Beyond heat baths: Generalized resource theories for small-scale thermodynamics
BBeyond heat baths: Generalized resource theories for small-scalethermodynamics
Nicole Yunger Halpern a1, 2 and Joseph M. Renes b31
Institute for Quantum Information and Matter,Caltech, Pasadena, CA 91125, USA Perimeter Institute for Theoretical Physics,31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5 Institute for Theoretical Physics, ETH Z¨urich, Switzerland (Dated: August 27, 2018)
Abstract
Thermodynamics has recently been extended to small scales with resource theories that modelheat exchanges. Real physical systems exchange diverse quantities: heat, particles, angular mo-mentum, etc. We generalize thermodynamic resource theories to exchanges of observables otherthan heat, to baths other than heat baths, and to free energies other than the Helmholtz freeenergy. These generalizations are illustrated with “grand-potential” theories that model move-ments of heat and particles. Free operations include unitaries that conserve energy and particlenumber. From this conservation law and from resource-theory principles, the grand-canonical formof the free states is derived. States are shown to form a quasiorder characterized by free oper-ations, d -majorization, the hypothesis-testing entropy, and rescaled Lorenz curves. We calculatethe work distillable from, and we bound the work cost of creating, a state. These work quantitiescan differ but converge to the grand potential in the thermodynamic limit. Extending thermo-dynamic resource theories beyond heat baths, we open diverse realistic systems to modeling withone-shot statistical mechanics. Prospective applications such as electrochemical batteries are hopedto bridge one-shot theory to experiments. PACS numbers: 05.70.Ce, 89.70.Cf, 05.70.-a, 03.67.-aKeywords: Resource theory, One-shot, Statistical mechanics, Thermodynamics, Information theory,Nonequilibrium a E-mail: [email protected] b E-mail: [email protected] a r X i v : . [ qu a n t - ph ] F e b . INTRODUCTION Advances in small-scale experiments and in quantum information have generated inter-est in “thermodynamics without the thermodynamic limit.” Recent experiments involvemolecular motors and ratchets [1, 2], optical thermal ratchets [3], the unfolding of one DNAor RNA molecule [4–7], and nanoscale walkers [8]. Analyses of these experiments featurethermodynamic concepts such as heat, work, and equilibrium. These concepts are notwell-defined outside the thermodynamic limit of n → ∞ particles. Hence the experimen-tal advances in single-molecule manipulations invite us to extend thermodynamics to smallscales.The resource-theory framework developed in quantum information theory has recentlybeen successfully applied to this problem. Resource theories have been used to calculatehow efficiently scarce quantities can be distilled and transformed via cheap, or “free,” op-erations [9]. Perhaps the most famous example is the resource theory of pure bipartite en-tanglement (which we will call “entanglement theory”) [10]. In entanglement theory, agentsdistill Bell pairs of maximally entangled qubits, usable to simulate quantum channels, frompartially entangled states via local operations and classical communications (LOCC). Otherresource theories quantify the values of asymmetry [11–13], quantum-computation tools [14],and information [15–17]. Benefits of the resource-theory framework include its operationalformulation and the explicit modeling of all resources with physical degrees of freedom.To an agent with access to a heat bath, nonequilibrium states have value because work canbe extracted from them and stored in a battery. Nonequilibrium states’ values have beenquantified with a family of equivalent resource theories, each associated with an inversetemperature β of the bath [18–23]. We call these resource theories Helmholtz theories , asthe central results involve variations on the Helmholtz free energy F := E − T S .Many experiments involve baths other than heat baths, involve interactions other heatexchanges, and are characterized by free energies other than the Helmholtz free energy. TheGibbs free energy G := E − T S + pV describes processes that occur at fixed temperaturesand pressures, such as tabletop chemical reactions. The grand potential Φ := E − T S − µN describes heat-and-particle exchanges; and other free energies describe electrochemistry,magnetic fields, mechanical stress and strain, etc. [24, 25]. Different types of baths (equiva-lently, different types of interactions, or different free energies) invite modeling by differentfamilies of resource theories. Each family’s constituents correspond to different values ofthe bath’s properties. For example, each member of the family of Helmholtz theories cor-responds to one value of the inverse temperature β . Altogether, the families describingdifferent baths form an extended family of thermodynamic resource theories amenable toexperimental investigation in the present or near future.We introduce this extended family in this paper, illustrating the formalism with heat-and-particle exchanges. In grand-potential resource theories, free operations conserve energyand particle number. The only states that, if free, prevent such resource theories from beingtrivial are shown to be grand canonical ensembles e − β ( H − µN ) /Z . We derive the grand canoni-cal ensemble upon establishing rigorously, using the resource-theory formalism, that the freestates in Helmholtz theories are canonical ensembles e − βH /Z . States are shown to form aquasiorder characterized by a variant of majorization called d -majorization , which is relatedto binary hypothesis testing. By exploiting the quasiorder, we calculate the work extractablefrom, and bound the work required to create, one copy of a state R = ( ρ, H, N ), even by pro-tocols that have a specified probability of failing. The work yield and work cost are shown to2iffer from each other in general, unlike in conventional thermodynamics, as observed in [19].In the limit as the number of copies of R extracted from or created approaches infinity, theaverage work yield and work cost approach the difference Φ( ρ, H, N ) − Φ( γ ρ , H, N ) betweenthe state’s grand potential and the corresponding equilibrium state’s grand potential.We have structured our results as follows. In the next section, we review the resource-theory framework and define the family of generalized thermodynamic resource theories. Wewill illustrate the family with grand-potential theories. In Sec. III, we deduce the uniqueform that free states can assume in these theories. In Sec. IV, we interrelate the quasiorderof states, the generalized notion of majorization, and binary hypothesis testing. In Sec. V,we define work in the resource-theory framework and use the quasiorder to determine thework yield and work cost of creating single instances of arbitrary states. In Sec. VI, we showthat these one-shot work quantities imply asymptotic results similar to traditional thermo-dynamics. We conclude by discussing possible applications of our generalized framework toreal physical systems.This work bridges the information-theoretic tool of thermodynamic resource theories tophysical reality. We pave the way for physical realizations, with experimental platforms, ofentropic predictions about small scales. II. THERMODYNAMIC RESOURCE THEORIES
First, we introduce the resource-theory framework. We define generalized thermodynamicresource theories, then illustrate them with grand-potential theories.
A. The resource-theory framework
The resource-theory framework models experimental situations in which some physicaltransformations between quantum states are difficult, while others are easy [9]. In quantumoptics, for instance, generating coherent states is easy (e.g., using a laser), while generatingnumber states is difficult. In entanglement theory, classical communication and physical op-erations on systems possessed by one party each (LOCC) are easy, while quantum operationson systems distributed amongst multiple parties are impossible. The states that are difficultto create can be regarded as resources since, with free operations, they can simulate difficultoperations. Given a maximally entangled state, separated parties restricted to LOCC canimplement a quantum channel.A resource theory is defined by physical operations assumed to be easy, or free . Freeoperations include the creation of free states ; all other states are resources. This definitionspecifies an ordering of states: States A and B are ordered as A (cid:55)→ B if free operationscan create B from A . The ordering (cid:55)→ is a quasiorder, satisfying reflexivity ( A (cid:55)→ A ) andtransitivity ( A (cid:55)→ B and B (cid:55)→ C implies A (cid:55)→ C ). The quasiorder differs from a partialorder: Even if A (cid:55)→ B and B (cid:55)→ A , A is not necessarily B [26].Functions of resources that respect the quasiorder, in the sense that A (cid:55)→ B implies f ( A ) ≥ f ( B ), are termed resource monotones [17, 21, 27]. Simple sets of monotones com-pletely characterize the quasiorders in some resource theories, including the resource theoriesin this paper. Many monotones have operational interpretations [17, 21].We are interested only in resource theories whose quasiorders are nontrivial—in whichsome transformations between some resources are impossible. When independently specify-3ng a resource theory’s free operations and free states, we must prevent free states and freeoperations from being able, together, to generate arbitrary states.Having introduced the resource-theory framework, we define generalized thermodynamicresource theories and illustrate them with grand-potential theories. B. Generalized thermodynamic resource theories
Requiring that free operations conserve particular physical quantities leads to the ex-tended family of thermodynamic resource theories. Which quantities are conserved dependson which physical systems are modeled, as explained in [28]. The Hamiltonian H is con-served in what we have termed Helmholtz theories . Janzing et al. first defined Helmholtztheories while investigating the resources required to cool systems, though those authorsdid not use the term “resource theory” [18]. More recently, Brand˜ao et al. studied conver-sions between resources in the asymptotic limit, as the number n of copies of the convertedresource diverges ( n → ∞ ) [20]. Horodecki and Oppenheim extended the analysis of con-versions beyond the asymptotic limit [19]. The literature about thermodynamic resourcetheories has exploded recently: Since the first draft of the present paper was released, co-herences [29, 30] and correlations have been explored [31]; connections have been drawn tofluctuation relations [32–34]; and free operations have been generalized [35].In the grand-potential theories focused on in this paper, free operations preserve totalenergy and total particle number. Free states in thermodynamic resource theories such asgrand-potential theories model baths such as heat-and-particle reservoirs. As we shall see inSection III, the free states must be equilibrium states, such as grand canonical ensembles if H and N are conserved. If nonequilibrium states are free, the resource theory’s quasiorderbecomes trivial.We associate different baths, (equivalently, different physical quantities that can be ex-changed, or, in anticipation of Sec. VI, different free energies) with different families ofthermodynamic resource theories. The family of grand-potential theories models heat-and-particle exchanges. The resource theories in each family differ only by the values of theintensive variables that characterize the bath. To specify a grand-potential resource theory,one specifies an inverse temperature β and a chemical potential µ . (For simplicity, we focuson systems that contain particles of only one type. To specify a grand-potential theory thatmodels exchanges of particles of k types, one specifies β, µ , µ , . . . , µ k .)Now, let us define the thermodynamic resource theories, their states, and their operationsmore precisely. Free operations preserve quantities represented, in conventional thermody-namics, by extensive variables. These variables are represented by operators that, witha density operator, define a state. To specify a state R in a grand-potential theory, onespecifies a density operator, a Hamiltonian, and a number operator: R = ( ρ, H, N ) . (1)These operators are defined on a quantum state space (Hilbert space) H R . How a physicalsystem’s bath and interactions translate into intensive and extensive variables that define ageneral family of thermodynamic resource theories detailed in [28].For simplicity, we specialize to states whose operators commute with each other ([ ρ, H ] =[ ρ, N ] = [ H, N ] = 0) and have discrete, finite spectra. (Since the initial release of this paper,noncommuting operators have been discussed in [28, 36–38].) However, we do not restrict4he forms of H and N further. We denote the dimension of H R by d R . The density operator ρ can be represented by a matrix diagonal relative to the eigenbasis shared by H and N .Such a quasiclassical density operator is fully specified by a vector r , which we shall callthe state vector , of its eigenvalues. Hence we also denote the state by R = ( r, H, N ). Theordering of the elements r i in r is discussed in Sec. IV.The composition of the state R = ( ρ, H R , N R ) on H R with the state S = ( σ, H S , N S ) on H S is defined as R + S = ( ρ ⊗ σ, H R + H S , N R + N S ) , (2)wherein H R + H S = H R ⊗ S + R ⊗ H S and N R + N S is defined similarly.As detailed in Section III, free states have density operators whose the probabilities equalBoltzmann factors. The free states in the grand-potential theory take the form γ = e − β ( H − µN ) /Z, (3)wherein β and µ are real numbers and Z is the normalization factor, or partition function.We denote free states by G = ( γ, H, N ) or G = ( g, H, N ). Each resource R = ( ρ, H, N ) isassociated with an equilibrium state G R = ( γ R , H, N ), or G = ( g R , H, N ).
1. Free operations
We call the free operations in thermodynamic resource theories equilibrating operations .They are defined in our grand-potential example as follows.
Definition 1 (Equilibrating operation) . In the grand-potential theory defined by ( β, µ ) , an equilibrating operation on a state R = ( ρ, H R , N R ) is any realization of the following threesteps:(a) the drawing of a free state G = ( γ, H G , N G ) from the bath;(b) the performing of a unitary transformation U on R + G , wherein [ U, H R + H G ] = [ U, N R + N G ] = 0 ; and(c) the discarding (tracing out) of any subsystem A associated with its own Hamiltonian andnumber operator.The operation is a completely positive trace-preserving linear map of the form R (cid:55)→ R (cid:48) = (Tr A ( U [ ρ ⊗ γ ] U † ) , H R + H G − H A , N R + N G − N A ) . (4)Free operations can mix levels whose energies equal each other and whose particle numbersequal each other. We call free operations equilibrating operations because (as shown inSec. IV) free operations monotonically evolve states toward equilibrium states. Equilibratingoperations induce on states a quasiorder that we denote by R β,µ (cid:55)−−→ R (cid:48) .In thermodynamic resource theories other than the grand-potential theories we focus on,free unitaries preserve operators associated with other extensive variables. For example, if asystem has N particles of Species 1 and N particles of Species 2, [ U, N tot ] = [ U, N tot ] = 0.Equilibrating operations idealize the operations easily performable by thermodynamicexperimentalists. Experimentalists cannot perform all unitaries that preserve energy, par-ticular number, etc. Thermal operations were “coarse-grained” to more realistic operationsin [39]. We expect similar coarse-graining to bridge equilibrating operations from idealizationto reality. 5 II. UNIQUE FORM OF FREE STATES
The form of equilibrating operations in Definition 1 implies that only grand-canonicalensembles can be free in grand-potential theories, else the quasiorder breaks down. Thebreakdown manifests in two ways.
Theorem 1.
Consider any grand-potential resource theory in which each pair ( H, N ) cor-responds to exactly one free state G = ( γ, H, N ) . If γ does not have the Boltzmann form ofEq. (3) , then(a) some resources R can be generated solely with equilibrating operations: G β,µ (cid:55)−−→ R , and(b) equilibrating operations can transform one copy of any state R into one copy of any state S : R β,µ (cid:55)−−→ S . The proofs appear in Appendix A, but we sketch the main ideas here. First, we derive theform of the free states in Helmholtz theories. Then, we bootstrap from Helmholtz theoriesto Theorem 1, which concerns grand-potential theories.To prove Claim (a), we apply the derivation of the forms of the free states in the resourcetheory of nonuniformity [16], which models closed isolated systems [28]. Consider someenergy-and-particle-number eigensubspace S E i ,N j . The free state γ has some weight on S E i ,N j . If that weight is distributed nonuniformly across the levels in S E i ,N j , free operationscan redistribute the weight arbitrarily across S E i ,N j , generating states not defined as free.Claim (b) follows from modifying an argument by Janzing et al . [18]. The argument concernsthe “effective temperatures” of the states that can be created from given resources in theabsence of any bath.These resource-theory derivations of equilibrium ensembles offer operational alternativesto assumptions such as the Ergodic Hypothesis. According to the Ergodic Hypothesis,uniform distributions represent equilibrated isolated systems’ states. Such assumptions havedrawn criticism [41, 42], lending operational replacements appeal. IV. QUASIORDER ON STATES
The quasiorder induced by equilibrating operations on quasiclassical states is equiva-lent to a generalization of majorization. Veinott defined this generalization first, callingit d -majorization [43]. Ruch and collaborators (who called d -majorization the mixing dis-tance ) [44, 45] applied d -majorization to physics, as did Uhlmann and colleagues [46, 47]. Wedub this quasiorder in general thermodynamic resource theories equimajorization , becauseit is d -majorization relative to equilibrium states. Definition 2.
Let R and S denote states in any grand-potential theory defined by ( β, µ ) . Let g R and g S denote the corresponding equilibrium states’ state vectors, which contain d R and In [21], the canonical form ( e − βH /Z, H ) of the free states in Helmholtz theories is argued to followfrom [40]. According to [40], only canonical ensembles are completely passive: No work can be extractedfrom canonical ensembles, even from infinitely many, in the absence of other resources. The work extrac-tion in [40], however, is not formulated as in Helmholtz resource theories. How to translate the resultfrom [40] into resource theories may not be obvious to all readers. To clarify this subtlety and others,we derive free states’ forms directly from the resource-theory framework. Around the time our paper wasreleased, an alternative approach appeared in [21]. S elements respectively. R equimajorizes S , written as R (cid:31) β,µ S , if there exists a d S × d R stochastic matrix M such that M r = s, M g R = g S , and d S (cid:88) i =1 M ij = 1 ∀ j = 1 , , . . . , d R . (5)In the resource theory of nonuniformity, which models closed isolated systems, equilibriumstates are microcanonical ensembles: g R = d R (1 , , . . . ,
1) [16, 17]. Relative to these uniformstates, equimajorization reduces to majorization [27].Janzing et al. established that the quasiorder on quasiclassical resources is equivalentto equimajorization in Helmholtz theories [18, Theorem 5]. An alternative proof appearsin [19]. In Appendix B, we extend the proof technique in [18] to grand-canonical theories,obtaining the following result.
Theorem 2.
Let R and S denote states in the grand-potential theory defined by ( β, µ ) .There exists an equilibrating operation that maps R to S if and only if R equimajorizes S : R β,µ (cid:55)−−→ S ⇐⇒ R (cid:31) β,µ S. (6)As mentioned above, sets of resource monotones completely characterize equimajorizationand so characterize the existence of equilibrating operations. One such set consists of the f -divergences [48–50]. Every convex function f corresponds to an f -divergence φ f ( R ) = d R (cid:88) i =1 g i f (cid:18) r i g i (cid:19) , (7)wherein g i denotes the i th element of the state vector g R of the equilibrium state as-sociated with R . Subsets of the f -divergences suffice to characterize equimajorization,as shown below. Various choices of f lead to well-known functions [51]. For example, f ( x ) = x log x and f ( x ) = − log x lead to the relative entropies D ( r || g R ) = (cid:80) d R i =1 r j log( r i /g i )and D ( g R || r ). The function f ( x ) = ( x α − / ( α −
1) leads to the R´enyi divergences D α ( r || g R ) = − α log (cid:16)(cid:80) d R i =1 r αi g (1 − α ) i (cid:17) for α ≥ Lorenz curve , introduced by Lorenz in economics [52], encodes another complete setof monotones. Lorenz curves were applied recently to Helmholtz resource theories [19]. In agrand-potential theory, the rescaled Lorenz curve L R : [0 , → [0 ,
1] represents the state R .The curve is the piecewise linear function that connects the points( t k , L R ( t k )) = (cid:40) (0 , k = 0 (cid:16)(cid:80) kj =1 g π ( j ) , (cid:80) kj =1 r π ( j ) (cid:17) k ∈ { , . . . , d R } , (8)wherein π denotes a permutation such that the sequence ( r π ( j ) /g π ( j ) ) j is non-increasing. Inaccordance with [19, 52], we define the rescaled Lorenz curve as a monotonically increasingconcave function. (Different conventions appear elsewhere [27].)Having defined the f -divergences and the rescaled Lorenz curve, we will state their rela-tionship with equimajorization. Ruch, Schranner, and Seligman first proved this relationshipfor continuous systems [44], using tools from measure theory. Uhlmann proved the relation-ship more directly, for discrete systems, which we address [46]. By following Uhlmann, wewill prove this proposition in Appendix B: 7 roposition 3. For any states R and S in the grand-potential resource theory defined by ( β, µ ) , the following are equivalent:(a) R (cid:31) β,µ S .(b) L R ( t ) ≥ L S ( t ) for all t ∈ [0 , .(c) φ f a ( R ) ≥ φ f a ( S ) for every function f a ( t ) = max { , t − a } associated with any a ∈ R .(d) φ f ( R ) ≥ φ f ( S ) for all continuous convex functions f . An illustration appears in Fig. 1. t L ( t ) R R R G FIG. 1: (Color online) Rescaled Lorenz curves for three resources ( R , R , R ) and anequilibrium state ( G ). The Lorenz curve encodes the quasiorder on states, as equilibratingoperations can transform R into R (cid:48) if and only if L R ( t ) ≥ L R (cid:48) ( t ) t ∈ [0 , R β,µ (cid:55)−−→ R and R β,µ (cid:55)−−→ R , but R and R are incomparable. The equilibrium state, havingthe linear Lorenz curve L G ( t ) = t , is at the bottom of the quasiorder.Having characterized L R in terms of eigenvalues, we explain its relationship with hypoth-esis testing. The rescaled Lorenz curve is equivalent to the minimal Type II error probability,cast as a function of the Type I error probability, in an asymmetric hypothesis test. Har-remo¨es noted the relationship between Lorenz curves and hypothesis tests [53]; we establishthe relationship more concretely.An asymmetric hypothesis test is used to distinguish whether a given state is ρ or σ . Asindicated by our notation, hypothesis testing can be defined in quantum contexts. A test canbe thought of as a two-outcome positive operator-valued measurement (POVM) { Q, − Q } .If the measurement yields the outcome Q , the state is likely ρ . If the measurement yields − Q , the state is likely σ . A Type I error occurs if the state is ρ but − Q obtains, so thestate seems likely to be σ . A Type II error occurs if the state is σ but seems likely to be ρ .The optimal test minimizes the Type II error probability while preventing the Type I errorprobability from exceeding some tolerance (cid:15) ∈ [0 , b (cid:15) ( ρ || σ ) := min Tr[ Qρ ] ≥ − (cid:15) ≤ Q ≤ Tr[ Qσ ] . (9)8he condition Tr[ Qρ ] ≥ − (cid:15) is called the constraint , and Tr[ Qσ ] is the objective function .Equation (9) defines a semidefinite program, a type of convex optimization, which has adual form: b (cid:15) ( ρ || σ ) = max µρ − σ ≤ τµ,τ ≥ { (1 − (cid:15) ) µ − Tr[ τ ] } . (10)The primal and dual forms’ equivalence follows from properties of semidefinite programs [54].In quasiclassical notation, Q is represented by a matrix, and traces are replaced by sums.Hypothesis testing can be related to rescaled Lorenz curves as follows. Consider distin-guishing between the state vector r in the quasiclassical state R = ( r, H, N ) and the g R inthe equilibrium state G R = ( g R , H, N ). Lemma 4.
The inverse of (cid:15) (cid:55)→ b (cid:15) ( r || g R ) is the piecewise linear function that connects thepoints ( L R ( t k ) , − t k ) , wherein t k and L R ( t k ) define the rescaled Lorenz curve for R . Thatis, ( t k , − L R ( t k )) = ( b (cid:15) ( r || g R ) , (cid:15) ) . (11)The proof appears in Appendix B. V. ONE-SHOT WORK YIELD AND COST
Let us quantify the work required to create, and the work extractable from, one copy ofa state R = ( r, H, N ) via protocols that can fail, as realistic protocols can. Upon motivat-ing the calculation, we introduce the hypothesis-testing entropy D (cid:15) H , incorporate a failureprobability into equilibrating operations, and define work in thermodynamic resource theo-ries. Finally, we calculate the extractable work and bound the work cost. Proofs appear inAppendix C.Conventional thermodynamics concerns the average work (cid:104) W gain (cid:105) extractable from, andthe average cost (cid:104) W cost (cid:105) of creating, states by infallible protocols in the asymptotic limit.In the asymptotic limit , or thermodynamic limit, infinitely many identical copies of R areextracted from or created. (cid:104) W gain (cid:105) and (cid:104) W cost (cid:105) depend on the Shannon entropy S S , itself anaverage: S S ( r ) := (cid:88) i r i ln r i = (cid:104) ln r i (cid:105) r . (12)If few copies of a state are extracted from or created, the average cost or yield quantifiesthe protocol’s efficiency poorly. Alternatives to S S , called one-shot entropies , quantify effi-ciencies in information-processing (e.g., [55–59]) and statistical-mechanics (e.g., [19, 21, 60–63]) problems that involve few systems or trials. In addition to involving finite numbers,realistic protocols have nonzero probabilities of failing to accomplish their purposes. Failureprobability has been incorporated into one-shot entropies as a parameter (cid:15) [54, 55].One alternative to S S is the hypothesis-testing entropy D (cid:15) H . D (cid:15) H is defined in terms ofthe hypothesis test quantified in Eq. (9). The work extractable from, and the work cost ofcreating, one copy of a state R will be quantified with D (cid:15) H .9 efinition 3. The hypothesis-testing relative entropy between quantum states ρ and γ isdefined as D (cid:15) H ( ρ || γ ) := − ln b (cid:15) ( ρ || γ ) (13) or, equivalently, by b (cid:15) ( ρ || γ ) = e − D (cid:15) H ( ρ || γ ) . Let us incorporate failure probability and work into thermodynamic resource theories.A faulty operation is defined as a transformation whose output approximates the desiredoutput. Operationally, a state R (cid:48) approximates a state R if no testing procedure consistentwith quantum mechanics can reliably distinguish the states. For simplicity, we focus on ap-proximations R (cid:48) that differ from R only because of their density operators: If R = ( ρ, H, N ),then R (cid:48) = ( ρ (cid:48) , H, N ). If R (cid:48) approximates R , we write R (cid:48) ≈ (cid:15) R and say that R (cid:48) is (cid:15) -close to R . (Equivalently, we write ρ (cid:48) ≈ (cid:15) ρ and say that ρ (cid:48) is (cid:15) -close to ρ .) Since density operators’distinguishability is related to the trace distance, we define R (cid:48) ≈ (cid:15) R by (cid:107) ρ (cid:48) − ρ (cid:107) ≤ (cid:15) .We define work in terms of the changing of the energy level occupied by a battery . Instatistical physics and mechanics, work is defined as an integral along a path in real space orin phase space. Quantum states can follow paths along which work integrals cannot easilybe calculated [64]. Work on and by quantum systems has been defined more operationally interms of a “work bit” that has a gap W [19] and in terms of a weight that stores gravitationalpotential energy [65]. We define work similarly to [65].Our battery B is any system that has the following qualities: (a) The energies in therange accessed by the agent are finely spaced. (b) The battery occupies an energy eigenstate,being a reliable energy reservoir. By B E , we denote the battery resource ( | E (cid:105)(cid:104) E | , H, N ). Weassume βE (cid:29)
1, for if βE ≈
1, agents can use the battery’s equilibrium state to driveprocesses that require energy E . Such a use would contradict our physical notion of usefulwork.Having defined B , we can define the work extractable from, and the work cost of, astate transformation. If the battery transitions from B E i to B E f while R transforms into S ,the transformation outputs the work E f − E i (which is negative if the transformation costswork). The (cid:15) -work value of a resource R is defined as the greatest W for which R + B E β,µ (cid:55)−−→ (cid:15) B E + W . (14)The work cost of (cid:15) -approximately creating R is the least W such that B E + W β,µ (cid:55)−−→ (cid:15) R + B E . (15)Formally, W (cid:15) gain ( R ) = max { W : R + B E β,µ (cid:55)−−→ (cid:15) B E + W , βE (cid:29) } (16) W (cid:15) cost ( R ) = min { W : B E + W β,µ (cid:55)−−→ (cid:15) R + B E , βE (cid:29) } . (17)The simplicity of our battery model facilitates calculations. Realistic features could beincorporated as follows. First, the battery could occupy a mixed state, or a superpositionof energy eigenstates, at any stage in either protocol. Second, the system and battery couldbegin or become entangled. Such entanglement could be analyzed as in [66]. Frenzel etal. point out that a classical field is often assumed to raise and lower a quantum system’senergy levels. But fields are not classical and become entangled with the system. A batterymight become entangled similarly.Having defined work, we state the work value, and bound the work cost, of R .10 heorem 5. The (cid:15) -work value of a state R = ( r, H, N ) associated with the free state G R =( g R , H, N ) is W (cid:15) gain ( R ) = β D (cid:15) H ( r || g R ) . (18) The work cost of creating an (cid:15) -approximation to a state R is bounded by max δ ∈ (0 , − (cid:15) ] (cid:104) β D − (cid:15) − δ H ( r || g R ) − β log (cid:0) δ (cid:1)(cid:105) ≤ W (cid:15) cost ( R ) ≤ β D − (cid:15) H ( r || g R ) − β log (cid:0) − (cid:15)(cid:15) (cid:1) . (19)A proof appears in Appendix C. Each expression in the theorem contains an entropy D ε H ( r || g R ), for some error probability ε . The factor β introduces dimensions of energy. Eachbound contains a logarithmic correction.Theorem 5 bounds optimal efficiencies. Thermodynamic optima tend to characterizephysically unrealizable processes. Example processes include quasistatic, or infinitely slow,evolutions. Experiments cannot proceed infinitely slowly. What implications can Theorem 5have for real physical processes? As a process is performed increasingly slowly, its efficiencyis expected approach our predictions. A similar approach has been reported in [67]. Koski et al. erased a bit of information repeatedly. As the erasure’s speed dropped, the amountof heat dissipated dropped to near the Landauer limit. VI. WORK YIELD AND COST OF MANY COPIES OF A RESOURCE
From the previous section’s one-shot work quantities, we can recover results reminis-cent of traditional thermodynamics and can compare how W (cid:15) gain differs from W (cid:15) cost as thethermodynamic limit is approached. We denote n copies of R = ( r, H, N ) by R ⊗ n =( r ⊗ n , (cid:80) ni =1 H i , (cid:80) ni =1 N i ). In the thermodynamic limit, or asymptotic limit, n → ∞ . Alsoin the limit, we show, W (cid:15) gain ( R ⊗ n ) and W (cid:15) cost ( R ⊗ n ) tend to the difference between the grandpotential of R and the grand potential of the associated equilibrium state G R . As thethermodynamic limit is approached, W (cid:15) gain ( R ⊗ n ) and W (cid:15) cost ( R ⊗ n ) differ by terms of order √ n .To derive the thermodynamic limits of Eq. (18) and Ineqs. (19), we invoke the AsymptoticEquipartition Property of D (cid:15) H [54]:lim n →∞ n D (cid:15) H ( r ⊗ n || s ⊗ n ) = D ( r || s ) ∀ (cid:15) ∈ (0 , , (20)wherein r = ( r , r , . . . , r d ) and s = ( s , s , . . . , s d ) denote probability distributions over thesame alphabet. We have used the definition D ( r || s ) := d (cid:88) i =1 r i ln (cid:18) r i s i (cid:19) (21)of the relative entropy, defining 0 ln 0 = 0 [68].Applying Eq. (20) to Eq. (18) and to both sides of Ineqs. (19) yieldslim n →∞ n W (cid:15) gain ( R ⊗ n ) = lim n →∞ n W (cid:15) cost ( R ⊗ n ) = 1 β D ( r || g R ) . (22)11n the asymptotic limit, the bounds in Ineqs. (19) converge. All strategies of work extractionand state formation, from risky ( (cid:15) ≈
1) to conservative ( (cid:15) ≈ β D ( r || g R ) = 1 β (cid:88) i r i (cid:18) ln r i − ln e − β ( E i − µn i ) Z (cid:19) (23)= (cid:104) H (cid:105) r − T · k B S S ( r ) − µ (cid:104) N (cid:105) r + k B T ln Z (24)= Φ β,µ ( R ) − Φ β,µ ( G R ) . (25)Recall that Φ := E − T S − µN denotes the grand potential, and − k B T ln Z denotes theequilibrium state’s free energy, in conventional thermodynamics. Using one-shot informationtheory, we have recovered the convergence, in the asymptotic limit, of a state’s average workcost and average work yield to a difference between free energies.Equation (22) implies that all resources can be reversibly converted into one another inthe asymptotic limit. For any states R and S and for fixed (cid:15) , there exists an n great enoughthat R ⊗ n β,µ (cid:55)−−→ (cid:15) S ⊗ m n for some m n ≥
1. To create m n copies of S from n copies of R , oneextracts all the work possible from R ⊗ n , then constructs ( S ⊗ m n ) (cid:48) ≈ (cid:15) S ⊗ m n from the work.We define the optimal asymptotic conversion rate R ( R (cid:55)→ S ) as the asymptotic limit ofthe supremum of the rates m n /n achievable by conversion protocols that approximate thedesired output arbitrarily well in the asymptotic limit (protocols for which (cid:15) → R ( R (cid:55)→ S ) = D ( r || g R ) D ( s || g S ) . (26)Thus, all nonequilibrium states can be reversibly converted into each other in the asymp-totic limit. This result may be surprising. One might have thought that resourcefulnesscan be “locked” into one form—energy, particle number, or information—preventing an R whose resourcefulness manifests in energy from transforming into an S whose resourcefulnessmanifests in particle number. Apparently, such locking does not occur. This asymptoticreversible convertibility resembles that in Helmholtz theories [20] and in the nonuniformitytheory [15, 16]. Asymptotic reversible convertibility in general resource theories is discussedin [69].The Asymptotic Equipartition Theorem dictates the leading order (order- n ) behavior of D (cid:15) H ; a more-refined analysis reveals the next-leading-order terms. Applying techniques frominformation theory, we show in Appendix D that the latter terms are of order √ n : W (cid:15) gain ( R ⊗ n ) = 1 β [ nD ( r || g R ) − O ( √ n )] , and (27) W (cid:15) cost ( R ⊗ n ) = 1 β [ nD ( r || g R ) + O ( √ n )] . (28)As one might expect, the work cost W (cid:15) cost ( R ⊗ n ) lies above the thermodynamic value, whereas W (cid:15) gain ( R ⊗ n ) lies below. This discrepancy contrasts with conventional thermodynamics, ac-cording to which a reversible cycle can extract work from R and use that work to recreate R .Outside the thermodynamic limit, such reversible cycles are impossible. The resource-theoryframework refines the Second Law of Thermodynamics, as discussed in [21].12 II. CONCLUSIONS
We have extended the resource-theory formulation of thermodynamics beyond heat baths.Earlier thermodynamic resource theories model heat exchanges; but many physical systemsexchange heat, particles, volume, magnetization, and other observables. We model these ex-changes with a set of families of thermodynamic resource theories. Each family correspondsto one one free energy, one type of interaction, and one type of bath.To illustrate mathematical results, we focused on grand-potential theories, whose freeoperations conserve energy and particle number. We showed, using resource-theory princi-ples, why free states must be grand canonical ensembles. We characterized the quasiorderon states by an extension of majorization, here termed equimajorization . We showed thatequimajorization can be formulated in terms of rescaled Lorenz curves and of the optimal er-ror probability in asymmetric hypothesis testing. The hypothesis-testing entropy was shownto be proportional to the amount of work extractable from a state R and to bound the workcost of creating R . In the asymptotic limit n → ∞ , W (cid:15) gain ( R ⊗ n ) and W (cid:15) cost ( R ⊗ n ) were shownto converge to a difference Φ( R ) − Φ( G R ) between grand potentials. The convergence rateswere shown to differ on the order of √ n . In the limit, all states were shown to be reversiblyinterconvertible.Opportunities for bringing these resource theories closer to experiments remain. Exam-ples include the finite sizes of heat baths, catalysts (ancillas that facilitate transformationswhile suffering no or little degradation), limitations on how much of a resource can be ex-changed, and the speeds with which transformations can be implemented. In the presence ofan infinitely large heat bath and enough work, every resource R can be converted into everyother. As the number n of copies R grows large, the size of the required bath scales onlysuperlinearly with n [20]. A finite-sized bath limits the resource hierarchy, possibly spoilingthe interconvertibility of all resource states. The effects of the bath’s finiteness might beincorporated as in [70], which concerns the cost of erasing with a small bath (albeit outsidethe resource-theory framework). Another open question concerns how finite-sized catalystsaffect the work cost of resource interconversion [21]. Finally, the optimal efficiencies in thepresent paper might characterize only quasistatic—infinitely slow—protocols. Realistic pro-tocols proceed at finite rates. Extensions of our results to finite speeds may draw inspirationfrom [71, 72]. As noted in the final reference, finite speeds relate directly to the density ofaccessible bath levels.Generalized thermodynamic resource theories open a host of realistic thermodynamicsystems to modeling with resource theories. Particular physical platforms call out for mod-eling: heat-and-particle exchanges, electrochemical batteries, chemical reactions, etc. Asthermodynamic potentials other than the Helmholtz free energy F characterize commonexperiments, generalized thermodynamic resource theories offer opportunities for realizingone-shot statistical mechanics experimentally. ACKNOWLEDGEMENTS
NYH is grateful for conversations with Tobias Fritz, Iman Marvian, Markus M¨uller,Brian Space, and Rob Spekkens. JMR acknowledges helpful conversations with MichaelWalter. This work was supported by a Virginia Gilloon Fellowship, an IQIM Fellowship,NSF grant PHY-0803371, the Perimeter Institute for Theoretical Physics, the Swiss NationalScience Foundation (through the National Centre of Competence in Research Quantum13cience and Technology and grant No. 200020-135048), and the European Research Council(grant No. 258932). The Institute for Quantum Information and Matter (IQIM) is anNSF Physics Frontiers Center that receives support from the Gordon and Betty MooreFoundation. Research at the Perimeter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontario through the Ministry of Researchand Innovation. NYH is grateful to Renato Renner for hospitality at ETH Z¨urich duringthe development of this paper.
VIII. APPENDICES
Below, we prove claims, presented above, about grand-potential theories. We derive thegrand-canonical forms of free state vectors; describe the quasiorder; calculate the work W (cid:15) gain extractable from, and bound the work W (cid:15) cost required to create, one copy of a state; and show,via second-order asymptotics, that W (cid:15) gain does not always equal W (cid:15) cost . Appendix A DERIVATIONS OF FREE STATES’ FORMSA Proof of Theorem 1(a)
The state vectors of the free states in grand-potential theories are shown to be grandcanonical ensembles. We first review the derivation, in [16], of the forms of the free states inthe resource theory of nonuniformity, which models closed isolated systems [28]. From thenonuniformity result, we deduce the canonical form of the free states in Helmholtz theories.From the Helmholtz result, we bootstrap to grand-potential theories. Free operations in the nonuniformity theory are called noisy operations . Each noisyoperation consists of three steps: Any free state u (whose form is to be derived) can becreated, any permutation π can be implemented, and any subsystem A can be discarded(marginalized) [15–17]: r (cid:55)→ (cid:88) A π ( r ⊗ u ) . (29)Resource states are defined here as states that are not free u ’s (or, in general thermodynamicresource theories, states that are not free G ’s) that appear explicitly in the definition of freeoperations. A resource theory is trivial if its free operations alone can generate resourcestates.As shown in [16], the free states must be uniform probability distributions, lest thenonuniformity theory be trivial. (Indeed, the quasiorder of resources becomes trivial: Fromenough copies of any state, free operations can generate any other state.) The free states’form is derived as follows [16]: Suppose that some nonuniform state u is free. By Shannoncompressing many copies of u [73], agents can create pure states (1 , , , . . . ,
0) for free.Via noisy operations, agents can create noise for free. Able to generate purity and noise,free operations can generate arbitrary states. Only if all free states are uniform is thenonuniformity theory nontrivial.General thermodynamic resource theories contain the nonuniformity theory as a specialcase. In grand-potential theories, for example, free operations can arbitrarily permute levelswithin each sector S E,N that corresponds to one energy E and one particle number N . Hencethe weight that each free state has on a sector S E,N is distributed uniformly across the levels14n S E,N . We call this uniformity the uniform-eigensubspace condition . The condition isdefined in Helmholtz theories as follows.
Definition 4.
Let R = ( r, H ) denote a state in any Helmholtz theory, wherein r =( r , . . . , r d ) . R obeys the uniform-eigensubspace condition if, for every degenerate eigenvalue E of H , all the r i associated with E equal each other. Let us apply the nonuniformity-theory argument to the uniform-eigensubspace condition.
Proposition 6.
The free states in each thermodynamic resource theory obey the uniform-eigensubspace condition. If the free states disobeyed the condition, there would exist resources R that equilibrating operations alone could generate: G (cid:55)→ R . Suppose that free states disobeyed the uniform-eigensubspace condition. Each free state’sstate vector g would have some weight p on each sector S that corresponds to some energy,some particle number, etc. Equilibrating operations could distribute p arbitrarily across thelevels in S but could not change the value of p .The uniform-eigensubspace condition implies the following three lemmas, which completeour derivation of the canonical form of the free states in Helmholtz theories. Lemma 7.
Let H and H denote any Hamiltonians that share an energy gap ∆ (and whosespectra are discrete). Let E and E + ∆ denote eigenvalues of H , and let E and E + ∆ denote eigenvalues of H . Define G = ( g , H ) as a Helmholtz-theory state whose weightson E and E + ∆ are g ( E ) and g ( E + ∆) . Define G = ( g , H ) , g ( E ) , and g ( E + ∆) analogously. If G + G satisfies the uniform eigensubspace condition, the ratio of the weightsdepends only on the gap: g ( E + ∆) g ( E ) = g ( E + ∆) g ( E ) . (30) Proof.
The eigenenergy E + E + ∆ of G + G has a twofold degeneracy. Since G + G satisfies the uniform eigensubspace condition, the weight of g ⊗ g on one degenerate levelequals the weight on the other: g ( E ) g ( E + ∆) = g ( E + ∆) g ( E ) . (31)Since E and E are arbitrary, each ratio of probabilities depends only on ∆; the otherdetails of H and H are irrelevant. Lemma 8.
Let G = ( g, H ) denote any free Helmholtz-theory state that has weights g ( E ) and g ( E + ∆) on either side of an energy gap ∆ . The ratio of the weights varies exponentiallywith the gap: g ( E + ∆) g ( E ) = e − β ∆ , (32) wherein β ∈ R .Proof. Consider a state G = ( g, H ) that has three energies separated by gaps ∆: H = E | (cid:105)(cid:104) | + ( E + ∆) | (cid:105)(cid:104) | + ( E + 2∆) | (cid:105)(cid:104) | + d (cid:88) i =4 E i | i (cid:105)(cid:104) i | . (33)15et us apply Lemma 7 to two copies of G , defining E = E and E = E + ∆. Equation (30)becomes g ( E + 2∆) g ( E + ∆) = g ( E + ∆) g ( E ) =: f (∆) , (34)which implies [ f (∆)] = g ( E + 2∆) g ( E + ∆) g ( E + ∆) g ( E ) = g ( E + 2∆) g ( E ) = f (2∆) . (35)This scaling implies that, if f is continuous, it is an exponential: f (∆) = e − β ∆ (36)for some β ∈ R . (The realness of β follows from the probabilities’ realness.)Let G (cid:48) denote any free state that has the same gap ∆ as G . The composition G + G (cid:48) obeys the uniform-eigensubspace condition. Hence G (cid:48) satisfies Eq. (36), by Lemma 7, evenif G (cid:48) does not have the form in Eq. (33). Lemma 9.
All the gaps in all the free states in any given Helmholtz theory correspond tothe same β .Proof. Let G = ( g, H ) and G (cid:48) = ( g (cid:48) , H (cid:48) ) denote free states in some Helmholtz theory. Let∆ denote a gap in H ; and let ∆ (cid:48) = n ∆, wherein n denotes a positive integer, denote a gapin H (cid:48) . We wish to show that g and g (cid:48) correspond to the same β . The proof will be extendedto rational proportionality constants, then to arbitrary constants.Some free state G (cid:48)(cid:48) = ( g (cid:48)(cid:48) , H (cid:48)(cid:48) ) has p > n equally spaced levels E, E + ∆ , . . . , E + p ∆.For example, G (cid:48)(cid:48) might denote a harmonic oscillator. This G (cid:48)(cid:48) will serve as a thermometerthat interrelates the temperatures of G and G (cid:48) . Let E = E and E = E + m ∆ for any m ∈ { , , . . . , p } . By an argument like the one used to prove Lemma 8, f ( m ∆) := g (cid:48)(cid:48) ( E + m ∆) g (cid:48)(cid:48) ( E ) (37)= (cid:20) g (cid:48)(cid:48) ( E + ∆) g (cid:48)(cid:48) ( E ) (cid:21) m (38)=: f (∆) m , (39)wherein g (cid:48)(cid:48) ( E + k ∆) denotes the weight on level k .Consider substituting m = n and ∆ = n ∆ (cid:48) into the left-hand side (LHS) of, f ( m ∆) = f (∆) m : f (∆ (cid:48) ) = f (∆) n . (40)We have related the ratio of the weights across the gap of G (cid:48) to the ratio of the weightsacross the gap of G . That is, we have related the temperature of G (cid:48) to the temperature of G . Now, suppose that ∆ (cid:48) = mn ∆. Consider substituting m ∆ = n ∆ (cid:48) into the LHS of f ( m ∆) = f (∆) m : f ( n ∆ (cid:48) ) = f (∆) m . This equation’s LHS also equals f (∆ (cid:48) ) n , by Eq. (39).Equating the two expressions for f ( n ∆ (cid:48) ) yields f (∆ (cid:48) ) = f (∆) m/n . (41)16e have related the temperature of G (cid:48) to the temperature of G , effectively by consideringmultiple copies of each state.Finally, suppose that ∆ (cid:48) = α ∆, wherein α denotes an irrational number. α can beapproximated arbitrarily well by a ratio m/n . Arbitrarily many copies of G and G (cid:48) relatethe temperature of G to that of G (cid:48) via Eq. (41).Lemmas 7-9, with the normalization condition, imply that the free states in Helmholtztheories are canonical ensembles. This result will facilitate our proof of Theorem 1 aboutgrand-potential theories. Theorem.
Consider any grand-potential resource theory in which each pair ( H, N ) corre-sponds to exactly one free state G = ( g, H, N ) . If g is not a grand canonical ensemble, someresources R can be generated solely with equilibrating operations: G β,µ (cid:55)−−→ R .Proof. First, we show that each element of g has the form e − β ( E i ) E i + α ( n j ) n j /Z , wherein β ( E i )and α ( n j ) denote functions of the energy and particle number. Second, by comparing thegrand-potential theory with Helmholtz theories, we will show that β and α are constantfunctions.Consider the most general state vector associated with H and N . Each element hasthe form e − β ( E i ) E i + α ( n j ) n j + f ( E i ,n j ) /Z , wherein f denotes some function and Z normalizesthe state. Recall that every Helmholtz-theory problem can be decomposed into single-energy lemmas that are equivalent to nonuniformity-theory problems. Likewise, every grand-potential–theory problem can be decomposed into lemmas that feature just one n j apieceand that are equivalent to Helmholtz-theory problems. Therefore, the elements of g thatcorrespond to the same n must form a canonical ensemble. (Rather, they would forma canonical ensemble if normalized appropriately.) These g elements could not form acanonical ensemble if f depended on energy nontrivially. Hence f ( E i , n j ) = f ( n j ). By ananalogous argument, f cannot depend on n j . Hence f is a constant, and each element of g has the form e − β ( E i ) E i + α ( n i ) n i /Z . G could feature in a problem in which every number operator is trivial: N = 0. Such aproblem is equivalent to a problem in a Helmholtz theory. In each Helmholtz theory, all freestates share a β that is a constant function of energy. This β must characterize the grand-potential theory. Analogously, all free states in the grand-potential theory share an α ∈ R .Hence each element of g has the form e − βE i + αn j /Z . Define µ ∈ R such that α = − βµ . B Proof Sketch of Theorem 1(b)
Here we provide a proof sketch for Theorem 1(b). For simplicity, we consider a Helmholtz-theory context. We must show that, if the free states do not have the Boltzmann form, thequasiorder on states is trivial: Any state can be created from any other by equilibratingoperations. We follow an argument by Janzing et al. [18] about the effective temperaturespresent in many copies of a non-Boltzmann state. These effective temperatures can be usedto cool a qubit.Consider the transformation of R = ( r, H ) into S = ( s, H ). It suffices to operate suc-cessively on pairs of levels, such as the first and the j th . An equilibrating operation of thefollowing form converts r j /r into s j /s . Let E j denote the gap between the two levels. Fol-lowing [18, Sec. 3], we suppose that we have access to a free state on three levels, separatedby gaps E j , the probabilities on which are not Boltzmann-weighted. As shown by Janzing17 t al. , the product of n → ∞ copies of this free state contains pairs of levels, separatedby E j , characterized by essentially any desired relative probability p j /p . Two levels of theresource-and-free-state composite are degenerate. One degenerate level is the product of theresource’s lower level and the free state’s upper level; the other degenerate level consists ofthe reverse. Swapping the degenerate levels is an equilibrating operation. Easy calculationshows that the swap transforms the resource’s probabilities into r (cid:48) j = r j + δ and r (cid:48) = r − δ, wherein δ = r p j − r j p . (42)If p j + p = 1, any relative probability r (cid:48) j /r could be reached by appropriate choice of p j /p .Generally, p j + p is very small (exponentially small in n in the Janzing et al. example).Hence the relative weights of the levels in R can be changed by only a tiny amount. Repeat-ing the procedure sufficiently many times, however, yields the desired relative probability s j /s . Unless the free state has Boltzmann weights, therefore, the quasiorder induced byequilibrating operations is trivial. Appendix B QUASIORDER PROOFS
Let us prove three statements, introduced in Sec. IV, about the quasiorder on states:Theorem 2, Proposition 3, and Lemma 4.
Theorem (2) . Let R and S denote states in the grand-potential theory defined by ( β, µ ) .There exists an equilibrating operation that maps R to S if and only if R equimajorizes S : R β,µ (cid:55)−−→ S ⇐⇒ R (cid:31) β,µ S. (43) Proof.
This proof is adapted from the proof of [18, Theorem 5], a Helmholtz-theory analogof our Theorem 2. Let R = ( r, H R , N R ) and S = ( s, H S , N S ). By d R and d S , we denote thenumbers of elements in r and s . By G R = ( g R , H R , N R ) and G S = ( g S , H S , N S ), we denotethe equilibrium states associated with R and S . We begin with the easier part of the proof,showing that the existence of an equilibrating operation implies equimajorization.Assume that some equilibrating operation maps R to S : R β,µ (cid:55)−−→ ( E ( R ) , H S , N S ) = ( S, H S , N S ) . (44)Let v = ( v , . . . , v d R ) denote any vector that contains d R elements. The d R × d S matrix M that implements E can be defined by M v = E ( v ) . (45)By the definition of E , M r = s . Since equilibrating operations map equilibrium states toequilibrium states, M g R = g S . We can see as follows that M is stochastic: If v represents a(normalized) state, (cid:80) i v i = 1. Equilibrating operations preserve normalization, so1 = d R (cid:88) i =1 [ E ( v )] i = d R (cid:88) i =1 [ M v ] i , (46) An alternative approach to the Helmholtz-theory analog appears in [19]. w ] i denotes the i th element of any vector w . Mapping normalized vectors to nor-malized vectors, M is stochastic. By Definition 2, R (cid:31) β,µ S .Now, we proceed to the converse claim. Assume that R (cid:31) β,µ S . One can prove that someequilibrating operation maps R to S by augmenting three lines in the proof of Theorem 5by Janzing et al. [18]. After outlining the latter proof, we explain how to augment it.Janzing et al. define a particular energy-preserving transformation implemented with aheat bath; consider the limit as the bath’s size approaches infinity; and show that, in thelimit, the transformation converts the initial state into the equimajorized state. Blendingtheir notation with ours, we denote the initial state by ( p, H p ), the final state by (˜ p, H ˜ p ), andthe associated equilibrium states by ( g, H p ) and (˜ g, H ˜ p ). The Hamiltonian H p has l levels,and H ˜ p has ˜ l levels.Janzing et al. consider the set S n of pure eigenstates of H p + H np + H n ˜ p + H ˜ p , wherein H n denotes n copies of H . Each state in S n is characterized by two length-( n + 1) strings.Each letter in the first (second) string is a number between 1 and l (˜ l ) that indicates onwhich energy level of H p ( H ˜ p ) the state’s weight lies. Denote by u i ∈ [0 , n + 1] the numberof times that i ∈ [1 , l ] appears in the first string; and by v j ∈ [0 , n + 1], the number of timesthat j ∈ [1 , ˜ l ] appears in the second string. If two states in S n correspond to the same pair u = ( u , u , . . . , u l ) and v = ( v , v , . . . , v ˜ l ) , (47)the states correspond to the same energy. ( u and v are called r and s in [18].)A permutation π n : S n → S n is defined in terms of the matrix assumed to map p to ˜ p .Because π n maps each input to an output that has the same ( u, v ), π n conserves energy. π n is applied to the probability distribution P n defined by p ⊗ g ⊗ n ⊗ ˜ g ⊗ n ⊗ ˜ g . A set T n of typical( u, v ) tuples is defined in terms of P n and the limit n → ∞ . In this limit, Janzing et al. show, π n maps P n to the distribution defined by g ⊗ g ⊗ n ⊗ ˜ g ⊗ n ⊗ ˜ p .To adapt this Helmholtz proof to grand-potential theories, replace p , ˜ p , g , and ˜ g with r , s , g R , and g S . If two states in S n correspond to the same ( u, v ), they correspond not only tothe same energy, but also to the same particle number. Just as π n conserves energy, it con-serves particle number. The rest of the proof in [18] shows that, from the equimajorizationcondition, an equilibrating operation can be constructed.The proof technique used above extends from grand-potential theories to thermodynamicresource theories in which extensive-variable operators other than H and N commute witheach other [28]. Before proceeding to Proposition 3, we establish Lemma 4 for convenience. Lemma (4) . The inverse of (cid:15) (cid:55)→ b (cid:15) ( r || g R ) is the piecewise linear function that connects thepoints ( L R ( t k ) , − t k ) , wherein t k and L R ( t k ) define the rescaled Lorenz curve for R : ( L R ( t k ) , − t k ) = ( b (cid:15) ( r || g R ) , (cid:15) ) . (48) Proof.
Let π denote a permutation such that the sequence ( r π ( k ) /g π ( k ) ) k is nonincreasing.Let R m := (cid:80) mk =1 r π ( k ) and G m := (cid:80) mk =1 g π ( k ) for all m ∈ { , , . . . , d R } , wherein d R denotesthe number of elements in r . For m = 0, define R = G = 0. The points that define therescaled Lorenz curve are ( G m , R m ) for m ∈ { , , . . . , d R } . To prove the claim, we firstshow that ( G m , − R m ) equals the ( b (cid:15) ( r || g R ) , (cid:15) ) associated with an optimal hypothesis testfor each m . Then, we show that optimal tests interpolate linearly between the points.We begin with m = 0. The optimal test for (cid:15) = 1 is Q = 0; thus, b = 0. Hence( b ,
1) = (0 ,
1) = ( G , − R ). Now, consider the hypothesis test for an (cid:15) m = 1 − R m for19hich m (cid:54) = 0. Define Q m as a d R × d R matrix that projects onto the m values of k for which r π ( k ) /g π ( k ) is greatest. Operation by Q m on a vector v preserves the part of the support of v that lies on these m values of k and maps all other elements of v to zero. As (cid:80) i [ Q m r ] i = R m , Q m is a feasible measurement element in the primal definition of b (cid:15) m [Eq. (9)]. Therefore, b (cid:15) m ( r || g R ) ≤ (cid:88) i [ Q m g R ] i = G m . (49)To show that equality holds, we consider the dual problem in Eq. (10). A feasible pair ( µ m , τ m ) that satisfies the constraint µ m r − g R ≤ τ m is given by defining µ m such that r π ( m +1) /g π ( m +1) ≤ /µ m < r π ( m ) /g π ( m ) and τ m = (cid:80) mk =1 (cid:0) µ m r π ( k ) − g π ( k ) (cid:1) e k e Tk , wherein e k denotes the unit vector that has exactly one nonzero element, which corresponds to the[ π ( k )] th energy–and–particle-number level, and the superscript T denotes the transpose.Evaluating Eq. (10) shows that the two contributions dependent on µ m cancel, by 1 − (cid:15) m = R m and by the definition of R m . Hence b (cid:15) m ( r || g ) ≥ G m . Combining this result withIneq. (49), shows that b (cid:15) m ( r || g R ) = G m . (50)Now, consider a Type I error for which (cid:15) m ≥ (cid:15) ≥ (cid:15) m +1 . Set λ ∈ (0 ,
1) such that1 − (cid:15) = (1 − λ )(1 − (cid:15) m ) + λ (1 − (cid:15) m +1 ) . (51)Note that (1 − (cid:15) ) = (1 − (cid:15) m ) + λr π ( m +1) . Since (cid:15) (cid:55)→ b (cid:15) ( r || g R ) is convex, b (cid:15) ( r || g R ) ≤ (1 − λ ) b (cid:15) m ( r || g R ) + λb (cid:15) m +1 ( r || g R ) . (52)Let us show that Ineq. (52) holds if the inequality is reversed. In the dual problem, if µ = g π ( m +1) /r π ( m +1) and τ = (cid:80) mk =1 (cid:0) µr π ( k ) − g π ( k ) (cid:1) e k e Tk , then b (cid:15) ( r || g R ) ≥ µ [(1 − (cid:15) m ) + λr π ( m +1) ] − µR m + G m (53)= λg π ( m +1) + G m (54)= (1 − λ ) G m + λG m +1 (55)= (1 − λ ) b (cid:15) m ( r || g R ) + λb (cid:15) m +1 ( r || g R ) . (56)The final equality follows from Eq. (50). Inequalities (52) and (56) show that interpolatinglinearly between ( G m , − R m ) and ( G m +1 , − R m +1 ) amounts to interpolating linearlybetween ( b (cid:15) m ( r || g R ) , (cid:15) m ) and ( b (cid:15) m +1 ( r || g R ) , (cid:15) m +1 ).Finally, we give a mostly self-contained proof of Proposition 3. Proposition (3) . For any states R and S in the grand-potential resource theory defined by ( β, µ ) , the following are equivalent:(a) R (cid:31) β,µ S .(b) L R ( t ) ≥ L S ( t ) for all t ∈ [0 , . Q is said to be feasible if the measurement { Q, − Q } corresponds to a Type I error probability of atmost (cid:15) . The feasible measurement that minimizes the Type II error probability is optimal . ( µ, τ ) is said to be feasible if it satisfies the constraints in Eq. (10). c) φ f a ( R ) ≥ φ f a ( S ) for every function f a ( t ) = max { , t − a } and every a ∈ R .(d) φ f ( R ) ≥ φ f ( S ) for all continuous convex functions f .Proof. We will show that ( a ) ⇒ ( b ) ⇒ ( c ) ⇒ ( d ) ⇒ ( a ). R , S , G R , and G S are defined asin the proof of Theorem 2.( a ) ⇒ ( b ) Let M denote the stochastic matrix from the equimajorization condition. Let Q definethe optimal test that distinguishes s from g S with a Type I error probability of at most (cid:15) . To distinguish r from g R , one can apply M and then measure { Q, − Q } . Thistest might distinguish between r and g R suboptimally. Hence b (cid:15) ( r || g R ) ≤ b (cid:15) ( s || g S ).Lemma 4 implies (b).( b ) ⇒ ( c ) The dual formulation of (cid:15) (cid:55)→ b (cid:15) ( r || g R ) can be written as b (cid:15) ( r || g R ) = max µ (cid:110) (1 − (cid:15) ) µ − (cid:88) i [ { µr − g R } + ] i (cid:111) . (57)Hence b − (cid:15) ( r || g R ) is the Legendre transform of (cid:88) i [ { µr − g R } + ] i = µ (cid:88) i g i f /µ (cid:18) r i g i (cid:19) = µ φ f /µ ( r, g R ) , (58)wherein f a ( t ) = max { , t − a } . Since b (cid:15) ( r || g R ) ≤ b (cid:15) ( s || g S ), φ f /µ ( r, g R ) ≥ φ f /µ ( s, g S ).( c ) ⇒ ( d ) In [46] (see also [47, Lemma 1.2.5]), Uhlmann shows that every continuous convexfunction f ( x ) can be approximated to arbitrary accuracy by a linear combination, thathas positive coefficients, of functions f a ( x ). [In Uhlmann’s phrasing, a concave f ( x )is approximated by positive linear combinations of − f a ( x )]. Because φ f a ( R ) ≥ φ f a ( S )for all f a , φ f ( R ) ≥ φ f ( S ).( d ) ⇒ ( a ) Following [47, Theorem 1.4.4], we prove the contrapositive. Assume that no stochasticmatrix M satisfies M r = s and M g R = g S . Since the set of stochastic matrices isconvex and compact (all entries being in the unit interval), the set of vectors M r ⊕ M g R is convex and compact. As this set does not contain s ⊕ g S , a hyperplane separates s ⊕ g S from { M r ⊕ M g R } [74, Theorem 3.5]. That is, a vector x ⊕ y satisfies x · s + y · g S >x · M r + y · M g R for all stochastic matrices M . Taking the maximum over M on theright-hand side (RHS) and denoting by [ g R ] k the k th element of g R gives x · s + y · g S > max M { x · M r + y · M g R } (59)= max M (cid:88) jk ( x j M jk r k + y j M jk [ g R ] k ) (60)= (cid:88) k max j { x j r k + y j [ g R ] k } . (61)Maximizing on the LHS produces (cid:88) k max j { x j s k + y j [ g S ] k } > (cid:88) k max j { x j r k + y j [ g R ] k } . (62)But f ( s, t ) = max j { x j s + y j t } is a convex function, so φ f ( S ) ≥ φ f ( R ). The contra-positive implies ( a ). 21 ppendix C ONE-SHOT WORK-YIELD AND WORK-COST PROOFS Equilibrating operations can extract work from one copy of a state R = ( r, H, N ) andcan store the work in a battery. From enough stored work, equilibrating operations cangenerate R . Each protocol can have a probability (cid:15) ∈ [0 ,
1] of failing to accomplish itspurpose. We calculate the maximum work W (cid:15) gain ( R ) extractable from, and bound the leastwork W (cid:15) cost required to create, R with error-prone protocols. First, we prove a helpful lemma.Anticipating applications of the lemma, we use the notation G and G (cid:48) associated with freestates. However, the lemma holds if G denotes an arbitrary quantum state and G (cid:48) denotesan arbitrary quasiclassical state.We use the following notation: By e E , we denote a vector of the eigenvalues of the purestate | E (cid:105)(cid:104) E | . The element associated with energy E is one, and the other elements arezeroes. By [ v ] i , we denote the i th element of any vector v . If r and s denote equal-sizedvectors, their scalar product is r · s := (cid:80) i r i s i . Lemma 10.
Let R = ( r, H, N ) denote any state; and ( e E , H (cid:48) , N (cid:48) ) , any pure state. Let G = ( g, H, N ) and G (cid:48) = ( g (cid:48) , H (cid:48) , N (cid:48) ) . The optimal hypothesis test between r and g is relatedto the optimal test between r ⊗ e E and g ⊗ g (cid:48) by b (cid:15) ( r ⊗ e E || g ⊗ g (cid:48) ) = ( g (cid:48) · e E ) b (cid:15) ( r || g ) . (63) Proof.
Consider any feasible measurement operator Q in b (cid:15) ( r ⊗ e E || g ⊗ g (cid:48) ). We can finda feasible Q (cid:48) that gives the same value of the objective function and that has the form Q (cid:48) = (cid:80) i Q E (cid:48) i ⊗ e E (cid:48) i e TE (cid:48) i . Here, Q E (cid:48) i = ( ⊗ e E (cid:48) i ) Q ( ⊗ e TE (cid:48) i ), and the superscript T denotes thetranspose. Without loss of generality, we focus on Q (cid:48) operators that have this form.The constraint becomes (cid:80) i [ Q E r ] i ≥ − (cid:15) , and the objective function becomes (cid:88) i [ Q (cid:48) ( g ⊗ g (cid:48) )] i = (cid:88) i (cid:40) ( g (cid:48) · e E (cid:48) i ) (cid:88) j [ Q E (cid:48) i g ] j (cid:41) . (64)The minimum follows from setting Q E (cid:48) i = 0 for all E (cid:48) i (cid:54) = E : b (cid:15) ( r ⊗ e E || g ⊗ g (cid:48) ) = min { ( g (cid:48) · e E ) (cid:88) j [ Q E g ] j (cid:12)(cid:12)(cid:12) (cid:88) j [ Q E r ] j ≥ − (cid:15), ≤ Q E ≤ } (65)= ( g (cid:48) · e E ) b (cid:15) ( r || g ) . (66)This lemma implies another lemma, associated with (cid:15) = 0, that will facilitate our work-bound proofs. Lemma 11.
Let R denote any state in a grand-potential resource theory defined by β and µ . Let W denote the work extractable from R , and let W (cid:48) denote the work cost of creating R , with error tolerance (cid:15) = 0 . There exist batteries B such that R + B E (cid:31) β,µ B E + W ⇔ R (cid:31) β,µ B W and (67) B E + W (cid:48) (cid:31) β,µ R + B E ⇔ B W (cid:48) (cid:31) β,µ R. (68)22 roof. Consider a battery B E + W that consists of two noninteracting parts (e.g., two batter-ies, B E and B W ). The total Hamiltonian is the sum of the subsystems’ Hamiltonians, andthe total number operator is a sum. Suppose that the first Hamiltonian has d eigenvaluesand the second has d . The joint-system state vector e E ⊗ e W is an energy-( E + W ) eigen-state. The joint system’s equilibrium state is the composition of the constituent systems’equilibrium states, whose state vectors we denote by g and g (cid:48) . That is, B E + W = B E + B W .Applying several results, we can prove the equivalences in (67): R + B E (cid:31) β,µ B W + B E ⇔ L R + B E ( t ) ≥ L B W + B E ( t ) ∀ t ∈ [0 ,
1] (69) ⇔ b (cid:15) ( r ⊗ e E || g ⊗ g (cid:48) ) ≥ b (cid:15) ( e W ⊗ e E || g ⊗ g (cid:48) ) (70) ⇔ b (cid:15) ( r || g ) ≥ b (cid:15) ( e W || g ) (71) ⇔ L R ( t ) ≥ L B W ( t ) ∀ t ∈ [0 ,
1] (72) ⇔ R (cid:31) β,µ B W . (73)The first equivalence follows from Proposition 3; the second, from Lemma 4; the third, fromLemma 10; the fourth, from Lemma 4; and the fifth, from Proposition 3. Similar reasoningjustifies the equivalence of Eqs. (68).Having simplified the model of work, we will calculate W (cid:15) gain . Theorem (5) . The (cid:15) -work value of a state R = ( r, H, N ) associated with the free state G R = ( g R , H, N ) is W (cid:15) gain ( R ) = β D (cid:15) H ( r || g R ) . (74) Proof.
In the converse part of the proof, we show that the RHS is an upper bound on theextractable work. In the direct part, we construct an equilibrating operation that attainsthe bound.For the converse, define an equilibrating operation by E ( r ⊗ e E ) ≈ (cid:15) e E ⊗ e W . The channel’soutput is (cid:15) -close, in the l norm, to the desired state:12 |E ( r ⊗ e E ) − e E ⊗ e W | ≤ (cid:15). (75)Since equilibrating operations map equilibrium states to equilibrium states, E ( g R ⊗ g (cid:48) ) = g (cid:48) .Using E , we can construct a hypothesis test between r ⊗ e E and g R ⊗ g (cid:48) . The test consistsof an application of E followed by an energy measurement. If the measurement yields E + W , we guess that the state is r ⊗ e E . Otherwise, we guess g R ⊗ g (cid:48) . By construction,the probability that we correctly guess r ⊗ e E is at least 1 − (cid:15) . The test is feasible for D (cid:15) H ( r ⊗ e E || g R ⊗ g (cid:48) ), and e − D (cid:15) H ( r ⊗ e E || g R ⊗ g (cid:48) ) ≤ E ( g R ⊗ g (cid:48) ) · e E + W (76)= ( g ⊗ g (cid:48) ) · e E + W (77)= e − β ( E + W ) Z . (78) The proof does not depend on how, or whether, measurements are defined in the resource theory. Be-cause the proof is not a protocol for extracting work, the resource-theory agent need not perform themeasurement.
23y Eq. (63), e − D (cid:15) H ( r || g R ) ≤ e − βW , (79)which is equivalent to the upper bound.For the proof’s direct part, we define the state vector˜ g (cid:48) := 11 − Z e − β ( E + W ) (cid:2) g (cid:48) − Z e − β ( E + W ) e E + W (cid:3) . (80)Using the optimal measurement Q in D (cid:15) H ( r ⊗ e E || g R ⊗ g (cid:48) ), we define the operation E by E ( s ⊗ u ) = (cid:40) − (cid:88) i [ Q ( s ⊗ u )] i (cid:41) ˜ g (cid:48) + (cid:40)(cid:88) i [ Q ( s ⊗ u )] i (cid:41) e E + W (81)for state vectors s and u . By construction, E ( r ⊗ e E ) has the form of the desired output.Since E is an equilibrating operation, E ( g R ⊗ g (cid:48) ) = g (cid:48) . This condition determines the possiblevalues of W and is equivalent to (cid:88) i [ Q ( g R ⊗ g (cid:48) )] i = e − β ( E + W ) Z . (82)This equation is equivalent to Ineq. (78), except for containing an equality. Therefore, freeoperations can distill at least the work W that satisfies e − D (cid:15) H ( r || g R ) = e − βW , (83)which is equivalent to the lower bound.Having calculated the work extractable from R , we bound the work cost of creating R [Ineqs. (19)]. Theorem (5, ctd.) . The work cost of creating an (cid:15) -approximation to a state R is boundedby max δ ∈ (0 , − (cid:15) ] (cid:104) β D − (cid:15) − δ H ( r || g R ) − β log (cid:0) δ (cid:1)(cid:105) ≤ W (cid:15) cost ( R ) ≤ β D − (cid:15) H ( r || g R ) − β log (cid:0) − (cid:15)(cid:15) (cid:1) . (84) Proof.
To derive the lower bound, we suppose that E is an equilibrating operation thatsatisfies E ( e E + W ) ≈ (cid:15) r ⊗ e E . Using E , we can transform the optimal dual program for e E + W and g (cid:48) into a feasible dual program for E ( e E + W ) and g R ⊗ g (cid:48) . This feasible program can berelated to the hypothesis-testing entropy of r relative to g R .Consider distinguishing between e E + W and g (cid:48) by hypothesis test. Let 1 denote the stateon a one-dimensional space. By Lemma 10, e − D − (cid:15) H ( e E + W || g (cid:48) ) = ( g (cid:48) · e E + W ) e − D − (cid:15) H (1 || (85)= (cid:15) ( g (cid:48) · e E + W ) . (86)The dual formulation of D (cid:15)H reads, e − D − (cid:15) H ( e E + W || g (cid:48) ) = max µ e E + W − g (cid:48) ≤ τµ,τ ≥ (cid:40) (cid:15)µ − (cid:88) i τ i (cid:41) , (87)24herein τ i denotes the i th element of τ . Comparing Eqs. (86) and (87) shows that µ = g (cid:48) · e E + W and τ = 0 are the optimal choices in the dual formulation.Acting on each side of the constraint, µ e E + W ≤ g (cid:48) , with E yields µ E ( e E + W ) ≤ g R ⊗ g (cid:48) .Therefore, µ = g (cid:48) · e E + W and τ = 0 are feasible for D − δ H ( E ( e E + W ) || g R ⊗ g (cid:48) ): e − D − δ H ( E ( e E + W ) || g R ⊗ g (cid:48) ) ≥ δ e − β ( E + W ) Z (88)for all δ ∈ [0 , (cid:107) r ⊗ e E − E ( e E + W ) (cid:107) ≤ (cid:15) , (cid:12)(cid:12)(cid:12) (cid:88) i [ Q { r ⊗ e E − E ( e E + W ) } ] i (cid:12)(cid:12)(cid:12) ≤ (cid:15) (89)for every Q . Suppose Q is the optimal choice in D − η H ( r ⊗ e E || g R ⊗ g (cid:48) ), such that (cid:80) i [ Q ( r ⊗ e E )] i = η . For this Q , (cid:80) i [ Q E ( e E + W )] i ≥ η − (cid:15) . Therefore, e − D − η + (cid:15) H ( E ( e E + W ) || g R ⊗ g (cid:48) ) ≤ (cid:88) i [ Q ( g R ⊗ g (cid:48) )] i (90)= e − D − η H ( r ⊗ e E || g R ⊗ g (cid:48) ) (91)= e − βE Z e − D − η H ( r || g R ) . (92)If η = (cid:15) + δ , δe − βW ≤ e − D − (cid:15) − δ H ( r || g R ) . (93)We have lower-bounded W (cid:15) cost ( R ) for every δ ∈ (0 , − (cid:15) ]. The tightest of these boundsfollows from a maximization over δ .To derive upper bound, we construct an equilibrating operation that maps e E + W to ˜ r ⊗ e E ,wherein ˜ r ≈ (cid:15) r , for a suitably chosen value of W . The work system is approximated, whereasthe battery is not. The associated work-cost bound may be suboptimal.By Condition (c) of Proposition 3, such an equilibrating operation exists if and only if K in ( a ) ≥ K out ( a ) (94)for all a ∈ R and for K R ( a ) defined as follows. In Proposition 3, the function f a ( t ) := max { , t − a } appears in φ f a ( R ) := d R (cid:88) i =1 g i f a (cid:18) r i g i (cid:19) (95)= d R (cid:88) i =1 g i max (cid:26) , r i g i − a (cid:27) (96)= d R (cid:88) i =1 max { , r i − g i a } . (97)25o simplify notation, we will relabel this sum as K R ( a ). Because r and g are normalized, K R ( a ) = (1 − a ) if a ≤
0. We can rewrite the LHS of Ineq. (94) as K in ( a ) = (cid:18) − a e − β ( E + W ) Z (cid:19) + . (98)This function is linear and satisfies K in (0) = 1 and K in ( Ze β ( E + W ) ) = 0. We can rewrite theRHS of Ineq. (94) as K out ( a ) = (cid:88) i (cid:18) ˜ r i − g i a e − βE Z (cid:19) + . (99)Just as for the input state, K out (0) = 1. As a sum of convex functions, K out ( a ) is convex.Thus, the condition K in ( a ) ≥ K out ( a ) for all t ∈ R reduces to K out ( Ze β ( E + W ) ) = 0 . (100)Let us find a value of W for which the transformation is possible. First, we construct asuitable ˜ r from the dual form of D − (cid:15) H ( r || g R ). Suppose that µ and τ are the optimal choices,so that e − D − (cid:15) H ( r || g R ) = (cid:15)µ − (cid:88) i τ i (101)and µr − g R ≤ τ . We define r (cid:48) = T rT † for T = g / R ( g R + τ ) − / , using the pseudoinverse (theinverse on the support). These definitions satisfy µr (cid:48) ≤ g R . Let us bound the trace distance (cid:107) r − r (cid:48) (cid:107) = (cid:88) k | r k − r (cid:48) k | (102)= (cid:88) k | r k − g k r k ( g k + τ k ) − | (103)= (cid:88) k r k | − g k ( g k + τ k ) − | (104)= (cid:88) k r k τ k g k + τ k (105) ≤ (cid:88) k µ τ k (106)= (cid:80) i τ i µ (107) ≤ (cid:15) . (108)To derive the first inequality, we used the inequality µr ≤ g R + τ ; to derive the second, (cid:15)µ − (cid:80) i τ i = e − D − (cid:15) H ( r || g R ) ≥
0. Let ˜ r = r (cid:48) / (cid:80) i r (cid:48) i . By the Triangle Inequality, (cid:107) r − ˜ r (cid:107) ≤ (cid:107) r − r (cid:48) (cid:107) + (cid:107) ˜ r − r (cid:48) (cid:107) (109) ≤ (cid:15) + (cid:18) (cid:80) i r (cid:48) i − (cid:19) (cid:107) r (cid:48) (cid:107) (110)= (cid:15) + (cid:32) − (cid:88) i r (cid:48) i (cid:33) (111) ≤ (cid:15). (112)26he fourth inequality follows from (cid:80) i r i − (cid:80) j r (cid:48) j ≤ (cid:107) r − r (cid:48) (cid:107) . We have constructed a state˜ r ≈ (cid:15) r .Moreover, µ ( (cid:80) i r (cid:48) i ) ˜ r ≤ g . Applying this inequality to (99) yields K out ( a ) ≤ (cid:88) i g i (cid:18) µ (cid:80) i r (cid:48) i − a e − βE Z (cid:19) + (113)= (cid:18) µ (cid:80) i r (cid:48) i − a e − βE Z (cid:19) + . (114)Hence K out ( Ze β ( E + W ) ) satisfies K out ( Ze β ( E + W ) ) ≤ (cid:18) µ (cid:80) i r (cid:48) i − e βW (cid:19) + (115) ≤ (cid:18) µ (1 − (cid:15) ) − e βW (cid:19) + (116) ≤ (cid:18) (cid:15) − (cid:15) e D − (cid:15) H ( r || g ) − e βW (cid:19) + . (117)The second inequality follows from (cid:80) r (cid:48) i ≥ − (cid:15) ; the third, from (cid:15)µ ≥ (cid:15)µ − (cid:80) i τ i = e − D − (cid:15) H ( r || g R ) . We can satisfy (100) by choosing W such that e βW = (cid:15) − (cid:15) e D − (cid:15) H ( r || g R ) . (118)Since W (cid:15) cost ( R ) ≤ W , the upper bound follows directly.One can show that the upper bound is a nonnegative quantity, using e − D − (cid:15) H ( r || g R ) ≤ (cid:15) .(This inequality follows from the choice Q = (cid:15) .) Because (cid:15) ≤ (cid:15) − (cid:15) , e − D − (cid:15) H ( r || g R ) ≤ (cid:15) − (cid:15) . Thelatter implies the upper bound’s nonnegativity. The lower bound is nonnegative in all thenumerical examples we tested. Appendix D COMPARISON OF ONE-SHOT WORK YIELD AND WORK COST
We will use second-order asymptotics to show that W (cid:15) gain ( R ⊗ n ) tends to differ from thebounds on W (cid:15) cost ( R ⊗ n ), and that the bounds lie arbitrarily close together, as the thermo-dynamic limit is approached. Consider distilling work from, or creating, n copies R ⊗ n of R = ( r, H, N ). The work involved depends on the normal approximation to the hypothesis-testing relative entropy [75, Theorem 5] (see also [76, 77]): D (cid:15) H ( r ⊗ n || g ⊗ nR ) = nD ( r || g R ) + √ n s ( r || g R ) Φ − ( (cid:15) ) + O (log n ) , (119)wherein g R denotes the state vector of the equilibrium state associated with R , the square-root of the relative entropy variance is s ( r || g R ) := (cid:112) V ( r || g R ) = (cid:112) Tr( r [log r − log g R ] ) − D ( r || g R ) , (120) The quantum version of Eq. (119) appears in [75, 78], but we have specialized to commuting densityoperators. − ( (cid:15) ) := sup (cid:26) z ∈ R (cid:12)(cid:12) √ π (cid:90) z −∞ e − t dt ≤ (cid:15) (cid:27) . (121)Equation (121) admits of the following interpretation. Suppose that, if a hypothesis testis performed on the null-hypothesis state R , the outcome is distributed normally. Theprobability that a Type I error occurs equals Φ − ( (cid:15) ). Let us apply Eq. (119) to Eq. (18)and to Ineqs. (19).To characterize the latter expressions’ approach toward D ( r || g R ), we evaluate the nor-malized differences between each W (cid:15),β (cid:0) R ⊗ n (cid:1) and β D ( r || g R ), assuming n is large. The actualdistillable work differs from the asymptotic distillable work aslim n →∞ √ n (cid:20) n β D ( r || g R ) − W (cid:15) gain (cid:0) R ⊗ n (cid:1)(cid:21) = lim n →∞ β (cid:20) − s ( R || g R )Φ − ( (cid:15) ) − O (log n ) √ n (cid:21) (122)= 1 β s ( R || g R )Φ − (1 − (cid:15) ) . (123)The final equation follows from Φ − ( (cid:15) ) = − Φ − (1 − (cid:15) ). If the Type I error probability is small( (cid:15) < ), Eq. (123) is positive because the work distilled at the optimal asymptotic efficiencyexceeds the work distilled at any sub-asymptotic efficiency [i.e., Eq. (122) is positive].The lower work-cost bound differs from the asymptotic cost bylim n →∞ √ n (cid:20) W (cid:15) cost (cid:0) R ⊗ n (cid:1) v − n β D ( r || g R ) (cid:21) ≥ lim n →∞ β max δ ∈ (0 , − (cid:15) ] (cid:20) s ( r || g R )Φ − (1 − (cid:15) − δ ) − log δ √ n (cid:21) = 1 β max δ ∈ (0 , − (cid:15) ] s ( r || g R )Φ − (1 − (cid:15) − δ ) (124)= 1 β s ( r || g R )Φ − (1 − (cid:15) ) . (125)The first equality holds if δ grows more slowly than e √ n . The last equality holds since, bythe definition and monotonicity of Φ − , the least possible δ -value maximizes Φ − (1 − (cid:15) − δ ).In the limit, this difference arising from the lower bound matches that of the upper bound,as the work cost’s upper bound differs from the asymptotic work cost aslim n →∞ √ n (cid:20) W (cid:15) cost (cid:0) R ⊗ n (cid:1) − n β D ( r || g R ) (cid:21) ≤ lim n →∞ β (cid:20) s ( r || g R )Φ − (1 − (cid:15) ) − √ n log (cid:18) − (cid:15)(cid:15) (cid:19)(cid:21) = 1 β s ( r || g R )Φ − (1 − (cid:15) ) . (126)The final equality holds if − (cid:15)(cid:15) grows more slowly than e √ n .Let us compare these normalized work differences. If n is large, the work-cost boundsexceed the optimal asymptotic work cost β D ( r || g R ) by an amount proportional to √ n . Incontrast, β D ( r || g R ) exceeds the work gain by an amount proportional to √ n . The workgain and work cost differ in general, unlike in the thermodynamic limit, as in [19, 21]. 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