Third Law of Thermodynamics as a Single Inequality
Henrik Wilming Rodrigo Gallego


 
 


The third law of thermodynamics in the form of the unattainability principle states that exact ground-state cooling requires infinite resources. Here we investigate the amount of non-equilibrium resources needed for approximate cooling. We consider as resource any system out of equilibrium, allowing for resources beyond the i.i.d. assumption and including the input of work as a particular case. We establish in full generality a sufficient and a necessary condition for cooling and show that for a vast class of non-equilibrium resources these two conditions coincide, providing a single necessary and sufficient criterion. Such conditions are expressed in terms of a single function playing a similar role for the third law to the one of the free energy for the second law. From a technical point of view we provide new results about concavity/convexity of certain Renyi-divergences, which might be of independent interest.

The third law of thermodynamics as a single inequality
    Rodrigo Gallego
    December 30, 2023
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§ INTRODUCTION

Pure quantum states are indispensable resources for any task in quantum information processing. However, the third law of thermodynamics (more precisely, the unattainability principle) states that cooling a system exactly to zero temperature requires an infinite amount of resources, being it in the form of time, space, work or some other resource <cit.>. Similarly, no-go theorems have been put forward for the task of bit erasure –which is closely related to ground-state cooling– showing that no unitary process on a system and a finite dimensional reservoir can bring the system from a mixed to a pure state <cit.>. However, these no-go results do not say much about the amount of resources needed for approximate cooling. Indeed, in recent times a sizable number of studies deal with different protocols to cool a small quantum system by unitarily acting on a heat bath and a certain number of systems out of equilibrium to be “used up”  (known under the name of algorithmic or dynamical cooling) <cit.> or studying particular models of refrigerating small quantum systems <cit.>, including ones that seem to challenge the unattainability principle in terms of required time <cit.>.

In this work we will focus on quantifying in full generality the expenditure of arbitrary systems out of equilibrium that are needed for approximate cooling while having access to a heat bath. Our scenario is similar to the one considered in algorithmic cooling, but here we treat the full thermodynamics of the problem by allowing for resources with non-trivial Hamiltonians and accounting for the energy conservation of the total process.
We will do this in the resource theoretic framework of quantum thermodynamics <cit.>, which has proven useful to answer a variety of fundamental questions in quantum thermodynamics, such as establishing an infinite family of second laws <cit.>, providing fundamental bounds to single-shot thermodynamics <cit.>, providing definitions of work for quantum systems <cit.>, generalizing fluctuation theorems <cit.>, elucidating the thermodynamic meaning of negative entropies <cit.> and elucidating the role of quantum coherence in thermodynamics <cit.>.

Recently, there have also been studies from this point of view on the problem of cooling <cit.>, however mostly focusing on providing necessary conditions in terms of resources such as time, space or Hilbert-space dimension.

The task of cooling that we are considering can be phrased as finding a cooling protocol between an arbitrary resource described by the state and Hamiltonian ρ_R and H_R, respectively, and a target system described by ρ_S and H_S so that ρ_S approximates the ground-state of H_S. We will later assume for simplicity that ρ_S is a thermal state – in this case the goal is to bring its final temperature T_S to a very low value. We will assume that the density matrix of the resource has full rank because otherwise the problem trivializes, since one can, for example, simply swap with a ground-state [Similar arguments can be made for other states without full rank. For example two copies of a rank-2 state in a 4-dimensional system can be written as a pure state in tensor product with a full-rank state. Such a resource therefore already contains a pure state which can then be mapped to the ground-state.]. We furthermore assume that the target system is initially in thermal equilibrium with some environment. Then the transition, i.e. the cooling protocol, can be performed by using a thermal bath at a fixed inverse temperature β and performing a global unitary that commutes with the total Hamiltonian, so that energy conservation is properly accounted for. This kind of transitions have been extensively studied and they can be characterized by families of functions M^α, the so-called monotones, so that a transition is possible if and only if <cit.>

    M^α(ρ_R,H_R) ≥ M^α(ρ_S,H_S)    ∀
     α.

Hence, the problem at hand is in principle hard to characterize since one needs to verify an infinite number of conditions to conclude that a given transition is possible. The main contribution of the present work is to show that in the limit where T_S is sufficiently close to zero –i.e. the regime where the (un)attainability problem is formulated– the infinite set of monotones appearing in (<ref>) can be essentially reduced to a single monotone. We call this monotone the vacancy and it is defined as

    V_β (ρ,H):=S(ω_β(H)ρ),

where ω_β(H) is the Gibbs state of at inverse temperature β and S is the relative entropy defined as

    S(ρσ) =(ρlogρ) - (ρlogσ),

if supp(ρ)⊆supp(σ) and equal to +∞ otherwise.


We find that sufficient and necessary conditions for cooling, respectively, are given by

    V_β(ρ_R,H_R) - K(ρ_R,H_R,ρ_S,H_S,β)    ≥V_β(ρ_S,H_S),
    V_β(ρ_R,H_R)    ≥V_β(ρ_S,H_S),

where K (ρ_R,H_R,ρ_S,H_S,β) → 0 as T_S → 0. Hence in the limit of very low temperature cooling V_β(ρ_S,H_S) is the key quantity that determines the fundamental limitations.
Importantly, V_β(ρ_S,H_S) diverges as T_S→ 0. The necessary condition (<ref>) therefore shows that an infinite amount of resources (as measured by V_β) is necessary for exact ground-state cooling. Furthermore we show that for a vast class of resource systems, for example thermal states of coupled harmonic oscillators, the function  K(ρ_R,H_R,ρ_S,H_S,β) vanishes identically. Hence

    V_β(ρ_R,H_R) ≥V_β(ρ_S,H_S)

becomes both a sufficient and necessary condition. That V_β plays an important role for the third law had been previously found in the setting of i.i.d. resources and qubits as target systems in the seminal work of Ref. <cit.>. Here, we extend the significance of the quantity V_β to arbitrary scenarios.

Usually, the unattainability principle is formulated with respect to time, arguing that an infinite amount of time (or infinitely many cycles of a periodically working machine) are needed to cool a system exactly to zero temperature. Our results show, for example, that if the non-equilibrium resources are simply hot thermal systems (as in the example of a thermal machine that operates between two heat baths), the system to be cooled and the cooling machine have to effectively interact with infinitely many such resource systems (or all parts of one infinitely large system). This implies that an infinite amout of time is needed, since each such interaction takes a finite time (see <cit.> for a thorough discussion of this point).

Our findings not only serve to pose limitations to protocols of algorithmic cooling, but also suggest a surprising symmetry between the second and third law of thermodynamics. The second law –in its averaged version or in the version of the Jarzynski equality <cit.>– can be expressed in terms of the free-energy difference defined as

    Δ F_β (ρ,H) = 1/β S(ρω_β(H)).

In analogy, we show that the third law can be expressed similarly in terms of V_β(ρ,H) which simply inverts the arguments of the relative entropy in Eq. (<ref>) and drops the pre-factor.
This symmetry between the second and third law is quite surprising and hints at the fact that the second and third law can be related to the errors of first and second kind in hypothesis testing <cit.>.
We leave the investigation of this deeper relation between the two for future work.

From a technical point of view, our results rely on certain convexity-properties of the function α↦ S_α(ρ || σ), where S_α are classical Renyi-divergences <cit.>. We believe that these results might be of independent interest.



§ SET-UP AND GENERAL NECESSARY CONDITION

In the following we will use the set-up of catalytic thermal operations <cit.> applied to the task of cooling. In this set-up we imagine to possess a resource given by the pair of state and Hamiltonian (ρ_R,H_R). We can then use an arbitrary thermal bath at inverse temperature β, that is, a system in a Gibbs state ω_β(H_B) of a Hamiltonian H_B, and finally an ancillary system, the so-called catalyst with arbitrary state and Hamiltonian (σ_C,H_C) in such a way the latter is returned in the same configuration and uncorrelated from the rest of the systems after implementing the protocol. The target system to be cooled is initially assumed to be in thermal equilibrium with the thermal bath and therefore described by a Gibbs state (ω_β(H_S),H_S). The total compound RSBC is transformed by a cooling protocol, which consists simply of a unitary transformation U which commutes with the total Hamiltonian.

More formally, we say that there exists a cooling protocol to ρ_S using the resource (ρ_R,H_R) if there exists a fixed catalyst (σ_C,H_C) and for any ϵ>0 there exists a unitary U and a bath Hamiltonian H_B such that

    ρ'_RS⊗σ^ϵ_C = _B(U ρ_R ⊗ω_β(H_S) ⊗ω_β(H_B)⊗σ_C   U^†)

with _R(ρ'_R S)=ρ_S  and σ_C - σ_C^ϵ_1≤ϵ. The only constraint on the unitary U is that it conserves the global energy, i.e.,

    [U,H_R+H_S+H_B+H_C]=0.


Note that this formulation of the cooling process contains as a particular case partial cooling in which we do not start with the target in a Gibbs state. In this case, the initial system of S, if it is partially cooled before starting the protocol, can be simply incorporated as a part of the resource R.

The problem of finding conditions for the existence of a transitions of the form (<ref>) has been studied in Ref. <cit.> for diagonal states, that is, with [ρ_R,H_R]=0 and [ρ_S,H_S]=0. Throughout this manuscript we will restrict to such diagonal states, but we emphasize that the the necessary condition (<ref>) also holds for non-diagonal states as we will see later.

Under the assumption that ρ_R and ρ_S are diagonal, one can show that cooling to a state ρ_S is possible if and only if <cit.>

    S_α(ρ_R || ω_β(H_R)) ≥ S_α(ρ_S || ω_β(H_S))  ∀α≥ 0,

where S_α are so-called Renyi-divergences. The proof of this statement relies simply on the results of Ref. <cit.> together with the additivity of the Renyi-divergences under tensor-products.

An important tool that appears in Eq. (<ref>) is the concept of a monotone of (catalytic) thermal operations <cit.>. This is any function f which can only decrease under (catalytic) thermal operations. The functions S_α appearing in (<ref>) are monotones under catalytic thermal operations and more generally under any channel that has the Gibbs state as a fixed point. Importantly, any monotone f, possibly different from S_α, allows us to construct necessary conditions for a given transition. We will now show that V_β is also a monotone under catalytic thermal operations and derive the corresponding necessary condition for cooling.
 The vacancy is an additive monotone under catalytic thermal operations. This has as an implication that
for any target (ρ_S,H_S) and resource (ρ_R,H_R) –not necessarily diagonal states–, the condition
  
    V_β(ρ_R,H_R)≥V_β(ρ_S,H_S).

  is necessary for cooling.

Let us first show that V_β is a monotone under catalytic thermal operations. Let us consider an arbitrary transition from state ρ to state ρ' –both with Hamiltonian H– by catalytic thermal operations, then we will now show that V_β(ρ,H)≥V_β(ρ',H).

First note that the vacancy diverges for a state ρ without full rank, thus the inequality V_β(ρ,H)≥V_β(ρ',H) is satisfied trivially for those states. Let us therefore assume that ρ is a full rank state.
As was shown in <cit.>, for any 0≤α≤ 2 the Renyi-divergences

    S_α(ρω_β(H)) := 1/α-1log(ρ^αω_β(H)^1-α)

are monotonoic under (catalytic) thermal operations for arbitrary states ρ. That is, we have the necessary condition

    S_α(ρ || ω_β(H)) ≥ S_α(ρ' || ω_β(H))  ∀ 0≤α≤ 2,

  By simple algebra, one can show that

    .∂/∂_α|_α=0 S_α(ρω_β(H)) = V_β(ρ,H) ≥ 0.

Taylor-expanding (<ref>) on both sides for any α>0 and dividing by α then yields

    V_β(ρ,H) + O(α)≥ V_β(ρ',H) + O(α),

where O(α) indicates that it is of first order on α. Taking α arbitrarily small then yields V_β(ρ,H) ≥ V_β(ρ',H). This proves monotonicity under catalytic thermal operations. Additivity follows directly from the additivity of the relative entropy under tensor products.









Once we established that the vacancy is a monotone under catalytic thermal operations, we can now derive the necessary condition for cooling by simply applying this condition to the transition that SR undergo in the cooling process:

    V_β(ρ_R ⊗ω_β(H_S), H_R+H_S ) 
         ≥V_β(ρ'_RS , H_R+H_S )
         ≥V_β( ω_β(H_R) ⊗ρ_S, H_R+H_S ).

The last inequality follows from the fact that one can always replace the state on any system by an uncorrelated thermal state at the heat bath's temperature using a thermal operation. Using additivity of the vacancy and the fact that V_β(ω_β(H),H)=0, we obtain the necessary condition (<ref>).


We emphasize that the necessary condition (<ref>) is derived in full generality and it applies to any full-rank state ρ_R and any state ρ_S, possibly not diagonal in the eigenbasis of H_S.

The monotone V_β was first introduced in Ref. <cit.>. Its relevance for the unattainability principle is clear since if ρ_S does not have full support, then the r.h.s. of (<ref>) diverges. Hence, exact cooling is impossible unless the resource ρ_R does not have full support either. In the particular case of ρ_R= ⊗ _i=1^n ϱ^i where ϱ^i has full support, the condition (<ref>) already tell us that we need infinite resources –infinite n in this case– for exact cooling. Hence, such a simple analysis already suggests that the quantity V_β plays a crucial role for the limitations on cooling.


To summarize, we have seen, building upon previous literature, that the vacancy V_β establishes completely general necessary conditions for cooling. However, for necessary and sufficient conditions one should in principle verify an infinite number of inequalities given by (<ref>). Our contribution will be to show that these infinite number of inequalities can be reduced to a single one, which can also be expressed in terms of the vacancy for a sufficiently cold final state ρ_S. Furthermore, we will show that for a large family of resource systems the single sufficient condition that we find coincides with the necessary condition (<ref>). Hence, the limits on cooling are entirely ruled by the function V_β. This holds for large classes of finite systems, with possibly correlated and interacting subsystems.



§ GENERAL SUFFICIENT CONDITIONS FOR COOLING

The process of cooling laid out in the previous section can in principle be applied to any final state ρ_S. We will now assume for simplicity that the final state, as it corresponds to a cooling process, is of the form ρ_S=ω_β_S(H_S) with β_S very large. We can then derive the following completely general sufficient condition for cooling.

  For every choice of β and H_S there is a critical β_cr>0 such that for any β_S > β_ cr and full-rank resource (ρ_R,H_R) the condition

    V_β(ρ_R,H_R) - K(β_S,β,ρ_R,H_R,H_S) ≥ V_β(ω_β_S(H_S),H_S)

is sufficient for cooling. The positive semidefinite function K has the property K(β_S,β,ρ_R,H_R) → 0 as β_S→∞ for any fixed β,H_R,ρ_R>0 and H_S.


The proof of the theorem is given in Sec. <ref>. Nonetheless we provide a sketch of the main ideas involved in such a proof at the end of this section. The function K is given by

    K(β_S,β,   ρ_R,H_R,H_S)=
       max{0,-δ(β_S) min_α≤δ(β_S)∂^2/∂α^2S_α(ρ_Rω_β(H_R))},

where

    δ(β_S):=log(Z_β)/V_β(ω_β_S(H_S),H_S)≥ 0,   Z_β = (^-β H_S).

The bound (<ref>) applies for any possible (diagonal) resource state, however finding K(β_S,β,ρ_R,H_R,H_S) involves a minimization, that although is feasible for low dimensional systems, might be an obstacle for practical purposes when dealing with large systems and for values of β_S so that K(β_S,β,ρ_R,H_R,H_S) cannot be neglected. That said, we will investigate kinds of resource systems ρ_R,H_R for which  K(β_S,β,ρ_R,H_R,H_S)= 0. In those cases, the general sufficient condition given by (<ref>) taken together with the necessary condition (<ref>) will imply that that a necessary and sufficient condition is given simply by

    V_β(ρ_R,H_R) ≥ V_β(ω_β_S(H_S),H_S),

In particular, we will see in section <ref>, that this holds true for large classes of thermal non-equilibrium resources. Let us also note that K(β_S,β,ρ_R,H_R,H_S), just as the vacancy, is additive over non-interacting and uncorrelated resources. We will use this property in the next section to investigate the setting of i.i.d. resources.

In the result given above, we have focused on thermal target states. This is in fact not necessary. We show in the appendix <ref> that a completely analogous result holds for states of the form

    ρ_ϵ = (1-ϵ) |0⟩⟨0| + ϵρ^⊥,  ϵ≪ 1.

where ρ^⊥ is any density matrix which has full rank on the subspace orthogonal to the ground-state |0⟩ and commutes with H_S.



 §.§ Sketch of the proof of Thm. <ref>


As we have seen in the previous section, a set of sufficient conditions for a transition with catalytic thermal operations is given by the infinite set of inequalities of (<ref>). The main idea behind the proof is that when the target system is sufficiently cold (β_S > β_cr) it suffices to check the conditions (<ref>) for very small α. This follows from the fact that for β_S > β_cr the r.h.s. of (<ref>), given by S_α(ρ_Sω_β(H_S)), rapidly saturates to its maximum value as we increase α and it is concave (see Fig. <ref>). Given that one only needs to consider small values of α it is possible to make a Taylor expansion around α=0 of S_α of the form

    S_α(ρ_Rω_β(H_R))   ≈    S_0(ρω_β(H_R))+ ∂ S_α(ρω_β(H_R))/∂α|_α=0α
       +    k α^2.

This reduces the infinite inequalities of (<ref>) to a single one that depends on the derivate of S_α and an error term k, that is related to the error term  appearing in Thm. <ref> denoted by K. This expansion can be further simplified noting that S_α=0(ρω_β(H))=0. The vacancy comes into play because of the identity

    ∂ S_α(ρσ)/∂α|_α=0=S_1(σρ):=S(σρ),

which inverts the arguments of the second term on the r.h.s. of (<ref>). Taking all these elements into account and accounting properly for the precision of the Taylor approximations we arrive at an inequality involving only the vacancy and a vanishing error term as determined by Thm. <ref>.






 §.§ A short comment on catalysts

As laid out in Sec. <ref>, we define the catalytic thermal operations by including the possibility that the catalyst changes during the transition, as long as this change can be made arbitrarily small, as is standard in recent literature on the resource theoretic approach to thermodynamics <cit.>. This formulation is a form of exact catalysis, in the sense that as the error has to be arbitrarily small, the catalyst is returned arbitrarily unchanged. However, it is a possible to consider other forms of catalytic thermal operations which are either more restrictive about the change of the catalyst, where no error –not even arbitrarily small– is allowed for; or less restrictive, in the sense the catalyst is allowed to change by a finite amount. We consider both alternatives on Appendix <ref> and <ref> respectively. First, we study the case in which one requires that the catalysts is always returned exactly in the same state, that is, taking ϵ=0 in the definitions laid out in Sec. <ref>. In this case it is no longer valid that a set of sufficient conditions is given by positive values of α on (<ref>), but one also has to consider Renyi-divergences for negative α. This case is analyzed in Appendix <ref>, where we show that in this scenario i) The general necessary condition of Theorem <ref> holds and ii) A general sufficient condition similar to the one of Theorem <ref> is derived. This general sufficient condition only differs on a multiplicative constant –independent of the final temperature to which one cools– from the one derived in Theorem <ref>.
Finally, in Appendix <ref> we furthermore discuss the case of approximate catalysts. We put forward a consistent method to allow for finite errors on the catalyst while maintaining the validity of the third law of thermodynamics.




§ I.I.D RESOURCES AND SCALING OF THE TARGET TEMPERATURE


Theorem <ref> together with the necessary condition (<ref>) provide completely general sufficient and necessary condition, respectively, for cooling a system to target temperature T_s=1/β_S (setting k_B=1) using a given resource (ρ_R,H_R). They thus characterize the possibility of cooling in full generality. To obtain results for concrete physical situations and find out how the target temperature T_S scales with physical key quantities of the resource –such as the system-size of the resource–  one has to choose a particular resource and calculate its vacancy as well as the error term K. Then one has to check how these quantities depend on the physical properties of interest.

We will now focus on the scaling between the size of the resource and the final temperature of the target system. For this we assume that the resource is given by a number of identically and independently distributed copies. Later we will also discuss other assumptions we can make about the resource.
 Thus, we consider the case where the resource state is given by ρ_R=ϱ_R^⊗ n and Hamiltonian H_R=∑_i H_R^i where H_R^i=𝕀_1 ⊗⋯⊗𝕀_i-1⊗ h_R ⊗𝕀_i+1⊗⋯⊗𝕀_n.

 Let us now consider the following task: Given fixed ϱ_R,H_R,β,H_S, find the minimum n so that it is possible to cool down the target state to inverse temperature β_S.

By using Thm. <ref> together with additivity of V_β we obtain that the necessary number of copies n^ nec(β_S) fulfills

    n^ nec(β_S) ≥V_β (ω_β_S(H_S),H_S)/V_β(ϱ_R,h_R) .

By using Thm. <ref> in the i.i.d. we also obtain a sufficient number of copies n^ suff. The condition (<ref>) takes the form

    n [ V_β(ϱ_R,h_R) - K(β_S,β,ϱ_R,h_R,H_S) ]≥V_β (ω_β_S(H_S),H_S).

Since this condition is sufficient, but not always necessary, we obtain

    n^ suff(β_S) ≤V_β (ω_β_S(H_S),H_S)/V_β(ϱ_R,h_R) - K(β_S,β,ϱ_R,h_R,H_S) .

Since the correction K goes to zero as β_S→∞ (the target temperature going to zero), we see that

    lim_β_S→∞n^ suff(β_S)/n^ nec(β_S) = 1.

It is also interesting to re-express the previous conditions to obtain a more transparent relation between the final achievable temperature and the number of copies. We will see next that n^ nec(β_S) and n^ suff(β_S) scale as β_S for large β_S. Thus the target temperature approaches zero as 1/n.



 §.§ Scaling of the target temperature


In the special case where the target state is a thermal state one can reformulate the vacancy in terms of non-equilibrium free energies. Indeed, the vacancy of a thermal state at temperature β_S simply takes the form

    _β(ω_β_S(H),H) = β_S Δ F_β_S(ω_β(H_S),H_S),

with Δ F_β(ρ,H) = ⟨ H⟩_ρ - ⟨ H ⟩_β - (S(ρ)-S(ω_β))/β.

In this case the condition (<ref>) reads:

    V_β(ρ_R,H_R) ≥β_S Δ F_β_S(ω_β(H_S),H_S).


From (<ref>) we see that for large β_S we have (assuming vanishing ground-state energy)

    _β(ω_β_S(H),H) = β_S E_β-S_β,  as  β_S→∞.

Assuming again a resource system of n non-interacting identical particles each described by (ϱ_R,h_R), we then obtain that the minimum achievable temperature T^(n)_S scales as

    T^(n)_S  = 1/nE^S_β/ V_β(ϱ_R,h_R),   n ≫ 1.

This result is similar to the asymptotic result of Janzing et al. <cit.>.

Lastly, let us point out that the above scaling relation implies that the probability p to find the system in the ground-state after the cooling procedure increases exponentially to 1 with n. For example, if the target system is a d+1 dimensional system with gap Δ above a unique ground-state, we have (for large n)

    p ≥ 1/(1+d^-β_S Δ) ≈ 1 - d ^-n  V_β(ω_β_R(h_R),h_R) Δ/E^S_β.

Thus, while an exact third law holds in the sense that n→∞ for T_S→ 0, the ground-state probability asymptotically converges very quickly to unity.

The above relations demonstrate how one can obtain quantitative expressions of the unattainability principle from Theorems <ref> and <ref> by making assumptions about the given resources.




 §.§ Scaling of the vacancy with system size

In the case of i.i.d. resources with non-interacting Hamiltonians the vacancy is an extensive quantity in the sense that it scales linearly with the number of particles. However, for arbitrary quantum systems with correlated and interacting constituents, it is in general difficult to calculate the vacancy and hence estimate directly how it scales with the number of particles. Nonetheless,  we can use the relation (<ref>) to argue that the vacancy will be extensive for large classes of many-body systems.

In particular, let us assume that the resource is in a thermal state of some local Hamiltonian. That is ρ_R=ω_β̃(H̃_R), where H̃_R is any local Hamiltonian (possibly differing form H_R) and β̃ is finite. In this case one can use Eq. (<ref>) to write

    V_β(ρ_R,H_R)= β̃Δ F_β̃(ω_β_R(H_R),H̃_R).

From the fact that the von Neumann entropy is sub-additive and from the locality of the Hamiltonians H_R and H̃_R it then follows that the vacancy V_β(ρ_R,H_R) scales (at most) linearly with the system size. As a consequence, the minimal final temperature T_S^(n) scales (at best) inversely proportional to the volume of the resource.


In the light of the previous considerations, it seems likely that a similar scaling holds for any resource (potentially under reasonable physical assumptions, such as clustering of correlations). We leave the general characterisation of many-body systems such that the vacancy is extensive as an interesting future research direction.





§ THERMAL RESOURCES


As discussed after the statement of Theorem <ref>, it is useful to find general conditions under which the error term K disappears and the sufficient condition coincides with the general necessary condition. Naturally, it is necessary to make additional assumptions about the resources to achieve this.

We now consider as resource state ρ_R a (possibly multi-partite) thermal state of some Hamiltonian H_R at inverse temperature β_R. In the following we will derive a simple expression that allows us to check whether

    K(β_S,β,ρ_R,H_R,H_S)= 0,   ∀β_S>0,H_S,

and hence (<ref>) becomes a necessary and sufficient condition.
The reasoning is based on showing that

    S_α(ρ_R=ω_β_R(H_R)||ω_β(H_R))
 is convex for a given range of α<1, which implies (<ref>). The convexity of (<ref>) can be determined by looking at the convexity of the average energy as a function of the inverse temperature of the resource

    x ↦ E_x^R :=  (ω_x(H_R)H_R).

In particular, we will see that if β_R< β and the function x↦ E^R_x is convex  for x∈[β_R,β], then the function S_α(ω_β_R(H_R)||ω_β(H_R)) is convex for all α<1.

 For resources of the form (ω_β_R(H_R),H_R) that are hotter than the bath, that is with β_R ≤β, if E^R_x=(ω_x(H_R) H_R) is convex in the range x∈ [β_R,β], then (<ref>) is a sufficient and necessary condition for low temperature cooling.

This Theorem simplifies considerably the task of formulating bounds on the third law, since the average energy is a much more accessible quantity than the Renyi-divergences. In Sec. <ref> we discuss several classes of physically motivated conditions that imply that E^R_x=(ω_x(H_R) H_R) is convex. We emphasize, however, that the convexity of the energy is not a necessary condition for the correction K to vanish: There are cases for which x ↦ E^R_x is not convex for the whole range of inverse temperatures [β_R,β] and (<ref>) is nevertheless a sufficient and necessary condition for cooling.

Lastly, let us mention that condition (<ref>) is fulfilled if, for a fixed β_R, the bath's inverse temperature β is sufficiently large, without any extra assumption on the convexity of E_x. This implies, that for sufficiently cold baths, (<ref>) is also a sufficient and necessary condition. This is shown in Appendix <ref>, together with several properties of the Renyi divergences for thermal states that might be of independent interest and also include a proof of Thm. <ref>.



 §.§ Systems for which the energy is convex



As implied by Cor. <ref>, Eq. (<ref>) becomes a sufficient and necessary condition for cooling if the resource is a thermal state hotter than the bath and its average energy is convex in the inverse temperature. We will now see that the convexity of the energy is fulfilled by a wide range of physical models.

We will first re-express the convexity in terms of the heat capacity. This allows to check for general systems whether E^R_x is convex, as the heat capacity as a function of the temperature is an intensively studied quantity for many-body systems. Using the definition of heat capacity C_x:=d  E^R_x/dT, with T=1/x, we find that the convexity of the energy, as formulated in the condition of Thm. <ref>, can be expressed as

    d^2  E^R_x/dx^2=1/x^2( C_x/x - d C_x/dx)≥ 0   with  x∈ [β_R,β].

Equivalently, this condition can be expressed as

    1/β_R^2 C_β_R - 1/β'^2 C_β'≥  0

for all β_R≤β' ≤β.
In most thermodynamics systems, the heat capacity is monotonically increasing with the temperature, hence d C_x/dx≤ 0 and (<ref>) is satisfied. A seminal exception to this case is given by the so-called Schottky anomaly, which is present in certain solid states at very low temperatures <cit.>.
We thus see, that for thermodynamic systems the convexity of the energy is a very natural property.
Nevertheless it can fail, in particular in finite systems. We will now show that even for large classes of finite systems the energy is convex.

This is due to the following Lemma, which we prove in the appendix.

Consider any Hamiltonian with equidistant and non-degenerate energy levels. Then the function β↦ E_β is convex.

Immediate examples of Hamiltonians with equidistant energy levels are two-level systems or harmonic oscillators. But in fact, the lemma covers a much wider class of models since the vacancy is unitarily invariant and additive over non-interacting subsystems.

 It follows that also any harmonic system and any system described by free fermions has a convex energy function, since free bosonic and fermionic systems can always be made non-interacting by a normal-mode decomposition. In such a normal-mode decomposition they simply correspond to a collection of non-interacting harmonic oscillators or two-level systems, respectively.

  These systems include highly-correlated (even entangled) systems and no thermodynamic limit needs to be taken. A particularly interesting resource that is included by these results is that of hot thermal light, which has been considered before as a valuable resource for cooling <cit.>.

It can furthermore be checked that for large but finite many-body systems whose density of states in the bulk is well approximated by μ(ϵ) ≃^γϵ - αϵ^2 the average energy E_β is convex in β [Such a density of states is typical for many-body systems in the bulk of the energy, but deviations typically do appear in the tails of the distribution.].

Finally, for every finite system there is a critical β_c such that E_β is convex for all β>β_c. Thus as soon as β>β_R>β_c, the sufficient condition (<ref>) holds for small enough target temperatures. This means that if an experimenter has a mechanism to pre-cool the environment to a very low temperature, well below 1/β_c, and the resource has a temperature, which is larger than that of the environment but still smaller than 1/β_c, then condition (<ref>) holds as a sufficient and necessary condition.




 §.§ A source of work


Our formalism can also incorporate a source of work as particular case of a resource for cooling. The limitations on cooling as a function of the input of work have been studied in Ref. <cit.>. There it is shown that the fluctuations of work, rather than its average value, have to diverge when the target state reaches vanishing final temperature and if the heat bath has finite heat capacity. We will here derive a result similar in spirit using Thm. <ref>, although we employ a different model for the work source. Importantly, our result implicitly allows for infinite heat capacity in the heat bath. It should thus be viewed as being complementary to the results in Ref. <cit.>.

Let us model the work source by a system R with Hamiltonian

    H^w_R=∑_x=-d/2^d/2Ex/d|Ex/d⟩⟨Ex/d|.

One can see this Hamiltonian as a d-dimensional harmonic oscillators with energies bounded between E/2 and -E/2. We are interested in the limit of d→∞. In this case R is similar to the model put forward in Ref. <cit.>, with the difference that we consider here a finite value of E. We enforce the condition that the battery acts as a energy reservoir and not as an entropy sink (that would make the task of cooling trivial) by assuming the work source to be in state ρ^w_R=𝕀/d (we can also interpret this as the work-source being at temperature +∞). These assumptions on the work source are justified by the fact that it fulfills the second law of thermodynamics.

By this statement we mean the following. Suppose we want to use a non-equilibrium state ρ of some system S with Hamiltonian H_S to extract work and put it as average energy into the work-source. To do this we implement a (catalytic) thermal operation on the heat bath, system S and the work-source R. Then the increase of energy on the work-source (i.e., the work) Δ E_R is bounded by the non-equilibrium free energy of the system as

    Δ E_R ≤Δ F_β(ρ,H_S).

This is shown in Appendix <ref>.

Now we will show that the third law can be obtained, in the sense that both E and d have to diverge in order to be able to use R to cool down a system to zero final temperature. Let us first recall that by Lemma <ref> a sufficient and necessary condition for cooling for such a resource is given by (<ref>). Furthermore, the vacancy of the work source is given by

    V_β(ρ_R^w,H_R^w)   =   S(ω_β(H_R^w)𝕀/d)
       =   ( ω_β(H_R^w)(log (ω_β(H_R^w) - log (𝕀/d)))
       =    -β( ω_β(H_R^w)H_R^w) + log(d)
    - log(Z_β (H_R^w))
       ≤    -β E/2 +log(d) - log(Z_β (H_R^w)).

The partition function can be upper bounded as

    Z_β(H_R^w)   =   ∑_x=-d/2^d/2 e^-βEx/d
       ≥    e^β E/2 + (d-1) e^-β E/2
       =    e^β E/2 d( 1/d + d-1/d e^-β E).

Hence, we find that

    V_β (ρ_R^w,H_R^w) ≤ - log( 1/d + d-1/d e^-β E).

Combined with (<ref>), this implies that a necessary condition for cooling to a state ρ_S is given by

    V_β ( ρ_S,H_S) ≤ - log( 1/d + d-1/d e^-β E).

Most importantly, note that in order to obtain a state ρ_S that is close to a Gibbs state at zero temperature the r.h.s. of (<ref>) has to diverge. For this to be possible both E and d have to diverge, since

    lim_d→∞V_β(ρ_R^w,H_R^w)    ≤   β E,
    lim_E→∞V_β(ρ_R^w,H_R^w)    ≤   log (d).

This implies the unattainability principle in the sense that an infinitely dense spectrum with unbounded energy is needed for cooling to absolute zero.

Before coming to the proof of Thm. <ref> and our conclusions, let us briefly comment on a different model of work and a possible source of confusion that might arise. A  model of work known as a work-bit has been used in the literature of thermal operations <cit.>. In this model, it is assumed that the work-source is a two-level system with energy-gap W that undergoes a transition from the excited state |W⟩ to the ground-state |0⟩ to implement a transition on a system S. Using the results of Ref. <cit.> one can show that it is possible to cool a system to the ground-state in this model as long as W> log Z_β, where Z_β is the partition function of the system S. This means that there exists a (catalytic) thermal operation that implements cooling in the sense that

    ω_β(H_S)⊗W↦0⊗0.

At first this seems to be in conflict with our results. However, using the vacancy, it is easy to show that the above process is extremely unstable: it only works for pure initial states of the work-bit. Indeed, one can use condition (<ref>) to establish limits on cooling if the initial state on the work-bit is any full-rank state that approximates the excited state W to arbitrary but finite precision. A simple calculation for a initial state of the work system as ρ=(1-ϵ)W +ϵ0 with Hamiltonian H_W=W W yields the bound

    V_β(ρ,H_W)≤ - log( ϵ(1-ϵ))   ∀  W.

This implies that perfect cooling is impossible for any value of W –even diverging– if the initial state of the work-bit is a full-rank state.










§ PROOF OF THM. <REF>

We will now proof Thm. <ref>. Before we go to the details, let us first  explain the general logic behind the result. It is clear that to obtain a single necessary and sufficient condition for cooling at low temperatures, we have to show that the infinite set of second laws in eq. (<ref>) collapse to a single condition.
The first important step in the proof is the following Lemma.

Let β>0 and a Hamiltonian H_S be given. There exists a critical inverse temperature β_ cr such that for all β_S>β_ cr and

    α↦ S”_α(ω_β_S(H_S)ω_β(H_S)) ≤ 0.

and

    S_∞ (ω_β_S(H_S)ω_β(H_S)) ≤log Z_β.

Here, the critical value δ(β_S) is given by

    δ(β_S) = log (Z_β)/_β(ω_β_S(H_S),H_S).


See appendix <ref>.


 Using this result, we can now upper bound the Renyi-divergence on the target by its linear approximation at the origin in this parameter regime.
Since S'_0(ρ || ω_β(H)) = S(ω_β(H) || ρ)=_β(ρ,H), we get

    S_α(ω_β_S(H_S)||ω_β(H_S)) ≤_β(ω_β_S(H_S),H_S)α,  ∀α≤α_c.

Secondly, for small enough target temperatures we also have S_∞(ω_β_S(H_S)||ω_β(H_S)) ≤_β(ω_β_S(H_S),H_S)α_c. Since α↦ S_α is monotonously increasing, the second laws in eq. (<ref>) are hence also satisfied if

    S_α(ρ_R || ω_β(H_R)) > _β(ω_β_S(H_S),H_S)α,  ∀α≤α_c.

For small temperatures, we can further restrict the range of α to the interval [0,δ(β_S)), where δ(β_S) is given by:

    δ(β_S) = S_∞(ω_β_S(H_S)||ω_β(H_S))/_β(ω_β_S(H_S),H_S).


The final step is now given by bounding the Renyi-divergence of the resource S_α(ρ_R || ω_β(H_R)). In particular, if we knew that it was convex (such as in the case of a thermal resource with E^R_β being convex), we could lower bound it by its linear approximation at the origin and obtain the necessary and sufficient condition (<ref>).

In the general case, S_α(ρ_R || ω_β(H_R)) will not be convex. But we can use that we only have to check small values of α<δ(β_S) and simply Taylor expand S_α(ρ_R || ω_β(H_R)). Using Taylor's theorem we then obtain

    S_α(ρ_R || ω_β(H_R)) ≥_β(ρ_R,H_R)α - k(β_S,β,ρ_R,H_R)α^2.

This yields as new sufficient condition

    _β(ρ_R,H_R)α - k(β_S,β,ρ_R,H_R)α^2 ≥_β(ω_β_S(H_S),H_S)α,

for all 0<α≤δ(β_S). The function k(β_S,β,ρ_R,H_R)≥ 0 is given by

    k(β_S,β,ρ_R,H_R) = max{0,-min_α≤δ(β_S) S”_α(ρ_R ω_β(H_R))}.

We can now divide the sufficient condition by α and, since k(β_S,β,ρ_R,H_R)≥ 0, replace α by δ(β_S) to arrive at the final sufficient condition

    _β(ρ_R,H_R) - K(β_S,β,ρ_R,H_R,H_S) ≥_β(ω_β_S(H_S),H_S),

with K(β_S,β,ρ_R,H_R,H_S) = k(β_S,β,ρ_R,H_R)δ(β_S). This finishes the proof. 








§ SUMMARY

In this work we have investigated the limits on low temperature cooling when arbitrary systems out of equilibrium are used as a resource. We provide sufficient and necessary conditions that establish novel upper and lower bounds on the amount of resources that are needed to cool a system close to its ground state. We found that the limitations are ruled by a single quantity, namely the vacancy. This is remarkable, since at higher temperatures there is an infinite family of “second laws” that need to be checked to determine whether a non-equilibrium state transition is possible.

We have only focused on the amount of non-equilibrium resources, as we assume access to an infinite heat bath and we leave considerations about the time and complexity of the cooling protocol aside. These other kind of resources have been explored in other complementary works on the third law <cit.>. It would be interesting to see if the the vacancy plays a role to express the limitations on the size of the heat bath or any other resources that diverge when cooling a system to absolute zero. More particularly, it is an interesting question for future work to obtain the optimal sufficient scaling of the size of the heat bath and the potential “catalyst” τ that is needed to cool the system to the final low temperature <cit.>.
In this work, we have required the catalyst to be returned exactly. The necessary condition (<ref>) and the resulting quantitative unattainability principle is, however, stable when one requires instead that the vacancy of the catalyst only changes little (see appendix <ref> for a discussion of approximate catalysts). We leave it as an open problem to study how the sufficient condition behaves in such an approximate scenario.



The results of Sec. <ref> suggests that for a large class of physically relevant systems the third law can be expressed simply as the monotonicity of the vacancy. It would be of interest to specify more general assumptions on a many-body system so that this is the case. On the other hand, there exist systems for which the vacancy is not a sufficient condition. This open the possibility to have families of resources that, although are out of equilibrium, are useless for cooling. We leave this as an open question for future work.

Lastly, we note that in this work we have focused on the expenditure of non-equilibrium resources for low temperature cooling, which are precisely the resources that are employed in laser cooling <cit.>. We leave as an interesting open research direction to analyse protocols of laser cooling in the light of the bounds obtained here.


Acknowledgments.
We thank Lluis Masanes, Joseph M. Renes and Jens Eisert for interesting discussions and valuable feedback. This work has been supported by the ERC (TAQ), the DFG (GA 2184/2-1) and the Studienstiftung des Deutschen Volkes.









§ PROOF OF CONCAVITY OF RENYI DIVERGENCE FOR LOW TEMPERATURES


In this section we proof the Lemma <ref> about concavity of the Renyi-divergence at low temperatures. The Lemma holds for any Hamiltonian with pure point spectrum, a gap above the ground-state and  the property that the partition sum exists for any positive temperature.

Let β>0 and a Hamiltonian H_S with ground-state degeneracy g_0 be given. There exists a critical inverse temperature β_ cr such that for all β_S>β_ cr and for all 0<α<δ(β_S) we have

    α↦ S”_α(ω_β_Sω_β) ≤ 0.

and

    S_∞ (ω_β_Sω_β) ≤log Z_β.

Here, the critical value δ(β_S) is given by

    δ(β_S) = log (Z_β)/_β(ω_β_S(H_S),H_S) <1.





Let us first prove that the max-Renyi-divergence is upper bounded by the partition function at the environment temperature. Suppose that β_S>β and let us write

    S_α(ω_β_S(H_S)ω_β(H_S))   =1/α-1log(∑_ig_i ^-α(β_S-β)E_i(Z_β/Z_β_S)^α^-β E_i/Z_β) 
       = 1/α-1log(^-α(β_S-β)E_0(Z_β/Z_β_S)^α∑_ig_i^-α(β_S-β)(E_i-E_0)^-β E_i/Z_β),

where E_i denote the different energies of H_S, with degeneracies g_i. Assuming w.l.o.g. E_0=0, we write this as

    S_α(ω_β_S(H_S)ω_β(H_S))   = α/α-1log( Z_β/Z_β_S) + 1/α-1log(1 + ∑_i>0^-α(β_S-β)E_i^-β E_i/Z_β).

It is now obvious that in the limit we obtain

    S_∞(ω_β_S(H_S)ω_β(H_S))   = lim_α→∞S_α(ω_β_S(H_S)ω_β(H_S)) = log(Z_β) - log(Z_β_S) ≤log Z_β.


As a second step let us find the condition for which δ(β_S)<1. To do that we express the vacancy as

    _β(ω_β_S(H_S),H_S) = β_S E_β - S_β + log Z_β_S,

where we write S_β := S(ω_β(H_S)). We thus need that

    β_S E_β - S(ω_β(H_S)) > log Z_β - log Z_β_S.

Relaxing to the sufficient criterion β_S E_β - S_β > log Z_β = S_β - β E_β we thus obtain

    β_S > 2S_β - β E_β/E_β.


Let us now turn to the concavity. We will use the representation of S”_α proven in the next section, which is given by

    S”_α(ω_β_Sω_β) = 2/(1-α)^3(log Z_β_S-log Z_β̃(α) + (β_S-β̃(α))E_β̃(α)-(β_S-β̃(α))^2 Var(H)_β̃(α)),

where β̃(α)=β(1-α)+αβ_S.  Since we are only interested in α<δ(β_S)<1, we have β≤β̃(α) < β_S. We therefore have to show that the terms in the parenthesis are negative. Let us use that the average energy is monotonic with β and that Z_β̃(α)>1 to bound these terms as

    parenthesis    ≤log Z_β_S + (β_S-β̃(α))E_β̃(α)-(β_S-β̃(α))^2 Var(H)_β̃(α)
       ≤log Z_β_S + (β_S-β)E_β-(β_S-β̃(α))^2 Var(H)_β̃(α)
       ≤log(d) + (β_S-β)E_ max-(β_S-β̃(α))^2 min_x∈ [β,β̃(α)]Var(H)_x.

Now we bound β̃(α).  To do that we use that β̃(α)≤β̃(δ(β_S))=: β̃^*(β_S). It is clear that if we can bound β̃^*(β_S) by a constant, the terms in the parenthesis become negative for some β_S since the second order term in β_S dominates. To see that β̃^*(β_S) is indeed upper bounded by a constant, we again write the vacancy as

    _β(ω_β_S(H_S),H_S)    = -S(ω_β)+ β_S E_β + log Z_β_S

to obtain

    β^*:= lim_β_S→∞β̃^*(β_S)    = lim_β_S→∞β(1-δ(β_S)) + δ(β_S)β_S
       = β + lim_β_S→∞log Z_β/β_S E_β +log Z_β_S-S(ω_β(H_S))β_S = β+ log Z_β/E_β.

This finishes the proof that β_ cr as claimed in the Lemma exists. We also note that the function β̃^*(β_S) is monotonically decreasing for all β_S such that β̃^*(β_S)<1.
Finally, note that eq. (<ref>) allows to give upper bounds on β_ cr once one has lower bounds on the energy variance for inverse temperatures in the interval [β,β^*].











§ RENYI DIVERGENCE BETWEEN THERMAL STATES

Here, we will specialize to the situation where the resource states are thermal, with inverse temperature β_R. We now calculate the Renyi divergence for α<1 in this case. We first write:

    S_α (ω_β_R || ω_β)    = -α/α-1log Z_β_R + log Z_β + 1/α-1log(^-β_R Hα^-β H(1-α)) 
       = -α/α-1log Z_β_R + log Z_β + 1/α-1log Z_(β_R-β)α + β
       = -α-1/α-1log Z_β_R + log Z_β + 1/α-1log(Z_(β_R-β)α + β/Z_β_R) 
       = log(Z_β/Z_β_R) + 1/α-1log(Z_(β_R-β)α + β/Z_β_R).


We will now show that the function is convex provided β_R<β and that the  function x↦ E_β_R+x is convex for 0≤ x ≤β-β_R.  For the second derivative (with β̃=(β_R-β)α + β) we obtain:

    S_α (ω_β_R || ω_β)”   = 2/(1-α)^3log Z_β_R -2/(1-α)^3log Z_β̃ - 21/(1-α)^2∂_αlog Z_β̃ + 1/α-1∂^2_αlog Z_β̃
       = 2/(1-α)^3log Z_β_R -2/(1-α)^3log Z_β̃ - 21/(1-α)^2(β-β_R)E_β̃ - 1/1-α(β-β_R)^2 Var(H)_β̃
       =2/(1-α)^3[log Z_β_R - log Z_β̃ -(1-α)(β-β_R)E_β̃ - (1-α)^2/2(β-β_R)^2Var(H)_β̃].

Utilizing (1-α)(β-β_R) = β̃ - β_R, we can write this as

    S_α (ω_β_R || ω_β)”   = 2/(1-α)^3[log Z_β_R - log Z_β̃ - (β̃-β_R)E_β̃ - (β̃-β_R)^2/2Var(H)_β̃].

Here, we have introduced the average energy E_β and the variance Var(H)_β = ⟨ H^2⟩_β - ⟨ H⟩_β^2, which fulfill ∂_x E_x = - Var(H)_x. With these expressions at hand, we will now show Thm. <ref> and another result about about the convexity of Renyi divergences for sufficiently large reference temperature β.



 §.§ Proof of Thm. <ref>



We need to show that the r.h.s. of (<ref>) is positive with the premise that x ↦ E_x is convex in x∈ [ β_R,β], β_R ≤β and α <1. This last condition on α implies that we need to show that


    log Z_β_R - log Z_β̃≥ (β̃-β_R)E_β̃ + (β̃-β_R)^2/2Var(H)_β̃.

We use an integral representation of the l.h.s:

    log Z_β_R - log Z_β̃   = -∫^β̃-β_R_0d/d xlog Z_β_R+x dx = ∫^β̃-β_R_0 E_β_R+xd x.

Hence, we conclude that what needs to be shown is

    ∫^β̃-β_R_0 E_β_R+xd x ≥ (β̃-β_R)E_β̃ + (β̃-β_R)^2/2Var(H)_β̃.

Whether this inequality is satisfied, and thus, S_α(ρ_Rω_β(H)) is convex, is entirely determined by the function x↦ E_x.  This is due to the fact that the derivative of E_x is given by -Var(H)_x, so that the right hand side can be seen as a linear approximation to the function E_x. A geometrical interpretation is provided in Fig <ref> showing that it is trivially satisfied when E_x is convex. This finishes the proof.

As a final remark, although not useful to obtain bounds on the third law, we note that a completely analogous argument implies that if α<1, E_x is convex but in contrast to the previous case β_R≥β, then it is fulfilled that

    ∫^β̃-β_R_0 E_β_R+xd x ≤ (β̃-β_R)E_β̃ + (β̃-β_R)^2/2Var(H)_β̃.

This shows that in the case of resources colder than the bath, the function S_α(ρ_Rω_β(H)) is concave.







 §.§ Very cold heat baths


We will now show that in the case of very cold heat baths (very large β) we also have that S_α(ω_β_Rω_β) is convex, and hence (<ref>) becomes sufficient and necessary.

 For any resource of the form (ω_β_R(H_R),H_R), given a fixed β_R there exist a sufficiently large value of β so that (<ref>) is a sufficient and necessary condition for low temperature cooling.


We only give a sketch and show that S_α(ω_β_Rω_β) is convex for values of α<α_c, where α_c<1 is chosen arbitrarily. Recalling Eq. <ref>, we then need to show that

    ∫^β̃-β_R_0 E_β_R+xd x ≥ (β̃-β_R)E_β̃ + (β̃-β_R)^2/2Var(H)_β̃.

Note that in the limit of large β the scaling of the r.h.s. of (<ref>) is such that β̃-β_R=(1-α)(β-β_R) scales proportionally to β, while E_β̃ and Var(H)_β̃ scale as e^-kβ. Therefore, the r.h.s. of (<ref>) approaches zero as β→∞ whereas the l.h.s. grows monotonically with β. Hence, (<ref>) is fulfilled which concludes the proof.





§ EQUIDISTANT LEVELS

Here, we consider the particular case of a system with M+1 equidistant levels and show that the function E_β is convex. The energy-gap between subsequent levels is Δ and we set the ground-state energy to zero. The energy E_β then takes the form

    E_β = 1/^βΔ-1Δ  - M+1/^(M+1)Δβ-1Δ.

In particular, for M→∞ we obtain results for the harmonic oscillator and for M=1 for a qubit. We have to prove that the second derivative is positive, i.e.,

    E”_β = 1/8Δ^3[sinh(βΔ)/sinh(βΔ/2)^4-(M+1)^3sinh((M+1)βΔ)/sinh((M+1)βΔ/2)^4_=:f(β,M+1)] ≥ 0.

For M=0 this is clearly true. We will set M+1=:γ and show that ∂_γ f(β,γ) ≤ 0. We have

    ∂_γ f(β,γ) = -γ^2 1/sinh(γβΔ/2)^4[γβΔ(2+cosh(γβΔ))-3sinh(γβΔ)].

In the following, set γβΔ=x. Due to the negative pre-factor, we are done if we can show

    x(2+cosh(x)) - 3sinh(x) ≥ 0.

We will show this using a Taylor-expansion:

    2x + x cosh(x) - 3sinh(x)    = 2x+∑_n=0^∞ x^2n+1(1/(2n)!-3/(2n+1)!)
       = 2x+∑_n=0^∞ x^2n+1(((2n+1)-3)(2n)!/(2n)!(2n+1)!)
       = 2x - 2x + ∑_n=1^∞ x^2n+1(((2n+1)-3)(2n)!/(2n)!(2n+1)!)
       ≥ 0.




§ THE WORK SOURCE MODEL


Here we show that a work source of the form (ρ_R^w, H_R^w) as given in the main text fulfills the second law of thermodynamics. Let us consider an arbitrary system (ρ_S,H_S) and let us consider catalytic thermal operations on SW. We will show that the maximum amount of mean energy that one can store on the work source is bounded by the initial non-equilibrium free energy of S. Let us recall, see for instance Ref. <cit.>, that the free energy difference is given by

    Δ F_β (ρ,H):=1/β S(ρω_β(H)) =F(ρ,H) - F(ω_β(H)),

where F(ρ,H)= (ρ H) - 1/β S(ρ) is the free energy.

The protocol of work extraction is a transition of the form
 
    ρ_SW^i:=ρ_S ⊗ρ_R^w →ρ_SW^f= E(ρ_SW^i)

 where E is any channel that has the Gibbs state as a fixed point. Monotonicity of Δ F_β under channels of the form E imply that

    Δ F^f_β := Δ F_β(ρ_SR^f,H_S+ H_R) ≤Δ F_β(ρ_SR^i,H_S+ H_R) := Δ F_β^i.

Combining this last equation with super-additivity and additivity of the relative entropy one can easily find, following a similar reasoning as in Ref. <cit.>, that

    Δ E_R ≤Δ F_β (ρ_S,H_S)

where Δ E_R= (𝕀⊗ H_R^w  ρ_SR^f)-  (𝕀⊗ H_R^w    ρ_SR^i) is the mean energy stored in the work source R.



§ ARBITRARY TARGET-STATES CLOSE TO THE GROUND-STATE

In this section we prove a result similar to our general sufficient condition for cooling, but where we consider target states of the form

    ρ_ϵ = (1-ϵ) |0⟩⟨0| + ϵρ^⊥,  ϵ≪ 1.

where ρ^⊥ is a density matrix which has full rank on the subspace orthogonal to the ground-state |0⟩ and commutes with the Hamiltonian H_S.

  For every choice of β, H_S and ρ^⊥ as above, there is a critical ϵ_cr>0 such that for any ϵ < ϵ_ cr the condition

    V_β(ρ_R,H_R) +K̃(ϵ,β,ρ_R,H_R,H_S,ρ^⊥) ≥ V_β(ρ_ϵ,H_S)

is sufficient for cooling. The function K̃ has the property K̃(ϵ,β,ρ_R,H_R,ρ^⊥) → 0 as ϵ→∞ for any fixed β,H_R,ρ_R,ρ^⊥ and H_S.


The proof of this theorem is essentially identical to the one of Theorem <ref>. The only difference is that instead of Lemma <ref>, we use the following Lemma:

Let H_S be a d-dimensional Hamiltonian with ground-state |0⟩ and H|0⟩=0. Let β>0 be fixed and consider the state

    ρ_ϵ = (1-ϵ) |0⟩⟨0| + ϵρ^⊥,

with rank(ρ^⊥)=d-1, ρ^⊥|0⟩=0 and [ρ^⊥,H_S]=0. Then there exists an ϵ_cr>0 such that for all α<δ(ϵ)

    d^2/dα^2 S_α(ρ_ϵ||ω_β(H_S)) <0,   ∀ϵ<ϵ_cr.

Here, δ(ϵ) fulfills

    δ(ϵ) = log Z_β/_β(ρ_ϵ,H_S) < 1,  ∀ϵ<ϵ_cr.



We will now proof this Lemma. Let us then express the Renyi-divergence as

    S_α(ρ_ϵ||ω_β(H_S))    = 1/α-1log((1-ϵ)^α + ϵ^α((ρ^⊥)^α^-β H_S(1-α))) + log(Z_S) 
       =: 1/α-1log (f(α)) + log(Z_S).

As is apparent from the expression, in the following we will often encounter the functions

    f̃(α)    := ((ρ^⊥)^α^-β H(1-α)),
    
    f_ϵ(α)    := (ρ_ϵ^α^-β H(1-α)) = (1-ϵ)^α + ϵ^αf̃(α).

It is useful to remember from the main text that ρ^⊥ is a normalized quantum state that commutes with H and has rank d-1.
In the following we will also often write S_α instead of S_α(ρ_ϵ||ω_β(H_S)) and simply f_ϵ or f_ϵ,α instead of f_ϵ(α) to simplify the notation (similarly for f̃). While f_ϵ and f̃ are structurally essentially the same, it is important to keep in mind that only f_ϵ, and not f̃, depends on ϵ.

We now have to prove that S_α is concave for small enough ϵ, i.e., have to show that there exists a ϵ_cr>0 such that  its second derivative is negative for ϵ < ϵ_cr.
The second derivative of S_α can be computed straightforwardly and gives

    S”_α   = -2/(1-α)^3log f_ϵ - 2/(1-α)^2f_ϵ'/f_ϵ + 1/1-α((f_ϵ'/f_ϵ)^2-f_ϵ”/f_ϵ).

To go on we need to establish a few properties of functions like f_ϵ and f̃. We will collect these properties in a series of Lemmata.


Let ρ be a quantum state and σ be a positive semi-definite operator with [ρ,σ]=0. Define f(α):=(ρ^ασ^1-α).
Then

    (f')^2 - f” f ≤ 0,   0<α<1.


A simple calculation shows that

    f'(α)    = (ρ^α (log(ρ) - log(σ)) σ^1-α),
    
    f”(α)    = (ρ^α (log(ρ) - log(σ))^2 σ^1-α).

We now use the Cauchy-Schwartz inequality |(A^† Bρ)|^2 ≤(A^† A ρ)(B^† Bρ) with A=ρ^-α/2 (log(ρ) - log(σ)) σ^α/2 and B = ρ^-α/2σ^α/2 to obtain (note the change from α to 1-α)

    f'(1-α)^2    = (ρ^-α/2 (log(ρ) - log(σ)) σ^α/2ρ^-α/2σ^α/2ρ)^2
       ≤(ρ^-α (log(ρ) - log(σ))^2 σ^αρ)(ρ^-ασ^αρ) 
       = (ρ^1-α (log(ρ) - log(σ))^2 σ^α)(ρ^1-ασ^α) 
       = f”(1-α) f(1-α).





Let H be a Hamiltonian with ground-state energy E_0=0 and let sigma be a quantum state with [σ,H]=0. Then

    f(α) := (σ^α^-β H(1-α)) ≤ Z,   0≤α≤ 1.


From the calculation of the previous Lemma we see that the second derivative of f is the trace of a product of positive commuting operators. Hence it is always positive and therefore f is convex. But since H≥ 0 we have f(0) = Z ≥ 1 = f(1) and from convexity we get f(α) ≤ Z.



Note that due to our assumption about the groundstate energy we have Z_S≥ 1 and from the above Lemma we know f_ϵ≤ Z_S. We will now show that for every 0 < α_c' < 1, we have 1≤ f_ϵ(α)≤ Z_S if ϵ is small enough and α < α_c'.


For any 0< α_c'< 1 there exists a critical ϵ'_cr>0, such that for all ϵ < ϵ'_cr we have

    f_ϵ(α)    := (ρ_ϵ^α^-β H(1-α)) ≥ 1,    0≤α < α_c'.


Assume some 0 < α < α_c'. Using that f̃ is independent of ϵ and positive, we can lower bound it by some f̃_min>0. Also, (1-α)^α is monotonically decreasing with α for 0<α < 1. We therefore get the lower bound

    f_ϵ(α) = (1-ϵ)^α + ϵ^αf̃(α) ≥ (1-ϵ) + ϵ^αf̃_min = 1 + ϵ(ϵ^α-1f̃_min-1).

Thus for ϵ < ϵ'_cr (α_c') := (f̃_min)^1/1-α_c we have f_ϵ(α) ≥ 1.



Due to the preceding Lemma, we will in the following take the (somewhat arbitrary) choice α_c' = 1/3 and only consider α < α_c' as well as values of ϵ < ϵ_cr'(α_c'). Since later we are anyway only interested in arbitrarily small values of ϵ and α≤δ(ϵ), this is no obstruction.



For all α < α_c' and ϵ < ϵ_cr' we have f_ϵ'(α) ≤ 0.

Follows from f_ϵ(α) ≥ 1 for all α≤α_c' together with the facts that f_ϵ(1)=1,f_ϵ(0)=Z_S and that f_ϵ is convex.





We are now in position to go on with the proof of the asymptotic concavity. First, we will further restrict the values of α by choosing arbitrarily α_c < α_c'=1/3 and restricting to α≤α_c. The reason for this will become clear later in the proof.

Considering eq. (<ref>) and using Z_S ≥ f_ϵ≥ 1 as well as Lemma <ref>, we can upper now bound the second derivative as

    S”_α   ≤ - 2/(1-α)^2f_ϵ'/f_ϵ + 1/(1-α)f_ϵ^2((f_ϵ')^2-f_ϵ”f_ϵ) 
       ≤ - 2/(1-α_c)^2f_ϵ'/f_ϵ + 1/Z_S^2((f_ϵ')^2-f_ϵ”f_ϵ).

One might be tempted to use Lemma <ref> and simply upper bound the second term by zero, but that bound would be too weak, since the first term diverges as log(1/ϵ). We will therefore now have to do a more detailed calculation. We first compute the derivatives of f_ϵ:

    f_ϵ,α'    = (1-ϵ)^αlog(1-ϵ) + log(ϵ)ϵ^αf̃_α + ϵ^αf̃'_α,
    
    (f_ϵ,α')^2    = (1-ϵ)^2αlog(1-ϵ)^2 + log(ϵ)^2ϵ^2αf̃^2_α + ϵ^2α (f̃'_α)^2 + 2(1-ϵ)^αlog(1-ϵ)log(ϵ)ϵ^αf̃_α
          + 2(1-ϵ)^αlog(1-ϵ)ϵ^αf̃'_α + 2 log(ϵ)ϵ^2αf̃_αf̃'_α.
    
    f_ϵ,α”   = (1-ϵ)^αlog(1-ϵ)^2 + log(ϵ)^2ϵ^αf̃_α+2log(ϵ)ϵ^αf̃'_α + ϵ^αf̃”_α.

These give

    (f_ϵ,α')^2 - f_ϵ,α”ϵ^αf̃_α   = (1-ϵ)^2αlog(1-ϵ)^2 + log(ϵ)^2ϵ^2αf̃^2_α + ϵ^2α (f̃'_α)^2 + 2(1-ϵ)^αlog(1-ϵ)log(ϵ)ϵ^αf̃_α
          + 2(1-ϵ)^αlog(1-ϵ)ϵ^αf̃'_α + 2 log(ϵ)ϵ^2αf̃_αf̃'_α
          - ϵ^αf̃_α((1-ϵ)^αlog(1-ϵ)^2 + log(ϵ)^2ϵ^αf̃_α+2log(ϵ)ϵ^αf̃'_α + ϵ^αf̃”_α) 
       = log(1-ϵ)^2 (1-ϵ)^α((1-ϵ)^α -ϵ^αf̃_α)  + ϵ^2α (f̃'_α)^2 
          + 2(1-ϵ)^αlog(1-ϵ)log(ϵ)ϵ^αf̃_α + 2(1-ϵ)^αlog(1-ϵ)ϵ^αf̃'_α - ϵ^2αf̃”_αf̃_α
       = log(1-ϵ)^2 (1-ϵ)^α((1-ϵ)^α -ϵ^αf̃_α)  + 2(1-ϵ)^αlog(1-ϵ)ϵ^α(log(ϵ) f̃_α + ϵ^αf̃'_α) 
          +  ϵ^2α((f̃'_α)^2- f̃”_αf̃_α).

Hence we have

    (f_ϵ,α')^2 - f_ϵ,α” f_α   = -(1-ϵ)^α f_ϵ,α” + log(1-ϵ)^2 (1-ϵ)^α((1-ϵ)^α -ϵ^αf̃_α) 
          + 2(1-ϵ)^αlog(1-ϵ)ϵ^α(log(ϵ) f̃_α + ϵ^αf̃'_α) +  ϵ^2α((f̃'_α)^2- f̃”_αf̃_α).

Note in particular that the last term is negative semi-definite due to Lemma <ref>.

Let us now also write 0< 1/k_c := (1-α_c)^2 < 1. Inserting the previous result into eq. (<ref>) we then obtain

    S”_α   ≤ - 2 k_c f_ϵ,α' + 1/Z^2((f_ϵ,α')^2-f_ϵ,α” f_α) 
       ≤ -2 k_c f_ϵ,α' + 1/Z^2( -(1-ϵ)^α f_ϵ,α”
      + log(1-ϵ)^2 (1-ϵ)^α((1-ϵ)^α -ϵ^αf̃_α) .
                     .+ 2(1-ϵ)^αlog(1-ϵ)ϵ^α(log(ϵ) f̃_α
      + ϵ^αf̃'_α) )
       ≤ -2 k_c f_ϵ,α' + 1/Z^2( -(1-ϵ) f_ϵ,α”
      + log(1-ϵ)^2 (1 -(1-ϵ)ϵf̃_α) .
                     .+ 2log(1-ϵ)(log(ϵ) f̃_α
      + ϵ^2α(1-ϵ)^αf̃'_α) )

We now lower bound f_ϵ' and f_ϵ” as

    f_ϵ,α'    ≥log(1-ϵ) + log(ϵ)f̃_α + f̃_α'
    
    f_ϵ,α”   ≥ (1-ϵ)log(1-ϵ)^2 + log(ϵ)^2 ϵ^α_cf̃_α + 2log(ϵ)ϵ^αf̃'_α + ϵf̃”_α.

Note that we cannot easily bound the terms involving f̃'_α since we do not know the sign of f̃'_α.  However, we emphasize again that f̃ is independent of ϵ and can hence essentially be treated as constant. Putting in the bounds then yields

    S”_α   ≤ -2k_c (log(1-ϵ) + log(ϵ)f̃_α + f̃_α') -1-ϵ/Z^2((1-ϵ)log(1-ϵ)^2 + log(ϵ)^2 ϵ^α_cf̃_α + 2log(ϵ)ϵ^αf̃'_α + ϵf̃”_α) 
         + 1/Z^2( log(1-ϵ)^2 (1 -(1-ϵ)ϵf̃_α) + 2log(1-ϵ)(log(ϵ) f̃_α
      + ϵ^2α(1-ϵ)^αf̃'_α) )
       = log(ϵ)f̃_α(2/Z^2log(1-ϵ)-1-ϵ/Z^22ϵ^αf̃'_α/f̃_α -1-ϵ/Z^2log(ϵ)ϵ^α_c - 2/(1-α_c)^2) 
          + log(1-ϵ)( - 2/(1-α_c)^2 - (1-ϵ)^2/Z^2log(1-ϵ) + 1-(1-ϵ)ϵf̃_α/Z^2log(1-ϵ) + 2/Z^2ϵ^2α(1-ϵ)^αf̃'_α)
          -2/(1-α_c)^2f̃'_α - (1-ϵ)ϵ/Z^2f̃”_α
       ≤log(ϵ)f̃_α(2/Z^2log(1-ϵ)-1-ϵ/Z^22ϵ^αf̃'_α/f̃_α -1-ϵ/Z^2log(ϵ)ϵ^α_c - 2/(1-α_c)^2) + M(ϵ, H,β,ρ^⊥) - K(α_c,H,β, ρ^⊥),

where M goes to zero as ϵ goes to zero and K is independent of ϵ. Also note that M is bounded and independent of α (due to the boundedness of f̃_α and its derivatives).
Let us define m(α_c) = max_α≤α_cf̃'_α/ f̃_α. Since α_c < 1/2 we can simplify the bound to

    S”_α   ≤log(ϵ)f̃_α/Z^2(2 log(1-ϵ)-2(1-ϵ)ϵ^α m(α_c) -(1-ϵ)log(ϵ)ϵ^α_c - 8 Z^2 ) + M(ϵ, H,β,ρ^⊥) - K(α_c,H,β, ρ^⊥).

Clearly S”_α can be made negative by taking ϵ and α_c arbitrarily small since the dominant term in the bracket goes as -log(ϵ). However, since our objective is to upper bound S_α by V_β(ρ_ϵ,H_S) α for all α≤α_c, we also need that V_β(ρ_ϵ,H_S) α_c ≥log Z_S and hence α_c ≥log(Z_S)/ V_β(ρ_ϵ,H_S). Hence we choose α_c = δ(ϵ) = log Z_S/ V_β(ρ_ϵ,H_S) and hope for the best. The vacancy is given by:

    V_β(ρ_ϵ,H_S)    = - log(1-ϵ)1/Z_S + Z_S-1/Z_Slog(1/ϵ) + C_1(ρ^⊥,β, H_S),

where C_1 does not depend on ϵ. Hence

    lim_ϵ→ 0ϵ^δ(ϵ) = 1/Z_S^Z_S/Z_S-1  and  lim_ϵ→ 0  - (1-ϵ)log(ϵ)ϵ^δ(ϵ) = +∞.

Since the other terms in the first bracket in S”_α go to zero as ϵ→ 0 and K is independent of ϵ, we can thus find a finite ϵ_cr such that

    S”_α≤ 0,  α≤δ(ϵ_cr).

This finishes the proof.





§ EXACTLY CONSERVED CATALYSTS


In this section we analyze the scenario where the catalyst always has to be returned without any error. That is, the cooling protocol considers a process like the one described in Sec. <ref> but taking ϵ=0. First, note that the vacancy is automatically also a monotone in this setting, since we are considering a subset of free operations. Hence, the inequality

    V_β(ρ_R,H_R) ≥ V_β(ρ_S,H_S)

is also a necessary condition for this set of free operations.
In the following, we will consider for simplicity only the case where the target system is thermal, ρ_S = ω_β_S(H_S).

We will now prove the following theorem, which provides a sufficient condition for cooling and which coincides with that in our general Theorem <ref> up to a multiplicative factor.

  Assume thermal operations with exact catalysts. Then for every choice of β and H_S there is a critical β_cr>0 such that for any β_S > β_ cr and full-rank resource (ρ_R,H_R) (diagonal in the energy eigenbasis) the condition

    V_β(ρ_R,H_R) - K(β_S,β,ρ_R,H_R,H_S) ≥ r(β,H) V_β(ω_β_S(H_S),H_S)

is sufficient for cooling. The positive semidefinite function K is identical to that in Theorem <ref> and the constant r(β,H_S) is independent of ρ_R,H_R and β_S and given by

    r(β,H_S)= 1+2E_max-E_β/E_β

where E_max is the largest eigenvalue of H_S and we assume that the ground-state energy of H_S is zero.

Before going into the proof, let us discuss briefly the implications that the correction given by r(β,H_S) has over the sufficient condition Thm. <ref>. This is better explained if we look at the scaling results of Sec. <ref>. There we showed that the sufficient condition of Thm. <ref> provides an upper bound on the number of copies of a resource that are sufficient to implement a cooling process, as given by n^suff in (<ref>). The sufficient condition laid out in Thm. <ref> implies simply that that we need r-times more systems to implement the cooling protocol, where r=r(β,H). Note importantly that r does not depend on the final temperature, so employing r(β,H_S) × n^suff is always sufficient for cooling. We emphasize that we believe that the factor r(β,H_S) can be made much closer to 1 by more elaborate proof techniques, but leave this as an open problem. 


It was shown in Ref. <cit.> that a transition ρ→ρ' between two diagonal states is possible with exact preservation of the catalyst if and only if the Renyi-divergences

    S_α(ρω_β(H)) = sign(α)/α-1log(ρ^αω_β(H)^1-α)

do not increase for all α∈ (-∞,+∞). The sufficient condition in Theorem <ref> covers all α≥ 0. We thus have to check that we can fulfill all the inequalities for α<0 using the multiplicative factor r(β,H_S). To do this we provide new lower- and upper-bounds for the Renyi-divergences for negative α. We begin with a lower-bound. Consider any state ρ with eigenvalues p_i in the energy-eigenbasis. Then we have

    S_-|α|(ρω_β(H)) = 1/|α|+1log(∑_i p_i^-|α| w_i^1+|α|),

where w_i = ^-β E_i/Z_β are the eigenvalues of the thermal state. Using concavity of the logarithm we can bound this as

    S_-|α|(ρω_β(H))    ≥1/|α|+1∑_i w_i log( p_i^-|α| w_i^|α|) = |α|/|α|+1∑_i(w_i log(w_i)- w_i log(p_i) )
       = |α|/|α|+1 V_β(ρ,H).

We can thus lower bound all the Renyi-divergences for negative α by a simple function. Later, we will apply this bound to the resource.

We will now derive a similar upper bound for the target system, i.e., assuming a system in a thermal state.
First, we rewrite the Renyi-divergences as

    S_-|α|(ω_β_S(H_S) ω_β(H_S)) = |α|/|α|+1log(Z_β_S) - log(Z_β) + 1/1+|α|log(^(β_S-β)|α|H_S^-β H_S),

which can be verified by direct calculation. We will now use the log-sum inequality. It states that for any two sets of d non-negative numbers {a_i} and {b_i} we have

    loga/b≤∑_i a_i/aloga_i/b_i,

with a=∑_i a_i and b=∑_i b_i. Let E_i be the energy-eigenvalues of H_S. Then we set

    a_i = ^(β_S-β)|α| E_i^-β E_i = ^-β̃(α)E_i,   b_i = ^-β E_i/Z_β,

where β̃(α):= β - (β_S -β)|α|. Using the log-sum inequality we then obtain

    S_-|α|(ω_β_S(H_S) ω_β(H_S))    ≤|α|/|α|+1log(Z_β_S) - log(Z_β) + 1/1+|α|∑_i ^-β̃(α)E_i/Z_β̃(α)log( ^(β_S-β)|α| E_i Z_β)
       = |α|/|α|+1(log(Z_β_S) - log(Z_β))  + |α|/|α|+1(β_S-β) ∑_i ^-β̃(α)E_i/Z_β̃(α) E_i.

Denoting by E_max the maximum energy, we then get the bound

    S_-|α|(ω_β_S(H_S) ω_β(H_S))    ≤|α|/|α|+1(log(Z_β_S) - log(Z_β) + (β_S-β) E_max).

Let us recall from Eq. (<ref>), that for thermal states, the vacancy can be expressed as a function of the non-equilibrium free energy as

    V_β(ω_β_S(H_S),H_S) = β_S Δ F_β_S(ω_β(H),H) = β_S E_β - S_β + log(Z_β_S),

where E_β and S_β denote the thermal energy expectation value and von Neumann entropy at inverse temperature β. Using this together with -logZ_β = β E_β - S_β, we can rewrite the upper bound on the Renyi-divergences as

    S_-|α|(ω_β_S(H_S) ω_β(H_S))    ≤|α|/|α|+1(  V_β(ω_β_S(H_S),H_S)[1+E_max-E_β/Δ F_β_S(ω_β(H),H)] - β(E_max-E_β)) 
       ≤|α|/|α|+1 V_β(ω_β_S(H_S),H_S)[1+E_max-E_β/Δ F_β_S(ω_β(H),H)].

Since we have

    Δ F_β_S(ω_β(H),H) = E_β - E_β_S - 1/β_S(S_β - S_β_S),

it is always possible find a critical inverse temperature β'_S such that Δ F_β_S(ω_β(H),H)≥ E_β/2 for all β_S>β'_S. Then, for all β_S larger than this critical temperature we can bound the Renyi-divergences as

    S_-|α|(ω_β_S(H_S) ω_β(H_S))    ≤|α|/|α|+1 V_β(ω_β_S(H_S),H_S)[1+2E_max-E_β/E_β] =: |α|/|α|+1 V_β(ω_β_S(H_S),H_S) r(β,H_S).

Using the lower bound (<ref>) for the resource, we then find that the inequalities for negative α are fulfilled if

    |α|/|α|+1 V_β(ρ_R,H_R) ≥|α|/|α|+1 V_β(ω_β_S(H_S),H_S) r(β,H_S)

Cancelling the prefactors, we get as sufficient condition for negative values of α the inequality

    V_β(ρ_R,H_R) ≥ V_β(ω_β_S(H_S),H_S) r(β,H_S).

Combining this with the sufficient condition for positive α, which is the sufficient condition provided by Thm. <ref>, then yields the claimed sufficient condition in the theorem.




  §.§.§ Catalysts can always be chosen with full rank

Before finishing this section, let us point out that in the case of exact catalysis, one can always choose the catalyst to have full rank. That is, the actual implementation of the cooling protocol, which is guaranteed to exist under the conditions of Thm. <ref>, never requires to employ a catalyst which is not full rank.

To see this, consider a bi-partite system with non-interacting Hamiltonian H_1+H_2.
Then consider the initial state ρ_SB⊗σ_C and apply an energy-preserving unitary operation U that results in the state ρ'_SBC with ρ'_C=σ_C.
Here, we imagine that ρ_SB also includes the state of the heat-bath and thus there can be a built-up of correlations between the catalyst and ρ_SB.
Furthermore assume that ρ_SB has full-rank and that σ_C is supported only on a subspace P⊂ H_2 with complement Q= - P (we identify the vector-space and the projector on the space). Thus P=∑_jj, where the sum is over the eigen-states of σ_C.
Let the spectrum of ρ_SB and σ_C be {p_α} and {q_j}, respectively. Then the final state

    ρ'_SBC = ∑_α,j p_α q_j Uα⊗jU^†

is a convex sum of the positive semi-definite operators Uα⊗jU^†. The sum has support only within ⊗ P since otherwise the reduced state ρ'_C would also be supported outside of P. Hence also every summand is supported within ⊗ P. Using ⊗ P = ∑_α,jα⊗j we then obtain

    (⊗ Q) U(⊗ P) U^† = ⊗ Q ∑_α,jUα⊗jU^† = 0.

In other words, we have (⊗ Q) U (⊗ P) = 0 and by a similar calculation also   (⊗ P) U(⊗ Q)=0.
Thus, the unitary U is block-diagonal. In particular, the operator V=(⊗ P)U(⊗ P) considered as an operator on the Hilbert-space H_1⊗ P is unitary.
Since U is energy-preserving by assumption, we can deduce that P=span{|E_j⟩} for some subset of energy-eigenstates |E_j⟩ of the Hamiltonian H_2 of the catalyst.

Then V commutes with the Hamiltonian H_1+ H_2|_P, where H_2|_P denotes the Hamiltonian of the catalyst, but restricted to the subspace P.

We can thus obtain an equivalent catalyst with full rank and a corresponding thermal operation by restricting σ_C and H_2 to the subspace P on which σ_C has full rank and using the thermal operation defined by V:

    V ρ_SB⊗σ_C|_P V^† = ρ'_SBC|_⊗ P.

In particular note that the above analysis also shows that, in the case of exact catalysis, pure catalysts are useless: If a transition can be done with a pure catalyst, it can also be done without a catalyst.





§ APPROXIMATE CATALYSIS


In this manuscript, we have assumed that catalysts are returned arbitrarily close to their initial state (or exactly, in the last section).
Here we will discuss possible relaxations of this assumption to include approximate catalysts.


First, we note that the problem of allowing for finite errors –in some suitable measure– between the initial and final state of the catalyst is a delicate one, specially in the context of the third law of thermodynamics.
The challenge is caused by the fact that already the statement of the unattainability principle is not stable under arbitrarily good approximations:
It compares the case where the state of the target system is exactly the ground-state with the case of approximating the  ground-state to arbitrary precision.
In the latter case, infinite resources are needed while in the latter case finite resources are needed (however diverging with the approximation precision).
This is the ultimate reason why a discontinuous measure of resources (like the vacancy) is necessary to capture the third law in the resource theoretic setting.

With this in mind, let us discuss the problem of approximate catalysts. If one demands that the catalyst is returned in approximately the same state,  it is crucial how one measures “approximately”. In the context of thermal operations, this problem has been studied in Refs. <cit.>. It has been shown in Ref. <cit.>, that if one requires only that the catalyst is returned up to an arbitrarily small but fixed error in trace-distance, any transition can be implemented using a thermal operation to arbitrary precision – without any resource. In particular, this implies that perfect cooling can be achieved without using any resource state. Therefore, it is clear that stronger conditions are necessary to not trivialize problem of cooling.

A second way to define approximate catalysts is to require that the catalyst is returned up to an error ϵ/log(d) in trace-distance, where d is the dimension of the catalyst, and ϵ>0 is arbitrarily small but fixed for all catalysts.
Intuitively, this definition requires that the error is small even when multiplied by the number of particles in the catalyst.
In this case, transitions can be implemented to arbitrary precision if the non-equilibrium free energy decreases <cit.>.
This would lead to a constant amount of resources needed to cool a given system to arbitrary low temperatures – hence the unattainability principle is also in this case violated.

Because of the arguments above, it seems allowing for a finite error –measured in trace-distance– in catalyst seems to be too forgiving.
However, Ref. <cit.> also hints at a solution to this problem: One should measure the error in terms of a quantity that is meaningful for the problem at hand. In Ref. <cit.> the authors consider the problem of work extraction and demand in turn that the catalyst is returned with approximately the same “work distance”, where the work distance measures the potential of one state to produce work. In our case, we are concerned with the task of low-temperature cooling. Indeed, the vacancy itself plays the role of the cooling potential, since the limitations for low temperature cooling of Thm. <ref> are expressed in terms of the vacancy. We can thus require that the catalyst has to be returned with a vacancy that differs only by an amount ϵ from the initial vacancy. If we adopt this definition of approximate catalysts, the general necessary condition (<ref>) is modified only slightly.
This can be seen in the following way. First note that this notion requires that catalysts all have finite vacancy, i.e., they must have full rank. In this case, we can simply evaluate the vacancy of the resource, system and target before and after the cooling protocol has been applied. Let us assume that the initial state of the catalyst is σ, while the final state is σ'. Since the vacancy is an additive monotone of thermal operations and vanishes on thermal states, we then obtain

    V_β(ρ_R⊗ω_β(H_S)⊗σ,H_R+H_S+H_C)    =  V_β(ρ_R,H_R) +  V_β(σ,H_C) 
       ≥ V_β(ω_β(H_R)⊗ρ_S ⊗σ', H_R+ H_S +H_C) = V_β(ρ_S,H_S) +  V_β(σ',H_C).

We hence obtain as a new necessary condition

    V_β(ρ_R,H_R) + ϵ≥ V_β(ρ_S,H_S),

with ϵ =  V_β(σ,H_C) -  V_β(σ',H_C) being the error in the catalyst measured by the vacancy.


Thus the necessary condition and hence the quantitative unattainability principle is stable under approximate catalysts if defined consistently: allowing a fixed but small error measured by the vacancy difference.

It seems plausible that under this definition of catalysts, also the sufficient condition in Theorem <ref> simplifies to (<ref>) for arbitrary resources – at least for low enough target temperatures. However, proving this statement rigorously seems to require further technical innovations beyond the scope of this work. We therefore leave this as an open problem.






