Ann. Henri Poincaré 24 (2023), 1725–1777 c 2022 The Author(s) 1424-0637/23/051725-53 published online November 30, 2022 https://doi.org/10.1007/s00023-022-01254-1 Annales Henri Poincaré Resource Theory of Heat and Work with Non-commuting Charges Zahra Baghali Khanian , Manabendra Nath Bera , Arnau Riera , Maciej Lewenstein and Andreas Winter Abstract. We consider a theory of quantum thermodynamics with mul- tiple conserved quantities (or charges). To this end, we generalize the seminal results of Sparaciari et al. (Phys. Rev. A 96:052112, 2017) to the case of multiple, in general non-commuting charges, for which we formulate a resource theory of thermodynamics of asymptotically many non-interacting systems. To every state we associate the vector of its ex- pected charge values and its entropy, forming the phase diagram of the system. Our fundamental result is the Asymptotic Equivalence Theorem, which allows us to identify the equivalence classes of states under as- ymptotic approximately charge-conserving unitaries with the points of the phase diagram. Using the phase diagram of a system and its bath, we analyze the first and the second laws of thermodynamics. In particu- lar, we show that to attain the second law, an asymptotically large bath is necessary. In the case that the bath is composed of several identical copies of the same elementary bath, we quantify exactly how large the bath has to be to permit a specified work transformation of a given sys- tem, in terms of the number of copies of the “elementary bath” systems per work system (bath rate). If the bath is relatively small, we show that the analysis requires an extended phase diagram exhibiting negative en- tropies. This corresponds to the purely quantum effect that at the end of the process, system and bath are entangled, thus permitting classically impossible transformations (unless the bath is enlarged). For a large bath, or many copies of the same elementary bath, system and bath may be left uncorrelated and we show that the optimal bath rate, as a function of how tightly the second law is attained, can be expressed in terms of the heat capacity of the bath. Our approach solves a problem from ear- lier investigations about how to store the different charges under optimal work extraction protocols in physically separate batteries. 1726 Z. B. Khanian et al. Ann. Henri Poincaré 1. Introduction Thermodynamics is one of the most successful physical theories and a pillar of modern science and technology. It was initially developed empirically to de- scribe heat engines, such as the steam engine and internal combustion engines that powered the industrial revolution of the eighteenth and nineteenth cen- tury. Later on, it has been founded on statistical mechanics with the assump- tion that the systems are composed of a large number of classical particles. The thermal baths, which the system interacts with, are even larger in size so that the temperature of the bath effectively does not alter after the interaction. The laws of thermodynamics find their applications in almost all branches of the exact sciences. The emergence of quantum mechanics in the last century, and the subsequent achievements in controlling and tuning of an individual or a finite number of quantum systems, led to the exploration of thermody- namics in the quantum regime. There, the system is made up of a single or moderate number of quantum particles interacting with a thermal bath. This regime is often termed the finite-size regime. The system may possess non- trivial quantum correlations, such as entanglement among the particles, and the bath can be finite or comparable in size with the system. In the quantum domain, another layer of difficulties arises when one considers more than one conserved quantities (charges) that do not commute with each other, as the simultaneous conservation of all the charges cannot be guaranteed. Recent studies of quantum thermodynamics [1,2] focus on systems of finite size and the cases where measurements are allowed only once. In addition to thermodynamic averages, there one is interested in values and bounds on fluctuations of thermodynamic quantities. One way to handle these problems is by the use of various fluctuations theorems [3–5]. Another way to deal with these regimes is exactly via the resource theory of thermodynamics that allows for rigorous treatment of second laws, optimal work extraction problem, etc. (cf. [6–8], see also [9–12]). The resource theory of quantum thermodynamics was recently extended to deal with quantum and nanoscale engines made up of a finite or a small number of quantum particles, and two baths at different temperatures [13]. Resource theory is a rigorous mathematical framework initially devel- oped to characterize the role of entanglement in quantum information pro- cessing tasks. Later the framework was extended to characterize coherence, non-locality, asymmetry and many more, including quantum Shannon theory itself, see [14–31]. The resource theory approach applies also to classical theo- ries. In general, the resource theories have the following common features: (1) a well-defined set of resource-free states, and any states that do not belong to this set has a non-vanishing amount of resource; (2) a well-defined set of resource-free operations (allowed operations), that cannot create or increases resource in a state. These allow one to quantify the resources present in the states or operations and characterize their roles in the transformations between the states or the operations. In particular, it enables the definition and rigorous calculation or bounding of resource measures; to determine which states can Vol. 24 (2023) Resource Theory of Heat and Work 1727 be transformed to others using allowed operation; how the resource content of states may be changed; and how these changes are bounded under the allowed operations, etc. In the present paper, we formulate a resource theory of quantum ther- modynamics with multiple conserved quantities, where the system and bath a priori are arbitrary in size. We adhere to the asymptotic regime where a system of many non-interacting particles with multiple conserved quantities or charges interacts with a bath. It is discussed in [32] that in the resource theory of thermodynamics with multiple non-commuting conserved quanti- ties, complete passivity and maximum entropy principle lead to incompatible sets of resource free states. Here we choose the maximum entropy principle, that is the resource-free states are the generalized Gibbs states (GGS), and allowed operations are the (average) entropy and (average) charge preserving operations. The thermodynamic resource is quantified by the Helmholtz free entropy. Clearly, the entropy and charge preserving operations cannot create thermodynamic resource in the resource-free GGSs. For any quantum state, we associate a vector with entries of the average charge values and entropy of that state. We call the set of all these vectors the phase diagram of a system. The concept of phase diagram in the present sense was originally pioneered in [33] for a system with energy as the only conserved quantity of the system where it has been shown that the phase diagram is a convex set. This termi- nology is motivated by traditional thermodynamics, where the phase diagram is a multi-dimensional map of the equilibrium states of a system according to the temperature and other relevant intensive or extensive quantities (such as pressure, volume, particle concentrations, etc.). The difference here is that we also allow non-equilibrium states, effectively decoupling the entropy from the dynamical parameters. The seminal results of [33] were generalized to multi- ple pairwise commuting conserved quantities by the present authors [12], and the further generalization to the case of multiple, in general non-commuting charges is the subject of the present paper. For an individual system with multiple charges the phase diagram is not necessarily convex. Interestingly, however, for a composition of two or more systems, the phase diagram be- comes convex. Moreover, for a composition of large enough systems, for any point in the phase diagram, there is a state with tensor product structure that realizes it. This implies that from the macroscopic point of view it is enough to consider states of a composite system with tensor product structure. This is an important feature when we study a traditional thermodynamics set-up considering only tensor product states, and it does not affect the generality of the laws of thermodynamics which only depend on the macroscopic prop- erties of a state rather than the state itself. We find that given the entropy and charge preserving operations as the allowed operations, the (generalized) phase diagram fully characterizes the thermodynamic transformations of the states and the role of thermodynamic resources in such processes. We further extend our study to situations where the system and bath become correlated after initially being independent. In such a case we use the conditional entropy instead of the entropy, to express the phase diagram and derive the laws of 1728 Z. B. Khanian et al. Ann. Henri Poincaré quantum thermodynamics when the final state exhibits possible system-bath correlations. The rest of the paper is organized as follows. In Sect. 2, we specify our resource theory, considering a quantum system Q with a finite-dimensional Hilbert space, together with a Hamiltonian H = A1 and other quantities (“charges”) A2 , . . . , Ac . We introduce here the concept of phase diagram and prove the fundamental Asymptotic Equivalence Theorem 4 (AET), which shows that the points in the phase diagram label asymptotic equivalence classes of sequences of states. This allows us to study asymptotic thermodynamics of sys- tems with multiple conserved quantities in Sect. 3. We start by describing the system model, comprising a work system, baths and batteries, which permits us to formulate and prove the first law in Sect. 3.1; the second law is discussed in Sect. 3.2; in Sect. 3.3 we characterize precisely which work transformations are possible on a system with a given bath, in terms of the extended phase dia- gram, which features negative entropies corresponding to the purely quantum effect of entanglement between system and bath; in Sect. 3.4 we introduce the thermal bath rate, and discuss the tradeoff between the bath rate and work extraction. We conclude in Sect. 4 with a discussion of our theory and an outlook. The paper also includes three appendices: “Appendix A” introduces technical notation and some auxiliary results; “Appendix B” gives an explicit construction of so-called approximate microcanonical subspaces (a.m.c.) for non-commuting observables [34]; “Appendix C” provides the full proof of the AET Theorem 4. 2. Resource Theory of Charges and Entropy A system in our resource theory is a quantum system Q with a finite-dimensional Hilbert space (denoted Q, too, without danger of confusion), together with a Hamiltonian H = A1 and other quantities (“charges”) A2 , . . . , Ac , all of which are Hermitian operators that do not necessarily commute with each other. We consider composition of n non-interacting systems, where the Hilbert space of the composite system Qn is the tensor product Q⊗n = Q1 ⊗ · · · ⊗ Qn of the Hilbert spaces of the individual systems, and the j-th charge of the composite system is the sum of charges of individual systems as follows, (n)  n Aj = 1⊗(i−1) ⊗ Aj ⊗ 1⊗(n−i) , j = 1, 2, . . . , c. (1) i=1 (Q ) For ease of notation, we will write throughout Aj i = 1⊗(i−1) ⊗ Aj ⊗ 1⊗(n−i) . We note that throughout the paper we label various subsystems or individual systems with subscripts whereas here subscript j denotes different charges of each subsystem. To avoid this confusion, note that various charges are always labeled by j. We wish to build a resource theory where the objects are states on a quan- tum system, which are transformed under thermodynamically meaningful op- erations. To any quantum state ρ is assigned the point (a, s) = (a1 , . . . , ac , s) = Vol. 24 (2023) Resource Theory of Heat and Work 1729   Tr ρA1 , . . . , Tr ρAc , S(ρ) ∈ Rc+1 , which is an element in the phase diagram that has been originally introduced, for c = 1, as energy-entropy diagram in [33]; there it is shown, for a system where energy is the only conserved quantity, that the diagram is a convex set. In the case of commuting multiple conserved quantities, the charge-entropy diagram has been generalized and further in- vestigated in [12]. Note that the set of all these vectors, denoted P (1) , is not in general convex (unless the quantities commute pairwise). An example is a qubit system with charges σx , σy and σz where charge values uniquely deter- mine the state as a linear function of the Tr ρσi , hence the entropy, while the von Neumann entropy itself is well-known to be strictly concave. Moreover, the set of these points for a composite system with charges (n) (n) A1 , . . . , Ac , which we denote P (n) contains, but is not necessarily equal (1) to nP (which however is true for commuting charges). Namely, consider  the point g = 12 Tr (ρ1 + ρ2 )A1 , . . . , 12 Tr (ρ1 + ρ2 )Ac , 12 S(ρ1 ) + 12 S(ρ2 ) , which does not necessarily belong to P (1) but belongs to its convex hull; however, 2g ∈ P (2) due to the state ρ1 ⊗ ρ2 . Therefore, we consider the convex hull of the set P (1) and call it the phase diagram of the system, denoted   (1)    P ≡ P := pi Tr ρi A1 , . . . , pi Tr ρi Ac , pi S(ρi ) : i i i   0 ≤ pi ≤ 1, pi = 1 . (2) i The interpretation is that the objects of our resource theory are ensembles of states {pi , ρi }, rather than single states. We define the zero-entropy diagram and max-entropy diagram, respec- tively, as the sets (1) P 0 = {(a, 0) : Tr ρAj = aj for a state ρ}, (1)   P max = a, S(τ (a)) : Tr ρAj = aj for a state ρ , where τ (a) is the unique state maximizing the entropy among all states with charge values Tr ρAj = aj for all j, which is called generalized thermal state, or generalized Gibbs state, or also generalized grand canonical state [35]. Note that, as a linear image of the compact convex set of states, the zero-entropy diagram is compact and convex. We similarly define the set P (n) , the phase (n) (n) (n) diagram P , zero-entropy diagram P 0 and max-entropy diagram P max for (n) (n) the composition of n systems with charges A1 , . . . , Ac . Figure 1 illustrates these concepts. Lemma 1. For an individual system Q and composite system Q⊗n with charges (n) Aj and Aj , respectively, the following holds: (n) 1. P , for n ≥ 1, is a compact and convex subset of Rc+1 . (n) (n) (n) 2. P , for n ≥ 1, is the convex hull of the union P 0 ∪ P max , of the zero-entropy diagram and the max-entropy diagram. 1730 Z. B. Khanian et al. Ann. Henri Poincaré Figure 1. Schematic of the phase diagrams P (1) , P (2) and P. As seen, P (1) is not convex, having a hollow on the underside (n) (1) 3. P = nP for all n ≥ 1. (n) (n) (1) 4. P is convex for all n ≥ 2, and indeed P (n) = P = nP . 5. Every point of P (n) is realized by a suitable tensor product state ρ1 ⊗· · ·⊗ ρn , for all n ≥ |Q| where  |Q| is the dimension of system Q. 6. All points a, S(τ (a)) ∈ P max are extreme points of P. (n) Proof. 1. The phase diagram is convex by definition. Further, Tr ρAj and S(ρ) are continuous functions defined on the set of quantum states which is a compact set; hence, the set P (n) is also a compact set. The convex hull of a finite-dimensional compact set is compact, so the phase diagram is a compact set. 2. Any point in the phase diagram according to the definition is a convex combination of the form      (a1 , . . . , ac , s) = pi Tr (ρi A1 ), . . . , pi Tr (ρi Ac ), pi S(ρi ) . i i i (1) The point (a1 , . . . , ac , 0) belongs toP0 because the state ρ = i pi ρi has charge values a1 , . . . , ac . Moreover, the state with charge values a1 , . . . , ac of maximum entropy is the generalized thermal state τ (a), so we have  S(τ (a)) ≥ S(ρ) ≥ pi S(ρi ), i where the second inequality is due to concavity of the entropy. Therefore, any point (a, s)can be written as the convex combination of the points (a, 0) and a, S(τ (a)) . (n) (1) (n) 3. Due to item 2, it is enough to show that P 0 = nP 0 , and P max = (1) nP max . The former follows from the definition. The latter is due to the fact that the thermal state for a composite system is the tensor power of the thermal state of the individual system. Vol. 24 (2023) Resource Theory of Heat and Work 1731 4. Let τ (a) = i pi |i i| be the diagonalization of the generalized thermal √ ⊗n state. For n ≥ 2, define |v = i pi |i . Obviously, the charge values of ⊗n the states τ (a) and |v v| are the same, since they have the same reduced states on the individual systems; thus, there is a pure state for any point in the zero-entropy diagram of the composite system. Now, consider the state ⊗n ⊗n λ|v v|+(1−λ)τ  (a) , which has the⊗n  charge values as τ (a) and |v v|. same The entropy S λ|v v| + (1 − λ)τ (a) is a continuous function of λ; hence, for any value s between 0 and S(τ (a)⊗n ), there is a state with the given values and entropy s. 5. For n ≥ |Q|, it is easy to see that any state ρ can be decomposed into a uniform convex combination of n pure states, i.e. ρ = n1 i=1 |ψi  ψi |. For n |Q| instance, consider the diagonalization of ρ = t=1 qt |t t|, and define |ψ  := √ 2πit/n t qt e |t. Due to the cyclotomic properties of the primitive n-root of unit, this satisfies the claim. Now observe that the state ψ n = |ψ1  ψ1 | ⊗ · · · ⊗ |ψn  ψn | has the same charge values as the state ρ⊗n , but as it is pure it has entropy 0. Further, consider the thermal state τ with the same charge values as ρ, but the maximum entropy consistent with them. Now let ρi := λ|ψi  ψi | + (1 − λ)τ , and observe that ρnλ = ρ1 ⊗ · · · ρn has the same charge values as ψ n , ρn and τ ⊗n . Since the entropy S(ρnλ ) is a continuous function of λ, thus interpolating smoothly between 0 and nS(τ ), there is a tensor product state with the same given charge values and prescribed entropy s in the said interval. 6. This follows from the strict concavity of the von Neumann entropy S(ρ) as a function of the state, which imparts the strict concavity on a → S(τ (a)).  The penultimate point of Lemma 1 motivates us to define a resource the- ory where the objects are sequences of states on composite systems of n → ∞ parts. Inspired by [33], the allowed operations in this resource theory are those that respect basic principles of physics, namely entropy and charge conserva- tion. We point out right here, that “physics” in the present context does not necessarily refer to the fundamental physical laws of nature, but to any rule that the system under consideration obeys. It is well-known that quantum op- erations that preserve entropy for all states are unitaries. The class of unitaries that conserve charges of a system are precisely those that commute with all charges of that system. However, it turns out that these constraints are too strong if imposed literally, when many charges are to be conserved, as it could easily happen that only trivial unitaries are allowed. Our way out is to consider the thermodynamic limit and at the same time relax the allowed operations to approximately entropy and charge conserving ones. As for the former, we couple the composite system to an ancillary system with corresponding Hilbert space K of dimension 2o(n) , where restricting the dimension of the ancilla en- sures that the entropy rate per system, that is the entropy of the composite system divided by n, does not change in the limit of large n. Here and in the following, we use standard little-oh notation, by which o(n) denotes a function such that limn→∞ o(n)n = 0. Moreover, as for charge conservation, we consider 1732 Z. B. Khanian et al. Ann. Henri Poincaré unitaries that are almost commuting with the total charges of the composite system and the ancilla. The precise definition is as follows. Definition 2. A unitary operation U acting on a composite system coupled to an ancillary system with Hilbert spaces H⊗n and K of dimension 2o(n) , re- spectively, is called an almost-commuting unitary with the total charges of a composite system and an ancillary system if the operator norm of the normal- ized commutator for all total charges vanishes asymptotically for large n: 1 (n) 1 (n) (n) lim [U, Aj + Aj ] = lim U (Aj + Aj ) − (Aj + Aj )U n→∞ n ∞ n→∞ n ∞ = 0 j = 1, . . . , c. (n) where Aj and Aj are respectively the charges of the composite system and the ancilla, such that Aj ∞ ≤ o(n). We stress that the definition of almost-commuting unitaries automatically implies that the ancillary system has a relatively small dimension and charges with small operator norm compared to a composite system. The first step in the development of our resource theory is a precise characterization of which transformations between sequences of product state are possible using almost commuting unitaries. To do so, we define asymptotically equivalent states as follows: Definition 3. Two sequences of product states ρn = ρQn = (ρ1 )Q1 ⊗ · · · ⊗ (ρn )Qn and σ n = σQn = (σ1 )Q1 ⊗ · · · ⊗ (σn )Qn of a composite system with (n) charges Aj for j = 1, . . . , c, are called asymptotically equivalent if 1 lim |S(ρn ) − S(σ n )| = 0, n→∞ n 1 (n) (n) lim Tr ρn Aj − Tr σ n Aj = 0 for all j = 1, . . . , c. n→∞ n In other words, two sequences of product states are considered equivalent if their associated points in the normalized phase diagrams n1 P (n) differ by a sequence converging to 0. We note that in the above definition, ρn and σ n are tensor products of possibly different states; a tensor power state is denoted ρ⊗n . The asymptotic equivalence theorem of [33] characterizes feasible state transformations via exactly commuting unitaries where energy is the only con- served quantity of a system, showing that it is precisely given by asymptotic equivalence. We prove an extension of this theorem for systems with multiple, possibly non-commuting conserved quantities, by allowing almost-commuting unitaries. Theorem 4. (Asymptotic (approximate) Equivalence Theorem—AET). Let ρn = ρ1 ⊗ · · · ⊗ ρn and σ n = σ1 ⊗ · · · ⊗ σn be two sequences of product (n) states of a composite system with charges Aj for j = 1, . . . , c. These two states are asymptotically equivalent if and only if there exist ancillary quantum Vol. 24 (2023) Resource Theory of Heat and Work 1733 systems with corresponding Hilbert space K of dimension 2o(n) and an almost- commuting unitary U acting on H⊗n ⊗ K such that lim U (ρn ⊗ ω  )U † − σ n ⊗ ω 1 = 0, n→∞ where ω and ω  are states of the ancillary system, and charges of the ancillary system are trivial, Aj = 0. The proof of this theorem is given in “Appendix C”, as it relies on a number of technical lemmas, among them the concept of an approximate mi- crocanonical subspace (a.m.c.) [34], of which we give a novel construction in “Appendix B”. 3. Asymptotic Thermodynamics of Multiple Conserved Quantities As a thermodynamic theory, or even as a resource theory in general, transfor- mations by almost-commuting unitaries do not appear to be the most fruitful: they are reversible and induce an equivalence relation among the sequences (1) of product states. In particular, every point (a, s) of the phase diagram P defines an equivalence class, namely of all state sequences with charges and entropy converging to a and s, respectively. To make the theory more interesting, and more resembling of ordinary thermodynamics, as expressed in its first and second laws (including irre- versibility), we now specialize to a setting considered in many previous pa- pers in the resource theory of thermodynamics, both with single or multiple conserved quantities. Specifically, we consider an asymptotic analogue of the setting proposed in [36] concerning the interaction of thermal baths with a quantum system and batteries, where it was shown that the second law con- strains the combination of extractable charge quantities. In [36], explicit pro- tocols for state transformations to saturate the second law are presented, that store each of several commuting charges in its corresponding explicit battery. However, for the case of non-commuting charges, one battery, or a so-called ref- erence frame, stores all different types of charges [34,37]. Only recently it was shown that reference frames for non-commuting charges can be constructed, at least under certain conditions, which store the different charge types in phys- ically separated subsystems [38]. Moreover, the size of the bath required to perform the transformations is not addressed in these works, as only the limit of asymptotically large bath was considered. We will address these questions in a similar setting but in the asymptotic regime, where Theorem 4 provides the necessary and sufficient condition for physically possible state transformations. In this new setting, the asymptotic second law constrains the combination of extractable charges; we provide explicit protocols for realizing transformations satisfying the second law, where each explicit battery can store its correspond- ing type of work in the general case of non-commuting charges. Furthermore, 1734 Z. B. Khanian et al. Ann. Henri Poincaré we determine the minimum number of thermal baths of a given type that is required to perform a transformation. 3.1. System Model, Batteries and the First Law We consider a system being in contact with a bath and suitable batteries, with a total Hilbert space Q = S ⊗ B ⊗ W1 ⊗ · · · ⊗ Wc , consisting of many non-interacting subsystems; namely, the work system, the thermal bath and c battery systems with Hilbert spaces S, B and Wj for j = 1, . . . , c, respectively. We call the j-th battery system the j-type battery as it is designed to absorb j-type work. The work system and the thermal bath have respectively the charges ASj and ABj for all j, but j-type battery has only one nontrivial charge AWj , and all its other charges are zero because it is meant to store only the j-th charge. We note that S, B and Wj s are different Hilbert spaces, so the charges of their corresponding systems can be different as well. The total charge is the sum of the charges of the sub-systems Aj = ASj + ABj + AWj for all j. Furthermore, for a charge A, let Σ(A) = λmax (A)−λmin (A) denote the spectral diameter, where λmax (A) and λmin (A) are the largest and smallest eigenvalues of the charge A, respectively. We assume that the total spectral diameter of the work system and the thermal bath is bounded by the spectral diameter of the battery, that is Σ(ASj ) + Σ(ABj ) ≤ Σ(AWj ) for all j; this assumption ensures that the batteries can absorb or release charges for transformations. As we discussed in the previous section, the generalized thermal state τ (a) is the state that maximizes the entropy subject to the constraint that the charges Aj have the values aj . This state is equal to Z1 e− j=1 βj Aj for real c numbers βj called inverse temperatures and chemical potentials; each of them is a smooth function of charge values a1 , . . . , ac , and Z = Tr e− j=1 βj Aj is the c generalized partition function. Therefore, the generalized thermal state can be equivalently denoted τ (β) as a function of the inverse temperatures, associated uniquely with the charge values a. We assume that the thermal bath is initially in a generalized thermal state τb (β), for globally fixed β. This is because in [34] it was argued that these are precisely the completely passive states, from which no energy can be extracted into a battery storing energy, while not changing any of the other conserved quantities, by means of almost-commuting unitaries and even when unlimited copies of the state are available. We assume that the work system with state ρs and the thermal bath are initially uncorrelated, and furthermore that the battery systems can acquire only pure states. Therefore, the initial state of an individual global system Q is assumed to be of the following form, ρSBW1 ...Wc = ρS ⊗ τ (β)B ⊗ |w1  w1 |W1 ⊗ · · · ⊗ |wc  wc |Wc , (3) and the final states we consider are of the form σSBW1 ...Wc = σSB ⊗ |w1  w1 |W1 ⊗ · · · ⊗ |wc  wc |Wc , (4) where ρS and σSB are states of the system and system-plus-bath, respectively, and wj and wj label pure states of the j-type battery before and after the Vol. 24 (2023) Resource Theory of Heat and Work 1735 transformation. The notation is meant to convey the expectation value of the () () () () j-type work, i.e. wj is a real number and Tr |wj  wj |AWj = wj . The established resource theory of thermodynamics treats the batteries and the bath as ‘enablers’ of transformations of the system S, and we will show first and second laws that express the essential constraints that any such transformation has to obey. We start with the batteries. With the notations W = W1 . . . Wc , |w = |w1  · · · |wc , and |w  = |w1  · · · |wc , let us look at a sequence ρn = ρS n = ρS1 ⊗ · · · ⊗ ρSn of initial system states, and a sequence |w w|n = |w1  w1 |W 1 ⊗· · ·⊗|wn  wn |W n of initial battery states, recalling that the baths are initially all in the same thermal state, τB n = τ (β)⊗n ; furthermore a sequence of target states σ n = σS n B n = σS1 B1 ⊗ · · · ⊗ σSn Bn of the system and bath, and a sequence |w  w |n = |w1  w1 |W 1 ⊗ · · · ⊗ |wn  wn |W n of target states of the batteries. Definition 5. A sequence of states ρn on any system Qn is called regular if its charge and entropy rates converge, i.e. if 1 (n) aj = lim Tr ρn Aj , j = 1, . . . , c, and n→∞ n 1 s = lim S(ρn ) n→∞ n exist. To indicate the dependence on the state sequence, we write aj ({ρn }) and s({ρn }). In the rest of the chapter we will essentially focus on regular sequences, so that we can simply identify them, up to asymptotic equivalence, with a point in the phase diagram. However, it should be noted that at the expense of clumsier expressions, most of our expositions can be extended to arbitrary sequences of product states or block-product states. According to the AET and the other results of the previous section, every (1) point (a, s) in the phase diagram P labels an equivalence class of regular sequences of product states under transformations by almost-commuting uni- taries. We emphasize that in AET by grouping the Q-systems into blocks of k, we do not of course change the physics of our system, except that now in the asymptotic limit we only consider n = kν copies of Q, but the state ρn is asymptotically equivalent to ρn+O(1) via almost-commuting unitaries according to Definition 2 and Theorem 4. But now we consider Qk with its (k) charge observables Aj as elementary systems, which have many more states than the k-fold product states we began with. Yet, Lemma 1 shows that the phase diagram for the k-copy system is simply the rescaled single-copy phase (k) (1) (1) diagram, P = kP , and indeed for k ≥ d, P (k) = kP . This means that we can extend the relation of asymptotic equivalence and the concomitant Asymptotic Equivalence Theorem (AET) 4 to any sequences of states that factor into product states of blocks Qk , for any integer k, which freedom we exploit in this thermodynamics setup. 1736 Z. B. Khanian et al. Ann. Henri Poincaré Now, for regular sequences ρS n of initial states of the system and final states of the system plus bath, σS n B n , as well as regular sequences of initial and final battery states, |w w|n and |w  w |n , respectively, define the asymptotic rate of j-th charge change of the j-type battery as   ΔAWj := aj |wj  wj |n − aj ({|wj  wj |n }) 1   (n) = lim Tr |wj  wj |n − |wj  wj |n AWj . (5) n→∞ n Where there is no danger of confusion, we denote this number also as Wj , the j-type work extracted (if Wj < 0, this means that the work −Wj is done on system S and bath B). Similarly, we define the asymptotic rate of j-th charge change of the work system and the bath as 1 (n) ΔASj := aj ({σS n }) − aj ({ρS n }) = lim Tr (σS n − ρS n )ASj , n→∞ n 1   (n) ΔABj := aj ({σB n }) − aj ({τ (β)B n }) = lim Tr σB n − τ (β)⊗n B ABj , n→∞ n where we denote σS n = TrB n σS n B n and likewise σB n = TrS n σS n B n . Theorem 6 (First Law). Under the above notations, if the regular sequences of the initial state ρS n B n W n = ρS n ⊗ τ (β)⊗n B ⊗ |w w| and the final state n   σS n B n W = σS n B n ⊗ |w  w | are equivalent under almost-commuting uni- n n taries, then s({σS n B n }) = s({ρS n }) + S(τ (β)) and Wj = −ΔASj − ΔABj for all j = 1, . . . , c. Conversely, given regular sequences ρS n and σS n B n of product states such that s({σS n B n }) = s({ρS n }) + S(τ (β)), and assuming that the spectral radius of the battery observables WAj is large enough (see the discussion at the start of this section), then there exist regular sequences of product states of the j-type battery, |wj  wj |n and |wj  wj |n , for all j = 1, . . . , c, such that ρS n B n W n = ρS n ⊗ τ (β)⊗n B ⊗ |w w| n and (6)   n σS n B n W n = σS n B n ⊗ |w  w | (7) can be transformed into each other by almost-commuting unitaries. Proof. The first part is by definition, since the almost-commuting unitaries asymptotically preserve the entropy rate and the work rate of all charges. In the other direction, all we have to do is find states |wj  wj | and |wj  wj | of the j-type battery Wj , such that Wj = ΔAWj = −ΔASj −ΔABj , for all j = 1, . . . , c. This is clearly possible if the spectral radius of WAj is large enough. With this, the states in Eqs. (6) and (7) have the same asymptotic entropy and charge rates. Hence, the claim follows from the AET, Theorem 4.  Vol. 24 (2023) Resource Theory of Heat and Work 1737 Remark 7. The second part of Theorem 6 says that for regular product state sequences, as long as the initial and final states of the work system and the thermal bath have asymptotically the same entropy, they can be transformed one into the another because there are always batteries that can absorb or release the necessary charge difference. Furthermore, we can even fix the initial (or final) state of the batteries and design the matching final (initial) battery state, assuming that the charge expectation value of the initial (final) state is far enough from the edge of the spectrum of AWj . For any such states, we say that there is a work transformation taking one to the other, denoted ρS n ⊗ τ (β)⊗n B → σS n B n . This transformation is always feasible, implicitly assuming the presence of suitable batteries for all j-type works to balance to books explicitly. Remark 8. As a consequence of the previous remark, we now change our point of view of what a transformation is. Of our complicated S-B-W compound, we only focus on SB and its state, and treat the batteries as implicit. Since we insist that batteries need to remain in a pure state, which thus factors off and does not contribute to the entropy, and due to the above first law Theorem 6, we can indeed understand everything that is going on by looking at how ρS n B n transforms into σS n B n . Note that in this context, it is in a certain sense enough that the initial states ρS n form a regular sequence of product states and that the target states σS n B n form a regular sequence. This is because the first part of the first law, Theorem 6, only requires regularity, and since the target state defines a unique point (a , s ) in the phase diagram, we can find a sequence of product states S n B n in its equivalence class, and use the second part of Theorem 6 to realize σ the work transformation ρS n ⊗ τ (β)⊗n B →σ S n B n . 3.2. The Second Law If the first law in our framework arises from focusing on the system-plus-bath compound SB, while making the batteries implicit, the second law comes about from trying to understand the action on the work system S alone, through the concomitant back-action on the bath B. Following [34,36], the sec- ond law constrains the different combinations of commuting conserved quan- tities that can be extracted from the work system. We show here that in the asymptotic regime, the second law similarly bounds the extractable work rate via the rate of free entropy of the system. The free entropy for a system with state ρ, charges Aj and inverse tem- peratures βj is defined in [36] as  c F(ρ) = βj Tr ρAj − S(ρ). (8) j=1 It is shown in [36] that the generalized thermal state τ (β) is the state that minimizes the free entropy for fixed βj . 1738 Z. B. Khanian et al. Ann. Henri Poincaré Figure 2. State change of the bath for a given work trans- formation under extraction of j-type work Wj , viewed in the phase diagram of the bath P B . The blue line represents the tangent hyperplane at the corresponding point of the gener- alized thermal state τ (β)B , R is the number of copies of the elementary baths in the proof of Theorem 9, and F is the point corresponding to the final state of the bath For any work transformation ρS n ⊗ τ (β)⊗n B → σS n B n between regular sequences of states, we define the asymptotic rate of free entropy change for the work system and the thermal bath respectively as follows: 1   ΔFS := lim F (σS n ) − F(ρS n ) , n→∞ n 1   (9) ΔFB := lim F (σB n ) − nF(τB ) , n→∞ n where the free entropy is with respect to the charges of the work system and the thermal bath with fixed inverse temperatures βj . Theorem 9 (Second Law). For any work transformation ρS n ⊗ τ (β)⊗n B → σS n B n between regular sequences of states, the j-type works Wj that are ex- tracted (and they are necessarily Wj = −ΔASj − ΔABj according to the first law) are constrained by the rate of free entropy change of the system: c βj Wj ≤ −ΔFS . j=1 Conversely, for arbitrary regular sequences of product states, ρS n and σS n , and any real numbers Wj with j=1 βj Wj < −ΔFS , there exists a bath c system B and a regular sequence of product states σS n B n with Tr B n σS n B n = σS n , such that there is a work transformation ρS n ⊗ τ (β)⊗nB → σS n B n with accompanying extraction of j-type work at rate Wj . This is illustrated in Fig. 2. Vol. 24 (2023) Resource Theory of Heat and Work 1739 Proof. We start with the first statement of the theorem. Consider the global system transformation ρS n ⊗τ (β)⊗n B → σS n B n by almost-commuting unitaries. We use the definition of work (5) and free entropy (8), as well as the first law, Theorem 6, to get   βj Wj = − βj (ΔASj + ΔABj ) j j = −ΔFS − ΔFB − ΔsS − ΔsB . (10) The second line is due to the definition in Eq. (9). Now observe that 1 1  ΔsS + ΔsB = lim (S(σS n ) − S(ρS n )) + S(σB n ) − nS(τ (β)B ) n→∞ n n 1   ≥ lim S(σSB n ) − S(ρS n ) − S τ (β)⊗n B = 0, (11) n→∞ n where the inequality is due to sub-additivity of von Neumann entropy, and the final equality is due to asymptotic entropy conservation. Further, the general- ized thermal state τ (β)B has the minimum free entropy [36], hence ΔFB ≥ 0. (12) For the second statement of the theorem, the achievability part of the second law, we aim to show that there is a work transformation ρS n ⊗τ (β)⊗n B → σS n ⊗ ξB n , with a suitable regular sequences of product states, and works W1 , . . . , Wc are extracted. This will be guaranteed, by the first law (Theorem 6) and the AET, Theorem 4, if s({ξB n }) = S(τ (β)B ) − ΔsS , (13) aj ({ξB n }) = Tr τ (β)B ABj − ΔASj − Wj for all j = 1, . . . , c. The left-hand side here defines a point (a, s) in the charges-entropy space of the bath, and our task is to show that it lies in the phase diagram, for which purpose  we have to define the bath characteristics  suitably. On the right-hand side, Tr τ (β)B AB1 , . . . , Tr τ (β)B ABc , S(τ (β)B ) is the point corresponding to the initial state of the bath, which due to its thermal nature is situated on the upper boundary of the region. At that point, the region has a unique tangent hyperplane, which has the equation j βj aj − s = F(τ (β)B ), and the phase diagram is contained in the half space j βj aj − s ≥ F(τ (β)B ), corresponding to the fact that their free entropy is larger than that of the thermal state. In fact, due to the strict concavity of the entropy, and hence of the upper boundary of the phase diagram, the phase  diagram, with the exception of the thermal point Tr τ (β)B AB , S(τ (β)B ) is contained in the open half space  j βj aj − s > F (τ (β)B ). One of many ways to construct a suitable bath B is as several (R  1) non-interacting copies of an “elementary bath” b: B = bR and charges ABj = (R) Abj , so that the GGS of B is τ (β)B = τ (β)⊗R b . We claim that for large enough R, the left-hand side of Eq. (13) defines a point in the phase diagram of B. 1740 Z. B. Khanian et al. Ann. Henri Poincaré Indeed, we can express the conditions in terms of b, assuming that we aim for a regular sequence of product states ξbnR : 1 s({ξbnR }) = S(τ (β)b ) − ΔsS , R (14) 1 aj ({ξbnR }) = Tr τ (β)b Abj − (ΔASj + Wj ) for all j = 1, . . . , c. R For all sufficiently large R, these  points (a, s) are arbitrarily close to where  the bath starts off, at (aβ , sβ ) = Tr τ (β)b Ab1 , . . . , Tr τ (β)b Abc , S(τ (β)b ) , while they always remains in the open half plane jβj aj − s > F(τ (β)b ). Indeed, they all lie on a straight line pointing from (aβ , sβ ) into the interior of that half plane. Hence, for sufficiently large R, (a, s) ∈ P, the phase diagram of b, and by point 5 of Lemma 1 there does indeed exist a regular sequence of product states corresponding to it.  In the next two subsections we study the achievability of the second law in a setting where the thermal bath is given. Namely, given a bath system with fixed charges and the thermal states τ (β)⊗nB , we aim to understand whether a specific work transformation is feasible and if so what is the minimum size of the thermal bath to implement such a work transformation? We rigorously state these questions as Q1 and Q2 in Sects. 3.3 and 3.4, respectively, and answer them in their corresponding subsections. 3.3. Finiteness of the Bath: Tighter Constraints and Negative Entropy In the previous two subsections we have elucidated the traditional statements of the first and second law of thermodynamics, as emerging in our resource theory. In particular, the second law is tight, if sufficiently large baths are allowed to be used. Here, we specifically look at the second statement (achievability) of the second law in the presence of an explicitly given, finite bath B. It will turn out that usually, equality in the second law cannot be attained, only up to a certain loss due to the finiteness of the bath. We also discover a purely quantum effect whereby the system and the bath remain entangled after effecting a certain state transformation, allowing quantum engines to perform tasks impossible classically (i.e. with separable correlations). The question we want to address is the following refinement of the one answered in the previous subsection: Q1: For a given bath B, regular sequences {ρS n } and {σS n } of the initial and final states of the product form, respectively, as well as real numbers W1 , . . . , Wc satisfying the second law, are there ex- tensions σS n B n of σS n forming a regular sequence of product states, such that the work transformation ρS n ⊗τ (β)⊗n B → σS n B n is feasible with the extracted works at rates W1 , . . . , Wc ? To answer it, we need the following extended phase diagram. For a give state σS of the system S, and a bath B, it is defined as the following set: Vol. 24 (2023) Resource Theory of Heat and Work 1741 Figure 3. Schematic of the extended phase diagram P |s0 . Depending on the value of s0 , whether it is smaller or larger than log |B|, the diagram acquires either the left-hand or the right-hand one of the above shapes   (1) (B) P|σS := Tr ξB A1 , . . . , Tr ξB Ac(B) , S(B|S)ξ : ξSB state with Tr B ξSB = σS } . (15) Furthermore its n-copy version, for a given product state σS n = (σ1 )S1 ⊗ · · · ⊗ (σn )Sn ,   (n) (B n ) Tr ξB n A1 , . . . , Tr ξB n A(B ) n P|σSn := c , S(B |S n n )ξ : ξS n B n  state with Tr B n ξS n B n = σS n . (16) These sets capture which combinations of charge value of the bath and condi- tional von Neumann entropy S(B|S) of the bath conditional on the system are consistent with quantum mechanics. Note that extended phase diagram con- tains the previously discussed phase diagram of the bath, since we can choose ξSB = σS ⊗ ξB as a product state, and then S(B|S)ξ = S(ξB ), but correlations between the system and the bath can reduce the conditional entropy below this quantity, in some cases not only to zero but to negative values. Finally, define the conditional entropy phase diagram as (1)   (B) P |s0 := P |s0 := a, s : aj = Tr ξB Aj , − min{s0 , S(τ (a))} ≤ s  ≤ S(τ (a)) for a state ξB , (17) (n) and likewise its n-copy version P |ns0 , for a number s0 (intended to be an entropy or entropy rate). These concepts are illustrated in Fig. 3. The relation between the sets, and the name of the latter, are explained in the following lemma. 1742 Z. B. Khanian et al. Ann. Henri Poincaré Lemma 10. With the previous notation, we have: (k) (k) 1. For all k, P|σ ⊂ P |S(σ ), and the latter is a closed convex set. Sk Sk (k) (1) 2. For all k, P |ks0 = kP |s0 . 3. For a regular sequence {σS k } of product states with entropy rate s0 = s({σS k }), every point in P |s is arbitrarily well approximated by points in 1 (k) (k) k P|σS k for all sufficiently large k, i.e., P |s0 = limk→∞ k1 P|S(σ ). Sk Proof. 1. We only have to convince ourselves that for a state ξS k B k with Tr B k ξS k B k = σS k , − min{S(σS k ), kS(τ (a))} ≤ S(B k |S k )ξ ≤ kS(τ (a)), (B k ) where a = (a1 , . . . , ac ) with ai = k1 Tr ξB k Ai . The upper bound follows from subadditivity, since S(B k |S k )ξ ≤ S(B k )ξ ≤ kS(τ (a)). The lower bound consists of two inequalities: first, by purifying ξ to a state |φ ∈ S k B k R and strong subadditivity, S(B k |S k )ξ ≥ S(B k |S k R)φ = −S(B k )ξ ≥ −kS(τ (a)). Secondly, S(B k |S k )ξ ≥ −S(S k )ξ = −S(σS k ). 2. Follows easily from the definition. 3. It is enough to show that the points of the minimum entropy diagram    (B) P min |s := a, − min{s0 , S(τ (a))} : Tr ξB Aj = aj for a state ξB can be approximated as claimed by an admissible k-copy state ξS k B k . This is (k) because the maximum entropy diagram P max is realized by states ϑS k B k := ⊗k σS k ⊗ τ (a)B , and by interpolating the states, i.e. λξ + (1 − λ)ϑ for 0 ≤ λ ≤ 1, we can realize the same charge values a with entropies in the whole interval [S(B k |S k )ξ ; kS(τ (a))]. The approximation of P min |s can be proved invoking results from quan- tum Shannon theory, specifically quantum state merging, the form of which that we need here is stated below as Lemma 11. For this, consider a tuple a ∈ P 0 and a purification |Ψ ∈ S k B k Rk of the state ϑS k B k = σS k ⊗ τ (a)⊗k B , which can be chosen in such a way as to be a product state itself: |Ψ = |Ψ1 S1 B1 R1 ⊗ · · · ⊗ |Ψk Sk Bk Rk . Our strategy is to find ξS k B k as correlated as possible, in the sense that we would like to minimize its entropy, subject to the constraint that its marginal on S k is σS k and that on B k shares the charge values with τ (a)⊗k B . As we do not know an explicit construction that achieves this, we resort to a random one that succeeds with high probability, which is what quantum state merging facilitates. We distinguish  two cases, depending on which of the entropies S(σS k ) and kS τ (a)B is the smaller.   (i) S(σSk ) ≥ S τ (a)B : We shall construct ξS k B k in such a way that ξS k =  ⊗k σS k and ξB k ≈ τ a B . To this end, choose a pure state φCR with en-   tanglement entropy S(φC ) = k1 S(σS k ) − S τ (a)B + 12 , and consider the state Ψ  S k B k C k Rk Rk = ΨS k B k Rk ⊗ φ⊗k  . Now we apply state merging CR Vol. 24 (2023) Resource Theory of Heat and Work 1743 (Lemma 11) twice to this state (which is a tensor product of k sys- tems), with a random rank-one projector P on the combined system Rk R : first, by splitting the remaining parties S k : B k C k , and second k by splitting them B k : S k C k . By construction, in both bipartitions it is the solitary system (S k and B k , resp.) that has the smaller entropy by at least 12 k, showing that the post-measurement state ξ(P  )S k B k C k with high probability approximates the marginals of ϑS k B k on S k and on B k simultaneously. Choose  a typical  subspace projector Π of φ⊗k C with log rank Π ≤ S(σS k ) − kS τ (a)B + k, and let 1  ), |ξ(P )S k B k C k := (1 k k Π k )|ξ(P c S B C with a normalization constant c. Merging and properties of the typical subspace imply that for sufficiently large k, 1 ξ(P )S k − σS k 1 ≤ , (18) 2 1 ξ(P )B k − τ (a)⊗kB 1 ≤ . (19) 2 Now, we invoke Uhlmann’s theorem applied to purifications of σS k and of ξ(P )S k B k , together with the well-known relations between fidelity and trace norm applied to Eq. (18),to obtain a state ξS k B k with ξS k = σS k and 12 ξ(P )S k B k − ξS k B k 1 ≤ (2 − ), thus by Eq. (19) 1  ξB k − τ (a)⊗kB 1 ≤ + (2 − ). 2 From the latter bound it follows that 1    (B k ) Tr ξB k Aj − aj ≤ ABj  + (2 − ) . k It remains to bound the conditional entropy: 1 1   1 S(B k |S k )ξ = S ξS k B k − S(ξS k ) k k k 1   1    ≤ S ξ(P )S k B k − S(σS k ) + + (2 − ) log(|S||B|) k  k  +h + (2 − ) 1 1    ≤ log rank Π − S(σS k ) + + (2 − ) log(|S||B|) k  k   +h + (2 − ) 1   1 ≤ S(σS k ) − kS τ (a) − S(σS k ) k  k     + 2 + (2 − ) log(|S||B|) + h + (2 − )      = −S τ (a) + 2 + (2 − ) log(|S||B|)    +h + (2 − ) , 1744 Z. B. Khanian et al. Ann. Henri Poincaré where in the second line we have used the Fannes inequality on the con- tinuity of the entropy [39,40], with the binary entropy h(x) = −x log x − (1−x) log(1−x); in the third line that ξ(P )S k B k has rank at most rank Π; and in the fourth  line  the upper bound on the latter rank by construction. (ii) S(σSk ) < S τ (a)B : We shall construct ξS k B k such that ξS k = σS k and (B k )   Tr ξB k Aj ≈ Tr τ a B ABj for all j = 1, . . . , c. Here, choose a pure state φCR with entanglement entropy S(φC ) = , and define Ψ  S k B k C k Rk R k = ΨS k B k Rk ⊗ φ⊗k CR . Now we apply state merging (Lemma 11) to this state (which is a tensor product of k systems), with a random rank-one pro- jector P on the combined system Rk R , by splitting the remaining par- k ties S : B C , which ensures that S k has the smaller entropy by at k k k least k, showing that the post-measurement state ξ(P  )S k B k C k with high probability approximates the marginal of ϑS k B k on S k . Proceed as be- fore with a typical subspace projector Π of φ⊗k C such that log rank Π ≤    ), S(σS k ) − kS τ (a)B + k, and let |ξ(P )S k B k C k := 1c (1S k B k ΠC k )|ξ(P with a normalization constant c. Merging and properties of the typical subspace thus imply that for sufficiently large k, 1 ξ(P )S k − σS k 1 ≤ . (20) 2 Next we need to look at the charge values of ξ(P )B k . Note that the expectation EP ξ(P )B k is approximately equal to EP ξ(P  )B k = τ (a)⊗k . It B follows from [41, Lemma III.5], that if k is sufficiently large, then with high probability   (B k ) Tr ξ(P )B k − τ (a)⊗k B Aj ≤ ABj  for all j = 1, . . . , c. (21) So we just focus on a good instance of P , where both Eqs. (20) and (21) hold. Now we proceed as in the first case  to find a state ξS k B k with ξS k = σS k and 12 ξ(P )S k B k − ξS k B k 1 ≤ (2 − ), using Uhlmann’s theorem. Thus, as before we find 1    (B k ) Tr ξB k Aj − aj ≤ ABj  + (2 − ) . k Regarding the conditional entropy, we have quite similarly as before, 1 1   1 S(B k |S k )ξ = S ξS k B k − S(ξS k ) k k k 1   1    ≤ S ξ(P )S k B k − S(σS k ) + + (2 − ) log(|S||B|) k  k  +h + (2 − ) 1 1    ≤ log 2 k − S(σS k ) + + (2 − ) log(|S||B|) k  k   +h + (2 − ) 1       ≤ − S(σS k ) + 2 + (2 − ) log(|S||B|) + h + (2 − ) . k Vol. 24 (2023) Resource Theory of Heat and Work 1745 Since in both caseswe knew the conditional entropy to be always ≥ − k1 min S(σS k ), kS τ (a) , this concludes the proof.  Lemma 11 (Quantum state merging [42,43]). Given a pure product state ΨAn B n C n = (Ψ1 )A1 B1 C1 ⊗· · ·⊗(Ψn )An Bn Cn , such that S(ΨAn )−S(ΨB n ) ≥ n, consider a Haar random rank-one projector P on C n . Then, it holds that the post-measurement state 1 ψ(P )An B n = TrC n Ψ(1An B n ⊗ P ) Tr ΨC n P satisfies 12 ψ(P ) − ΨAn B n 1 ≤ , except with arbitrarily small probability for sufficiently large n.  Remark 12. While we have seen that the upper boundary of the extended (k) (k) phase diagram P |S(σ k ) is exactly realized by points in P|σ k , namely those S S corresponding to the tensor product states σS k ⊗τ (a)⊗k B , it seems unlikely that we can achieve the analogous thing for the lower boundary: this would entail finding, for every (sufficiently large) k a tensor product state, or a block tensor product state,  ξS k B k with prescribed charge vector a on B k , and S(B k |S k )ξ = − min{kS τ (a) , S(σS k )}.   Now, for concreteness, consider the case that kS τ (a)  ≤ S(σ  S k ), so that the conditional entropy aimed for is S(B k |S k )ξ = −kS τ (a)B , which is the ⊗k value  of apurification of τ (a)B . In particular, it would mean that S(ξB k ) = kS τ (a)B , and so—recalling the charge values and the maximum entropy principle—it would follow that ξB k = τ (a)⊗k B . However, from the equality conditions in strong subadditivity [44], this in turn would imply that ξS k B k is a probabilistic mixture of purifications of τ (a)⊗k B whose restrictions to S k are pairwise orthogonal. This would clearly put constraints on the spectrum of σS k that are not generally met.   In the other case that kS τ (a) > S(σS k ), the conditional entropy should be S(B k |S k )ξ = −S(σS k ), and since ξS k = σS k , this would necessitate a pure state ξS k B k . Looking at the proof of Lemma 10, however, we see that it leaves quite a bit of maneuvering space, so it may or may not be possible to satisfy (B k ) all charge constraints Tr ξB k Aj = aj (j = 1, . . . , c). Coming back to our question, if a work transformation ρS n ⊗ τ (β)⊗nB → σS n B n is feasible for regular sequences on the left-hand side, by the first law this implies that s({σS n B n }) = s({ρS n }) + S(τ (β)) and Wj = −ΔASj − ΔABj = aj ({ρS n }) − aj ({σS n }) + aj ({τ (β)B n }) − aj ({σB n }). When σS n and the Wj are given, this constrains the possible states σS n B n as follows: for each n, 1 S(B n |S n )σ ≈ S(τ (β)) − ΔsS , n 1746 Z. B. Khanian et al. Ann. Henri Poincaré 1 (n) Tr σB n ABj ≈ Tr τ (β)B ABj − ΔASj − Wj , for all j = 1, . . . , c. n Since by Lemma 10 the left-hand sides converge to the components of a point in P |s({σSn }) , meaning that a necessary condition for the feasibility of the work transformation in question is that (a, t) ∈ P |s({σSn }) , with aj := Tr τ (β)B ABj − ΔASj − Wj , (22) t := S(τ (β)) − ΔsS . Again by Lemma 10, this is equivalent to all aj to be contained in the set of joint quantum expectations of the observables ABj , and     − min s({σS n }), S τ (a) ≤ t ≤ S τ (a) . The following theorem shows that this is also sufficient, when we allow block- ings of the asymptotically many systems. Theorem 13 (Second Law with fixed bath). For arbitrary regular sequences ρS n and σS n of product states, a given bath B, and any real numbers Wj , if there exists a regular sequence of block product states σS n B n with Tr B n σS n B n = σS n , such that there is a work transformation ρS n ⊗ τ (β)⊗n B → σS n B n with accompanying extraction of j-type work at rate Wj , then Eq. (22) defines a point (a, t) ∈ P |s({σSn }) . Conversely, assuming additionally that σS n = σS⊗n is an i.i.d. state, if 0 Eq. (22) defines a point (a, t) ∈ P |S(σS ) in the interior of the extended phase diagram, then for every > 0 there is a work transformation ρS n ⊗ τ (β)⊗n B → σS n B n with block product states σS n B n such that Tr B n σS n B n = σS n , and with accompanying extraction of j-type work at rate Wj ± . This is illustrated in Fig. 4. Proof. We have already argued the necessity of the condition. It remains to show its sufficiency. Using Lemma 10, this is not hard: Namely, by its point 3, (k) for sufficiently large k, (a, t) ∈ P |s is -approximated by k1 P|σ⊗k , i.e. there exists S (B k ) a σS k B k with TrB k σS k B k = σS⊗k with 1 k k ≤ t− and k1 Tr σB k Aj k S(B |S )σ ≈  ⊗k aj for all j = 1, . . . , c. By mixing σ with a small fraction of τ (a)B ⊗ σS , we (B k ) can in fact assume that k1 S(B k |S k )σ = t while preserving 1 Tr σB k Aj ≈ aj .  ⊗ nk k Now our target block product states will be σS n B n := σS k B k for n a multiple of k. By construction, this sequence has the same entropy rate as the initial regular sequence of product states ρS n ⊗ τ (β)⊗n B , so by the first law, Theorem 6, and the AET, Theorem 4, there is indeed a corresponding work transformation with j-type work extracted equal to Wj ± .  Remark 14. One might object that tensor power target states are not general enough in Theorem 13, as we had observed in Sect. 2 that such states do not generate the full phase diagram P of the system S. However, by considering blocks of  systems S  , we can apply the theorem to block tensor power target Vol. 24 (2023) Resource Theory of Heat and Work 1747 Figure 4. State change of the bath for a given work trans- formation under the extraction of j-type work Wj , viewed in the extended phase diagram of the bath, which initially is in the thermal state τ (β)B , the blue line at the corresponding point in the diagram representing the tangent hyperplane of the diagram. The final states {σS n B n } give rise to the point F in the extended diagram, whose charge values are those of {σB n }, while the entropy is n1 S(B n |S n )σ  ⊗ n states σS n = σ1 ⊗ · · · ⊗ σ  , and these latter are in fact a rich enough class to exhaust the entire phase diagram P , when  ≥ dimS (point 5 of Lemma 1). More generally, we can allow as target uniformly regular sequences of product states σS n , by which we mean the following strengthening of the con- N +n dition in Definition 5. Denoting BN +1 := BN +1 . . . BN +n , we require that for all > 0 and uniformly for all N , it holds that for sufficiently large n, 1 (n) aj − Tr σB N +n Aj ≤ for all j = 1, . . . , c, n N +1 1 and s − S(σB N +n ) ≤ . n N +1 Remark 15. We conclude this subsection with a reflexion on the peculiar role of entanglement played in the quantum advantage implied by Theorem 13. Indeed, whereas in many quantum tasks entanglement is the fuel requisite at the beginning to perform super-classically, here it is the possibility of leaving the system and bath in an entangled state which allows to reach points in the extended phase diagram outside the usual phase diagram, i.e. with negative conditional entropy S(B|S). Note that no separable state can achieve this, as by the result of [45,46] then S(B|S) ≥ 0. 1748 Z. B. Khanian et al. Ann. Henri Poincaré Evidently, demonstrating such an effect would require phenomenal con- trol of the quantum degrees of freedom of both S and B, so in a macro- scopic system that would presumably be impossible. But we believe it not completely beyond the bounds of the recent demonstrations of thermal ma- chines in mesoscopic and nanoscopic systems. While we cannot indicate any concrete references, a well-designed experiment would be feasible with any of the contemporary platform for quantum simulations (QS), such as • Superconducting qubits, used by Google [47] or D-Wave [48], are often employed as digital QSs (cf. [49]) and/or in circuit QED systems [50]; • Ultracold atoms, which offer analog quantum simulation, can be realized in the continuum or in optical lattices [51]. They are very flexible and they allow to simulate complex Hubbard models, as well as spin systems; • Trapped ions can also be used as perfect analog or digital QSs [52,53]. They typically simulate spin- 21 systems, but very recently a qudit quan- tum computer/simulator was realized with ions [54]; • Rydberg atoms are atoms where the electron has been excited to a high principal quantum number, and which are trapped in optical tweezers. They mimic spin systems with long-range interactions [55–57]; • Light and cavity materials: Quantum simulators based on cavity QED take advantage of the coupling between quantum system and the coherent light field of the cavity in which such system has been placed. Experiments are mainly conducted in the scope of Jaynes-Cummings and Dicke models [58]. Recent studies concern also engineering materials entirely from light with resulting photon-photon interactions [59–62]; • Twistronics systems: Twistronics deals with twisted bilayer graphene or other two-dimensional materials [63,64]. For small “magic” angle, such systems lead to periodic Moiré patterns at a length scale much larger than the typical scale of condensed matter systems: in this sense, they can themselves be considered as condensed matter quantum simulators of condensed matter [65]. Twisted bilayer materials can, however, also be mimicked by ultracold atoms in a two-dimensional lattice with synthetic dimensions [66]; • Polaritons are especially useful for non-equilibrium systems and quantum hydrodynamics simulation, as well as relativistic effects thanks to dual (half light half particle) nature of the polaritonic quasi-particles [67–69]. 3.4. Tradeoff Between Thermal Bath Rate and Work Extraction Here we consider a different take on the question of the work deficit due to finiteness of the bath. Namely, we still consider a given fixed finite bath system B, but now as which state transformations and associated generalized works are possible when for each copy of the subsystem S, R ≥ 0 copies of B are present. It is clear what that means when R is an integer, but below we shall give a meaning to this rate as a real number. We start off with the observation that “large enough bath” in Theorem 9 can be taken to mean B R , for the given elementary bath B and sufficiently large integer R. Vol. 24 (2023) Resource Theory of Heat and Work 1749 Theorem 16. For arbitrary regular sequences of product states, ρS n and σS n , and any real numbers Wj with j=1 βj Wj < −ΔFS , there exists an integer c R ≥ 0 and a regular sequence of product states σS n B nR with Tr B nR σS n B nR = σS n , such that there is a work transformation ρS n ⊗ τ (β)⊗nR B → σS n B nR with accompanying extraction of j-type work at rate Wj . Proof. This was already shown in the achievability part of Theorem 9.   n To give meaning to a rational rate R = k, group the systems of S , for  n = νk, into blocks of k, which we denote S = S , and consider ρS n ≡ ρSν as k a ν-party state, and likewise σS n ≡ σSν . For each S = S k we assume  copies of the thermal bath, τ (β)⊗   , with B = B . If {ρS n } and {σS n } are  B = τ (β)B regular sequences of product states, then evidently so are {ρSν } and {σSν }. With this definition of the rate, the question that we address in this subsections is: Q2: For a given bath B, regular sequences {ρS n } and {σS n } of the initial and final states of the product form, respectively, as well as real numbers W1 , . . . , Wc satisfying j βj Wj = −ΔFS − δ, δ ≥ 0, what is the infimum over all rates R = k such that there is a work transformation ρS n ⊗ τ (β)B nR ≡ ρSν ⊗ τ (β)⊗ν  B → σSν B ν ≡ σS n B nR , with the extracted works at rates W1 , . . . , Wc and the final state satisfying Tr B ν σSν B ν = σSν . We first observe that if S(ρS n ) = S(σS n ), then j βj Wj = −ΔFS can hold without using any thermal bath, which follows from Eq. (10). That is, the thermal bath is not necessary for the work transformation and extracting work if the entropy of the work system does not change. Conversely, the role of the thermal bath is precisely to facilitate changes of entropy in the work system. To answer the above question about the minimum bath rate R∗ , we first show the following lemma. Lemma 17. Consider regular sequences of product states, ρS n and σS n , and real numbers Wj , and assume that for large enough rate R there is a work transformation ρS n ⊗ τ (β)⊗nR B → σS n B nR , with σS n as the reduced final state on the work system, and works W1 , . . . , Wc are extracted. Then there is another work transformation ρS n ⊗ τ (β)⊗nR B → σS n ⊗ ξB nR , in which the final state of the work system and the thermal bath are uncorrelated, ξB nR is a regular sequence of product states, and the same works W1 , . . . , Wc are extracted. Proof. Assuming that ρS n ⊗ τ (β)⊗nR B → σS n B nR is a work transformation, the second law implies that j βj Wj = −ΔFs − δ for some δ ≥ 0, and we obtain the following coordinates for the bath system for 0 ≤ δ  ≤ δ: 1 δ s({σB nR }) = S(τ (β)B ) − ΔsS + , R R 1750 Z. B. Khanian et al. Ann. Henri Poincaré 1 aj ({σB nR }) = Tr τ (β)B ABj − (ΔASj + Wj ) for all j = 1, . . . , c. R (23) To obtain the first equality, which is the expansion of ΔsS + ΔsB = δ  , we use the fact that j βj Wj = −ΔFS −ΔFB − ΔsS − ΔsB , i.e. ΔFB +ΔsS +ΔsB =    −δ δ which follows from Eq. (10). Due to positivity of the entropy rate change, i.e. ΔsS + ΔsB ≥ 0 from Eq. (11) and ΔFB ≥ 0 from Eq. (12), we infer that 0 ≤ ΔsS + ΔsB ≤ δ. The second equality, which is the expansion of    δ ΔABj + ΔASj = −Wj , follows from the first law, Theorem 6, and the AET, Theorem 4. If R is large enough, due to the convexity of the phase diagram of (1) the thermal bath P B , the following coordinates belong to the phase diagram as well 1 s({ξB nR }) = S(τ (β)B ) − ΔsS , R 1 aj ({ξB nR }) = Tr τ (β)B ABj − (ΔASj + Wj ) for all j = 1, . . . , c. R (24) We can observe this in Fig. 5; the new coordinates have the same charge values,  but the entropy is δR smaller than the entropy of the coordinates in Eq. (23). Therefore, as long as S(τ (β)B ) − R1 ΔsS ≥ 0, the new coordinates are inside the phase diagram as well. Hence, due to points 3 and 5 of Lemma 1, there (1) is a tensor product state ξB nR with coordinate of Eq. (24) on P B . Hence the first law, Theorem 6, implies that the desired transformation exists, and works W1 , . . . , Wc are extracted.  Theorem 18. For regular sequences of product states, ρS n and σS n , and real numbers Wj satisfying j βj Wj = −ΔFs − δ, let R∗ be the infimum of rates such that there is a work transformation ρS n ⊗ τ (β)⊗nRB → σS n ⊗ ξB nR un- der which works W1 , . . . , Wc are extracted, and ξB nR is a regular sequence of product states. Then, this minimum R∗ is achieved for a state ξB nR on the boundary of the phase diagram P B of the thermal bath. Indeed, it is the point where the line given by Eq. (14) intersects the boundary of the phase diagram; see Fig. 5. Equivalently, it is the smallest R such that the point in Eq. (14) is contained in P B . For δ  1, the minimum rate can be written as 1  ∂βj R≈− (ΔASi + Wi )(ΔASj + Wj ), (25) 2δ ij ∂ai where ΔASj = a({σS n }) − a({ρS n }). Proof. We notice that the initial and final states of the work system as well as the initial state of the bath and the extracted work rates are known. Also, Vol. 24 (2023) Resource Theory of Heat and Work 1751 Figure 5. Graphical illustration of R∗ , the minimum bath rate for a work transformation {ρS n } → {σS n } satisfying the second law, according to Theorem 18. The initial state is the generalized thermal state τ (β), its corresponding point marked on the upper boundary of the phase diagram. The final bath states correspond to points on the line denoted f , and they are feasible if and only they fall into the phase di- agram. Consequently, F ∗ is the point corresponding to the minimum rate the final state of the thermal bath ξB nR is a tensor product state, therefore, the first law (Theorem 6), and the AET( Theorem 4) imply that the entropy and the charges rates of the global system are preserved; hence, we obtain the following entropy and charge rates for the final state of the bath: 1 s({ξB nR }) = S(τ (β)B ) − ΔsS , R 1 aj ({ξB nR }) = Tr τ (β)B ABj − (ΔASj + Wj ) for all j = 1, . . . , c, R (26) where ΔsS = s({σS n }) − s({ρS n }). The above quantities on the left member are rates of the entropy and charge changes, therefore, they must belong to (nR) PB the diagram nR . Hence, due to point 3 of Lemma 1, the above coordinates (nR) (1) PB belong to P B = nR . Now, for R = R∗ assume that the above coordinates (1) belong to the point (a, s) on the boundary of the phase diagram P B . Then, for R > R∗ the point of Eq. (26) is a convex combination of the points (a, s) and the corresponding point of the state τ (β)B , so it belongs to the phase diagram due to its convexity. Therefore, all points with R > R∗ are inside the diagram. 1752 Z. B. Khanian et al. Ann. Henri Poincaré To approximate the minimum R for small δ, define the function S(a) := S(τ (a)B ) for a = (a1 , . . . , ac ). Its Taylor expansion  around  the point corre- sponding to the initial thermal state τ (β)B ≡ S τ (a0 )B of the bath gives the approximation    1  ∂βj    S(a) ≈ S(a0 ) + βj aj − a0j + aj − a0j ai − a0i , (27) j 2 ij ∂ai ∂S where we have used the well-known relation ∂ai = βi . From Eq. (26), we obtain ΔsS S(a) − S(a0 ) = − , R 1 aj − a0j =(−ΔASj − Wj ), R and by substituting these values in the Taylor approximation (27), using the definition of the free entropy and of the deficit δ, we arrive at the claimed Eq. (25).  Remark 19. For a single charge, c = 1, which we traditionally interpret as the internal energy E of a system, Eq. (25) takes on the very simple form 1 ∂β R≈− (ΔES + W )2 . 2δ ∂E 1 ∂β ∂ Here we can use the usual thermodynamic definitions to rewrite ∂E = ∂E T = 1 1 − T 2 C , with the heat capacity C = ∂T , all derivatives taken with respect to ∂E corresponding Gibbs equilibrium states. Thus, 1 1 1 R ≈ 2 · (ΔES + W )2 , (28) T C 2δ resulting in a clear operational interpretation of the heat capacity in terms of the rate of the bath to approach the second law tightly.  ∂β   2S  For larger numbers of charges, the matrix ∂aji ij = ∂a∂i ∂a is actually   j ij the Hessian of the entropy S τ (a)B with respect to the charges, and the r.h.s. 1 side of Eq. (25) is 2δ times the corresponding quadratic form evaluated on the vector (ΔAS1 + W1 , . . . , ΔASc + Wc ). Note that by the strict concavity of the generalized Gibbs entropy, this is a negative definite symmetric matrix, thus explaining the minus sign in Eq. (25). In the same vein as the single-parameter discussion before, the Hessian matrix can be read as being composed of gener- alized heat capacities, which likewise receive their operational interpretation in terms of the required rate of the bath. The heat capacity has made appearances in previous results in the re- source approach to thermodynamics: Chubb et al. [70] have found it to show up in the optimal interconversion rate between states in a resource theory of Gibbs-preserving transformations and with unlimited baths at temperature T . Their setting is the finite-copy regime, and in contrast to our result of finite bath where the heat capacity affects the first order term (scaling linear with n), the heat capacity determines the so-called second order term, scaling with Vol. 24 (2023) Resource Theory of Heat and Work 1753 √ n. While it is thus amusing to contemplate the separate appearance of the heat capacity in the two results, the settings seem too different to allow for a meaningful comparison. 4. Discussion We have presented a resource theory in which the objects are sequences of tensor product states, and thermodynamically meaningful allowed transforma- tions, namely operations which preserve the entropy and charges of a system asymptotically. The allowed operations classify the objects into equivalence classes of state sequences that are interconvertible under allowed operations. The basic result on which our theory is built is that the objects are inter- convertible via allowed operations if and only if they have the same average entropy and average charge values in the asymptotic limit. The existence of the allowed operations between the objects of the same class is based on two pillars: First, for objects with the same average entropy there are states with sublinear dimension which can be coupled to the objects to make their spectrum asymptotically identical. Second, objects with the same average charge values project onto a common subspace of the charges of the system which has the property that any unitary acting on this subspace is an almost-commuting unitary with the corresponding charges. Therefore, the spectrum of the objects of the same class can be modified using small ancillary systems and then they are interconvertible via unitaries that asymptotically preserve the charges of the system. The notion of a common subspace for different charges, which are Hermitian operators, is introduced in [34] as ap- proximate microcanonical (a.m.c.) subspace. In this paper, for given charges and parameters, we construct a permutation-symmetric a.m.c., something not guaranteed by the construction in [34]. We then applied this resource theory to understand quantum thermo- dynamics with multiple conserved quantities. We specifically consider an as- ymptotic generalization of the setting proposed in [36] where there are many copies of a global system consisting of a main system, called a work system, a thermal bath with fixed temperatures and various batteries to store the differ- ent charges of the system. Our approach allows us, in our setting, to resolve affirmatively a question from [34,36], which asks about the possibility of con- structing physically separate batteries for all the involved charge numbers, be they commuting or not (cf. [38]). Therefore, the objects and allowed operations of the resource theory apply quantum states of a thermodynamics system and thermodynamical transformations, respectively. It is evident that the allowed operations can transform a state with a tensor product structure to a state of a general form; however, we show that restricting the final states to the specific form of tensor product structure does not reduce the generality and tightness of the bounds that we obtain, which follows from the fact that for any point of the phase diagram there is a state with tensor product structure realizing it. As discussed in [36], for a system with multiple charges, the free entropy is a conceptually more meaningful quantity than the free energy, which is 1754 Z. B. Khanian et al. Ann. Henri Poincaré originally defined when energy is the only conserved quantity of the system. Namely, the free energy bounds the amount of energy that can be extracted (while conserving the other charges); however, for a system with multiple charges there are not various quantities that bound the extraction of indi- vidual charges. Rather, there is only a bound on the trade-off between the charges that can be extracted which is precisely the free entropy defined with respect to the temperatures of the thermal bath. We show that indeed this is the case in our scenario as well and formulate the second law: the amount of charge combination that is extracted is bounded by the free entropy change of the work system per number of copies of the work system, i.e. the free en- tropy rate change. Conversely, we show that all transformations with given extracted charge values, with a combination strictly bounded by the free en- tropy rate change of the work system, are feasible. In particular, any amount of a given charge, or the so-called work type, is extractable providing that sufficient amounts of other charges are injected to the system. This raises the following fundamental question: for given extractable charge values, with a combination saturating the second law up to a deficit δ, what is the minimum number of the thermal baths per number of the copies of the work system. We define this ratio as the thermal bath rate. We find that for large thermal bath rates the optimal value is inversely proportional to the deficit δ, and there is always a corresponding transformation where the final state of the work system and the thermal bath are uncorrelated. However, in general this is not true: the minimum rate might be obtained where the final state correlates the work system and the thermal bath. This is a purely quan- tum mechanical effect, making certain work transformations possible with a smaller size of the thermal bath than would be possible classically; it relies on work system and bath becoming entangled. In order to describe precisely the possible work transformations with a fixed bath, we define and analyze the ex- tended phase diagram of the bath, which depends on a given conditional state of the work system and records the conditional, rather than plain, entropy. Our results paint a broad picture of thermodynamics as a resource theory, which ultimately relies only on conservation laws, namely the conservation of information (i.e. entropy)—cf. [12]—, and the conservation of extensive phys- ical quantities. At the microscopic level, the former means that the allowed transformations are (approximate) unitaries, the latter that they (approxi- mately) commute with the conserved quantities. Amazingly, after these simple premises give rise to the phase diagram, the supposed deep distinction between entropy and the conserved charges disappears: they both are simply extensive conserved quantities. Following our development of thermodynamics, with its batteries for the distinct charges, we could augment this with an entropy bat- tery, which carries no charges and is only there to absorb or release entropy. This is a very general picture that in some respect includes as a special case our treatment of the second law: namely, the role of the bath is largely as such an entropy battery, although the fact that it also carries charges complicates things compared to this abstract vantage point. Vol. 24 (2023) Resource Theory of Heat and Work 1755 We leave several open questions to be addressed. Not to dwell on the overly technical ones, which will be evident to readers of the detailed claims and proofs, a fundamental problem is whether it is possible to prove the AET Theorem 4 with unitaries that exactly commute with the conserved quantities, rather than approximately? This would require the construction of a subexpo- nential reference frame to take care of the conservation laws; this is known to be possible for a single conserved quantity (energy) [33], and more generally for pairwise commuting charges [12]. If it were possible in the non-commuting setting, it would give our theory a much stronger appeal, since at a funda- mental level, conservation laws in nature are considered to hold strictly, rather than only approximately. There is a whole plethora of open questions concerning practical and ex- perimental implications of our results (similar experimental settings for ther- modynamics with non-commuting charges have been characterized recently in [71,72]). The most straightforward, and perhaps most interesting is this one: can one design a system and a bath of small to moderate size, such that a con- crete work transformation will necessarily leave the system and the bath in a final entangled state? The impossibility of such a transformation in a classical system could be interpreted as a thermal machine entanglement witness. Acknowledgements The authors thank Michal Horodecki for pointing out Ref. [70]. ZBK is grate- ful to Paul Skrzypczyk and Tony Short for valuable discussions. AW thanks Glen and Ella Runciter for invaluable hermetic and immanent remarks on the ubiquity of the second law. ZBK and AW acknowledge financial support by the Spanish MINECO (Projects FIS2016-86681-P and PID2019-107609GB- I00) with the support of FEDER funds, and the Generalitat de Catalunya (Project CIRIT 2017-SGR-1127). MNB acknowledges financial support from SERB-DST (CRG/2019/002199), Government of India. ML acknowledges the support from ERC AdG NOQIA, Spanish Ministry of Economy and Competi- tiveness (Severo Ochoa program for Centres of Excellence in R&D (CEX2019- 000910-S), Plan National FISICATEAMO and FIDEUA PID2019-106901GB- I00/10.13039/501100011033, FPI), Fundació Privada Cellex, Fundació Mir- Puig, and from Generalitat de Catalunya (AGAUR Grant no. 2017-SGR-1341, CERCA program, QuantumCAT U16-011424, co-funded by ERDF Opera- tional Program of Catalonia 2014–2020), MINECO-EU QUANTERA MAQS (funded by State Research Agency (AEI) PCI2019-111828-2/10.13039/501100 011033), EU Horizon 2020 FET-OPEN OPTOLogic (Grant no. 899794), and the National Science Centre, Poland-Symfonia Grant no. 2016/20/W/ST4/ 00314. Funding Information Open Access Funding provided by Universitat Autonoma de Barcelona. 1756 Z. B. Khanian et al. Ann. Henri Poincaré Open Access. 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Appendices Here we collect mathematically involved arguments that would detract from the exposition of the main article. These are the proof of Theorem 4 (in “Ap- pendix C”), which requires the construction of approximate microcanonical (a.m.c.) subspaces [34], for which we give a self-contained proof in “Appen- dix B”. We start with the collection of miscellaneous definitions and facts in “Appendix A”. Appendix A: Miscellaneous Definitions and Facts Definition 20. Let ρ1 , . . . , ρn be quantum states on a d-dimensional Hilbert space H with diagonalizations ρi = j pij πij and one-dimensional projectors πij . For α > 0 and ρn = ρ1 ⊗· · ·⊗ρn define the set of entropy-typical sequences as    n √ Tα,ρ n n = j n = j1 j2 . . . jn : − log piji − S(ρi ) ≤ α n . i=1 Define the entropy-typical projector of ρn with constant α as  Πnα,ρn = π1j1 ⊗ · · · ⊗ πnjn . j n ∈Tα,ρ n n Lemma 21 (Cf. [73]). There is a constant 0 < β ≤ max{(log 3)2 , (log d)2 } such that the entropy-typical projector has the following properties for any α > 0, n > 0 and arbitrary state ρn = ρ1 ⊗ · · · ⊗ ρn :   β Tr ρn Πnα,ρn ≥ 1 − 2 , √ α √ 2− i=1 S(ρi )−α n Πnα,ρn ≤ Πnα,ρn ρn Πnα,ρn ≤ 2− i=1 S(ρi )+α n Πnα,ρn , and n n   β √   √ 1 − 2 2 i=1 S(ρi )−α n ≤ Tr Πnα,ρn ≤ 2 i=1 S(ρi )+α n . n n α Vol. 24 (2023) Resource Theory of Heat and Work 1757 Lemma 22 (Gentle operator lemma [74–76]). If a quantum state ρ with di- agonalization ρ = j pj πj projects onto a POVM element Λ with probability 1 − , i.e. Tr (ρΛ) ≥ 1 − for 0 ≤ Λ ≤ 1, then  √ √ √ pj πj − Λπj Λ ≤ 2 . 1 j Lemma 23 (Cf. Bhatia [77]). For operators A, B and C and for any p ∈ [1, ∞], the following holds ABCp ≤ A∞ Bp C∞ . Lemma 24 (Hoeffding’s inequality, cf. [78]). Let X1 , X2 , . . . , Xn be indepen- dent random variables with ai ≤ Xi ≤ bi , and define the empirical mean of these variables as X = X1 +···+X n n . Then, for any t > 0,   2n2 t2 Pr X − E(X) ≥ t ≤ exp − n 2 , i=1 (bi − ai )   2n2 t2 Pr X − E(X) ≤ −t ≤ exp − n 2 . i=1 (bi − ai ) Appendix B: Approximate Microcanonical (A.M.C.) Subspace In this section, we recall the definition of the notion of approximate micro- canonical (a.m.c.) and give a new proof that it exists for certain explicitly given parameters. Definition 25. An approximate microcanonical (a.m.c.) subspace, or more pre- cisely a ( , η, η  , δ, δ  )-approximate microcanonical subspace, M of H⊗n , with projector P , for charges Aj and values vj = Aj  is one that consists, in a certain precise sense, of exactly the states with “very sharp” values of all the (n) Aj . Mathematically, the following has to hold: 1. Every state ω with support contained in M satisfies Tr ωΠηj ≥ 1 − δ for all j.  2. Conversely, every state ω on H⊗n such that Tr ωΠηj ≥ 1 − δ  for all j, satisfies Tr ωP ≥ 1 − . (n) Here, Πηj := nvj −nηΣ(Aj ) ≤ Aj ≤ nvj +nηΣ(Aj ) is the spectral projector (n) of Aj of values close to nvj , and Σ(A) = λmax (A) − λmin (A) is the spectral diameter of the Hermitian A, i.e. the diameter of the smallest disc covering the spectrum of A. Remark 26. It is shown in [34, Thm. 3] that for every > cδ  > 0, δ > 0 and η > η  > 0, and for all sufficiently large n, there exists a nontrivial ( , η, η  , δ, δ  )-a.m.c. subspace. However, there are two (related) reasons why one might be not completely satisfied with the argument in [34]: First, the proof uses a difficult result of Ogata [79] to reduce the non-commuting case to the seemingly easier of commuting observables; while this is conceptually nice, it makes it harder to perceive the nature of the constructed subspace. 1758 Z. B. Khanian et al. Ann. Henri Poincaré Secondly, despite the fact that the defining properties of an a.m.c. subspace are manifestly permutation symmetric (w.r.t. permutations of the n subsystems), the resulting construction does not necessarily have this property. Here we address both these concerns. Indeed, we shall show by essentially elementary means how to obtain an a.m.c. subspace that is by its definition permutation symmetric. Theorem 27. Under the assumptions of Definition 25, for every > 2(n + 2 1)3d δ  > 0, η > η  > 0 and δ > 0, for all sufficiently large n there exists an approximate microcanonical subspace projector. In addition, the subspace can be chosen to be stable under permutations of the n systems: U π M = M, or equivalently U π P (U π )† = P , for any permutation π ∈ Sn and its unitary action U π . More precisely, given η > η  > 0 and > 0, there exists a α > 0 such that there is a non-trivial ( , η, η  , δ, δ  )-a.m.c. subspace with 2 δ = (c + 3)(5n)5d e−αn and 2 δ = − (c + 3)(5n)2d e−αn . 2(n + 1)3d2 (η−η  )2 Furthermore, we may choose α = 8(cd+1)2 . Proof. For s > 0, partition the state space S(H) on H into Cs (v) = σ : ∀j | Tr σAj − vj | ≤ sΣ(Aj ) , (B1) Fs (v) = σ : ∃j | Tr σAj − vj | > sΣ(Aj ) = S(H)\Cs (v), (B2) which are the sets of states with Aj -expectation values “close” to and “far” from v. Note that if ρ ∈ Cs (v) and σ ∈ Ft (v), 0 < s < t, then ρ − σ1 ≥ t − s. Choosing the precise values of s > η  and t < η later, we pick a universal distinguisher (P, P ⊥ ) between Cs (v)⊗n and Ft (v)⊗n , according to Lemma 28 below: 2 ∀ρ ∈ Cs (v) Tr ρ⊗n P ⊥ ≤ (c + 2)(5n)2d e−ζn , (B3) 2 ∀σ ∈ Ft (v) Tr σ ⊗n P ≤ (c + 2)(5n)2d e−ζn , (B4) 2 with ζ = 2c2(t−s) (2d2 +1) . Our a.m.c. subspace will be M := supp P ; by Lemma 28, P and likewise M are permutation symmetric. It remains to check the properties of the definition. First, let ω be sup- ported on M. Since we are interested in Tr ωΠηj , we may without loss of gen- erality assume that ω is permutation symmetric. Thus, by the “constrained de Finetti reduction” (aka “Postselection Lemma”) [80, Lemma 18],  3d2 ω ≤ (n + 1) dσ σ ⊗n F (ω, σ ⊗n )2 , (B5) with a certain√universal probability measure dσ on S(H), and the fidelity √ F (ρ, σ) =  ρ σ1 between states. We need the monotonicity of the fidelity under cptp maps, which we apply to the test (P, P ⊥ ):  2 F (ω, σ ⊗n )2 ≤ F (Tr σ ⊗n P, 1 − Tr σ ⊗n P ), (1, 0) ≤ Tr σ ⊗n P, Vol. 24 (2023) Resource Theory of Heat and Work 1759 which holds because Tr ωP = 1. Thus,  η ⊥ 3d2   Tr ω(Πj ) ≤ (n + 1) dσ Tr σ ⊗n (Πηj )⊥ (Tr σ ⊗n P ). (B6) Now we split the integral on the right-hand side of Eq. (B6) into two parts, σ ∈ Ct (v) and σ ∈ Ft (v): If σ ∈ Ft (v), then by Eq. (B4) we have 2 Tr σ ⊗n P ≤ (c + 2)(5n)2d e−ζn . On the other hand, if σ ∈ Ct (v), then because of t < η we have 2 Tr σ ⊗n (Πηj )⊥ ≤ 2e−2(η−t) n , which follows from Hoeffding’s inequality [78]: Indeed, let Z be the i.i.d. ran- dom variables obtained by the measurement of Aj on the state σ. They take values in the interval [λmin (Aj ), λmax (Aj )], their expectation values satisfy EZj = Tr σAj ∈ [vj ± tΣ(Aj )], while   1  Tr σ ⊗n (Πηj )⊥ = Pr Z ∈ [vj ± ηΣ(Aj ) n    1 ≤ Pr Z ∈ [Tr σAj ± (η − t)Σ(Aj ) , n  so Hoeffding’s inequality applies. All taken together, we have 2  2 2  Tr ω(Πηj )⊥ ≤ (n + 1)3d (c + 2)(5n)2d e−ζn + 2e−2(η−t) n 2 2 ≤ (c + 3)(5n)5d e−2(η−t) n , because we can choose t such that t−s t−s η−t= √ ≥ . (B7) 2 2c 2d + 1 4cd Secondly, let ω be such that Tr ωΠηj ≥ 1 − δ  ; as we are interested in Tr ωP , we may again assume without loss of generality that ω is permutation symmetric, and invoke the constrained de Finetti reduction [80, Lemma 18], Eq. (B5). From that we get, much as before,  ⊥ 3d2 Tr ωP ≤ (n + 1) dσ (Tr σ ⊗n P ⊥ )F (ω, σ ⊗n )2 , and we split the integral on the right-hand side into two parts, depending on 2 σ ∈ Fs (v) or σ ∈ Cs (v): In the latter case, Tr σ ⊗n P ⊥ ≤ (c + 2)(5n)2d e−ζn , by Eq. (B3). In the former case, there exists a j such that Tr σAj = wj ∈ [vj ± sΣ(Aj )], and so     F (ω, σ ⊗n )2 ≤ F (1 − δ  , δ  ), (Tr σ ⊗n Πηj , 1 − Tr σ ⊗N Πηj )   2 √ η ≤  δ + Tr σ Πj ⊗n  ≤ 2δ  + 2 Tr σ ⊗n Πηj  2 ≤ 2δ  + 4e−2(s−η ) n , 1760 Z. B. Khanian et al. Ann. Henri Poincaré the last line again by Hoeffding’s inequality; indeed, with the previous nota- tion,   ⊗n η  1  Tr σ Πj = Pr Z ∈ [vj ± η Σ(Aj ) n    1  ≤ Pr Z ∈ [wj ± (s − η )Σ(Aj ) . n  All taken together, we get 2  2  2  Tr ωP ⊥ ≤ (n + 1)3d (c + 2)(5n)2d e−ζn + 4e−2(s−η ) n + 2δ  2 2  2 2 ≤ (n + 1)3d (c + 3)(5n)2d e−2(s−η ) n + 2(n + 1)3d δ  , because we can choose s such that t−s t−s s − η = √ ≥ . (B8) 2c 2d2 + 1 4cd From Eqs. (B7) and (B8) we get by summation    t−s 1 η−η =t−s+ √ ≤ (t − s) 1 + , c 2d2 + 1 cd from which we obtain η − η s − η = η − t ≥ , 4(cd + 1) concluding the proof.  Lemma 28. For all 0 < s < t there exists ζ > 0, such that for all n there exists a permutation symmetric projector P on H⊗n with the properties 2 ∀ρ ∈ Cs (v) Tr ρ⊗n P ⊥ ≤ (c + 2)(5n)2d e−ζn , (B9) ⊗n 2d2 −ζn ∀σ ∈ Ft (v) Tr σ P ≤ (c + 2)(5n) e , (B10) where Cs (v) and Ft (v) are defined in Eqs. (B1) and (B2), respectively. The 2 constant ζ may be chosen as ζ = 2c2(t−s) (2d2 +1) . Proof. We start by showing that there is a POVM (M, 1 − M ) with (t−s)2 ∀ρ ∈ Cs (v) Tr ρ⊗n (1 − M ) ≤ ce− 2c2 n , (B11) 2 − (t−s) ∀σ ∈ Ft (v) Tr σ ⊗n M ≤ e 2c2 n . (B12) Namely, for each  = 0, . . . , n choose j ∈ {1, . . . , c} uniformly at random and measure Aj on the -th system. Denote the outcome by the random variable Zj and let Zj = 0 for j = j . Thus, for all j, the random variables Zj are i.i.d. with mean EZj = 1c Tr ρAj , if the measured state is ρ⊗n . Outcome M corresponds to the event   1 j 1 s+t ∀j Z ∈ vj ± Σ(Aj ) ; n c 2  Vol. 24 (2023) Resource Theory of Heat and Work 1761 outcome 1 − M corresponds to the complementary event   1 j 1 s+t ∃j Z ∈ vj ± Σ(Aj ) . n c 2  We can use Hoeffding’s inequality to bound the traces in question. For ρ ∈ Cs (v), we have |EZj − vj | ≤ sc Σ(Aj ) for all j, and so:    ⊗n 1 j 1 s+t Tr ρ (1 − M ) = Pr ∃j Z ∈ vj ± Σ(Aj ) n c 2     c 1 j 1 s+t ≤ Pr Z ∈ vj ± Σ(Aj ) j=1 n c 2     c 1 j 1 s+t ≤ Pr Z ∈ vj ± Σ(Aj ) j=1 n c 2    c 1 j j t−s ≤ Pr Z − EZ1 > Σ(Aj ) j=1 n 2c  2 − (t−s) ≤ ce 2c2 n . For σ ∈ Ft (v), there exists a j such that |EZj − vj | > ct Σ(Aj ). Thus,   1  j 1 s + t Tr σ ⊗n M ≤ Pr Z ∈ vj ± Σ(Aj ) n c 2    1 j j t−s ≤ Pr Z − EZ1 > Σ(Aj ) n 2c  2 − (t−s) ≤e 2c2 n . This POVM is, by construction, permutation symmetric, but M is not a projector. To fix this, choose λ-nets NCλ in Cs (v) and NFλ in Ft (v), with 2 λ = e−ζn , with ζ = 2c2(t−s) (2d2 +1) . This means that every state ρ ∈ Cs (v) is no farther than λ in trace distance from a ρ ∈ NCλ , and likewise for Ft (v). By [41, Lemma III.6] (or rather, a minor variation of its proof), we can find such  2d2 nets with |NCλ |, |NFλ | ≤ 5n λ elements. Form the two states 1  ⊗n Γ := ρ , |NCλ | λ ρ∈NC 1  ⊗n Φ := σ , |NFλ | λ σ∈NF and let P := {Γ − Φ ≥ 0} 1762 Z. B. Khanian et al. Ann. Henri Poincaré be the Helstrom projector which optimally distinguishes Γ from Φ. But we know already a POVM that distinguishes the two states, hence (P, P ⊥ = 1−P ) cannot be worse: (t−s)2 Tr ΓP ⊥ + Tr ΦP ≤ Tr Γ(1 − M ) + Tr ΦM ≤ (c + 1)e− 2c2 n , thus for all ρ ∈ NCλ and σ ∈ NFλ ,  2d2 5n (t−s)2 ⊗n Tr ρ ⊥ P , Tr σ ⊗n P ≤ (c + 1) e− 2c2 n . λ So, by the λ-net property, we find for all ρ ∈ Cs (v) and σ ∈ Ft (v),  2d2 5n (t−s)2 2 Tr ρ⊗n P ⊥ , Tr σ ⊗n P ≤ λ + (c + 1) e− 2c2 n ≤ (c + 2)(5n)2d e−ζn , λ by our choice of λ.  Corollary 29. For charges Aj , values vj = Aj  and n > 0, Theorem 27 implies that there is an a.m.c. subspace M of H⊗n for any η  > 0, with the following parameters: η = 2η  , c+3 2 − nη 2 δ = (5n)2d e 8c2 (d+1)2 , 2 2 2 − 8c2nη δ = (c + 3)(5n)2d e (d+1)2 , 2 2 2 − 8c2nη = 2(c + 3)(n + 1)3d (5n)2d e (d+1)2 . 1 (n) Moreover, let ρn = ρ1 ⊗· · ·⊗ρn be a tensor product state with n Tr (ρn Aj ) − vj ≤ 12 η  Σ(Aj ) for all j. Then, ρn projects onto the a.m.c. subspace with proba- bility : Tr (ρn P ) ≥ 1 − .  Proof. For simplicity of notation we drop the subscript j from Aj , vj and Πηj , d so let =1 E | | be the spectral decomposition of A. Define independent random variables Xi for i = 1, . . . , n taking values in the set {E1 , . . . , Ed } with probabilities Pr{Xi = E } = pi (E ) = Tr ρi | |. Furthermore, define the random variable X = n1 (X1 + · · · + Xn ) which has the expectation value 1 E(X) = Tr ρn A(n) . n Therefore, we obtain   1 − Tr ρn Πη = 1 |ρ1 |1  · · · n |ρn |n  1 ,...,n : |E1 +···+En −nv|≥nη Σ(A) = Pr X − v ≥ η  Σ(A) = Pr X − E(X) ≥ η  Σ(A) + v − E(X) or X − E(X) ≤ −η  Σ(A) +v − E(X) Vol. 24 (2023) Resource Theory of Heat and Work 1763   2n(η  Σ(A) + v − E(X))2 ≤ exp − (Σ(A))2   2n(η  Σ(A) − v + E(X))2 + exp − (Σ(A))2   nη 2 ≤ 2 exp − ≤ δ, 2 where the second line follows because random the variables X1 , . . . , Xn are in- dependent, and as a result Pr {Xi = Ei ∀i = 1, . . . , n} = 1 |ρ1 |1  · · · n |ρn |n ; the fourth line is due to Hoeffding’s inequality, Lemma 24; the fifth line is due to assumption E(X) − v ≤ 12 η  Σ(A). Thus, by the definition of the a.m.c. subspace, Tr ρn P ≥ 1 − .  Appendix C: Proof of the AET Theorem 4 In this section, we first review the notion of the entropy-typical subspace de- fined in Definition 20, which we refer to it as the typical subspace for sim- plicity. Lemma 21 summarizes the properties of this subspace which we use in the proofs of this section. Intuitively, the typical subspace of the support of a tensor product state ρn = ρ1 ⊗ · · · ⊗ ρn with projector Πnα,ρn , for a positive constant α, is a high probability subspace for ρn of dimension approximately equal to 2S(ρ ) = 2 i=1 S(ρi ) . Moreover, eigenvalues of ρn inside this n n √ subspace belong to a tight interval around 2−S(ρ ) with the radius of 2−α n . We use n these properties to prove Lemma 30 of which we will use points 3 and 4 to prove the AET. In this lemma, we show that if ρn projects onto a subspace M with high probability, then we can find a state ρ inside this subspace which is close to the state ρn (in trace distance) and has useful properties. In particular, similar to the typicality properties, the eigenvalues of ρ belong to an interval around 2−S(ρ ) with a small radius. We use this to show that the state ρ can n be decomposed as the tensor product of a maximally mixed state of dimension almost equal to 2−S(ρ ) and another state with significantly smaller dimension. n Lemma 30. Let M ⊂ H⊗n with projector P be a high-probability subspace for the state ρn = ρ1 ⊗ · · · ⊗ ρn , i.e. Tr ρn P ≥ 1 − . Then, for α > 0 and all sufficiently large n, there exist a subspace M ⊆ M with projector P, and a state ρ with support in M, such that the following holds: √   1. Tr Πnα,ρn ρn Πnα,ρn P ≥ 1 − 2 − O α1 . √ √ 2. 2− i=1 S(ρi )−2α n P ≤ PΠn n ρn Πn n P ≤ 2− i=1 S(ρi )+α n P. n n α,ρ α,ρ 3. There is a unitary U such that U ρU√† = τ ⊗ ω, where τ is a maximally state of rank 2 i=1 S(ρi )−O(α n) , and ω is a state of dimension n mixed √ 2O(α n) .  √ √     ρ − ρn 1 ≤ 2 + O α1 + 2 2 + O α1 . 4.  1764 Z. B. Khanian et al. Ann. Henri Poincaré Proof. In point 1 and 2 of the lemma, we first construct the subspace M with projector P. To this end, we project the typical subspace of ρn with projec- tor Πnα,ρn onto the space M, i.e. P Πnα,ρn P , and define P as a projector onto √ the support of P Πnα,ρn P with corresponding eigenvalues bigger than 2−α n . Since ρn project onto M with high probability, therefore the unnormalized state Πnα,ρn ρn Πnα,ρn ≈ ρn projects onto M with high probability as well. We use this to show that they both project onto M with high probability. More- over, from Lemma 21, we know that the eigenvalues of the unnormalized state Πnα,ρn ρn Πnα,ρn are inside a tight interval around 2− i=1 S(ρi ) . We use this to n show that the new unnormalized state PΠnα,ρn ρn Πnα,ρn P has the same prop- erty. In point 3 of the lemma, we further trim the unnormalized state PΠnα,ρn ρn Πnα,ρn P to obtain a new state which has a degeneracy of the order of √ multiples of 2− i=1 S(ρi )−10α n . By trimming, we mean discarding some parts n of an unnormalized state in such a way that the trace of the new unnormalized state is almost the same. We use this property to decompose the state into the tensor product of a maximally mixed state and another state of smaller dimension. Lastly, in point 4, we show that the new state is close to state ρn . 1. Let E ≥ 0 and F ≥ 0 be two positive operators √ such that E + F = n P Πα,ρn P , where all eigenvalues of F are smaller than 2 −α n , and define P to be the projection onto the support of E. In other words, P is the projection √ onto the support of P Πnα,ρn P with corresponding eigenvalues greater 2−α n . Also, notice that all eigenvalues of E and F are smaller than 1. Thus, we obtain Tr (Πnα,ρn ρn Πnα,ρn P) ≥ Tr (Πnα,ρn ρn Πnα,ρn E) = Tr (Πnα,ρn ρn Πnα,ρn P Πnα,ρn P ) − Tr (Πnα,ρn ρn Πnα,ρn F ) √ ≥ Tr (Πnα,ρn ρn Πnα,ρn P Πnα,ρn P ) − 2−α n √ ≥ Tr (ρn P Πnα,ρn P ) − Πnα,ρn ρn Πnα,ρn − ρn 1 − 2−α n ≥ Tr (ρn Πnα,ρn ) − P ρn P − ρn 1 − Πnα,ρn ρ Πnα,ρn − ρn n 1 √ −α n −2 √ β √ β √ ≥1− 2 −2 −2 − 2−α n , α α where the first line follows from the definition of E which implies P ≥ E. The third line follows from Hölder’s inequality in the following form: Tr (Πnα,ρn ρn √ Πnα,ρn F ) ≤ Tr (Πnα,ρn ρn Πnα,ρn ) · F ∞ ≤ 2−α n . The fourth and fifth lines are due to Hölder’s inequality in the following form: for any two states ρ and σ and any operator 0 ≤ Λ ≤ 1, Tr (ρΛ) ≥ Tr (σΛ) − ρ − σ1 holds which is obtained by rearranging terms in the following Hölder’s inequality Tr ((ρ − σ)Λ) ≤ ρ − σ1 · Λ∞ ≤ ρ − σ1 . The last line follows from Lemmas 21 and 22. Vol. 24 (2023) Resource Theory of Heat and Work 1765 2. By the fact that in the typical subspace the eigenvalues of ρn are bounded (Lemma 21), we obtain √ PΠnα,ρn ρn Πnα,ρn P ≤ 2− S(ρi )+α n PΠnα,ρn P n i=1 √ ≤ 2− S(ρi )+α n P. n i=1 For the lower bound notice that √ PΠnα,ρn ρn Πnα,ρn P ≥ 2− S(ρi )−α n PΠnα,ρn P n i=1 √ = 2− S(ρi )−α n PP Πnα,ρn P P n i=1 √ ≥ 2− i=1 S(ρi )−2α n P, n where the equality holds because P ⊆ M, therefore PP = P. The last in- equality follows because P is the projection onto support of P Πnα,ρn P with √ eigenvalues greater 2−α n . 3. In this point, we construct ρ. Consider the unnormalized state PΠnα,ρn ρn Πα,ρn P with support inside M. From point 2, we know that all the eigenvalues n ! √ √ " of this state belongs to the interval 2− i=1 S(ρi )−2α n , 2− i=1 S(ρi )+α n := n n √ [pmin , pmax ]. We divide this interval into b = 2 5α n many intervals (bins) of equal length Δp = pmax −p b min . Now, we trim the eigenvalues of this unnormal- ized state in three steps as follows. (a) Each eigenvalue belongs to a bin which is an interval [pk , pk+1 ) for some 0 ≤ k ≤ b − 1 with pk = pmin + Δp × k. For example, eigenvalue λl is equal to pk + ql for some k such that 0 ≤ ql < Δp. We throw away the ql part of each eigenvalue λl . The sum of these parts over all eigenvalues is very small, |M|  √ ql ≤ Δp|M| ≤ 2−2α n+1 , l=1 where the dimension √of the subspace M is bounded as |M| ≤ 2 i=1 S(ρi )+2α n , which follows from point 2 of the lemma. n √ (b) We throw away the bins which contain less than 2 i=1 S(ρi )−10α n many n eigenvalues. The sum of all the eigenvalues that are thrown away is bounded by √ √ √ √ S(ρi )−10α n × 25α × 2− S(ρi )+α n ≤ 2−4α n n n n 2 i=1 i=1 , where the first number in the product is the number of eigenvalues in such a bin, the second is the number of bins, and the third is the maximum eigenvalue. (c) If the k-th bin is not thrown away in the previous step, it contains Mk many equal eigenvalues, where Mk is bounded as follows: √ √ n S(ρi )−10α n n S(ρi )+2α n 2 i=1 ≤ Mk ≤ 2 i=1 . (C1) 1766 Z. B. Khanian et al. Ann. Henri Poincaré Let √ n S(ρi )−10α n L=2 i=1 (C2) and for the kth bin, let mk be an integer number such that mk L ≤ Mk ≤ (mk + 1)L. (C3) Then, mk is bounded as follows √ mk ≤ 212α n . (C4) From the k-th bin, we keep mk L number of eigenvalues and throw away the rest, where there are Mk − mk L ≤ L many of them; the sum of the eigenvalues that are thrown away in this step is bounded by  b−1  b−1 √ pk (Mk − mk L) ≤ L pk ≤ 2−4α n . k=0 k=0 Hence, for sufficiently large n the sum of the eigenvalues thrown away in the last three steps is bounded by √ √ √ √ 2−2α n+1 + 2−4α n + 2−4α n ≤ 2−α n (C5) Therefore, there are only left b different eigenvalues where the k-th eigenvalue has degeneracy of mk L for k = 0, 1, . . . , b − 1. In other words, the eigenvalues of all bins that are not thrown away in these three steps, form an L-fold de- b−1 generate unnormalized state of dimension k=0 mk L because each eigenvalue has at least degeneracy of the order of L. Thus, up to a unitary U † , it can be factorized into the tensor product of a maximally mixed state τ and an unnor- malized state ω  of dimensions L and k=0 mk , respectively. From Eq. (C4), b−1  the dimension of ω is bounded by  b−1 √ √ √ mk ≤ 212α n × 25α n = 217α n . k=0 ω Then, let ω = Tr (ω  ) and define ρ := U τ ⊗ ωU † . 4. In point 3 of the lemma, we trimmed PΠnα,ρn ρn Πnα,ρn P to obtain the state ρ, i.e. ρ ≈ PΠnα,ρn ρn Πnα,ρn P. Moreover, from point 1 of the lemma we know that the unnormalized state Πnα,ρn ρn Πnα,ρn projects onto P with high probability. Therefore, Lemma 22 and Lemma 21 imply that the new unnor- malized state PΠnα,ρn ρn Πnα,ρn P ≈ Πnα,ρn ρn Πnα,ρn ≈ ρn . Hence, we obtain that ρ ≈ ρn . In the following, we prove this in detail. From points 3 and 1, we obtain Tr (ω  ) = Tr (τ ⊗ ω  ) (C6)   √ ≥ Tr PΠnα,ρn ρn Πnα,ρn P − 2−α n (C7) √ √ β β √ ≥1−2 −2 − 2 − 2−α n+1 . (C8) α α Vol. 24 (2023) Resource Theory of Heat and Work 1767 Thereby, we get the following ρ − ρn 1 ≤ ρ − U τ ⊗ ω  U †  1 + U τ ⊗ ω  U † − PΠnα,ρn ρn Πnα,ρn P 1 + PΠnα,ρn ρn Πnα,ρn P − ρn 1 ≤ 1 − Tr (ω ) + U τ ⊗ ω U − PΠnα,ρn ρn Πnα,ρn P   † 1 + PΠnα,ρn ρn Πnα,ρn P − ρn 1 √  ≤ 1 − Tr (ω ) + 2 −α n + PΠnα,ρn ρn Πnα,ρn P − ρn 1 # √ √ √ β β √ ≤ 1 − Tr (ω  ) + 2−α n + 2 2 + 2 + 2 + 2−α n √ #α α √ √ β β √ √ β β √ =2 +2 + 2 + 2−α n+1 + 2 2 + 2 + 2 + 2−α n , α α α α where the first line is due to triangle inequality. The second, third and fourth lines are due to Eqs. (C6) and (C5), and Lemma 22, respectively.  Proof of Theorem 4. We first sketch the proof in this paragraph and later pro- vide rigorous steps of the proof. The approximate microcanonical (a.m.c.) sub- space for charges Aj and average values vj which is basically a common sub- (n) space for the spectral projectors of Aj with corresponding values close to nvj ; that is, a subspace onto which a state projects with high probability if and only if it projects onto the spectral projectors of the charges with high probability. We show in Theorem 27 that for a large enough n such a subspace exits. An interesting property of an a.m.c. subspace is that any unitary acting (n) on this subspace is an almost commuting unitary with charges Aj . (n) (n) In Corollary 29, we show that assuming n1 Tr(ρn Aj ) ≈ n1 Tr(ρn Aj ) ≈ vj the states ρn and σ n project onto the a.m.c. subspace with high probability. Hence, in Lemma 30, we show that one can find states ρ and σ  with support inside the a.m.c. subspace which are very close to the original states in trace norm, that is, ρ ≈ ρn and σ  ≈ σ n , and there are unitaries V1 and V2 that factorizes these states to the tensor product of maximally mixed states τ and τ  and some other state of very small dimension: V1 ρV1† = τ ⊗ ω and V2† = τ  ⊗ ω  . V2 σ Further, assuming that the states ρn and σ n have very close entropy rates, i.e. 1 1  n S(ρ ) ≈ n S(σ ), one can find states τ and τ with the same dimension that n n is τ = τ . Thus, we observe that two states ρ ⊗ ω  and σ   ⊗ ω have exactly the same spectrum, so there is unitary acting on the a.m.c. subspace and the ancillary system taking one state to another. Based on the properties of the a.m.c. subspace, we show that this unitary is an almost commuting unitary (n) with the charges Aj . We first prove the if part. If there is an almost-commuting unitary U and an ancillary system with the desired properties stated in the theorem, then we 1768 Z. B. Khanian et al. Ann. Henri Poincaré obtain 1 1 |S(ρn ) − S(σ n )| = |S(ρn ⊗ ω  ) − S(σ n ⊗ ω) − S(ω  ) + S(ω)| n n 1 1 ≤ |S(ρn ⊗ ω  ) − S(σ n ⊗ ω)| + |S(ω  ) − S(ω)| n n 1 2 ≤ |S(ρn ⊗ ω  ) − S(σ n ⊗ ω)| + log 2o(n) n n 1 = S(U (ρn ⊗ ω  )U † ) − S(σ n ⊗ ω) + o(1) n 1 1 ≤ o(1) log(dn × 2o(n) ) + h (o(1)) + o(1) = o(1), n n where the first line follows from the additivity of the von Neumann on tensor product states and adding and subtracting S(ω) and S(ω  ). The second line is due to the triangle inequality. The third line is due to the fact that von Neu- mann entropy of a state is upper bounded by the logarithm of the dimension (assuming that the dimension of the ancillary system is bounded by 2o(n) ). The fourth line follows because unitaries do not change the entropy. The last line follows because the trace distance between the two states U (ρn ⊗ ω  )U † and σ n ⊗ ω converges to zero, therefore we can apply the continuity of von Neumann entropy [39,40] where h(x) = −x log x − (1 − x) log(1 − x) is the binary entropy function. Moreover, we obtain 1     (n) (n) Tr ρn Aj − Tr σ n Aj n 1     (n) (n) = Tr ρn ⊗ ω  (Aj + Aj ) − Tr σ n ⊗ ω(Aj + Aj ) n 1     (n) (n) ≤ Tr ρn ⊗ ω  (Aj + Aj ) − Tr U ρn ⊗ ω  U † (Aj + Aj ) n 1     (n) (n) + Tr U ρn ⊗ ω  U † (Aj + Aj ) − Tr σ n ⊗ ω(Aj + Aj ) n 1    (n) (n) = Tr ρn ⊗ ω  Aj + Aj − U † (Aj + Aj )U (C9) n 1    (n)  + Tr U ρn ⊗ ω  U † − σ n ⊗ ω (Aj + Aj ) n 1 (n) (n) ≤ Tr (ρn ⊗ ω  ) U (Aj + Aj )U † − (Aj + Aj ) (C10) n ∞ 1 (n) + U ρn ⊗ ω  U † − σ n ⊗ ω 1 Aj + Aj n ∞ = o(1), (C11) the second line follows because Aj = 0 for all j. The third and fifth lines are due to triangle inequality and Hölder’s inequality, respectively. Now we turn to the proof of the only if part. That is, assuming that ρn and σ n are asymptotically equivalent, we construct the ancillary system and the almost commuting unitaries. We apply Theorem 27 to construct a non- trivial a.m.c. subspace for ρn . Since σ n has average entropy and charges values very close to those of ρn , both ρn and σ n project to this a.m.c. subspace with Vol. 24 (2023) Resource Theory of Heat and Work 1769 high probability. Then we apply Lemma 30 to find states ρ ≈ ρn and σ  ≈ σn where these states (up to unitaries) are decomposed as the tensor product of a maximally mixed state τ of very large dimension and another state of very small dimension, i.e. V1 ρV1† = τ ⊗ ω and V2 σ V2† = τ ⊗ ω  . Now, we consider   the states τ ⊗ ω ⊗ω and τ ⊗ ω ⊗ω which have exactly the same eigenvalues,       ≈ρn ≈σ n hence the states ρn ⊗ ω  and σ n ⊗ ω have very similar eigenvalues. Therefore, ρn ⊗ω  and σ n ⊗ω are approximately equal up to a unitary. In the end, we show that such a unitary, with support inside a.m.c subspace, almost commutes with all charges of the total system. Namely, assume that for the sates ρn and σ n the following holds: 1 |S(ρn ) − S(σ n )| ≤ γn n 1 (n) (n) Tr (Aj ρn ) − Tr (Aj σ n ) ≤ γn j = 1, . . . , c, n for vanishing γn and γn as n goes to ∞. According to Theorem 27, for charges (n) Aj , values vj = n1 Tr (ρn Aj ), η  > 0 and any n > 0, there is an a.m.c. subspace M of H⊗n with projector P and the following parameters: η = 2η  , c+3 2 − nη 2 δ = (5n)2d e 8c2 (d+1)2 , 2 2 2 − 8c2nη δ = (c + 3)(5n)2d e (d+1)2 , 2 2 2 − 8c2nη = 2(c + 3)(n + 1)3d (5n)2d e (d+1)2 .   Choose η as the following such that δ, δ and vanish for large n: ⎧ √ ⎪ 8c(d+1) ⎨ 14 if γn ≤ 11  n Σ(A)min n4 η = √ ⎪  ⎩ 8c(d+1)γn if γ  > 11 Σ(A)min n n4 where Σ(A)min is the minimum spectral diameter among all spectral diam- 1 n (n) 1 n (n) eters of charges Σ(Aj ). Since n Tr (ρ Aj ) = vj and n Tr (σ Aj ) − vj ≤ 1  n n 2 η Σ(Aj ), Corollary 29 implies that states ρ and σ project onto this a.m.c. subspace with probability : Tr (ρn P ) ≥ 1 − , Tr (σ n P ) ≥ 1 − . Moreover, consider the typical projectors Πnα,ρn and Πnα,σn of states ρn and σ n , 1 respectively, with α = n 3 . Then point 3 and 4 of Lemma 30 implies that there are states ρ and σ  with support inside the a.m.c. subspace M and unitaries V1 and V2 such that  ρ − ρn 1 ≤ o(1),  σ − σ n 1 ≤ o(1), 1770 Z. B. Khanian et al. Ann. Henri Poincaré V1 ρV1† = τ ⊗ ω, V2† = τ  ⊗ ω  , V2 σ (C12) where τ and τ  are maximally mixed states; since |S(ρn ) − S(σ n )| ≤ nγn , one may choose the dimension of them in Eq. (C2) to be exactly the same √ as L = 2 i=1 S(ρi )−10z with z = max{α n, nγn }, hence, we obtain τ = τ  . n Then, ω and ω  are states with support inside Hilbert space K of dimension 2o(z) = 2o(n) . Then, it is immediate to see that the states ρ ⊗ ω  and σ  ⊗ ω on Hilbert space Mt = M ⊗ K have exactly the same spectrum; thus, there is a unitary U on subspace Mt such that  ρ ⊗ ω  U U † = σ  ⊗ ω. (C13)  to U = U We extend the unitary U  ⊕ 1M⊥ acting on H⊗n ⊗ K and obtain t U ρn ⊗ ω  U † − σ n ⊗ ω 1 ≤ U ρn ⊗ ω  U † − U ρ ⊗ ω  U † 1 + σ n ⊗ ω − σ  ⊗ ω1 + U ρ ⊗ ω  U † − σ ⊗ω 1 = U ρn ⊗ ω  U † − U ρ ⊗ ω  U † 1 + σ n ⊗ ω − σ  ⊗ ω1 ≤ o(1), where the second and last lines are due to Eqs. (C13) and (C12), respectively. As mentioned before, Mt = M ⊗ K is a subspace of H⊗n ⊗ K with pro- jector Pt = P ⊗ 1K where P is the corresponding projector of a.m.c. subspace. (n) We define total charges Atj = Aj + Aj and let Aj = 0 for all j and show that every unitary of the form U = UMt ⊕ 1M⊥ t asymptotically commutes with all total charges: U Atj U † − Atj ∞ = (Pt + Pt⊥ )(U Atj U † − Atj )(Pt + Pt⊥ )∞ ≤ Pt (U Atj U † ⊥ t † − Aj )Pt ∞ + Pt (U Aj U − Atj )Pt ∞ t + Pt (U Atj U † − Atj )Pt⊥ ∞ + Pt⊥ (U Atj U † − Atj )Pt⊥ ∞ = Pt (U Atj U † − Atj )Pt ∞ + 2 Pt⊥ (U Atj U † − Atj )Pt ∞ ≤ 3 (U Atj U † − Atj )Pt ∞ = 3 (U Atj U † − nvj 1 + nvj 1 − Atj )Pt ∞ ≤ 3 (U Atj U † − nvj 1)Pt ∞ + 3 (Atj − nvj 1)Pt ∞ = 6 (Atj − nvj 1)Pt ∞ = 6 max (Atj − nvj 1)|v 2 |v∈M t = 6 max (Atj − nvj 1)(Πηj ⊗ 1K + 1 − Πηj ⊗ 1K )|v 2 |v∈Mt ≤ 6 max (Atj − nvj 1)Πηj ⊗ 1|v 2 |v∈Mt + 6 max (Atj − nvj 1)(1 − Πηj ⊗ 1)|v 2 |v∈Mt Vol. 24 (2023) Resource Theory of Heat and Work 1771 ≤ 6nΣ(Aj )η + 6 max (Atj − nvj 1)(1 − Πηj ⊗ 1)|v 2 , |v∈Mt where the first line is due to the fact that Pt + Pt⊥ = 1H⊗n ⊗ 1K . The fourth line follows because U Atj U † − Atj is a Hermitian operator with zero eigenvalues in the subspace Pt⊥ . The fifth line is due to Lemma 23. The twelfth line is due to the definition of the a.m.c. subspace. Now, bound the second term in the above: 6 max (Atj − nvj 1)(1 − Πηj ⊗ 1)|v 2 |v∈Mt ≤ 6 max Atj − nvj 1 (1 − Πηj ⊗ 1)|v 2 ∞ |v∈Mt  = 6 Atj − nvj 1 ∞ max Tr ((1 − Πηj ⊗ 1)|v v|) |v∈Mt  = 6n Aj − vj 1∞ max Tr ((1 − Πηj )v) v∈M √ ≤ 6n Aj − vj 1∞ δ, where the first line is due to Lemma 23. The last line is by definition of the a.m.c. subspace. Thus, for vanishing δ and η we obtain 1 U Atj U † − Atj ∞ ≤ o(1), n concluding the proof.  References [1] Gemmer, J., Michel, M., Mahler, G.: Quantum Thermodynamics, Lecture Notes in Physics, Vol. 784. Springer, Berlin (2009). https://doi.org/10.1007/ 978-3-540-70510-9 [2] Binder, F., Correa, L. A., Gogolin, C., Anders, J., Adesso, G. (eds.) Thermo- dynamics in the quantum regime: fundamental aspects and new directions. 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Theory 62, 5260–5277 (2016). https://doi.org/10.1109/TIT.2016.2562580 Zahra Baghali Khanian and Andreas Winter Grup d’Informació Quàntica, Departament de Fı́sica Universitat Autònoma de Barcelona 08193 Bellaterra Barcelona Spain Zahra Baghali Khanian, Manabendra Nath Bera, Arnau Riera and Maciej Lewenstein ICFO–Institut de Ciències Fotòniques Barcelona Institute of Science and Technology 08860 Castelldefels Barcelona Spain Manabendra Nath Bera Department of Physical Sciences Indian Institute of Science Education and Research (IISER) Mohali Punjab 140306 India Arnau Riera Institut el Sui Carrer Sant Ramon de Penyafort, s/n 08440 Cardedeu Barcelona Spain Maciej Lewenstein and Andreas Winter ICREA–Institució Catalana de Recerca i Estudis Avançats 08010 Barcelona Spain e-mail: andreas.winter@uab.cat Zahra Baghali Khanian Munich Center for Quantum Science and Technology & Zentrum Mathematik Technische Universität München Boltzmannstrasse 3 85748 Garching bei München Germany Communicated by Matthias Christandl. Received: July 11, 2021. Accepted: November 15, 2022.