial and final times). Thus, all that happenshere is that eigenstates of fixed energy evolve to other eigenstates of the same energy, and this can be accomplishedby means of a fixed Hamiltonian H and a unitary V which commutes with it. We thus see that also the picture ofadding interaction terms is equivalent to having a fixed Hamiltonian H , and operations V which commute with it.Similarly, the application of a unitary during some time period can be made via application of a fixed Hamiltonian.One can include an internal clock τ which merely acts as a catalyst and thus have some fixed Hamiltonian H tot = H + H int g ( τ ) + Π τ (H3)which effectively implements e − iH int t over some time interval determined by the function g ( τ ). Here Π τ is conjugateto τ and one can verify via the Heisenberg equations of motion that τ depends linearly on t . Since [ H int , H ] = 0 onecan also verify via the Heisenberg equations of motion for Π τ that there is no backreaction or energy exchange to theclock at late times provided g ( τ ) is chosen such that (cid:82) t f t i g (cid:48) ( τ ) = 0 and g ( τ ) and τ ( t ) chosen such that g ( τ ) = 0 before t = t i and after t = t f .One might be concerned that if we perturb the Hamiltonian slightly, our work extraction will not be robust. To seethis, let us consider the case where we don’t succeed in implementing our unitary exactly, but rather some H int whichdoes not completely commute with the original Hamiltonian [ H, H int ] = − iδ . Viewed internally, will see that this isequivalent to allowing a violation of conservation of energy by amount δ – something which is interesting to study inits own right. Now the equations of motion for τ are unchanged, and thus the unitary e − iH int t is still implemented.However there is some backreaction on the clock. Solving the Heisenberg equations of motion for Π, we find thatthere is a small momentum kick to the clockΠ( t f ) − Π( t i ) = (cid:90) t f t i g (cid:48) ( t ) H int ( t )= H int ( t f ) − H int ( t i ) (H4)7where we have taken g ( τ ) to be 1 between t i and t f and 0 everywhere else. At each cycle some amount of energy getsstored in the clock, depending on the initial and final states of the system. If we allow these to fluctuate, then thetransfer is some δ and the clock undergoes a random walk.We thus find that if we run the extraction process as a cycle, where we repeat the process over several cycles, thenat each cycle we still exactly implement e − iH int t , it’s just that it now has some tiny non-commuting part with theoriginal Hamiltonian, resulting in some energetic backreaction to the clock, and thus some entropy being stored in theclock. However, the extracted work grows linearly, and δ can be made arbitrarily small. After n cycles, the momentof the clock has undergone a random walk, of order √ n , an amount which is negligible compared to the extractablework in the case of many cycles.Let us now turn to the second question. It might appear that an experimenter who wished to implement ourprotocols would need to very carefully manipulate all the many degrees of freedom of the n systems and the heatbath. However, this is not the case, as we will now demonstrate explicitly using the example of work distillation.There, we were mapping eigenstates which had a type lq on the heatbath γ , and pn on the resource ρ to microstateswhich had type rk on the garbage σ , and m any mapping will do. The only important thing which is required is just that the unitary operation map the initialtypes to the final types. Thus an experimenter who wishes to implement the protocol, does not need fine-grainedcontrol over the mapping of microstates within one type to microstates within another type. She only needs to knowthat the unitary maps one type into another. In other words, there are an exponentially large number of possibleimplementations of our protocols each of which map particular strings within the initial types to particular strings inthe final types. However, it doesn’t matter which implementation is chosen, and the experimenter thus does not needthe fine degree of control that is required to achieve a particular implementation.We can think of the type as being like a macroscopic variable such as the total magnetisation of a compositesystem, or its total energy (indeed it is the latter). In the distillation protocol, we map the macroscopic variables ofenergy on two large systems ( γ ⊗ l and ρ ⊗ n ) to the macroscopic variable of energy on the final system. Any unitarywhich accomplishes this will successfully implement our protocol. Thus, the experimenter only needs control over themacroscopic variables, not the microscopic ones.Equivalence of these paradigms is discussed in more detail, and in the case of finite systems, in [7]. [1] K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, Phys. Rev. Lett. , 200501 (2005).[2] W. Van Dam and P. Hayden, Phys. Rev. A , 060302, (2003).[3] H-K. Lo and S. Popescu. arXiv:quant-ph/9707038.[4] P. Hayden and A. Winter. Phys. Rev. A , 012326 (2003).[5] A. Harrow and H-K. Lo. IEEE Trans. Inf. Theory50