Axiomatic characterization of the quantum relative entropy and free energy
Henrik Wilming Rodrigo Gallego Jens Eisert







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.



Axiomatic characterization of the quantum relative entropy and free energy
    Henrik Wilming, Rodrigo Gallego, Jens Eisert
    December 30, 2023
==========================================================================



§ 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

    S(ρ || σ) = (ρlogρ - ρlogσ),

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 <cit.>,
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 <cit.>.
In the  relative entropy of entanglement it quantifies the entanglement content of a general quantum state <cit.>.
More generally, it appears in conversion rates in so-called resource theories <cit.>.
Relatedly, it takes center stage in the
 problem of (approximately) recovering quantum information <cit.>. 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 <cit.>. In quantum thermodynamics <cit.>, 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 <cit.>. Not the least,
 it has appeared in the context of the AdS/cft correspondence <cit.>, 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 <cit.>, we will show that the quantum relative entropy (<ref>) is (up to a constant factor) the only function featuring the following four properties:


  *  Continuity: For fixed σ, the map ρ↦ S(ρ || σ) is
continuous <cit.>.

  *  Data-processing inequality: For any quantum channel T we have,

    S(T(ρ)|| T(σ)) ≤ S(ρ || σ).


  *  Additivity:

    S(ρ_1⊗ρ_2 || σ_1⊗σ_2) = S(ρ_1||σ_1) + S(ρ_2||σ_2).


  *  Super-additivity: For any bipartite state ρ_1,2 with marginals ρ_1,ρ_2 we have

    S(ρ_1,2 || σ_1⊗σ_2) ≥ S(ρ_1||σ_1) + S(ρ_2||σ_2).


No subset of these properties characterizes the relative entropy uniquely, but the Properties <ref> to <ref> are, for example, also fulfilled by the Renyi-divergences <cit.>.

The uniqueness of the quantum relative entropy under Properties <ref>-<ref> 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 <cit.>, 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 <cit.>. They share the property that they are
all based of generalizations of the quantum relative entropy (<ref>).

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.




§ AXIOMATIC DERIVATION OF QUANTUM RELATIVE ENTROPY


We start by formally stating the main technical result.


Let f be a function on pairs of quantum states acting on the same finite dimensional Hilbert space, with the second argument having full rank. Suppose f fulfills Properties <ref>-<ref>. Then it is given by

    f(ρ,σ)= C (ρlogρ - ρlogσ) := C S(ρσ),

for some constant C>0.


The proof relies on a characterization of the relative entropy in terms of different properties laid out in Ref.<cit.>. 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 semi-continuous (l.a.s.) with respect to
σ if

    lim_n→∞ρ^⊗ n-ρ'_n_1 =0

implies

    lim inf_n→∞1/n(f(ρ'_n,σ^⊗ n)-f(ρ^⊗ n,σ^⊗ n))≥ 0.

Then Matsumoto's theorem <cit.> for the relative entropy can stated in the following way.

Let f fulfill the data-processing inequality, additivity and be lower asymptotically semi-continuous with respect to all σ.
Then f∝ S.

The proof of Theorem <ref> follows from the subsequent Lemma, which in turn implies that the Properties <ref>-<ref> give rise to the conditions of Theorem <ref>.

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: f(ρ_1⊗ρ_2,σ_1⊗σ_2)=∑_i=1^2
        f(ρ_i,σ_i).
    
  * Super-additivity:
    
    f(ρ_1,2,σ_1⊗σ_2)≥ f(ρ_1⊗ρ_2,σ_1⊗σ_2).


Then f is lower asymptotically semi-continuous with respect to any
σ.


    Let {ρ'_n} be a sequence of states such that
    ρ'_n-ρ^⊗ n_1→ 0. Since the trace norm
    fulfills the data-processing 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
    
    1/n(f(ρ'_n,σ^⊗ n)-f(ρ^⊗ n,σ^⊗ n) )
       ≥1/n∑_i(f(ρ'_n,i,σ)-f(ρ,σ))
       ≥min_i{f(ρ'_n,i,σ)} - f(ρ,σ) n→∞⟶ 0,

    where the limit follows from continuity and the second line from additivity and super-additivity.





§ UNIQUENESS OF THE FREE ENERGY


The results of the previous section, in particular Theorem <ref>, 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

    ρ≠ω_β,H

where ω_β,H is the Gibbs state for the Hamiltonian H and inverse temperature β>0, given by

    ω_β,H:=e^-β H/ (e^-β H).

In this way, the resource theory of β-atermality is concerned with the absence of thermal equilibrium at temperature 1/β <cit.>. 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 Gibbs-preserving maps (GP) which are quantum channels that have Gibbs state (<ref>) as a fixed point. More formally, a GP-channel is defined as a trace-preserving completely positive map G_β
with the property that

    G_β(ω_β,H)= ω_β,H  ∀  H.

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,H) ↦ (σ,K)=G_β(ρ,H) such that

    G_β(ω_β,H,H)=(ω_β,K,K).

This condition can equivalently be cast into the following form: One may define the set of GP channels
as G_β^H (ω_β,H)=ω_β,K(H) for all H, and the map between Hamiltonians as
G̅(H)=K so that

    G(ρ,H)=(G_β^H(ρ),G̅(H)).

With this notation, condition (<ref>) is simply given by

    G_β^H(ω_β,H)=ω_β,G̅(H).

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. <cit.>.


Any map G_β fulfilling (<ref>) 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

    G_β(ρ_S,H_S):=(σ,K)=(_S (G_β(ρ⊗ω_β,K)),K).


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 <cit.>. In the following, we will also frequently drop the β-subscript from GP-maps for simplicity of notation.



 §.§ 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.


We say that the transition

    (ρ_S,H_S) → (σ_S,K_S)

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

    G( (ρ_S,H_S) ⊗ (γ_A,R_A) )= (σ_S,K_S) ⊗ (γ_A,R_A).

We will in this case simply denote it by

    (ρ_S,H_S)c> (σ_S,K_S).


Here, we are employing the convenient notation

    (ρ_S,H_S) ⊗ (γ_A,R_A) :=(ρ_S ⊗γ_A , H_S ⊗𝕀_A+ 𝕀_S ⊗ R_A)

 to describe tensor products of objects. In the remainder of this work, we will simply write H_S ⊗𝕀_A+ 𝕀_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. <cit.>. 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.

We say that the transition

    (ρ_S,H_S) → (σ_S,K_S)

is a marginal-catalytic free transition if there exist a GP map G and systems A_1,…,A_k described by the object (γ_A,R_A)=⊗_i=1^k (γ^i,R^i) such that

    G( (ρ_S,H_S) ⊗ (γ_A,R_A) )= (σ_S,K_S) ⊗ (γ̃_A,R_A),

where _|A_i(γ̃_A)=_|A_i(γ_A) for all i ∈ (1,…,k). We will in this case simply denote it by

    (ρ_S,H_S)mc> (σ_S,K_S).


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 <ref>, 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. <cit.>, correlated-catalytic free transitions:


We say that the transition

    (ρ_S,H_S) → (σ_S,K_S)

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

    G( (ρ_S,H_S) ⊗ (γ_A,R_A) )= (η,K_S+ R_A),

where _A(η)=σ_S and _S(η)=γ_A. We will in this case simply denote it by

    (ρ_S,H_S)cc> (σ_S,K_S).


 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 <ref> and <ref>.



 §.§ 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

    Δ F_β(ρ,H):= 1/βS(ρω_β,H) =F_β(ρ,H)- F_β(ω_β,H,H)

where F_β(ρ,H)=(ρ H) -β^-1 S(ρ) with S being the von Neumann (and not the relative) entropy. The measure Δ F_β fulfills  the following properties that we express here for a generic measure denoted by M_β:


  *  Continuity: For fixed Hamiltonian H, the map ρ↦ M_β(ρ,H) is continuous.

  *  Additivity:

    M_β(ρ_1⊗ρ_2, H_1+ H_2) = M_β(ρ_1,H_1) +M_β(ρ_2,H_2).


  *  Monotonicity:


  * Monotonicity: 
 
M_β(ρ,H)≥ M_β(σ,K) if G(ρ,H) = (σ,K).

  * Catalytic monotonicity: 
 
M_β(ρ,H)≥ M_β(σ,K) if (ρ,H) c> (σ,K).

  * Marginal-catalytic monotonicity: 
 
M_β(ρ,H)≥ M_β(σ,K) if (ρ,H) mc> (σ,K).

  * Correlated-catalytic monotonicity: 
 
M_β(ρ,H)≥ M_β(σ,K) if (ρ,H) cc> (σ,K).


All those properties apply for all states and Hamiltonians involved. The fact that Δ F_β fulfills <ref> and <ref> 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 <ref>. 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 Gibbs-state of the modular Hamiltonian

    H_σ := -1/βlogσ+ C,

for any C∈. 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 <ref> and <ref> imply that M_β(ρ,H) = M_β(ρ,H+C) for any C∈ (see Appendix <ref> 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(_2(ρ)⊗_1(ρ),σ_1⊗σ_2)

and additive if it fulfills

    M_β(ρ_1⊗ρ_2,σ_1⊗σ_2)= M_β(ρ_1,σ_1) + M_β(ρ_2,σ_2).

Also, M_β is said to fulfill the data processing inequality if

    M_β(T(ρ),T(σ))≤M_β(ρ,σ)

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:

Let T:B( H)→B( H') be a quantum channel and σ any full-rank state. If T(σ) is only supported on a subspace P⊆ H', then T(ρ) is supported only within P for any ρ.

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),

    M_β(ρ,σ) =  M_β(ρ|_supp(σ),σ|_supp(σ)).

We can then show the following equivalence between properties 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 measure M_β fulfills additivity <ref> and marginal-catalytic monotonicity <ref>  M_β is super-additive <ref>, additive <ref> and fulfills the data-processing inequality <ref>.

  * The measure M_β fulfills Additivity <ref> and correlated-catalytic monotonicity <ref>  M_β is super-additive <ref>, additive <ref> and fulfills the data-processing inequality <ref>.


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 marginal-catalytic operations (Definition <ref>) or also under correlated-catalytic operations (Definition <ref>) 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 Δ F_β fulfills indeed Properties <ref>-<ref>. More importantly, using our re-formulation of Matsumoto's result of Theorem <ref>, we can show that Δ F_β is, up to a constant factor, the only measure of athermality that fulfills <ref>-<ref>. This is the content of our main result, which follows from Theorem <ref>.

[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 Δ F_β.


The implications of this result are that the free energy difference Δ F_β is the only bona fide quantifier of athermality under the most general set of free operations that do not create the resource.




§ 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 non-equilibrium 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. <cit.> 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 <ref> and the correlated-catalytic one of Definition <ref>. 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. <cit.> we have seen that indeed the operations of Definition <ref> are more powerful than the ones of <ref>. At the same time, it has been recently been shown in Ref. <cit.> that a variant of Definition <ref> allows to extract work from passive states. The question remains whether all the transitions that do not increase the free energy difference Δ F_β are possible, as they indeed are for the ones of Definition <ref>, as shown in Ref. <cit.>. If this is indeed true also for correlated-catalysts, 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. 

Acknowledgments: We acknowledge funding from the DFG (GA 2184/2-1), the BMBF, the
EU (COST, AQuS), and the ERC (TAQ) and the Studienstiftung des Deutschen Volkes.













25
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Hiai and Petz(2005)]Hiai2005
  author author F. Hiai and author D. Petz,  in 10.1142/9789812563071_0004 booktitle
  Asymptotic Theory of Quantum Statistical Inference (publisher World Scientific Pub Co Pte Lt, year 2005) pp.  pages 43–63NoStop
[Ogawa and Nagaoka(2005)]Ogawa2005
  author author T. Ogawa and author H. Nagaoka, in 10.1142/9789812563071_0003 booktitle Asymptotic Theory of Quantum Statistical Inference  (publisher World Scientific Pub Co Pte Lt, year
  2005) pp. pages 28––42NoStop
[Brandao and Plenio(2010)]GeneralStein
  author author F. G. Brandao and author M. B. Plenio, @noop  journal journal
  Commun. Math. Phys. volume 295, pages
  791 (year 2010)NoStop
[Vedral(2002)]Vedral2002
  author author V. Vedral, 10.1103/revmodphys.74.197 journal
  journal Rev. Mod. Phys. volume 74,  pages 197–234 (year 2002)NoStop
[Vedral et al.(1997)Vedral,
  Plenio, Rippin, and Knight]Vedral1997
  author author V. Vedral, author M. B. Plenio,
  author M. A. Rippin,  and  author P. L. Knight, 10.1103/physrevlett.78.2275 journal journal Phys. Rev. Lett. volume 78, pages 2275 (year 1997)NoStop
[Brandao et al.(2013)Brandao, Horodecki, Oppenheim,
  Renes, and Spekkens]Brandao2013
  author author F. G. S. L.  Brandao, author M. Horodecki, author J. Oppenheim, author J. M. Renes,  and author R. W. Spekkens, @noop  journal journal
  Phys. Rev. Lett. volume 111, pages
  250404 (year 2013)NoStop
[Brandao and Gour(2015)]Brandao2015
  author author F. G. S. L.  Brandao and author G. Gour, @noop  journal journal Phys. Rev. Lett. volume 115, pages 070503 (year 2015)NoStop
[Junge et al.(2016)Junge,
  Renner, Sutter, Wilde, and  Winter]Junge2016
  author author M. Junge, author R. Renner,
  author D. Sutter, author M. M. Wilde,  and author A. Winter, in 10.1109/isit.2016.7541748 booktitle 2016 IEEE
  International Symposium on Information Theory (ISIT) (publisher Institute of Electrical and Electronics Engineers (IEEE),  year 2016) pp. pages 2494–2498NoStop
[Kastoryano and Eisert(2013)]Kastoryano2013
  author author M. J. Kastoryano and author J. Eisert, @noop  journal journal J.
  Math. Phys. volume 54, pages 102201
  (year 2013)NoStop
[Bernigau et al.(2015)Bernigau, Kastoryano, and Eisert]Bernigau2015
  author author H. Bernigau, author M. J. Kastoryano,  and author J. Eisert, 10.1088/1742-5468/2015/02/p02008 journal journal J. Stat. Phys. volume
  2015, pages P02008 (year 2015)NoStop
[Goold et al.(2016)Goold,
  Huber, Riera, del Rio, and  Skrzypczyk]Goold2016
  author author J. Goold, author M. Huber,
  author A. Riera, author L. del Rio,  and author P. Skrzypczyk, 10.1088/1751-8113/49/14/143001 journal journal
  J. Phys. A volume 49, pages 143001
  (year 2016)NoStop
[Gallego et al.(2016)Gallego, Eisert, and Wilming]Gallego2015
  author author R. Gallego, author J. Eisert,   and author H. Wilming, 10.1088/1367-2630/18/10/103017 journal journal New J. Phys. volume 18, pages 103017 (year 2016)NoStop
[Lashkari and Van Raamsdonk(2016)]Lashkari2015
  author author N. Lashkari and author M. Van Raamsdonk, @noop  journal journal
  J. High En. Phys. volume 2016 (year
  2016)NoStop
[Matsumoto(2010)]Matsumoto2010
  author author K. Matsumoto, @noop  title Reverse test and
  characterization of quantum relative entropy,  (year 2010),  http://arxiv.org/abs/1010.1030 arXiv:1010.1030 NoStop
[Audenaert and Eisert(2005)]AudenaertContinuous
  author author K. M. R.  Audenaert and author J. Eisert, @noop  journal journal J. Math. Phys. volume 46, pages 102104 (year 2005)NoStop
[Tomamichel(2016)]Tomamichel2016
  author author M. Tomamichel, 10.1007/978-3-319-21891-5 title Quantum Information Processing with Finite Resources, series SpringerBriefs in Mathematical Physics, Vol. volume
  5 (publisher Springer International Publishing, year 2016)NoStop
[Janzing et al.(2000)Janzing, Wocjan, Zeier, Geiss, and Beth]Janzing00
  author author D. Janzing, author P. Wocjan,
  author R. Zeier, author R. Geiss,  and author
  T. Beth, @noop  journal journal Int. J. Th. Phys. volume 39, pages 2717 (year
  2000)NoStop
[Horodecki and Oppenheim(2013)]Horodecki2013
  author author M. Horodecki and author J. Oppenheim, @noop  journal journal
  Nature Comm. volume 4, pages 2059
  (year 2013)NoStop
[Brandao et al.(2015)Brandao, Horodecki, Ng, Oppenheim, and Wehner]Brandao15
  author author F. G. S. L.  Brandao, author M. Horodecki, author N. H. Y. Ng, author J. Oppenheim,   and author S. Wehner,  @noop  journal journal PNAS volume 112, pages 3275 (year
  2015)NoStop
[Gour(2016)]Gour16
  author author G. Gour, @noop  title Quantum resource
  theories in the single-shot regime,  (year 2016), http://arxiv.org/abs/1610.04247 arXiv:1610.04247 NoStop
[Buscemi and Gour(2016)]Buscemi16
  author author F. Buscemi and author G. Gour,  @noop  title Quantum relative Lorenz
  curves,  (year 2016), http://arxiv.org/abs/1607.05735 arXiv:1607.05735 NoStop
[Lostaglio et al.(2015)Lostaglio, Müller, and Pastena]Lostaglio15
  author author M. Lostaglio, author M. P. Müller,  and author M. Pastena, 10.1103/PhysRevLett.115.150402 journal journal Phys. Rev. Lett. volume 115, pages 150402 (year
  2015)NoStop
[Ng et al.(2015)Ng,
  Mančinska, Cirstoiu, Eisert, and Wehner]Ng15
  author author N. H. Y.  Ng, author L. Mančinska, author C. Cirstoiu, author J. Eisert,   and author S. Wehner, http://stacks.iop.org/1367-2630/17/i=8/a=085004 journal
  journal New J. Phys. volume 17,  pages 085004 (year 2015)NoStop
[Sparaciari et al.(2016)Sparaciari, Oppenheim, and Fritz]Sparaciari2016
  author author C. Sparaciari, author J. Oppenheim,  and author T. Fritz, @noop  title A resource theory for
  work and heat,  (year 2016), http://arxiv.org/abs/1607.01302 1607.01302 NoStop
[Sparaciari et al.(2017)Sparaciari, Jennings, and Oppenheim]Sparaciari2017
  author author C. Sparaciari, author D. Jennings,  and author J. Oppenheim, @noop  title Energetic
  instability of passive states in thermodynamics,  (year
  2017), http://arxiv.org/abs/1701.01703 arXiv:1701.01703
  NoStop






§ GAUGE INVARIANCE OF M_Β

Here, we show that any measure of athermality fulfilling properties <ref> – <ref> is gauge-invariant in the sense that M_β(ρ,H) = M_β(ρ,H+C) for all c∈. To see this, first note that since tracing out and adding a thermal ancilla are free transitions, M_β(ω_β,H,H)=0 for any H.

A simple calculation using additivity then also shows gauge-invariance:

    M_β(ρ,H+C)    = M_β((ρ,H+C)⊗(ω_β,K,K))
       = M_β(ρ⊗ω_β,K,H⊗ + C⊗ + ⊗ K)
       =M_β((ρ,H)⊗(ω_β,K,K+C))
       =M_β((ρ,H)⊗(ω_β,K+C,K+C))
       = M_β(ρ,H),

where we made use of the gauge invariance of Gibbs states.






§ RANK-DECREASING QUANTUM CHANNELS

In this appendix we prove the validity of Lemma <ref> of the main-text. We have to show that given a channel T: B( H)→ B( H') and a full-rank state σ such that supp(T(σ))⊂ P, we also have supp(T(ρ))⊂ P for all states ρ. Here, P is an arbitrary subspace of the total Hilbert space H'.
Let σ = ∑_i q_i i be the eigen-decomposition of σ. Since T maps positive operators to positive operators, and the support of the sum of positive operators is the union of the supports of the operators we conclude that T(i) is supported in P for all i. We thus only need to show that also operators of the form T(|i⟩⟨j|) are supported on P.
Now consider any density operator ρ = d+r where d is the diagonal part of ρ (in the eigenbasis of σ) and r the rest. We know that (T(d)) = 1 since T is trace-preserving. Hence (T(r))=0. Let us now assume (to arrive at a contradiction)
that T(r) has support within the subspace Q=-P. Since T maps positive operators to positive operators,

    0 ≤ Q T(ρ) Q  = Q T(r) Q.

Thus we conclude on the one hand that Q T(r) Q≥ 0. On the other hand, we know that

    1=(T(ρ)) ≥(P T(ρ)) = 1 + (PT(r)).

Hence, (P(T(r)))=0. Since T is trace-preserving we also have

    (PT(r))= - (Q T(r)) =0.

 Hence Q T(r)Q=0 and also Q T(ρ) Q=0. By positivity and Hermiticity of T(ρ) we also get PT(ρ)Q=0 and Q T(ρ)P=0. We thus conclude that T(ρ)=PT(ρ)P, which finishes the proof.



§ PROOF OF THEOREM <REF> AND OTHER EQUIVALENCES


We will show a more complete set of equivalences than the ones of Theorem <ref>, which corresponds simply to iii)
and iv).


The following properties are equivalent:


  i)M_β fulfills monotonicity <ref>  M_β fulfills the data-processing inequality (DPI) (<ref>).

  ii) M_β fulfills catalytic monotonicity <ref> and additivity <ref>  M_β fulfills additivity (<ref>) and the DPI (<ref>).

  iii) M_β fulfills marginal-catalytic monotonicity <ref> and additivity <ref>  M_β fulfills super-additivity (<ref>), additivity (<ref>) and the DPI (<ref>).

  iv) M_β fulfills correlated-catalytic monotonicity <ref> and additivity <ref>  M_β fulfills super-additivity (<ref>), additivity (<ref>) and the DPI (<ref>).



Let us first show i) (⇒). Let T be any given quantum-channel. We have to show that M_β(T(ρ),T(σ))≤ M_β(ρ,σ). But by the previous discussion, T(σ) can always be considered to be full-rank. Therefore, the Hamiltonian H_T(σ) exists and the map (ρ, H_σ) ↦ (T(ρ),H_T(σ)) is automatically a GP-map. We therefore obtain

    M_β(T(ρ),T(σ))    = M_β(T(ρ),H_T(σ)) 
       <ref>≤ M_β(ρ,H_σ) =  M_β(ρ,σ).

i) (⇐) follows as

    M_β(σ,K)   =   M_β(G(ρ,ω_β,H))
       =   M_β(G_β^H(ρ),ω_β,G̅(H))
       (<ref>)=   M_β(G_β^H(ρ),G_β^H(ω_β,H))
       (<ref>)≤     M_β(ρ,H).

The proof of ii) (⇒) is trivial given i), since <ref>⇒<ref> and it follows straightforwardly that (<ref>)⇒<ref>. The proof of ii) (⇐) follows from noting that (ρ,H)c>(σ,K) implies that there exist G so that

    G(ρ⊗γ, H+R)=(σ,K)⊗ (γ,R).

Hence, we find that

    M_β(σ,ω_β,K)   +   M_β(γ,ω_β,R)(<ref>)=M_β(σ⊗γ,ω_β,K⊗ω_β,R)
       (<ref>)≤   M_β(ρ,ω_β,H)+M_β(γ,ω_β,R),

which implies straightforwardly M_β(ρ,H)≥ M_β(σ,K), that is, (<ref>).

Now we show iii) (⇒). Note that <ref> implies <ref>, since a correlated catalyst is a particular case of using a catalyst. Together with the equivalences i) and ii), we should only show super-additivity of Eq. (<ref>). This follows from the fact that

    (ρ_1,2,H_1+H_2)mc>(ρ_1⊗ρ_2,H_1+H_2).

	To show this, let us choose as catalyst γ=ρ_1 ⊗ρ_2. The GP map performing the transition is just a swap between the initial system and the catalyst. Hence, the final system is (σ,K)=(ρ_1 ⊗ρ_2,H_1+H_2) and the final catalyst γ̃=(ρ_1,2,H_1+H_2), which clearly fulfills the conditions of Definition <ref>.
To see iii) (⇐), we first note that (<ref>) and (<ref>) already imply <ref> and <ref>. It thus remains to show that adding (<ref>) also implies <ref>. This follows since  super-additivity of M_β together with additivity implies

    M_β(γ̃_A,R_A)    =  M_β (γ̃_A, ⊗_i ω_β, R^i)
       ≥∑_i  M_β(γ^i,R^i) 
       = M_β(⊗_i (γ^i,R^i) = ∑_i M_β(γ^i,R^i).

Finally, let us turn to iv). Again, since correlated catalytic transitions include catalytic transitions, the only non-trivial property left to show here is that <ref> and <ref> also imply (<ref>). To see this consider an initial object (ρ_1,2,H_1+H_2) together with the catalyst (ρ_2,H_2) and use a similar trick as for marginal catalytic transitions. Since a swap between the second system of the initial object and the catalyst is a Gibbs-preserving transition which leaves the catalyst correlated but otherwise unchanged,  we know that M_β(ρ_1,2,H_1+H_2) ≥ M_β(ρ_1⊗ρ_2,H_1+H_2). But then we have

    M_β(ρ_1,2,ω_β,H_1⊗ω_β,H_2)    = M_β(ρ_1,2,H_1+H_2) 
       ≥ M_β(ρ_1⊗ρ_2,H_1+H_2)
        =  M_β(ρ_1⊗ρ_2,H_1+H_2),

which completes the argument.






