Catalytic transformations with finite-size environments: applications to cooling and thermometry

The laws of thermodynamics are usually formulated under the assumption of infinitely large environments. While this idealization facilitates theoretical treatments, real physical systems are always finite and their interaction range is limited. These constraints have consequences for important tasks such as cooling, not directly captured by the second law of thermodynamics. Here, we study catalytic transformations that cannot be achieved when a system exclusively interacts with a finite environment. Our core result consists of constructive conditions for these transformations, which include the corresponding global unitary operation and the explicit states of all the systems involved. From this result we present various findings regarding the use of catalysts for cooling. First, we show that catalytic cooling is always possible if the dimension of the catalyst is sufficiently large. In particular, the cooling of a qubit using a hot qubit can be maximized with a catalyst as small as a three-level system. We also identify catalytic enhancements for tasks whose implementation is possible without a catalyst. For example, we find that in a multiqubit setup catalytic cooling based on a three-body interaction outperforms standard (non-catalytic) cooling using higher order interactions. Another advantage is illustrated in a thermometry scenario, where a qubit is employed to probe the temperature of the environment. In this case, we show that a catalyst allows to surpass the optimal temperature estimation attained only with the probe.

is the study of transformations where the inclusion of the catalyst is crucial to cool the system. In addition, realistic systems interact with environments of limited size, or at most with finite portions of very large environments.
Cooling has been studied using techniques of open quantum systems [25] and information theory [23], with traditional approaches that include the analysis of quantum refrigerators [26][27][28][29][30] and algorithmic cooling [31][32][33][34][35]. Recently, an important effort has been devoted to understand and formalize the fundamental limits for this task [36][37][38][39][40]. These limits are intimately connected with the resources at hand. For example, in the framework of thermal operations, catalytic cooling is possible only in combination with an additional system that starts in a non-equilibrium state [13]. Within the context of energy-preserving interactions, it has also been shown that finite environments limit the probability for (noncatalytic) transitions to the ground state [41].
On the other hand, we are interested in transformations where work exchange is possible and the key physical constraint is related to the finite character of the environment. It is important to stress that by "work" we refer to the energy injected (or extracted) by a classical driving, which differs from the definition adopted in the case of thermal operations [16,42,43]. We also allow for the arbitrary generation of correlations between the catalyst and the other systems involved in the transformation (cf. Fig. 1(b)). This contrasts with the assumption, made in previous works [10,11], of a final state where the catalyst is uncorrelated from the main system (see Ref. [14] for results concerning the removal of this constraint).
As illustrated in Fig. 1(a), cooling may be forbidden in situations where the environment is too small. More formally, such a limitation implies that joint unitary operations on the system and the environment cannot lower Figure 1. General framework for the studied catalytic transformations. (a) Illustrative example: a three-level system in the initial state ρ h is used as (hot) environment to cool a (cold) qubit in the initial state ρc. The eigenvalues of the joint state ρc ⊗ ρ h are obtained by rescaling the eigenvalues of ρ h (orange bars) with the eigenvalues of ρc. When cooling is possible a swap |0c2 h |1c0 h (black arrows) suffices to perform it. If T h is too high or Tc is too low, cooling is impossible with any global unitary U ch . (b) The inclusion of a catalyst in an appropriate initial state ρv allows to lift this restriction. If ρs = ρc ⊗ ρ h , cooling is enabled only if the corresponding final state ρ s is not majorized by ρs (non-unital transformation). (c) The transformations are implemented by global unitaries of the form U = U cool ⊕ Vres, where U cool is employed to cool the system and Vres returns the catalyst to its initial state. the average energy of the system. A catalyst that enables to circumvent this restriction plays the role of an additional environment, which not only allows cooling but also remains unaltered by the global interaction in which it takes part. This kind of catalytic transformation belongs to a broader class that we term "non-unital transformations", which are illustrated in Fig. 1(b). From a physical viewpoint, non-unital transformations represent state transitions that cannot be driven by classical electromagnetic fields. Cooling is an example of transformation that satisfies this property, as it requires reducing the energy of a thermal state, and any thermal state is passive [44,45]. The example in Fig. 1(a) depicts the conditions that prevent to cool a two-level system using a three-level environment. If ρ c = 1 i=0 p c i |i c i c | and ρ h = 2 j=0 p c j |j h j h | denote respectively the states of the ("cold") system and the ("hot") environment, with p c 0 ≥ p c 1 and p h j ≥ p c j+1 , cooling is possible if and only if p c 0 p h 2 < p c 1 p h 0 . When this inequality is not fulfilled, any joint unitary U ch has the effect of a mixture of local unitary operations (corresponding to the probabilistic application of different unitaries) on the system. The usefulness of catalytic non-unital transformations is not limited to cooling. Their applicability is further illustrated by considering an example where the use of a catalyst yields an advantage for thermometry. In thermometry [46], and metrology in general [47,48], various results refer to the optimization of the so called the Cramer-Rao bound [49,50], which constitutes a lower bound for the estimation error of some physical parameter. In the case of thermometry this error quantifies the precision of a temperature estimation [51][52][53][54][55]57]. If the environment interacts with a probe, measurements on the probe contain information about the temperature of the environment [57][58][59]. In this context, we show that a catalytic transformation allows to reduce the minimum estimation error achieved through optimal interactions using only the probe.
Our results differentiate from previous works in two key aspects. On the one hand, we provide explicit protocols for the construction of catalytic transformations, which include the explicit global unitary U and the initial states of the catalyst and the environment. The unitaries investigated have the structure indicated in Fig. 1(c). While some results exist about explicit catalyst states [11,12], no general methods to construct catalytic transformations are known beyond very specific cases [12]. On the other hand, we consider transformations where the initial and final catalyst states exactly coincide. In the context of thermodynamics it has been shown that inexact catalysis can lead to extreme physical consequences, even when the final state of the catalyst deviates little from its initial configuration. Essentially, under inexact catalysis the catalyst can become a source of energy or purity that allows any state transition [10,11]. By considering exact transformations, we prevent this possibility and also guarantee that the catalyst is never degraded (within practical limitations), irrespective of the number of times it is used.
The paper is structured as follows. After some general comments on notation and definitions (Sect. II), Sect. III characterizes the limitations on cooling for systems and environments of finite, but otherwise arbitrary dimension. In Sect. IV we introduce the formal tools that are employed in the rest of the paper. Our first main finding consists in the establishment of necessary and sufficient conditions for a class of catalytic non-unital transformations. In the same section, we also develop a graphical method that provides an intuitive picture for this result and subsequent derivations. The applications to catalytic and cooling transformations are addressed in Sect. V. This section is based on two fundamental results, which refer to: • Catalytic cooling with arbitrarily small environments.
• Catalytic cooling where the catalyst enhances the cooling, even if the environment is large enough to cool the system.
Such results imply that a sufficiently large catalyst enables the aforementioned transformations, if the initial states of the system and the environment satisfy certain conditions. As one of the main applications we derive the catalyst state that maximizes the cooling of a qubit using another qubit as environment, for catalysts of arbitrary (finite) dimension. In addition, we study a catalytic transformation that enhances the optimal cooling obtained by coupling a two-level system to a three-level environment. In Sect. VI we extend our findings to a scenario where the environment can be arbitrarily large. We show that, even without the size constraint, larger cooling can be achieved with less control on the environment, by employing a two-level catalyst. In Sect. VII we demonstrate that a two-level catalyst provides a thermometric advantage when the temperature of a three-level environment is probed by a two-level system. Finally, we present the conclusions and perspectives in Sect. VIII.

II. BRIEF REMARKS ON NOTATION AND DEFINITIONS
In what follows we will term the system to be cooled and the environment "cold object" and "hot object", respectively. Moreover, the ground state of these systems will be denoted using the label "1" instead of "0". This choice is convenient to simplify the notation of other physical quantities that will be defined later. States that describe the total system formed by the catalyst, cold and hot objects are written without labels, as well as the corresponding unitary operations. This also simplifies notation and does not generate ambiguity, since this is the only three-partite setup considered.

III. GENERAL PASSIVITY AND COOLING
The fundamental limits for cooling can be understood using the notion of passivity. Passivity is essentially a condition whereby applying unitary transformations to a system cannot decrease the mean value of certain observables [60,61]. While traditionally it has been associated to the Hamiltonian and the impossibility of work extraction [12,45], passivity can be extended to any hermitian operator that represents an observable. Consider a bipartite system in the initial state For inverse temperatures β c and β h such that β c ≥ β h , cooling occurs if the average value of H c is reduced. This process requires an interaction with the hot object and possibly an additional external driving, that results in a joint unitary evolution U ch . However, it is possible that for any global unitary U ch (e.g. if the the cold temperature β −1 c is very low). In this case we say that the state ρ c ⊗ ρ h is passive with respect to the local Hamiltonian H c . Conversely, if ∆ H c < 0 for some U ch then ρ c ⊗ ρ h is non-passive with respect to H c , and the cold object can be cooled down using a hot object in the state ρ h . Throughout this text passivity will always refer to initial states that satisfy Eq. (1), unless otherwise stated. Moreover, we note that this kind of passivity is more stringent than the traditional one, since not only ρ c but also the "extended" state ρ c ⊗ ρ h is passive with respect to H c .
A question that follows up naturally is how can we characterize passive states ρ c ⊗ ρ h . Let {p c i } 1≤i≤dc and {p h j } 1≤j≤d h denote respectively the eigenvalues of ρ c and ρ h , being d c (d h ) the dimension of the Hilbert space of the cold (hot) object. Using the standard convention of non-decreasing eigenenergies, ε c i ≤ ε c i+1 and ε h i ≤ ε h i+1 , passivity is easily expressed by means of the inequalities From this expression we see that passivity is essentially determined by the ratio between the highest and smallest populations of the hot object. In particular, p h 1 /p h d h = 1 in the limit of infinite temperature, and the inequalities hold regardless the populations of the cold object. If the hot object is composed of a large of number N of identical subsystems, where q h max and q h min denote respectively the highest and smallest populations of each subsystem. Since q h max /q h min > 1 for finite temperature, in the limit N → ∞ the ratio p h 1 /p h d h tends to infinity and it is always possible to violate at least one of the inequalities (2). This explains why cooling is always allowed given unlimited access to a sufficiently large hot bath.
In a more general context, two hermitian operators A and B are said to be passive with respect to each other if [A, B] = 0 and the eigenvalues of A are non-increasing with respect to those of B [60]. Equation (2) simply translates this condition to the operators A = ρ c ⊗ ρ h and B = H c . According to Eq. (2), the eigenvalues of ρ c ⊗ ρ h are non-increasing with respect to the index i, while by construction the eigenvalues of H c are nondecreasing with respect to the same index. Moreover, for i fixed all the eigenstates |i c j h yield the same eigenvalue ε c i when H c is applied on them. This implies that the eigenvalues of ρ c ⊗ ρ h are non-decreasing with respect to those of H c .

A. Catalytic transformations and cooling
Given the passivity condition (2), our goal is to introduce a third system that enables cooling and works as a catalyst. This means that if the catalyst is initially in a state ρ v = dv k=1 p v k |k v k v |, at the end of the transformation it must be returned to the same state. In addition, we assume that the catalyst starts uncorrelated from the cold and objects, i.e. the initial total state is ρ = ρ c ⊗ ρ h ⊗ ρ v . The transformation on the cold object is implemented through a global unitary map U that acts on the total system. Denoting the final total state as ρ , a generic catalytic (C) transformation satisfies Note that Eq. (4) guarantees "catalysis" (i.e. the restoration of the catalyst to its initial state) but does not say anything about the final correlations between the catalyst and the rest of the total system. Contrary to previous works on catalysts, we do not impose any restriction on the nature and strenght of these correlations. This additional degree of freedom naturally extends the set of transformations that become possible once the catalyst is introduced [14]. Moreover, the access to a broader set of transformations is not the only motivation for using catalysts. Given the condition (4), we can imagine a situation where a fresh copy of the state ρ c ⊗ ρ h is brought into contact with the catalyst, allowing to repeat exactly the same process performed with the old copy. This is possible because the initial total state with the new copy is identical to that with the old one, i.e. ρ c ⊗ ρ h ⊗ ρ v . In this way, the catalyst can be harnessed with as many copies as desired, through repeated interactions of the form (3). In Section VI we will see how this possibility can be highly advantageous in a cooling scenario involving many cold objects.
Given a passive state ρ c ⊗ ρ h , the inclusion of the catalyst allows to reduce the mean energy H c as long as the total state ρ is non-passive with respect to H c . Let {p v k } 1≤k≤dv denote the eigenvalues of ρ v or "catalyst eigenvalues", for a catalyst of dimension d v , and let Assuming the ordering p v k ≥ p v k+1 for all k, the catalyst breaks down the initial passivity (2) if and only if there exists i such that Since the ratio between the highest and smallest eigenvalues of the composite state ρ h ⊗ ρ v (r.h.s. of Eq. (5)) is always larger than p h 1 /p h d h , by a factor of p v 1 /p v dv , we can understand why passivity with respect to ρ c ⊗ ρ h ⊗ ρ v can be violated, even if all the inequalities (2) are satisfied. In particular, this violation always takes place if ρ v = |1 v 1 v |. However, we will see later that the catalysis condition (4) requires the use of catalysts in initial mixed states.
By definition of passivity, if Eq. (5) is satisfied there exists a global unitary that cools down the cold object.
We can explicitly consider a two-level unitary U swap that swaps the states |i c d h d v and |(i + 1) c 1 h 1 v , while acting as the identity on any other eigenstate of ρ. This uni- thereby reducing the energy of the cold object by ∆ H c = −δp(ε c i+1 −ε c i ). However, the same operation modifies the state ρ v by transferring population δp from the eigenstate |1 v towards the eigenstate |d v . This illustrates how the possibility of cooling is accompanied by an initial alteration of the catalyst. The restoration of the catalyst involves an additional unitary operation, which in turn potentially spoils the cooling accomplished through U swap .
The characterization of the most general catalytic transformations that also perform cooling is a complex problem that is not addressed in the present work. Instead of that, we shall concentrate on a subset of this class of transformations, determined by sufficient conditions that will be specified later on. Meanwhile, motivated by our previous discussion, we present the general structure of the global unitaries U we will be interested in. These unitaries are composed of two independent unitaries U cool and V res , whose functions are respectively cooling and restoration of the catalyst (cf. Fig. 1). Hence, we will refer to them as the "cooling unitary" and the "restoring unitary". An example of cooling unitary is the swap U swap described before. The restoring unitary is chosen in such a way that the subspaces where U cool and V res are defined are orthogonal. Let H cool ⊂ H and H res ⊂ H denote such subspaces, being H = span{|i c j h k v } the total Hilbert space. We consider global unitaries that satisfy Eq. (4) and have the form V res : H res → H res , where ⊕ stands for the direct sum. For the sake of clarity we will always describe the action of U as a sequence V res U cool , where V res is preceded by U cool . However, due to the direct sum structure (6) we have that V res U cool = U cool V res . Crucially, the commutativity between U cool and V res also allows us to study the effect of V res using directly the initial state ρ, instead of the state obtained after the application of U cool . Majorization defines a preorder between quantum states [62,63], and is intimately connected with different cooling criteria [38]. Let and σ denote two generic quantum states defined on some Hilbert space of dimension d, with respective eigenvalues {r i } 1≤i≤d and {q i } 1≤i≤d . Moreover, let {r ↓ i } and {q ↓ i } be the same eigenvalues arranged in non-increasing order, i.e. r ↓ i+1 ≤ r ↓ i and q ↓ i+1 ≤ q ↓ i . It is said that " majorizes σ", formally written as for all 1 ≤ j ≤ d. Physically, majorization is useful to compare the degree of purity between two quantum states. For example, according to Eq. (9) a pure state majorizes any other state, while a fully mixed state is majorized by any other state. However, it is possible that for some pair of states and σ none of the conditions σ or σ takes place, whereby majorization does not constitute an order relation.
An important result of information theory [63,64] states that where E un is a unital map defined by the condition E un (I) = I [64]. Based on this observation, we define a "unital transformation" as a transformation → σ such that σ, since it can be implemented through the application of a unital map to .
The description of majorization in terms of the relation σ = i λ i U i U † i establishes a link between majorization and the standard definition of cooling. Consider a transformation on the cold object ρ c → ρ c , such that ρ c ρ c . Since this implies that ρ c can be written as i , this transformation can only increase the value of H c because unitary operations cannot lower the mean energy of a thermal state. Therefore, a necessary condition to have ∆ H c < 0 is that ρ c does not majorize ρ c , or equivalently that ρ c → ρ c is a "non-unital transformation". This occurs in particular if the populations of ρ c in the eigenbasis of ρ c satisfy where p l c = Tr(|l c l c |ρ c ) and ρ c |l c = p c l |l c . Let us see why a transformation satisfying Eq. The catalytic and cooling transformations that we will study are based on a more general class of catalytic transformations that satisfy Eq. (10). We denote a transformation of this kind as ρ CNU −→ ρ , with the abbreviation CNU standing for catalytic (with respect to the catalyst) and non-unital (with respect to the cold object). A characterization of these transformations shall be provided in Theorem 1.

Cooling currents and restoring currents
The question we want to answer can be precisely stated in the following way: given an initial state of the form ρ = ρ c ⊗ ρ h ⊗ ρ v , such that ρ c ⊗ ρ h is passive with respect to H c , is there a unitary U that satisfies Eq. (6) and that allows to implement a CNU transformation? We will provide sufficient conditions for a positive answer, expressed entirely in terms of the eigenvalues of ρ. To that end we will explicitly construct unitaries U that perform the transformation, which are conveniently characterized using the notion of "population currents", or simply "currents".
A population current is the population transferred between two eigenstates |i and |j of ρ, due to the action of a two-level unitary U (2) : span{|i , |j } → span{|i , |j } . Any population transferred in this way can be described through a unitary of the form where 0 ≤ a ≤ 1. Importantly, our definition of current refers to a net population transfer between two states, rather than a rate of population exchanged per unit of time. However, we will see that such a denomination is helpful in the construction of an intuitive picture for the transformations that will be studied throughout this paper. Keeping in mind that the initial populations are p i = Tr(|i i|ρ) and p j = Tr(|j j|ρ), after the application of U (2) the state |j acquires population In this way, the population current from |i to |j is defined as The definition (14) may at first look a bit artificial, given that it only describes the transferred population p j − p j if the population of |j increases (cf. Eq. (13)). However, this convention of positive currents has the advantage that a current J |i →|j unambiguously indicates a population flow from |i to |j , and consequently that p i > p j . If p i < p j , the flow occurs in the opposite direction and is characterized by the current J |j →|i = a 2 (p j − p i ). Clearly, J |i →|j varies between 0 and p i − p j , with its maximum value attained when U (2) performs a swap between the states |i and |j . This maximum is termed "swap current" and is denoted as Eventually, we will also employ the notation J |i → for a current that describes a population flow from |i to some unknown eigenstate of ρ. Likewise, J →|i will denote a population flow from an unknown eigenstate towards |i . Depending on the states |i and |j and their initial populations, the two-level unitaries described by Eqs. (11) and (12) generate different types of currents. In particular, the violation of majorization (10) (necessary for cooling) is possible if there exist a current that we term "cooling current". In the following the "existence of a current J |i →|j " signifies that J |i →|j = 0, which in turn implies that there exists a two-level unitary transferring population from |i to |j . Moreover, we note that J |i →|j exists iff J |i →|j exists.
Definition 1 (cooling currrent). A cooling current is a current J |(i+n)cj h kv →|icj h k v (n ≥ 1), whose function is to transfer population from an eigenstate of ρ c ⊗ I hv with eigenvalue p c i+n , towards an eigenstate with larger or equal eigenvalue p c i . From Eq. (15) it is straightforward to check that where the second inequality follows by bounding Noting that Eqs. (16) and (5) are equivalent, we conclude that the inclusion of a catalyst breaks down the passivity with respect to H c iff the eigenvalues of ρ v are such that there exists a cooling current J |(i+n)cj h kv →|icj h k v . Such a current has opposite effects on the cold object and the catalyst. First, it increases the partial sum i l=1 p c l by transferring population to |i c , which yields a nonunital transformation of the form (10). On the other hand, notice that the r.h.s. of Eq. (16) must be larger or equal than one due to passivity without the catalyst. As a consequence p v k > p v k , whereby the cooling current also reduces the population of the catalyst eigenstate |k v , and increases the (smaller) population of |k v by the same amount. This has a mixing effect on such states, which can be readily reproduced through a local two-level unitary acting on the subspace span{|k v , |k v }. By definition, the resulting transformation is unital. The purpose of restoring currents is to counter this effect by transferring population in the opposite sense. That is, from a low-population eigenstate of ρ v towards a higherpopulation eigenstate.
Definition 2 (restoring current). A general restoring current is a current J |icj h (k+n)v →|i c j h kv (n ≥ 1), whose function is to transfer population from an eigenstate of I ch ⊗ ρ v with eigenvalue p v k+n , towards an eigenstate with larger or equal eigenvalue p v k . However, the study of cooling transformations will be mainly based on restoring currents of the kind J |j h (k+n)v →|j h kv , which are associated with two-level unitaries that do not involve the cold object. From Eq. (15), it is straightforward to where the second inequality follows by bounding

C. CNU transformations with a single cooling current
Now we specialize to CNU transformations that contain a single cooling current, which amounts to impose that U cool = U (2) cool is given by a single two-level unitary. We denote these transformations as ρ CNU1 −→ ρ . Moreover, we consider restoring unitaries V res of the form where each V (2) k is a two-level unitary giving rise to a restoring current.
Given a cooling current J |(i+n)cj h kv →|icj h k v , the goal of V res is to return the populations of the states |k v and |k v to their initial values. The most direct way to do that would be through a single restoring current J |l c m h k v →|lcm h kv . However, in general such a current may not exist. The following theorem provides necessary and sufficient conditions for the existence of a CNU1 transformation, by characterizing the currents that lead to a proper restoring unitary V res . The proof of this theorem is given in Appendix A.
be an initial density matrix of the total system, with There exists a cooling current or restoring currents Remark 1 (Generalization of Theorem 1). The proof given in Appendix A is applicable to general states of the form ρ s ⊗ρ v , where ρ s represents an arbitrary state on which the non-unital transformation is performed. From this general proof, the conditions (19)-(21) follow by choosing ρ s = ρ c ⊗ ρ h , and requiring that the transformation is non-unital not only on ρ s , but also on the state ρ c .
On the other hand, we also note that in Theorem 1 no reference is made to the thermal character of the states ρ c and ρ h . Similarly, the theorems 2 and 3 that will be presented later are formulated only in terms of the eigenvalues of general states ρ c and ρ h .

Graphical characterization of CNU1 transformations
In order to provide an intuitive understanding of Theorem 1 we introduce the following graphical method to describe cooling currents and catalytic currents: Consider an horizontal axis where the values {ln(p c i p h j )} 1≤i≤dc,1≤j≤d h are arranged in decreasing order, i.e. the larger the value the more to the left it is placed on this axis. Similarly, we arrange the values {ln(p v k )} 1≤k≤dv in a vertical axis, with larger values at the top and smaller ones at the bottom. A "row" k v is an horizontal line that intersects the value ln(p v k ), and represents also the catalyst eigenstate |k v . A "column" i c j h is a vertical line that passes through the value ln(p c i p h j ), and represents the eigenstate |i c j h . The intersection between a row k v and a column i c j h is associated with the pair (i c j h , k v ), which corresponds to the global eigenstate |i c j h k v .
The main purpose of the ln(p ch ) × ln(p v ) diagram, illustrated in Fig. 2, is the depiction of cooling currents and catalytic currents. In addition, the non-overlap between gray rectangles (energy eigenspaces of the cold object) stands for the condition of passivity without catalyst. This means that the largest element of a any subset in the low gray boxes is always upper bounded by the smallest element in the next subset at the left. By applying the natural logarithm to the second inequality in Eq. (16), we obtain the relation ln( . Therefore, the existence of a cooling current (downward-oriented blue arrow) means that }i,j is associated with a "column" icj h (vertical line), and each element ln(p v k ) ∈ {ln(p v k )} k is associated with a "row" kv (horizontal line). The corresponding interesection yields the pair (icjc, kv) (purple box). The elements ln(p c i p h j ) are arranged in non-increasing order, from left to right, and ln(p v k ) are non-increasing, from top to bottom. The left-most green arrow represents a current that takes place inside an energy eigenspace of the cold object, and is the primary kind of restoring current involved in cooling transformations. The right-most green arrow connects different eigenspaces of ρc, and is involved in more general (not necessarily cooling) non-unital transformations on the composite state ρc ⊗ ρ h . the height of the ln(p ch ) × ln(p v ) diagram must be larger than the distance between two consecutive columns i c d h and (i + 1) c 1 h , for some value of i. In the diagram this relation is represented by enclosing the cooling current inside a vertical rectangle. Analogously, the application of the natural logarithm to Eq. (17) yields the inequality , meaning that the corresponding restoring current is enclosed by an horizontal rectangle of width ln(p h 1 /p h d h ) and height ln(p v k /p v k+1 ). The left-most green arrow in Fig. 2 illustrates this type of restoring current for k = 1. Moreover, generic restoring currents are always upward-oriented.
The diagrams for the conditions of Theorem 1 are given in Fig. 3. The sides of the vertical cyan rectangle in both diagrams have lenghts obtained from the application of the natural logarithm to Eq. (19). Specifically, its height is given by ln(p v l /p v l +1 ), and its width is given by ln(p c Hence, a cooling current exists iff we can identify a vertical rectangle with vertical sides living on consecutive columns i c d h and (i + 1) c 1 h . The restoring currents are enclosed by a set of adjacent horizontal rectangles (light green rectangles). When taken together, they compose a (not necessarily horizontal) total rectangle joining the rows l v and (l +1) v . The rectangles shown in Fig. 2 19) and (20). (b) Depiction of the conditions described by Eqs. (19) and (21).

Effect of currents on the the catalyst
The total population variation for a catalyst eigenstate |k v is given by (see Appendix B) where is the sum of all the currents that transfer population to eigenstates with catalyst eigenvalue p v k . Similarly, corresponds to the sum of all the currents that take population from these eigenstates. It is important to stress that, in the most general case, a current J →|icj h kv could connect |i c j h k v with another eigenstate that also has catalyst eigenvalue p v k . If this occurs such a current cannot contribute to J →|kv , since it leaves invariant the population of |k v . However, we always deal with currents J →|icj h kv and J |icj h kv → connecting eigenstates with different catalyst eigenvalues, which implies that the quantities J →|kv and J |kv → are properly characterized by Eqs. (23) and (24). In the following J →|kv and J |kv → will be termed "catalyst currents", as they describe population flows within the catalyst.

Restoration of the catalyst
Equation (22) implies that the population of the state |k v remains unchanged as long as the corresponding catalyst currents satisfy J →|kv = J |kv → . In the following lemma we provide necessary and sufficient conditions for the existence of a restoring unitary V res , given a two-level unitary U (2) cool that generates a single cooling current. Accordingly, the condition J →|kv = J |kv → holds under the action of the total unitary U = U (2) cool ⊕ V res . Here we will prove sufficiency, leaving the proof of necessity for Appendix C.
Lemma 1 (Existence of restoring unitaries). Let J |(i+1)c1 h lv →|icd h (l +1)v be a cooling current, which incresases (decreases) the population of the catalyst eigenstate |(l + 1) v (|l v ). A restoring unitary V res that reverses the effect of this current on the catalyst exists iff there exists a set of restoring currents Fig. 4).
To following definitions will be useful to prove this lemma: Definition 3 (loop and uniform loop). A loop is a set of currents, such that for any catalyst state with an incoming current J →|kv there is an outgoing current J |kv → . A "uniform loop" is a loop with the additional property that all the catalyst currents satisfy J →|kv = J |kv → . According to Eq. (22), a uniform loop keeps the state of the catalyst unchanged.
Definition 4 (chain). A chain is a set of currents, such that only two of the connected eigenstates are not connected by both types of currents (incoming and outgoing). Let us call such states the "outer links of the chain", while the other eigenstates (connected by an incoming and an outgoing current) will be called "inner links". To understand the role of chains in the existence of restoring unitaries, and at the same time keep consistency with previously introduced notation, it is convenient to denote the outer links as |l v and |(l + 1) v . In this way, a chain is a set is the total catalyst current connecting the states |l v and |k Since J |lv →|k (1) v constitutes an incoming current for , Eq. (25) indicates that the inner link |k (1) v contains both an incoming current and an outgoing current )v constitutes the outgoing current for the inner link |k . On the other hand, the outer link |l v only has an outgoing current, and the outer link |(l + 1) v only has an incomming current. If {|k } denotes the set of all the remaining inner links, then the ellipsis in Eq. (25) } 2≤i≤n−1 of incoming and outgoing currents connecting these links. We also note that the simplest chain has the form ch |lv →|(l +1)v = {J |lv →|(l +1)v }, with a single catalyst current and no inner links.
Definition 5 (restoring chain). When a chain ch |(l +1)v →|lv is joined with a chain ch |lv →|(l +1)v the resulting set of currents is a loop, since ch |(l +1)v →|lv provides an incomming current for the outer link |l v , and an outgoing current for the outer link |(l + 1) v . In this way, |l v and |(l + 1) v become inner links in the set ch |(l +1)v →|lv ∪ ch |lv →|(l +1)v (note that by definition all the links are inner links in a loop). If the currents contained by ch |(l +1)v →|lv and ch |lv →|(l +1)v have all the same magnitude, their union yields also a uniform loop. Thus, we can say that ch |(l +1)v →|lv is a "restoring chain" for ch |lv →|(l +1)v and vice versa, since the populations changes in the outer links are cancelled out once they are joined.
If the swap intensity of the cooling unitary, denoted by a 2 cool , and the intensities a 2 k , satisfy Eq. (14) implies that all the currents generated byṼ res and U (2) cool have the same magnitude J min loop . Therefore, a unitary U = U (2) cool ⊕Ṽ res that satisfies the previous equations is catalytic (or equivalentlyṼ res is restoring), since it generates a uniform loop.
The necessity condition for Lemma 1 is proven in Appendix C. The essential idea is that if the chain {J |1c1 h (k+1)v →|dcd h kv } l≤k≤l does not exist, no general restoring chain ch |(l +1)v →|lv exists either. Accordingly, it is impossible to form a loop with the cooling current and the catalyst cannot be restored.

Relation between restoring chains and restoring unitaries
The connection between restoring unitaries and restoring chains, previously elucidated, allows us to study the structure of restoring unitaries that yield CNU1 and cooling transformations. First, note that the sets of restoring currents characterized by Eqs. (20) and (21) are also restoring chains of the form Ch |(l +1)v →|lv , each of which forms a loop with the cooling current J |(i+1)c1 h lv →|icd h (l +1)v . Accordingly, they also have the loop structure shown in Fig. 4. The existence of these chains ensures that there exist unitaries where V res,L is a "left" restoring unitary, derived from Eq. (20), and V res,R is a "right" restoring unitary, derived from Eq. (21). Essentially, the partial swaps in V res,L generate the restoring currents illustrated in Fig.  3(a), and the partial swaps in V res,R generate the restoring currents shown in Fig. 3(b). By suitably adjusting the intensities of these swaps, it is possible to obtain a uniform loop that guarantees the restoration of the catalyst. In addition, it is worth remarking that either V res,L or V res,R exist only if the chain described in Lemma 1 exists. A restoring chain of the form {J |1 h (k+1)v →|d h kv } l≤k≤l also allows to reverse the effect that a cooling current J |(i+1)c1 h lv →|icd h (l +1)v has on the catalyst. The currents in this chain exist iff the inequalities hold for l ≤ k ≤ l . Crucially, these inequalities guarantee a unitary V res = I c ⊗ V hv , which restores the catalyst using only the hot object. In particular, a direct sum of appropriate partial swaps where all the currents have the same magnitude of the cooling current. Since I c ⊗ V hv does not interfere with the cooling effect of U (2) cool , in the rest of the paper we will mainly deal with transformations based on unitaries U = U (2) cool ⊕(I c ⊗V hv ).

V. CATALYTIC COOLING (CC) TRANSFORMATIONS
In the preceding section we established necessary and sufficient conditions for a CNU1 transformation. Here we show some examples where Theorem 1 can be applied to characterize catalytic cooling (CC) transformations, i.e. transformations that are catalytic and also obey some standard criterion for cooling. To this aim we start by presenting our second main result, which addresses the existence of CNU1 transformations in terms of the dimension of the catalyst. The corresponding proof is given in Appendix D.
Theorem 2 (Catalyst size and CNU1 transformations). Let d v denote the dimension of the catalyst Hilbert space H v . For d v large enough, a CNU1 transformation exists if any of the following conditions hold: 1. The initial state of hot object is not fully mixed, i.e. p h j = p h j for some pair j, j . 2. The Hilbert space of the cold object has dimension d c ≥ 3, and ρ c is not fully mixed.
According to Theorem 2, CNU1 transformations are possible for almost any initial state ρ c ⊗ ρ h . In particular, condition 1 implies that any hot object with nondegenerate energy spectrum and finite temperature suffices. It is also worth pointing out that a harmonic oscillator constitutes an example of universal catalyst, in two complementary aspects. On the one hand, for a harmonic oscillator d v → ∞, which makes it suitable to implement a CNU1 transformation on any state ρ c ⊗ ρ h that adheres to the previous conditions. On the other hand, any CNU1 transformation that can be realized with a catalyst of finite dimension d v , can also be performed with a harmonic oscillator. This is possible by simply populating d v levels of the harmonic oscillator with the eigenvalues of the (finite-catalyst) state ρ v . In addition, note that the pivotal property behind this advantage is the dimension of the catalyst, whereby any infinite-dimensional system is universal irrespective of its Hamiltonian.
A. Catalytic cooling by reducing the mean energy Hc Now we are ready to present some examples of cooling transformations. First, we consider cooling transformations that decrease the mean energy of the cold object. If condition 1 holds we can implement a cooling transition whose restoring effect relies on the non-disturbing (with respect to the cold object) unitary V hv . According to the proof given in Appendix C, if d v is sufficiently large there exists a cooling unitary U  |(i+1)c1 h 1v ↔|icd h dv and V hv , by using equations analogous to Eqs. (27) and (28).
Since the only effect of U = U This cooling transformation is illustrated in Fig. 5(a). Importantly, a two-level system with eigenenergies ε h 1 = ε h 2 and thermalized at finite temperature β h > 0 serves as hot object. Moreover, it can be used to cool down any cold object by transferring population between some pair of consecutive eigenstates |(i + 1) c and |i c , regardless of its size. In Subsection C we will study the optimization of this transformation, when both ρ c and ρ h describe two-level systems.
We also point out that any transformation that reduces the energy of the cold object automatically decreases its   Another approach for cooling consists of increasing the ground state population of the cold object. Based on our previous discussion, if condition 1 of Theorem 2 holds we can generate a cooling current J |1c1 h 1v →|1cd h dv that has this effect. However, it turns out that we can construct a CNU1 transformation that performs this kind of cooling, even if ρ h is fully mixed. This transformation also has the particularity that the hot object is not needed. That is, U = U  For the sake of generality, let us consider that the ground energy ε c 1 is degenerate and that the goal is to increase the population in the corresponding eigenspace. In other words, we aim at increasing the average value of the projector Π c 1 = g i=1 |i c i c |, where {|i c } 1≤i≤g are all the eigenstates with eigenenergy ε c 1 . The nondegenerate case is recovered for g = 1. By choosing i = g, the cooling current J |(g+1)c1v →|gcdv yields the increment It is important to stress that J |(g+1)c1v →|gcdv always exists if d v is large enough. On the other hand, suppose that d c ≥ g + 2, which means that the cold object possesses more than one excited eigenstate, and that p c dc < p c g+1 , which means that its highest eigenenergy is larger than the first excited eigenenergy ε c g+1 . In this case condition 2 of Theorem 2 holds, and there exists a restoring unitary This unitary gives rise to a restoring chain Ch |dv →|1v = {J |(g+1)c(k+1)v →|dckv }, which in turn forms a loop with J |(g+1)c1v →|gcdv . The physical mechanism behind the cooling effect is easy to understand. First, the cooling current transfers population from |(g + 1) c to |g c , which yields the increment ∆ Π c 1 = ∆ |g c g c | . Since V cv transfers population from |(g +1) c to |d c , the restoring unitary does not interfere with this increment. The corresponding transformation is illustrated in Fig. 5(b).
C. Optimal catalytic cooling of a qubit using another qubit as hot object Previously we mentioned that a system of infinite dimension constitutes a universal catalyst, in the sense that it enables CNU1 transformations for any state ρ ch that complies with conditions 1 or 2 of Theorem 2. Moreover, we have seen that CC transformations can also be performed for almost any initial state ρ ch . If this state is fixed, it is natural to ask which is the optimal catalytic cooling that can be achieved by using catalysts of different sizes. Here we address this question, regarding the simplest scenario of cooling of a qubit using another qubit as hot object. The cold and hot qubits start in states and satisfy the inequality p c 2 ≤ p h 2 (no cooling condition without catalyst). In this case, all the cooling criteria considered before are equivalent to the increasing of the ground population of the cold object.
Without loss of generality, we can focus on the optimization of cooling for an infinite-dimension catalyst. Specifically, we consider the maximization of the cooling current with respect to the eigenvalues of a state ρ v with fixed rank 2 ≤ n < ∞, which has support on a subspace of the infinite Hilbert space H v . Since a state of rank n is equivalent to a full-rank state for a catalyst of (finite) dimension n, the optimization for a given value of n yields also the maximum cooling using this catalyst. Accordingly, the optimal cooling using a finite catalyst can always be performed with one of infinite dimension. The inset in (a) shows the cooling regions corresponding to 2 ≤ n ≤ 5, where states having these ranks allow to cool. Since the maximum of J max cool is achieved for p c 2 = p h 2 and n = 2, 3, in (d) we consider the cooling using a two-level catalyst (n = 2) and a hot qubit such that p h 2 = p c 2 . The black dashed line depicts the initial ground population of the cold qubit and the blue curve is the corresponding final population.
On the other hand, we will see that for certain values of p c 2 and p h 2 larger cooling currents can be obtained if n is small. Such a result is remarkable, as it implies that in some cases small catalysts can be as effective as larger ones. In what follows we assume that if ρ v has rank n then p v k = 0 for all k ≥ n + 1. That is, only the levels 1 ≤ k ≤ n are populated.
In Appendix E we show that for n fixed the optimal CC unitary is given by where U cool = U |2c1 h 1v ↔|1c2 h nv is the swap between the states |2 c 1 h 1 v and |1 c 2 h n v , and V |1 h (k+1)v ↔|2 h kv is the swap between the states |1 h (k + 1) v and |2 h k v . If n = 4 and we adopt the view that ρ v describes a catalyst of dimension four, the currents generated by U are illustrated in Fig. 5(a). This structure is characterized by two restoring chains inside the energy eigenspaces of the cold qubit, and describes the effect of U for general values of n. The maximization of the cooling current J cool = J |2c1 h 1v ↔|1c2 h nv also yields catalyst eigenvalues that satisfy where From Eq. (36), we find that the maximum cooling current reads (see Appendix E) where r h ≡ p h 2 p h 1 and we have added parentheses to superscripts to distinguish them from powers. Moreover, the optimal eigenvalues {p v k } are characterized by the equation depicts the maximum cooling current corresponding to a different rank of ρ v . Moreover, J max cool in Eq. (37) is plotted as a function of 0 ≤ p c 2 ≤ p h 2 , which constitutes the interval where cooling without the catalyst is not possible. In Fig. 5(a) we can see that as n increases the interval of p c 2 where J max cool is positive also increases. Since J max cool < 0 means that population would be transferred from the ground state to the excited state of the cold qubit, the "cooling region" is described by the condition J max cool ≥ 0. The inset in Fig. 5(a) shows more clearly the cooling regions (blue bars) corresponding to states of ranks 2 ≤ n ≤ 5. The enlargement of these regions as n increases indicates that larger catalysts may allow cooling in regimes not accesible to small catalysts, characterized by p c 2 p h 2 . On the other hand, for p c 2 = p h 2 it is remarkable that J max cool is maximized by n = 2 and n = 3, and decreases for larger values of n. This implies that in such a case the smallest possible catalyst, corresponding to a two-level system, is enough to achieve maximum cooling. Moreover, it is also worth noting that the cooling current corresponding to n = 3 always surpasses the current corresponding to n = 2 (except for p c 2 = p h 2 ). Figures 5(b) and 5(c) display the same pattern that characterizes Fig. 5(a). In particular, notice that in both cases a catalyst state of rank n = 10 allows to cool for almost any value of p c 2 . In Fig. 5(c) we also see that a state of rank n = 3 (black curve) is essentially as effective as any state with rank 4 ≤ n ≤ 10. Accordingly, in this case a three-level catalyst is optimal for almost any value of p c 2 . Figure 5(d) shows the initial and final ground populations as a function of p c 2 , if the populations of the hot and cold qubits always coincide. The final population is computed as p c 1 = p c 1 −J max cool , where J max cool is the cooling current attained for n = 2 or n = 3.

D. Catalyst-aided enhancement of cooling
The usefulness of catalysts is not restricted to the implementation of transformations that are forbidden without the utilisation of these systems. Here we show that cooling can be catalytically enhanced, even if the hot object is sufficient to achieve a certain level of cooling. This is formally stated in the following theorem, which constitutes our third main result. The proof is given in Appendix F.
Theorem 3 (cooling enhancement with a catalyst). Let ρ h be the state of a hot object of dimension d h ≥ 3, and ρ c the state of a cold qubit. If d h is odd and the largest d h +1 2 eigenvalues of ρ h or the smallest d h +1 2 eigenvalues of ρ h are not fully degenerate (i.e. p h j = p h j for some pair of the referred largest or smallest eigenvalues), or d h is even and the largest d h 2 eigenvalues or the smallest d h 2 eigenvalues are not fully degenerate, then a large enough catalyst increases the optimal cooling achieved with the hot object alone.
To exemplify the catalytic improvement of cooling consider the minimal hot object that adheres to the hypothesis of Theorem 3, i.e. a three-level system. Let Tr(e −β h H h ) satisfies the hypothesis of the aforementioned theorem, since the two smallest eigenvalues p h 2 and p h 3 are non-degenerate. In Fig. 7(a) we show the maximum cooling attainable using ρ h , as well as an additional cooling through a CC transformation that employs a qubit as catalyst. The optimal cooling transformation without the catalyst is characterized in Appendix F, for a generic state ρ h . The parameter e −β h ε h 3 is set to e −β h ε h 3 = 0.01, which also fixes the eigenvalues of ρ h due to the degeneracy of ε h 1 and ε h 2 . The blue dashed-dotted curve depicts the ground population of the cold qubit after the optimal cooling without catalyst, associated with the cooling current J cool = J |2c1 h →|1c3 h . Moreover, the black solid curve stands for the final population after applying a suitable catalytic and cooling transformation. The left rectangle in Fig. 6(b) is a ln(p c ) × ln(p h ) diagram employed to illustrate the optimal transformation without the catalyst. In this diagram, we keep a small gap β h ω h 1,2 = β h (ε h 2 − ε h 1 ) > 0 that allows to distinguish the degenerate levels. However, it is indicated that β h ω h 1,2 tends to zero, to comply with the degeneracy condition. Assuming that ε c 1 = 0, in the limit β h ω h 1,2 → 0 the condition of cooling without the catalyst amounts to impose that ln(p h 2 /p h 3 ) = β h ω h 2,3 > β c ε c 2 = ln(p c 1 /p c 2 ). For β c ≥ β h , this inequality holds if the energy gap ω h 2,3 is sufficiently large.
The right diagram of Fig. 7(b) illustrates the CC transformation that yields the black curve in Fig. 7(a). In this diagram the columns represent the eigenstates of the state U cool (ρ c ⊗ ρ h )U † cool , where U cool = U |2c1 h ↔|1c3 h is the swap that maximizes the cooling with the hot object. By arranging the columns according to decreasing eigenvalues, we can apply the rules that determine cooling currents and restoring currents in a ln(p ch ) × ln(p v ) diagram, even if the state U cool (ρ c ⊗ ρ h )U † cool has not a product form. The only feature that we need to keep in mind is that now the values associated with the columns 2 c 1 h and 1 c 3 h are, respectively, ln(p c 1 p h 3 ) and ln(p c 2 p h 1 ) (see Fig. 7(b)). In this way, the depicted currents are generated by the unitary U = U cool ⊕ V res,R ⊕ V res,L , where U cool and V res,X=R,L are swaps between the connected eigenstates.
It is important to mention that U does not commute with U cool , since [U cool , U cool ] = 0. Therefore, the total transformation U U cool cannot be written in the direct sum form. This explains also why we require two independent diagrams for the representation of each transformation.
The restoring chain for the CC transformation contains the total current J res = J res,L + J res,R , where From the condition J cool = J res (uniform loop) and the degeneracy p h 1 = p h 2 , it follows that Remarkably, Fig. 7(a) shows that for low temperatures (β c large) the increment of p c 1 due to the catalytic transformation is comparable to that achieved via optimal cooling without the catalyst. Moreover, the cooling enhancement provided by the catalyst is significant in all the temperature range.
In Fig. 8 we plot the cooling currents J cool and J cool + J cool , where J cool + J cool is the current obtained from the total transformation U U cool . In these plots e −βcε c 2 is fixed, and we vary instead the parameter e −β h ε h 3 . The condition β c ε c 2 > β h ε h 3 (for cooling without the catalyst) implies that the maximum value of e −β h ε h 3 must coincide with e −βcε c 2 . When e −β h ε h 3 increases, the hot object is hotter and consequently the cooling current J cool decreases, reaching its minimum value J cool = 0 at e −β h ε h 3 = e −βcε c 2 . On the other hand, we see again that the catalytic contribution is more significant the lower the cold temperature. To conclude this section, we remark that CC transformation considered here could be suboptimal, and therefore the advantage derived from the catalyst could be even larger.

VI. CATALYTIC ADVANTAGE WHEN COOLING A LARGE NUMBER OF COLD OBJECTS
In quantum thermodynamics and related research areas, the possibility to implement otherwise forbidden transformations has been one of the main motivations for the introduction of catalysts [1,10,14]. However, the restoration of the catalyst per se is not mandatory to achive that goal, and, on the contrary, it is expected that a potentially larger number of transformations can be reached if the restoration constraint is removed. While it is true that such transformations are not technically catalytic, there is no a priori reason for not to consider the catalyst simply as an ancilla, and implement a global unitary that optimizes the transformation on the system of interest.
A practical motivation to preserve the state of the catalyst is that it can be reused when necessary. For example, the repeated use of a chemical catalyst can substantially increase the rate of a chemical reaction. In the context of cooling, we can also imagine a situation where a large number of cold objects are cooled down by the repeated application of a catalytic transformation. When taken together, the cold objects and the corresponding hot objects employed for each transformation can bee seen as environments of big size. A question that arises naturally in this scenario is how catalytic cooling compares to cooling strategies that do not use a catalyst. In particular, the passivity restriction that prevents cooling always breaks down for large enough environments, as explained in Sect. III. Here, we will show that catalytic cooling can outperform a cooling strategy that uses arbitrary manybody interactions between cold objects and the hot environment (formed by the hot objects). It is important to stress that the catalytic transformations involve at most three-body interactions. Therefore, the catalytic advantage is two-fold, since larger cooling is achieved with a lower degree of control on the environments.

A. Catalytic cooling vs. cooling using many-body interactions
Consider the scenario schematically depicted in Fig.  9. The goal is to cool as much as possible a group of N c qubits, using a group of N h qubits that play the role of a hot environment. All the qubits start at the same inverse temperature β and have identical energy spectrum. Therefore, the Hamiltonians of the ith cold and hot qubits are respectively H The total Hamiltonian for the X = C, H group is H X = N X i=1 |1 x=c,h i 1 x=c,h |, and the global initial state is a product of thermal states Tr(e −βH X ) . Assuming that the total number of qubits N = N c + N h is fixed, we now describe two cooling strategies, illustrated in Fig. 9.
1. Many-body cooling (MBC) strategy: subsets of 2 ≤ k ≤ N h qubits from the hot group are used to optimally cool individual qubits in the cold group, through optimal unitary transformations. Each qubit is cooled down only one time and the hot qubits pertaining to different subsets are all different (this implies that hot qubits are also used only once). Note also that k ≥ 2, since all the qubits have identical states and therefore cooling is forbidden for k = 1.
2. Catalytic cooling (CC) strategy: a catalyst is employed to cool down single qubits from the cold group, using only one hot qubit per cold qubit. As with the MBC strategy, there is no reusage of hot qubits and each cold qubit is cooled down only one time. In the MBC strategy the optimal cooling with a subset of k hot qubits involves (k + 1)-body interactions between these qubits and the corresponding cold qubit. More specifically, such couplings are described by an interaction Hamiltonian that contains products of the form ⊗ k+1 i=1 B i , where B i is a non-trivial (i.e. different from the identity) operator on the Hilbert space of the ith qubit. On the other hand, the CC strategy is based on the repeated application of the unitary U in Eq. (34), for the case n = 2. This means that each cycle implements the optimal cooling of a single qubit using a two-level catalyst and one hot qubit. Importantly, the corresponding restoring unitary involves only a two-body interaction between the catalyst and the hot qubit, while U cool requires a three-body interaction. In what follows we show that even for large values of k, the CC strategy always outperforms the MBC strategy if N c ≥ 3N/7.
The purpose of any of the described strategies is to reduce as much as possible the total average energy H C of the cold qubits. Depending on the value of N c , the number of qubits that can be cooled may be smaller than N c . This limitation is directly associated with the amount of hot qubits available to perform the cooling. For example, if N c = N − 2 only two hot qubits are available. In this case, two qubits can be cooled using the CC strategy and only one qubit can be cooled through the MBC strategy. That being said, it is important to remark that the follow-ing analysis covers all the possible values 1 ≤ N c ≤ N −1. Therefore, it provides a full picture of the task at hand, including also the situations where all theN c qubits can be cooled. Taking this into account, the total heat extracted is given by where n c ≤ N c .

B. Characterization of MBC
In the case of MBC, the maximum extractable heat Q C can be conveniently addressed by introducing a coefficient that characterizes how efficient is the cooling of a single qubit, with respect to the number of hot qubits employed. This is a natural figure of merit in our scenario, taking into account that the hot qubits constitute a limited resource. Specifically, we define the "k-cooling where Q (k) C is the heat extracted by using a subset of k ≤ N h hot qubits.
In the MBC strategy there are many ways in which the N h hot qubits can be divided into cooling subsets. Two of such possibilities are illustrated in Fig. 9, for the case N c = 3 and N h = 12. Each dash in the leftmost dashed line represents one qubit, with cold qubits occupying the blue region and hot qubits the purple region (recall that both groups of qubits have the same temperature and therefore color difference is only used to distinguish them). One option is to cool each cold qubit using subsets of four hot qubits (darker gray ellipses), through global unitaries that are depicted by the lines joining these subsets with dashes in the cold region. Instead of that, we could use all the hot qubits (lighter gray ellipse) to cool down a single cold qubit. In general, the heat extracted from this single qubit should be larger than the heat extracted by each four-qubit subset, keeping in mind that interactions with more hot qubits are allowed. However, a larger number of qubits are cooled down when several cooling subsets are employed. Since we are interested in the total heat Q C , and not necessarily on maximizing the cooling for single qubits, it is not immediately clear which strategy wins.
By resorting to the cooling coefficient (43), we can express the total extracted heat as Figure 10. Cooling coefficient (43) curves for the cooling of a cold qubit using 2 ≤ k ≤ 14 hot qubits. Blue solid (red dashed) curves stand for k even (odd). The highest and lowest curves correspond respectively to ξ (2) cool and ξ (14) cool . Since ξ (k) cool < ξ (2) cool for any value of p c 2 , this plot shows that the conjecture (46) is true for 2 ≤ k ≤ 14.
where K = {k 0 , k 1 , ...} describes a certain partition of the hot group into cooling subsets. In particular, we note that k∈K k = N h , and that it is perfectly legitimate to have subsets of different sizes k i = k j . Given a fixed partition, we also have the bound C is by construction a non-decreasing function of k, Fig. 10 provides numerical evidence that ξ (k) cool is maximum for k = 2. For very large values of k it is also naturally expected that ξ (k) cool tends to zero, since otherwise Q (k) C would be an unbounded quantity. Therefore, we conjecture that for all k ≥ 2 and for any β, which is satisfied for 2 ≤ k ≤ 14 (see Fig. 10). The explicit expression for ξ (2) cool is derived in Appendix G.
Although our conjecture and Eq. (45) seem to indicate that to maximize Q C one should always choose minimal cooling subsets, composed of two qubits, this choice is actually optimal if N c is above certain value. In this respect, we note that the maximization in Eq. (45) involves values of k characterizing a specific partition K, and that only partitions such that all the cooling subsets are employed are meaningful. For example, to cool only one qubit (N c = 1) it is clear that the best strategy consists of using k = N h qubits, which excludes any partition into cooling subsets. On the other hand, for N c ≥ N h /2 (equivalently N c ≥ N/3) we can use N h /2 cooling subsets of two qubits to cool n c = N h /2 ≤ N c cold qubits. In this case all the cooling subsets are harnessed and the bound (45) is saturated with the maximum coefficient ξ (2) cool . More generally, we have that 2ξ (2) where Eq. (47) indicates that partitions into subsets having more than two qubits are suboptimal. Importantly, this assertion depends on the validity of the conjecture ξ cool , for all k ≥ 2. The upper bound in Eq. (48) generalizes the bound (45) to all the possible partitions of N h ≥ 2N/3 + 1 hot qubits. This bound is in general not saturable, as already exemplified with the case N c = 1. The corresponding lower bound follows from the fact that N h ≥ 2N c for N c ≤ N/3 − 1, whereby N c subsets containing at least 2 qubits can be used to cool all the cold qubits. In such a case the heat extracted per cold qubit equals 2ξ (2) cool . However, it is clearly more profitable to employ larger cooling subsets that allow to cool more each individual qubit, as illustrated with the four-qubit subsets in Fig. 9. This implies that the left inequality in (48) is in general estrict, and clarifies why in this regime minimal cooling subsets are not the optimal choice.

C. Advantage of the CC strategy
In contrast to the MBC strategy, the CC strategy has a more direct characterization. Let us denote as Q (CC) C the total extracted heat in this case, to distinguish it from the heat Q C considered before. First, note that the CC strategy is by construction based on a cyclic operation where each cycle is optimized to maximize the cooling of a qubit, using a single hot qubit and a catalyst. This procedure is depicted in Fig. 9, where the four hot qubits (dashes) in the right-most dashed line are employed in a sequence of four cycles that cool four cold qubits. Since p for all 1 ≤ i, j ≤ N , we consider another qubit as catalyst (green circle in Fig. 9). This choice is based on the plots (a), (b) and (c) in Fig. 6, which show that when the hot qubit and the cold qubit have the same populations a two-level catalyst maximizes the cooling. Accordingly, the maximum heat extracted per cycle is given by the cooling current (37), with n = 2 and p h 2 = p c 2 = e −β 1+e −β . The total extracted heat is thus i.e. n c times the aforementioned cooling current, being n c the maximum number of cycles that can be implemented. For N c ≤ N/2, all the cold qubits can be cooled down using N c ≤ N h hot qubits (which corresponds to n c = N c cycles). On the other hand, for N c ≥ N/2 + 1 only n c = N h < N c qubits are cooled down but all the N h qubits are consumed. Therefore, from Eq. (49) and the aforementioned conditions it follows that To perform the comparison between CC and MBC we introduce the relative performance ratio where both the numerator and the denominator refer to a fixed number of cold qubits and population p c 2 (which in turn characterizes the inverse temperature β). For N c ≤ N/3 − 1, the lower bound in Eq. (48) and Eq. (51) lead to Clearly, γ Nc≤ N 3 −1 is bounded from above by unity and therefore MBC outperforms CC in this regime.
If N c ≥ N/3, Eqs. (47) and (50) yield the following expressions for γ: The lower bound at the r.h.s. of Eq. (53) is tight only for N c = N/3, which implies that the performance ratio is strictly larger if N c > N/3. In particular, we are interested in values of N c such that γ N/3≤Nc≤N/2 > 1, since this means that the CC strategy is better than the MBC strategy. This condition leads to the equivalent inequality The r.h.s. of such inequality varies between 1/3, for p c 2 = 0, and 3/7, for p c 2 = 1/2. Accordingly, in the regime 3N/7 < N c ≤ N/2 the performance ratio (53) satisfies γ 3N/7≤Nc≤N/2 > 1. For the remaining interval N/3 ≤ N c ≤ 3N/7, Eq. (55) provides an upper bound on p c 2 to have γ > 1.
On the other hand, from Eq. (54) it follows that 4/3 ≤ γ N/2+1≤Nc≤N ≤ 2, which implies that in this regime the CC strategy outperforms the MBC cooling strategy, for any value of the inverse temperature β. We also stress that in the definition of γ the heat Q C is optimized with respect to all the many-body interactions involving N h qubits. Hence, even allowing arbitrary control over the available N h hot qubits, the CC strategy with low control is more powerful in this regime. Figure 11. Performance ratio γ (cf. Eq. (51)), as a function of the fraction of cold qubits Nc/N . The upper and lower boundaries of the blurred region correspond respectively to the limits β → ∞ and β → 0. For β finite, γ is described by a curve inside this region that has the same shape as those describing the boundaries, and whose vertical position increases with β. For Nc/N ≥ 1/3 (continuous segment) the curve gives the exact value of γ, and for Nc/N < 1/3 (dashed segment) it yields an upper bound. The light blue (light red) area covers the regime where CC outperforms (is outperformed by) MBC. For Nc/N ≥ 3/7, γ > 1 for all β. For 1/3 ≤ Nc/N ≤ 3/7, γ > 1 iff β is contained between the upper boundary and the dash-dotted segment.
The darker blue region in Fig. 9 depicts the regime (in terms of N c ) where the CC advantage takes place, irrespective of the inverse temperature β. The darker purple region corresponds to the regime (in terms of N h ) where MBC outperforms CC, and the gray-like region in between is the interval where the CC advantage is restricted to temperatures that obey Eq. (55). In particular, we note that in the limit β → ∞ the fraction N c /N satisfies this equation in all the interval (1/3, 3/7]. Therefore, for very low temperatures and in the limit N → ∞, where such a fraction behaves approximately as a continuous variable, the CC advantage can be extended to the interval (1/3, 1]. The quantitative assessment of the performance ratio γ is provided in Fig. 11, following Eqs. (52)- (54). Importantly, we see once more that the CC advantage is strengthened as the temperature decreases, reaching a maximum value of γ = 2 for β → ∞.
Finally, note that even if only n c < N c qubits can be cooled down for a given value of N c (e.g. if N c ≥ N/2), the remaining N c − n c qubits are not heated up either. This condition guarantees that the temperature of all the N c qubits remains below a certain treshold (in this case below β −1 ), and may be important for some applications. In other words, it provides a justification for choosing a fixed value of N c . On the other hand, one may alternatively be interested in optimizing the extracted heat with respect to N c . In the case of the CC strategy it readily follows from Eq. (50) that such a maximum is attained for N c = N/2. Regarding the MBC strategy, we can resort to the bound max K Q C ≤ ξ (2) cool N h (cf. Eqs. (47) and (48)) to determine if there are values of N c such that max K Q C potentially surpasses the maximum Q which after a simple algebra leads to the inequality The maximum value of the fraction N c /N that satisfies this bound is achieved in the high temperature limit β → 0, where the bound tends to 1/3. Conversely, as β increases the bound becomes tighter, and in the limit β → ∞ we have that the fraction must go to zero. This implies that for MBC to outperform the optimal (with respect to N c ) CC the number of cold qubits must be lower than N/3, and that for very low temperatures optimal CC outperforms MBC for almost any value of N c . In particular, using Eqs. (46) and (47) we have that, for any temperature,

VII. CATALYTIC THERMOMETRY
In this section we study an example where a catalyst is applied for precision enhancement in thermometry [51], where the goal is to estimate the temperature of a certain environment at thermal equilibrium. Let ρ e = e −βHe Tr(e −βHe ) denote the state of an environment with Hamiltonian H e = j ε j |j e j e |, equilibrated at inverse temperature β. Essentially, a temperature estimation consists of assigning temperature valuesT i to the different outcomes of a properly chosen observable O. In this way, the set {T i } defines a temperature estimatorT , and the precision is assesed through the estimation error where T = β −1 is the actual temperature and p i is the probability of measuring the outcome i.
The traditional approach to characterize the thermometric precision and also the precision in the estimation of more general physical parameters is based on the Fisher information [49]. This quantity determines a lower bound on the estimation error, known as the Cramer-Rao bound. In the case of thermometry, it is known that the Cramer-Rao bound is always saturated if O = H e . That is, if the temperature estimation is carried out by directly performing energy measurements on the environment. Here we consider a different scenario, where an auxilliary ancilla or "probe" is used to extract temperature information via an interaction with the environment. Such a technique may be useful for example if the environment is very large and direct energy measurements are hard to implement. However, our main motivation is to show that the estimation error can be reduced below the minimum value attained only with the probe, by including an additional interaction with a catalyst. We consider a three-level environment with degeneracy ε 1 = ε 2 = 0, which is probed by a two-level system in the initial state = p 1 |1 P 1 P | + p 2 |2 P 2 P |, p 1 > p 2 (for simplicity we only use the subindex P for the eigenstates of ). Moreover, the catalyst is also a two-level system in the initial state ρ v . This setup is illustrated in Fig.  12(a). It also corresponds to the physical configuration studied in Sect. V-D, with the probe and the environment taking respectively the roles of the cold qubit and the hot object. As we will see, under suitable conditions the same catalytic transformation that allows cooling enhancement also allows precision enhancement in the temperature estimation.
We assume thatT is an unbiased estimator, which means that its expectation value coincides with the actual temperature: T = T . It is important to mention that the assumption of unbiased estimators is common not only in thermometry but also for metrology in general [Refs.]. In particular, the Cramer-Rao bound limits the precision attained with this kind of estimators. If Var(T ) = T 2 − T 2 is the variance ofT . Moreover, it can be shown [51] that if the temperature to be estimated belongs to a small interval (T − δT, T + δT ), the estimation error using the observable O reads For the sake of covenience, we shall consider an "inverse temperature estimator"β instead ofT . The errors σ(T ) and σ(β) are connected by the simple relation σ(T ) = T 2 σ(β), which follows from the chain rule A. Optimal precision using only the probe and catalytic enhancement In our example the observable O P = o 1 |1 P 1 P | + o 2 |2 P 2 P | describes a projective measurement on the probe, with eigenvalues o 1 and o 2 . Information about β is encoded in the probe state = Tr e U P e ( ⊗ ρ e )U † P e , which results after a unitary evolution U P e that couples the probe with the environment. It is straightforward to check that in this case the estimation error reads where p 1 = Tr(|1 P 1 P | ). The ratio in the r.h.s. of Eq. (60) constitutes the figure of merit in our analysis. On the one hand, under certain conditions one can find a unitary U P e that minimizes the product p 1 p 2 , and at the same time maximizes the quantity ∂ β p 1 . In such a case, the inequality min U P e σ (β) ≥ min U P e p 1 p 2 max U P e ∂ β p 1 (61) guarantees that the same operation minimizes the error σ (β). This implies that if a unitary U P e saturates the previous bound it also optimizes the temperature estimation by measuring only the probe.
On the other hand, we will see that when the bound (61) is saturable it is possible to apply a catalytic transformation such that where p 1 = Tr(|1 P 1 P | ), and is the probe state obtained after an interaction that involves a two-level catalyst. This means that such catalytic transformation further reduces the estimation error. Denoting the corresponding unitary evolution as U , the total transformation → is implemented by a global unitary of the form U U P e . Importantly, the optimal U P e and U satisfy U P e = U cool and U = U , being U cool and U the cooling and catalytic cooling unitaries defined in Sect. V-D. This is not a coincidence, as we show below that the bound (61) can be saturated by maximally cooling the probe with the environment. In addition, the fact that U yields a cooling enhancement for the probe (which here takes the role of the cold qubit) implies that it also reduces the product p 1 p 2 . Since we also show that ∂ β p 1 > ∂ β p 1 , Eq. (62) follows.

B. Maximization of the population sensitivity in terms of passivity
In what follows we will refer to ∂ β p 1 as the "population sensitivity", as it quantifies how the final population p 1 varies with respect to temperature changes. Defining ρ P e = ⊗ ρ e and ρ P e = U P e ρ P e U † P e , we can use the fact that U P e is independent of β to write the population sensitivity as The operator ∂ β ρ P e = ⊗ ∂ β ρ e has real eigenvalues Figure 12. (a) The thermometric setup. Initially a twolevel system optimally probes the temperature of a threelevel environment. Afterwards, a joint interaction with a twolevel catalyst (green triangle) reduces the minimum estimation error previously achieved. (b) ln(p) × ln(p e ) diagram for the initial probe-environment state ⊗ ρe. If the condition β(ε3 − ε2) ≥ ln(p1/p2) (cooling of the probe using only the environment) holds, the swap U |1 P 3e ↔|2 P 1e is an optimal unitary U op P e that minimizes the error σ (β).
and H e = Tr(H e ρ e ). As we show next, this property allows us to analyze the maximization of the population sensitivity by applying the tools of passivity. Since ρ P e is hermitian, the operator A ≡ ∂ β ρ P e − min{λ P e i,j }I is positive semidefinite. In this way, we can rewrite Eq. (63) in the form where A Tr(A) represents a density matrix (i.e. its eingevalues describe a probability distribution). Accordingly, maximizing ∂ β p 1 is equivalent to maximize the expectation value of |1 P 1 P | over global unitaries that act on this (effective) initial state. Moreover, it is not difficult to see that this maximization is achieved by transforming A Tr(A) in a passive state with respect to −|1 P 1 P |. The definition of A Tr(A) also implies that the corresponding passive state is obtained by a permutation that transfers the three largest eigenvalues of ∂ β ρ P e to the eigenstates {|1 P p e j } j . Consequently, the application of such a permutation yields an operator where {λ P e i,j } is a rearrangement of the eigenvalues {λ P e i,j } that satisfies min j λ P e 1,j ≥ max j λ P e 2,j .

C. Results
For an initial state such that βε 3 > ln(p 1 /p 2 ) (cf. Fig. 12(b) and the cooling transformation U cool in Sect. V-D), the swap U |1 P 3e ↔|2 P 1e optimally cools the probe, which amounts to maximize the value of p 1 p 2 . On the other hand, it is easy to check that U |1 P 3e ↔|2 P 1e is also a permutation that satisfies Eq. (65). This implies that in such a case U |1 P 3e ↔|2 P 1e is an optimal unitary U op P e that saturates the bound (61). However, we stress that while this swap always maximizes the population sensitivity ∂ β p 1 , it also minimizes the error σ (β) as long as cooling is possible with the environment, see Fig. 12(b). Otherwise, any unitary U P e increases simultaneously the product p 1 p 2 and the population sensitivity, and we cannot be certain that the maximization of ∂ β p 1 is accompanied by a minimization of σ (β).
The dashed red curves in Fig. 13 show the estimation error obtained with U |1 P 3e ↔|2 P 1e , for three different initial states . In the same figure, the black (dasheddotted) curves stand for the corresponding Cramer-Rao bound, which characterizes the minimum error that can be attained under POVMs (positive operator valued measurements) on the environment. Therefore, these curves are below the red ones, as expected. The estimation error achieved after the subsequent interaction with the probe and the catalyst is depicted by the blue curves in Fig. 13. The catalytic transformation has exactly the same form of the one illustrated in Fig. 7(b), if the labels c and h are substituted respectively by P and e. The corresponding final population sensitivity is given by ∂ β p 1 = Tr [|1 P 1 P |∂ β ρ P e ] , where ρ P e = U (ρ P e )U † .
To understand why the population sensitivity is increased through U it is convenient to write explicitly the sensitivity attained before the catalyst is employed. Specifically, where Tr e ∂ β ρ P e is computed from Eq. (65). Keeping in mind that U is composed of (total) swaps between the states connected by the currents in Fig. 7(b), the final sensitivity reads where the contribution p 2 [p v 1 (λ e 2 ) − p v 2 (λ e 1 )] is due to the swap that generates the cooling current. Crucially, ∂ β p 1 > ∂ β p 1 iff this contribution is positive. Noting that λ e 1 ≥ 0 (cf. Eq. (64)), it follows that the catalytic transformation increases the sensitivity iff λ e 2 > 0 and The degeneracy condition ε 1 = ε 2 implies that λ e 1 = λ e 2 , and consequently this inequality holds for any catalyst whose intial state is not fully mixed. In this way, the same catalytic transformation studied in Sect. V-D cools down the probe and simultaneously enhances the population sensitivity. Accordingly, the final estimation is such that σ (β) < σ (β), which is illustrated by the fact that the blue solid curves are always below the red dashed ones in Fig. 13. The insets stand for the restricted intervals e −βε3 ≤ p 2 /p 1 , where the swap that maximizes the population sensititvity (cf. Fig. 12) also minimizes σ (β) (by optimally cooling the probe). Therefore, in this region Eq. (62) is satisfied, i.e. σ (β) < min U P e σ (β). From Fig. 13 we see that both the precision without the catalyst and the catalytic advantage are more pronounced the purer is the initial state of the probe. In particular, for p 2 /p 1 = 0.1 the catalytic transformation yields an error very close to the Crammer-Rao bound.

VIII. CONCLUSIONS AND OUTLOOK
In this paper, we introduced tools for the systematic construction of catalytic transformations on quantum systems of finite size. Size limitations constrain tasks such as cooling using a finite environment or thermometry with a very small probe. In the case of cooling, we showed that the introduction of a catalyst lifts cooling restrictions in two complementary ways: catalysts enable cooling when it is impossible using only the environment, and enhance it when the environment suffices to cool. These results were illustrated with several examples regarding the cooling of a single qubit. In particular, we found that small catalysts such as three-level systems allow maximum cooling in wide temperature ranges. We also demonstrated that to cool a system of any dimension a large enough catalyst and any environment that starts in a non fully mixed state are sufficient. Moreover, the ground population of the system can be catalytically increased, without requiring any interaction with an environment. Another advantage of catalytic cooling was shown in a setup consisting of many qubits prepared in identical states, where a subset of qubits is employed as environment to cool another subset. In this system, we found that it is possible to outperform the cooling achieved through many-body interactions with the environment, by including a two-level catalyst that cools using at most three-body interactions.
An application to thermometry was illustrated by considering a three-level environment whose temperature is probed by a two-level system, where we demonstrated that the inclusion of a two-level catalyst enhances the precision of the temperature estimation. It is worth remarking that this is the smallest possible setup where a catalyst may provide an advantage with respect to optimal interactions between the probe and the environment. For example, a two-level environment can be directly swaped with a two-level probe, which allows to saturate the thermal Crame-Rao bound by performing appropriate measurements on the probe. A similar argument also implies that for larger environments probes that have at least the same dimension allow optimal thermometry. Hence, in contrast to cooling, a catalyst yields a thermometric enhancement by circumventing size limitations that do not refer to the environment, but rather to the probe. Beyond the example mentioned above, an interesting direction for future work is to determine more general conditions for catalytic advantages in thermom-etry. This includes to further study the roles of initially mixed probes and the sizes of the involved systems.
We have seen that the dimension of the catalyst is crucial to bypass thermodynamic restrictions imposed by the finiteness of the other systems. This observation is related to a question recently posed in [15], where the authors ask if certain transformations achieved with multipartite catalysts can be performed with a single, and sufficiently large catalyst. The findings presented here may contribute to elucidate this puzzle, since they are based on single-copy catalysts. To that end, the first step is to examine how our results can be extended to include the possibility of energy-preserving interactions, which is the framework considered in [15]. We also remark that the characterization of catalytic transformations provided here is valid for systems that do not necessarily start in thermal states. Hence, it can be useful for studying the role of catalysts in scenarios beyond thermodynamics. Finally, we hope that our explicit description of catalytic transformations paves the way to experimental realisations in the near future.

ACKNOWLEDGMENTS
RU is grateful for support from Israel Science Foundation (Grant No. 2556/20).
ing currents connect eigenstates of ρ s , with maximum and minimum eigenvalues restricted by G s . In other words, let I G = {j 0 , j 1 , ...} be the set of indices that label the eigenstates spanning the subspace G s , i.e. G s = span{|j s } j∈I G . Taking into account that p s j+1 ≤ p s j , a restoring unitary acting on G s ⊗span{|k v } l≤k≤l exists iff the chain {J |min j∈I G js(k+1)v →|max j∈I G jskv } l≤k≤l exists. In the case of the left subspace, G s = span{|j s } 1≤j≤i , and the corresponding chain is {J |1s(k+1)v →|iskv } l≤k≤l . Since we are also assuming that the right chain {J |(i+1)s(k+1)v →|dskv } l≤k≤l does not exist (i.e. that Eq. (A3) is also violated), there is no restoring unitary in the subspace span{|j s } i+1≤j≤ds ⊗ span{|k v } l≤k≤l either.
The preceding discussion implies that the only way to have a catalytic transformation is by means of a restoring unitary on a subspace span{|j s } j∈I G ⊗ span{|k v } l≤k≤l , such that max j∈I G ≥ i + 1 and min j∈I G ≤ i. Note that a subspace determined by the single condition max j∈I G ≥ i+1 (min j∈I G ≤ i) includes the possibility of a right (left) restoring unitary, which has already been discarded. If max j∈I G ≥ i + 1 and min j∈I G ≤ i, at least one of the partial swaps composing the restoring unitary (cf. Eq. (8)) must connect an eigestate of ρ s ⊗ ρ v belonging to span{|j s } 1≤j≤i ⊗ span{|k v } l≤k≤l , with one eigenstate belonging to span{|j s } i+1≤j≤ds ⊗span{|k v } l≤k≤l . Otherwise, the restoring unitary could be defined exclusively in the left or in the right subspace. This partial swap has an associated restoring current that transfers population from the (system) subspace span{|j s } 1≤j≤i towards span{|j s } i+1≤j≤ds . Let us denote this current simply as J. Since the cooling current J |(i+1)slv →|is(l +1)v transfers population in the opposite direction, we have that where the condition J = J |(i+1)slv →|is(l +1)v is necessary for the restoring unitary to generate a uniform loop.
In this way, we conclude that the potential increment of the partial sum i j=1 p s j is spoiled if we also demand that the transformation is catalytic. Hence, the transformation must be unital. Finally, we note that if ρ s = ρ c ⊗ ρ h , the substitutions (in Eqs. (A1)-(A3)) 1 ≤ i, i ≤ d c and 1 ≤ j, j ≤ d h ). The non existence (i.e. its nullity) of J |icj h k v →|i c j h k v means that it is impossible to connect any pair catalyst eigenstates |k v ≥ (K + 1) v and |k v ≤ K v using a restoring current. Since any chain of the form ch |(l +1)v →|lv must contain this type of current, the non existence of J |icj h k v →|i c j h k v also implies the non existence of ch |(l +1)v →|lv .
Optimal CC unitary. The optimal unitary that yields the current (E4) is composed of swaps between the states connected by the cooling current and by the restoring currents. The reason is that, as already proven, J max cool is maximized through a uniform loop formed by swap currents (cf. Eq. (E1)). Besides the swap for the cooling current, which exchanges the states |2 c 1 h 1 v and |1 c 2 h n v , the swaps that give rise to the restoring currents (E2) exchange the states |1 h (k + 1) v and |2 h k v .
. The eigenstates of ρ ch that possess eigenvalue ε c 1 with respect to H c have eigenvalues The unitaryŨ ch is a permutation that exchanges the eigenvalues A > and A < , whence the final eigenvalues corresponding to eigenstates {|1 c j h } j and {|2 c j h } j are respectively given by A > ∪A c < and A < ∪ A c > . Keeping in mind that the final state ρ ch =Ũ ch (ρ ch )Ũ † ch commutes with H c , this state is also passive with respect to H c if max(a ∈ A < ∪ A c > ) ≤ min(a ∈ A > ∪ A c < ). If this is the case the mean energy of the cold object cannot be further reduced andŨ ch performs optimal cooling. This inequality holds iff max(a ∈ A < ) ≤ min(a ∈ A > ), min(a ∈ A c < ), max(a ∈ A c > ) ≤ min(a ∈ A > ), min(a ∈ A c < ). The inequality max(a ∈ A < ) ≤ min(a ∈ A c < ) holds because A < is a subset of the smallest elements of A < ∪A c < , and max(a ∈ A c > ) ≤ min(a ∈ A > ) holds because A > is a subset of the largest elements of A > ∪ A c > . Moreover, we note that the maximum index j = j max in the sets A < and A > determines the corresponding maximum and minimum elements, and that by definition p c 1 p h jmax ↑ < p c 2 p h jmax ↓ . Therefore, max(a ∈ A < ) ≤ min(a ∈ A > ). This leaves us with the verification of the inequality max(a ∈ A c > ) ≤ min(a ∈ A c < ). Noting that we can write max(a ∈ A c > ) = p c 2 p h jmax+1 ↓ and min(a ∈ A c < ) = p c 1 p h jmax+1 ↑ , the assumption max(a ∈ A c > ) > min(a ∈ A c < ) is contradictory because it would imply that max(a ∈ A c > ) and min(a ∈ A c < )) also belong to A > and A < , respectively. In this way, it follows that max(a ∈ A c > ) ≤ min(a ∈ A c < ), which completes the proof of the inequality max(a ∈ A < ∪A c > ) ≤ min(a ∈ A > ∪A c < ) and the passivity of ρ ch with respect to H c .
Proof of Theorem 3. Let us see that, after the application ofŨ ch , a subsequent cooling can always be achieved through a CC transformation. Such a transformation operates with the eigenstates of ρ ch whose eigenvalues belong to the sets A c > and A c < , on whichŨ ch acts as the identity (cf. Eqs. (F1) and (F2)). Let us denote the eigenstates with eigenvalues in A c > as {|2 c j h,> } and the eigenstates with eigenvalues in A c < as {|1 c j h,< }. What we show below is that there exists a catalyst state ρ v ∈ B(H v ) that allows to implement a CC transformation, through a global unitary U that maps the enlarged subspacespan({|1 c j h,> } ∪ {|0 c j h,< }) ⊗ H v into itself. This is a consequence of the hypothesis concerning the non-degeneracy of the largest or the smallest eigenvalues of ρ h , referred to in Theorem 3. Specifically, we will show that there exists a restoring chain that forms a loop with the cooling current are respectively determined by the smallest and largest eigenvalues of ρ h , referred to in Theorem 3. Accordingly, if the hypothesis of this theorem holds, at least two elements of either of the sets A c > or A c < are different. Suppose first that such elements belong to A c < , and let us denote as |1 c j max h,< and |1 c j min h,< two eigenstates with eigenvalues max(a ∈ A c < ) and min(a ∈ A c < ) = max(a ∈ A c < ), respectively. By choosing a sufficiently mixed catalyst, we can construct a restoring chain Likewise, if the set A c > contains at least two different elements, the loop can be closed with the chain where |2 c j min h,> is an eigenstate of ρ ch with eigenvalue min(a ∈ A c > ) = max(a ∈ A c > ). Since any of the previous chains is composed of currents that take place inside energy eigenspaces of the cold qubit, they do not spoil the cooling effect due to J cool . In addition, note that the cooling current is determined by the inequality p c 2 max(a ∈ A c > )p v 1 > p c 1 min(a ∈ A c < )p v dv , which can always be satisfied if the d v is large enough, even for a very mixed state ρ v .