Axiomatic characterization of the quantum relative entropy and free energy

Building upon work by Matsumoto, we show that the quantum relative entropy with full-rank second argument is determined by four simple axioms: i) Continuity in the first argument, ii) the validity of the data-processing inequality, iii) additivity under tensor products, and iv) super-additivity. This observation has immediate implications for quantum thermodynamics, which we discuss. Specifically, we demonstrate that, under reasonable restrictions, the free energy is singled out as a measure of athermality. In particular, we consider an extended class of Gibbs-preserving maps as free operations in a resource-theoretic framework, in which a catalyst is allowed to build up correlations with the system at hand. The free energy is the only extensive and continuous function that is monotonic under such free operations.


I. INTRODUCTION
The quantum relative entropy captures the statistical distinguishability of two quantum states. For two states ρ and σ supported on the same Hilbert space it is defined as whenever supp(ρ) ⊆ supp(σ) and set to infinity otherwise. This quantity has a clear interpretation in the statistical discrimination of ρ from σ, appearing as an error rate in quantum hypothesis testing [1][2][3], a result commonly known as Stein's Lemma. It is hence no surprise that this quantity appears in a plethora of places in contemporary quantum physics. This is particularly true in the context of quantum information theory [4]. In the relative entropy of entanglement it quantifies the entanglement content of a general quantum state [5]. More generally, it appears in conversion rates in so-called resource theories [6,7]. Relatedly, it takes center stage in the problem of (approximately) recovering quantum information [8]. But the applications are not confined to quantum information theory. In many-body physics, it provides bounds on the clustering of correlations in space in terms of the mutual information [9,10]. In quantum thermodynamics [11], which is the context that is in the focus of attention in this note, its interpretation as the non-equilibrium free energy gives an upper bound to how much work can be extracted from a non-equilibrium system and is important in answering how to operationally define work in the quantum regime in the first place [12]. Not the least, it has appeared in the context of the AdS/cft correspondence [13], again drawing from and building upon the above mentioned applications.
In this note, we restrict to the case where the second argument σ has full rank and only consider finite dimensional Hilbert spaces. Essentially by re-interpreting and building upon a theorem by Matsumoto [14], we will show that the quantum relative entropy (1) is (up to a constant factor) the only function featuring the following four properties: 1. Continuity: For fixed σ, the map ρ → S(ρ||σ) is continuous [15].
2. Data-processing inequality: For any quantum channel T we have, 3. Additivity: 4. Super-additivity: For any bipartite state ρ 1,2 with marginals ρ 1 , ρ 2 we have No subset of these properties characterizes the relative entropy uniquely, but the Properties 1 to 3 are, for example, also fulfilled by the Renyi-divergences [16]. The uniqueness of the quantum relative entropy under Properties 1-4 has significant implications for quantum thermodynamics (QT), which we elaborate upon. The formalism of QT has recently been recast in within the framework of a resource theory [6,17,18], so one in which quantum states that are different from Gibbs states (at a fixed environment temperature) are considered resources. We will refer here to this kind of resource as athermality. Within this resource theory one is, among other problems, interested in finding bona fide measures of athermality. These are functions that quantify the amount of athermality of a given system. A requirement for a function reasonably quantifying the degree of athermality is that the it does not increase under the free operations of the resource theory. The problem of identifying such functions has been studied intensively in the last years for different classes of free operations, providing families of valid measures that are regarded as generalizations of the free energy [7,[19][20][21]. They share the property that they are all based of generalizations of the quantum relative entropy (1).
In this work, we will use the uniqueness result on the quantum relative entropy to show that the usual non-equilibrium free energy emerges as the unique continuous and extensive measure of athermality under a certain meaningful choice of free operations. In this sense, we also provide a fresh link of resource-theoretic considerations in quantum thermodynamics to more traditional descriptions of thermodynamic processes in the quantum regime.

II. AXIOMATIC DERIVATION OF QUANTUM RELATIVE ENTROPY
We start by formally stating the main technical result.
for some constant C > 0.
The proof relies on a characterization of the relative entropy in terms of different properties laid out in Ref. [14]. To state it, we first require a definition: Let (ρ, σ) be a pair of states on a finite-dimensional Hilbert space H and {ρ ′ n } be a sequence of states on the Hilbert spaces H ⊗n . We define a function f on pairs of quantum states to be lower asymptotically semicontinuous (l.a.s.) with respect to σ if Then Matsumoto's theorem [14] for the relative entropy can stated in the following way.
Theorem 2 (Matsumoto) Let f fulfill the data-processing inequality, additivity and be lower asymptotically semicontinuous with respect to all σ. Then f ∝ S.
The proof of Theorem 1 follows from the subsequent Lemma, which in turn implies that the Properties 1-4 give rise to the conditions of Theorem 2.
Lemma 3 (Lower asymptotically semi-continuity) Let f be a function on pairs of quantum states with the following properties, • The map ρ → f (ρ, σ) is continuous for any fixed σ.
• Additivity: Then f is lower asymptotically semi-continuous with respect to any σ.
Since the trace norm fulfills the dataprocessing inequality, we know that ||ρ ′ n,i − ρ|| 1 → 0, where ρ ′ n,i denotes the marginal of ρ ′ n on the i-th tensor-factor. Hence, the marginals converge to ρ. From the properties of f , we furthermore see that where the limit follows from continuity and the second line from additivity and super-additivity.

III. UNIQUENESS OF THE FREE ENERGY
The results of the previous section, in particular Theorem 1, can be applied to the resource theory of β-athermality. We formulate it as a resource theory of pairs of a quantum state and a Hamiltonian (ρ, H) that we call object. An object (ρ, H) is said to have the resource of β-athermality if it fulfills where ω β,H is the Gibbs state for the Hamiltonian H and inverse temperature β > 0, given by .
In this way, the resource theory of β-atermality is concerned with the absence of thermal equilibrium at temperature 1/β [6]. Concerning the set of free operations, we will be considering here the most general set of operations that do not create resourceful states from states featuring fewer resources.
In order to progress, let us first define the so-called Gibbspreserving maps (GP) which are quantum channels that have Gibbs state (13) as a fixed point. More formally, a GP-channel is defined as a trace-preserving completely positive map G β with the property that Note that formulated as above, GP channels only induce transitions that change the quantum state but not the Hamiltonian. This can be extended by simply considering functions G that act on the object, possibly changing also the Hamiltonian, but which at the same time do not create β-athermality. In this way, we define a GP-map as a function (ρ, This condition can equivalently be cast into the following form: One may define the set of GP channels as G H β (ω β,H ) = ω β,K(H) for all H, and the map between Hamiltonians as G(H) = K so that With this notation, condition (15) is simply given by GP-maps G are not only a natural extension of GP-channels G for the case where Hamiltonians are modified, but one can also see that any GP-map can be implemented if one is given access to a GP-channels and an ancillary system in a Gibbs state. This is formalized by the following Lemma taken from Ref. [12].
Lemma 4 (Implementation of GP maps [12]) Any map G β fulfilling (15) acting on a system S can be implemented by adding an ancillary system A in the Gibbs state (ω β,K , K) and applying a GP channel G to the entire compound. More formally, we find that Once we have established the set of GP maps for the objects, we will now introduce the notion of catalyst in this framework. This is done analogously to the case of catalysts for other sets of operations such as thermal operations [7,18,22,23]. In the following, we will also frequently drop the β-subscript from GP-maps for simplicity of notation.

A. Catalysts and correlations
We will now turn to defining the transitions between objects that can be performed with GP maps and the use of what is called a "catalyst" in this context. This is simply an ancillary quantum system that is left in the same state (in a sense that will be made precise later) after the transition is performed, rendering the metaphor of an actual catalyst quite appropriate.

Definition 5 (Catalytic free transition) We say that the transition
is a catalytic free transition if there exist a GP map G and a system A described by the object (γ A , R A ) such that We will in this case simply denote it by Here, we are employing the convenient notation to describe tensor products of objects. In the remainder of this work, we will simply write H S ⊗½ A +½ S ⊗R A := H S +R A . Importantly, we are assuming that the catalyst A is left in the same state and Hamiltonian and also uncorrelated from S. The role of the correlations of the catalysts and its role in quantum thermodynamics has been first noted in Ref. [22]. There, one considers a catalysts consisting of k subsystems and one merely demands that the marginal state of each subsystem is left untouched. We define it here formally for the case of GP maps.
Definition 6 (Marginal-catalytic free transition [22]) We say that the transition is a marginal-catalytic free transition if there exist a GP map G and systems A 1 , . . . , A k described by the object where tr |Ai (γ A ) = tr |Ai (γ A ) for all i ∈ (1, . . . , k). We will in this case simply denote it by Note that in this case the system A does not remain unchanged, but only its local marginals. In this sense, it is not truly a catalyst, but a catalyst on its reduced states. It is natural to expect that this indeed allows for a larger set of transitions, since the system A is "used up" by employing the initial lack of correlations as a resource.
We will now consider a family of transitions that also introduces correlations, but for which the catalyst is, unlike in Definition 6, left entirely untouched. In this case, correlations are built up between the system and the catalyst. In this way, the catalyst is re-usable as long as it is employed in order to implement a transition on a new system. We call this transitions, originally introduced in Ref. [12], correlated-catalytic free transitions: Definition 7 (Correlated-catalytic free transition) We say that the transition is a correlated-catalytic free transition if there exist a GP map G and a system A described by the object (γ A , R A ) such that where tr A (η) = σ S and tr S (η) = γ A . We will in this case simply denote it by We will now show that the non-equilibrium free energy is the only function, under reasonable assumptions, that does not increase under operations of the form of Definitions 6 and 7.

B. Free energy as a unique measure of non-equilibrium
We will call a measure of non-equilibrium a function that quantifies how far a given object (ρ, H) is from its equilibrium object (ω β,H , H). The minimal requirement on such a measure is that it is non-increasing under free transitions. The larger the set of free transitions, the more restricted is the allowed set of measures. One of the most well-studied measures of non-equilibrium is based on the quantum relative entropy. It is related to the free energy as III Monotonicity: (c) Marginal-catalytic monotonicity: (d) Correlated-catalytic monotonicity: All those properties apply for all states and Hamiltonians involved. The fact that ∆F β fulfills I and II follows from the continuity and additivity properties of the quantum relative entropy. The other properties can be related to the data processing inequality and super-additivity as we will see in Theorem 9. Before that, let us note that for any function M β on objects, we can define a function M β on pairs of quantum states as M β (ρ, ω β,H ) = M β (ρ, H). At the same time, it is true that any full-rank state σ can be thought of as the Gibbsstate of the modular Hamiltonian for any C ∈ R. With this notation, all objects of the form (σ, H σ ) are Gibbs-objects. Importantly, the modular Hamiltonian H σ is only defined up to an additive constant. It turns out however, that the properties II and III imply that M β (ρ, H) = M β (ρ, H + C½) for any C ∈ R (see Appendix A for a proof). Any additive measure of athermality is hence automatically gauge-invariant in this sense. Thus, the functions M β and M β are in a one-to-one correspondence. With this equivalence, we say that a measure M β is super-additive if, for any bipartite quantum states, it fulfills M β (ρ 1,2 , σ 1 ⊗ σ 2 ) ≥ M β (tr 2 (ρ) ⊗ tr 1 (ρ), σ 1 ⊗ σ 2 ) (31) and additive if it fulfills Also, M β is said to fulfill the data processing inequality if for all ρ, all σ being full-rank and for all quantum channels T . At this point, a note of caution is appropriate. We have previously defined the function M β only in the specific case where the second argument has full rank. There clearly are quantum-channels T that reduce the rank of full-rank states, in which case M β (T (ρ), T (σ)) may at first seem undefined. This is not a problem, however. To see this, we make use of the following fact about quantum channels: PROOF The proof is given in the appendix.
By the previous lemma, we see that any quantum channel that maps a full-rank state σ into a state T (σ) without full rank simply maps all states to the smaller Hilbert space P = supp(T (σ)) and should be considered as a map from states on H to states on P instead. Since the function M β is defined on all finite-dimensional Hilbert spaces, we can simply assume that it acts on B(P )×B(P ) in this case. In yet other words, the function M β (ρ, σ) is always defined if supp(ρ) ⊆ supp(σ) by restricting it to supp(σ), We can then show the following equivalence between properties of M β and M β .
Theorem 9 (Equivalence of M β and M β ) There exist the following two equivalences between the properties of the measures of athermality M β (ρ, H) and the corresponding function M β (ρ, σ).
The proof of this theorem, together with a more detailed set of the implications between the properties of M β and the corresponding function M β , is provided in the appendix. The previous theorem simply tells us that any additive measure of athermality M β that does not increase under marginalcatalytic operations (Definition 6) or also under correlatedcatalytic operations (Definition 7) is in one to one correspondence with a function M β that is additive, super-additive and fulfills the data processing inequality. This has as a first consequence that the measure ∆F β fulfills indeed Properties I-III. More importantly, using our re-formulation of Matsumoto's result of Theorem 1, we can show that ∆F β is, up to a constant factor, the only measure of athermality that fulfills I-III. This is the content of our main result, which follows from Theorem 9.
Result 10 (Uniqueness of monotones) Any monotone for marginal-catalytic transitions or correlated-catalytic transitions at environment temperature β that is additive and depends continuously on the density matrix is proportional to ∆F β .
The implications of this result are that the free energy difference ∆F β is the only bona fide quantifier of athermality under the most general set of free operations that do not create the resource.

IV. DISCUSSION AND OUTLOOK
In this work, we have investigated the question which properties uniquely determine the quantum relative entropy among all function on pairs of quantum states. Our re-formulation of Matsumoto's result highlights the role of super-additivity as a key property in the axiomatic derivation of the quantum relative entropy. The role of super-additivity in the arena of quantum thermodynamics has been shown to be related to the build up of correlations between the system at hand and a catalyst, which in turn represents the components of the machine that come back to their initial state after the cyclic process.
We have shown how the relative entropy and nonequilibrium free energy uniquely emerge from considerations about how to treat catalysts and their correlations in the resource theoretic approach to quantum thermodynamics. Usually, notions of relative entropy are employed to capture asymptotic weakly correlated settings (thermodynamic limit), thus when acting on many uncorrelated copies of a system (see Ref. [24] for a recent discussion of asymptotic thermodynamics from the point of view of resource theories). Importantly, and in contrast, in our approach they emerge without having to invoke any thermodynamic limit, but rather follow from properties of monotones in the single-shot setting. However, they are precisely singled out by the fact that we disregard correlations in the setting of marginal-catalytic and correlated-catalytic free transitions. It thus seems that the crucial feature of for the emergence of free energy is thus the disregarding of correlations. Note that this fits well to how these quantities appear in notions of macroscopic thermodynamics: Macroscopic equilibrium thermodynamics usually emerges in large systems which are well within thermodynamic phases. In such phases, correlations decay exponentially in space. Hence, the correlations of an object with its surrounding scale like its surface-area and not like their volume. Macroscopic objects are then essentially uncorrelated with other objects due to their small surface-to-volume ratio.
In this work, we have distinguished two ways of creating correlations with the catalyst, the marginal-catalytic one of Definition 6 and the correlated-catalytic one of Definition 7. The first represents the situation where the components of the machine become correlated among themselves, while the second represents the case where the machine builds correlation with the system upon which the machine induces a transition. Although these two sets both give rise to the free energy difference as a unique measure of athermality, we consider the latter as a much more adequate set of operations to incorporate correlations in thermodynamics. The reason for this is that the correlations build up between the catalyst and the system do not prevent one from re-using the catalyst to implement again a transition of the same kind on another system. It is an additional contribution of this work to flesh out this difference.
We end the discussion by posing an interesting open question. This is to investigate how to characterize all the possible thermodynamic transitions that can be implemented with correlated-catalysts. In Ref. [12] we have seen that indeed the operations of Definition 7 are more powerful than the ones of 5. At the same time, it has been recently been shown in Ref. [25] that a variant of Definition 7 allows to extract work from passive states. The question remains whether all the transitions that do not increase the free energy difference ∆F β are possible, as they indeed are for the ones of Definition 6, as shown in Ref. [22]. If this is indeed true also for correlatedcatalysts, one would have found an interpretation of the free energy as a unique criterium for the second law of thermodynamics. If it is not true, then it is necessary to consider genuinely new monotones, which are not additive or not continuous. Both options would be interesting from the perspective of the further development of quantum thermodynamics.