Continuous thermomajorization and a complete set of lawsfor Markovian thermal processes

Matteo Lostaglio∗,1, 2 and Kamil Korzekwa∗,3

1Korteweg-de Vries Institute for Mathematics and QuSoft, University of Amsterdam, The Netherlands2QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands

3Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Krakow, Poland

The standard dynamical approach to quantum thermodynamics is based on Markovian masterequations describing the thermalization of a system weakly coupled to a large environment, andon tools such as entropy production relations. Here we introduce a new framework overcoming thelimitations that the current dynamical and information theory approaches encounter when applied tothis setting. More precisely, based on a newly introduced notion of continuous thermomajorization,we obtain necessary and sufficient conditions for the existence of a Markovian thermal processtransforming between given initial and final energy distributions of the system. These lead to acomplete set of generalized entropy production inequalities including the standard one as a specialcase. Importantly, these conditions can be reduced to a finitely verifiable set of constraints governingnon-equilibrium transformations under master equations. What is more, the framework is alsoconstructive, i.e., it returns explicit protocols realizing any allowed transformation. These protocolsuse as building blocks elementary thermalizations, which we prove to be universal controls. Finally,we also present an algorithm constructing the full set of energy distributions achievable from a giveninitial state via Markovian thermal processes and provide a Mathematica implementation solvingd = 6 on a laptop computer in minutes.

I. INTRODUCTION

It is well-known that thermodynamics can be formu-lated in the resource theory language of information the-ory [1–6]. Since it focuses only on input-output relationsunder a class of quantum operations, the resource theoryis unable to discuss how the process is realized in time,nor does it involve the notions of entropy production ormaster equations, even though the latter are common-place in standard approaches. As such, the standard andthe resource-theoretic frameworks are for the most partdisconnected. Somewhat surprisingly, despite all its pow-erful theorems, the resource theory has not affected theanalysis of a thermodynamics practitioner following themore explicit dynamical approaches based on the masterequation formalism [7].

The aim of this work is to overcome this situationby showing how to unify the master equations and in-formation theory tools. We first summarize the ba-sic notions of both approaches to thermodynamics inSec. II, and demonstrate the insufficiency of each of them,when taken separately, to capture relevant thermody-namic constraints. Then, in Sec. III, we propose a hybridapproach that can overcome these limitations. In con-structing our solution we highlight the necessity of find-ing a finitely verifiable set of thermodynamic laws spec-ifying when one state can be thermodynamically trans-formed into another one. Furthermore, we wish to gobeyond the question of whether a thermodynamic trans-

∗ These authors contributed equally to this work.Emails: lostaglio@protonmail.com, korzekwa.kamil@gmail.com

formation exists, and ask how it should be realized. Theresource theory approaches are for the most part silentabout the latter, which is arguably a central obstacle totheir application to concrete problems. Our framework,on the contrary, will be constructive.

In Sec. IV we introduce our main novel technical tool,continuous thermomajorization, which extends the con-cept of thermomajorization introduced in Refs. [3, 8]. Weprove that this partial order of energy distributions pro-vides necessary and sufficient conditions for the existenceof a thermalization process generated by a Markovianmaster equation. More precisely, we show that an out ofequilibrium energy distribution can be transformed intoanother one by a Markovian thermal process if and only ifthe former continuously thermomajorizes the latter. Wethen connect with the notion of entropy production [9] inSec. V, where we show that continuous thermomajoriza-tion allows one to identify a complete set of generalizedentropy production relations, including the standard oneas a special case.

Our ultimate goal, however, is to find a finitely-verifiable set of such conditions. To achieve this, wefirst prove in Sec. VI that elementary thermalizations,generated by simple reset master equations on two-levelsubmanifolds, form universal controls in the Markovianregime. In other words, every state transformation thatcan be achieved by a Markovian thermal process can alsobe achieved by a sequence of elementary thermalizations.Employing this simplified set of controls, in Sec. VII weshow that continuous thermomajorization indeed can bechecked in a finite number of steps. Remarkably, we canalso return the exact sequence of elementary controls re-quired to realize any allowed transformation. As a finalresult of our paper, in Sec. VIII we provide an explicit

arX

iv:2

111.

1213

0v1

[qu

ant-

ph]

23

Nov

202

1

2

algorithm verifying the continuous thermomajorizationrelation between any two vectors, and offer a Mathemat-ica implementation [10] that checks the conditions forsystems of dimension up to 7 in a matter of hours on astandard laptop computer.

While this work focuses on developing the technicalmachinery, in the accompanying paper [11] we illustratehow one can employ it to design provably optimal ther-modynamic protocols. There, we apply our results tostudy the effects of memory on work fluctuations, to ex-plicitly construct optimal cooling protocols, and to show-case the important role played by catalysts in practicalscenarios. We also report on a recent work which usedour results to show that non-Markovianity boosts the ef-ficiency of thermal bio-molecular switches [12]. All thesebuild up encouraging evidence that the framework is suit-able both for providing model-independent bounds, aswell as for algorithmically constructing new thermody-namics protocols when a complete analysis is unattain-able by either analytic or numerical methods.

II. SETTING THE SCENE

A. Thermodynamic frameworks

1. The traditional approach

A general open dynamics of a d-dimensional quantumsystem is described by a quantum channel, i.e., a com-pletely positive trace-preserving map acting on the quan-tum state ρ [13]. However, in a thermodynamic setting,we are often interested in the evolution of such a quan-tum system interacting with a large thermal bath at in-verse temperature β = 1/(kBT ), where kB is Boltzmannconstant and T is the temperature of the bath. Typi-cal microscopic derivations employing the weak couplinglimit [7, 14, 15] then lead to a master equation with thegeneral form [16, 17]:

dρ(t)

dt= H(ρ(t)) + Lt(ρ(t)). (1)

The first term, H, is the generator of a closed (reversible)quantum dynamics,

H(ρ) = −i[H, ρ], (2)

with [·, ·] denoting the commutator, [A,B] = AB − BA,and H being the (dressed) Hamiltonian of the system.The second term, Lt, is known as the Lindbladian ordissipator and generates an open (irreversible) quantumdynamics. It has the following general form

Lt(ρ) =∑i

ri(t)

(LiρL

†i −

1

2{L†iLi, ρ}

), (3)

with {·, ·} denoting the anticommutator,{A,B} = AB +BA, Li being jump operators, andri being non-negative jump rates.

While a general Lindbladian only requires the rates rito be non-negative, Lindbladians arising from the inter-action of a quantum system with a large heat bath havetwo standard properties [7, 14, 15]:

(P1) Stationary thermal state. The Gibbs thermalstate of the system,

γ =e−βH

Tr (e−βH), (4)

is a stationary solution of the dynamics, i.e.,

∀t : Ltγ = 0. (5)

(P2) Covariance. The Lindbladian Lt commutes withthe generator of the Hamiltonian dynamics H at alltimes t, i.e.,

∀ρ : Lt(H(ρ)) = H(Lt(ρ)). (6)

For brevity, we will refer to the quantum dynamicsgenerated by master equations in the form of Eq. (1)and satisfying properties (P1)-(P2) as Markovian thermalprocesses:

Definition 1. A channel T is a Markovian thermalprocess (MTP) if it results from integrating a Marko-vian master equation, Eq. (1), between time 0 andtf ∈ [0,+∞], where the Lindbladian Lt satisfies prop-erties (P1)-(P2).

These form a standard description of thermalization inthe field of quantum thermodynamics and beyond (see,e.g., Sec. 3.1 of Ref. [15]), and will be the main focus ofthis work. A typical example of a Markovian thermal pro-cess is the quantum optical master equation [14], whichdescribes the evolution of a two-dimensional quantumsystem with H = ~ω

2 (|1〉〈1| − |0〉〈0|) interacting weaklywith a thermal radiation field. The quantum opticalmaster is of the form given in Eq. (1), with the jumpoperators and the corresponding rates given by

L1 = |0〉〈1| , r1(t) = r(N + 1), (7a)

L2 = |1〉〈0| , r2(t) = rN, (7b)

whereN is the average number of photons at the resonantfrequency ω and r is the spontaneous emission rate.

We will prove that any transformation realized by ar-bitrary Markovian thermal process can be achieved bysequential thermalization of 2-level subsets of energy lev-els of the system. Hence, the time dependence of Ltcan be limited to switching which levels are coupled tothe bath, a standard set of controls in the context ofquantum thermodynamics [18]. We highlight, however,that our results apply well-beyond this setup. It suf-fices to say that any model that can be formally writ-ten as Eq. (1) with properties (P1)-(P2) (with or with-out time-dependence on Lt) falls within the scope ofthis work. These include incoherent noise in quantum

3

computers [19] and effective models describing fluores-cence and other non-radiative decay channels in atoms,molecules and nanostructures. The formal equivalencebetween thermodynamic and other models of dissipationis well-known and it is in fact leveraged as a standardtechnique to realize effective heat baths [20]. In quantuminformation terms, depolarization and amplitude damp-ing can be seen as limiting cases of Markovian thermalprocesses when β → 0 and β →∞, respectively. Here,we study Markovian thermal processes independently ofwhat application one has in mind. In the accompanyingpaper [11], we present examples of how the formalism canbe applied in practice.

The most well-known and important constraint on theallowed thermodynamic transitions takes the form of asort of H-theorem [15]:

dΣ(t)

dt:=− d

dtS(ρ(t)‖γ) ≥ 0, (8)

where

S(ρ‖γ) = Tr (ρ(log ρ− log γ)) (9)

is the quantum relative entropy and dΣ/dt is known asthe entropy production (relative to γ) [21]. One can rec-ognize in this equation the standard second law of ther-modynamics:

dΣ(t)

dt=dS(t)

dt− βJ(t) ≥ 0, (10)

with

S(t) := −Tr (ρ(t) log ρ(t)) , (11a)

J := Tr

(Hdρ(t)

dt

), (11b)

being the von Neumann entropy and the heat current,respectively.

2. The resource-theoretic approach

The notion of a Markovian thermal process should becontrasted with the notion of a thermal operation [3] orthe closely related notion of a thermal process [22], usedin the resource theory of quantum thermodynamics:

Definition 2. Thermal processes (TP) are all channelsE that satisfy the two conditions analogous to (P1)-(P2):

E(γ) = γ, (12a)

∀ρ, t : E(e−iHtρeiHt) = e−iHtE(ρ)eiHt, (12b)

with γ defined in Eq. (4).

Examples of thermal processes include all thermaloperations, i.e. dynamics induced by generic energy-preserving unitary interactions between the system and

an environment E described by an arbitrary HamiltonianHE and prepared in a thermal Gibbs state γE at fixedinverse temperature β [1, 4, 23]. When such a transfor-mation exists between the initial and final states, ρ(0)and ρ(tf ), we will write

ρ(0)TP7−→ ρ(tf ). (13)

The resource theory is concerned with giving neces-sary and sufficient conditions for Eq. (13). Because of thesymmetries inherent in the thermodynamic processes un-der consideration, it is convenient to represent the stateof the system ρ(t) at time t by a vector p(t) of populations(energy distributions) and a matrix C(t) of coherences,defined as

pi(t) = 〈Ei|ρ(t)|Ei〉, (14a)

Cij(t) = 〈Ei|ρ(t)|Ej〉 for i 6= j, (14b)

where |Ei〉 is the eigenstate of H corresponding to energyEi (for simplicity we consider non-degenerate H). Whenρ(0) and ρ(tf ) are diagonal (i.e., C(0) = C(tf ) = 0), theconditions for Eq. (13) are the well-known thermoma-jorization constraints of Ref. [3].1 The problem for gen-eral states was formally solved by the remarkable workin Ref. [22], where an extremely complex but completeset of entropy conditions was derived.

One can readily see, e.g. by taking a thermal opera-tion with E being a small environment, that the resourcetheory and the standard framework apply to differentregimes. For a concrete and relevant example, considera single qubit system and take the environment definingthe thermal operation to be given by a single bosonic os-cillator in resonance with it. Take the energy-preservingunitary interaction U(t) = exp(−itHint) with the inter-action Hamiltonian from the Jaynes-Cummings model:

Hint = g(|1〉〈0| ⊗ a+ |0〉〈1| ⊗ a†), (15)

with a† and a denoting the bosonic creation and anni-hilation operators, and g being a coupling constant. NoMarkovian master equation like Eq. (1) can be derived.In fact, the dynamics is fully solvable, so one can com-pute ρ(t) and verify that the entropy production relationfrom Eq. (8) is violated due to non-Markovian effects.The resource theory approach, in allowing fine controlover memory effects and system-bath correlations, doesnot incorporate the Markovianity condition of standardthermalization processes.

B. Insufficiency of current approaches

Quantum thermodynamics aims at deriving laws hold-ing independently of the particular dynamics. In other

1 In fact, the case C(0) = 0, C(tf ) 6= 0 is trivial, since thesetransformations are always forbidden.

4

words, based on minimal assumptions (such as assumingthat the environment the system interacts with is ther-mal), one wants to constrain possible state transforma-tions of the system. In what follows we discuss how boththe resource-theoretic and the standard approaches farein this regard, and highlight some important limitationsthat we wish to overcome with the present contribution.We will consider the stereotypical situation of a systemput into a weak thermal contact with a thermal bath atinverse temperature β. The crucial question is: given aquantum system initially described by a known energydistribution p(0), what general conditions determine thepossible p(t) achievable at some later time t?

1. Insufficiency of the traditional approach

The second law in the form of Eq. (8) provides a set ofconditions that can help us answer the question posed.In fact, the second law provides a functional on the set ofstates that must be monotonically decreasing along thedynamics. To explain its usefulness and limitations, letus focus on a simple example of an incoherent three-levelsystem, i.e., with d = 3 and C(0) = 0. Due to the co-variance of Markovian thermal processes (property (P2)),we necessarily have C(t) = 0 for all t > 0.2 Thus, theentropy Σ reads

Σ(t) = −3∑i=1

pi(t)(log pi(t)− log γi), (16)

where γ denotes the vector of thermal populations. Weplot the values of Σ in the simplex of all 3-dimensionalprobability distributions in Fig. 1a, for one particularchoice of γ. Then, we can verify whether there exists acontinuous path p(t) connecting a given initial p(0) withthe final p(tf ), with constantly non-decreasing Σ(t) alongthe path (as required by the second law). In particular,for two states with equal entropy Σ there is a uniqueisoentropic path connecting them, and the second law inthe form of positive entropy production does not forbidsuch a transition.

However, one can prove that for most pairs of isoen-tropic states there is no Markovian thermal process thatcan map between them, independently of the chosen con-trols and details of the environment. What the standardsecond law is missing is that there exist generalized (non-equilibrium) entropy production relations that must besatisfied for a thermodynamic transformation to be al-lowed. As we will discuss later in Sec. V, one of them isthe non-negativity of the collisional entropy production

2 This is because C(0) = 0 is equivalent to H(ρ(0)) = 0, and (P2)then implies H(Lt(ρ(0))) = 0. Thus, the state at time δt isstationary, C(δt) = 0, and the argument extends to all t > 0.

(a)

p(0)

p(tf )

γ

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

p1

p2

−2

−1

0

(b)

p(0)

p(tf )

γ

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

p1

p2

−2

−1

0

FIG. 1. Thermodynamic entropy landscape for athree-level incoherent system. The state of an inco-herent system is represented by a vector of populationsp = (p1, p2, 1− p1 − p2), and the chosen thermal state isγ = (0.7, 0.2, 0.1). (a) Entropy functional Σ. The blackdashed trajectory is the path of constant Σ. The constraintof positive entropy production cannot exclude that the ther-modynamic transition from p(0) to p(t) is allowed. (b) En-tropy functional Σ2. The black dashed trajectory is the pathof constant Σ. The constraint of positive collisional entropyproduction implies that one cannot transform p(0) into p(t).

(with respect to γ):

dΣ2(t)

dt:= − d

dtS2(p(t)‖γ) ≥ 0, (17)

where

S2(p‖γ) := log

(d∑i=1

p2iγi

), (18)

is the relative Renyi entropy of order two [24]. Eq. (17)provides an independent monotonically increasing func-tional for the thermodynamic system. As before, we plotthe values of Σ2 in Fig. 1b for the same choice of γ, sothat we can clearly see that the path which kept Σ con-stant is decreasing Σ2. Decreasing the collisional entropy(relative to γ) is forbidden, hence the transition is impos-sible under any Markovian thermal process. This rulesout, by means of an explicit counterexample, that thestandard entropy production functionals (often referredto as the second law of thermodynamics in the literature)faithfully characterizes thermalization out of equilibrium.

2. Insufficiency of the resource-theoretic approach

If the standard entropy production constraint is insuf-ficient, the information theory-minded reader could won-der whether the results of the resource theory approachcan come to the rescue. Here we will argue why thesetools, in their present form at least, are too weak to cap-ture the relevant limitations of quantum thermodynamicsin the Markovian regime.

For an explicit example, consider a two-level incoher-ent system, i.e., with d = 2 and C(0) = 0. Again, dueto covariance property (P2), we have C(t) = 0. Let us

5

choose

p(tf ) =

[(1− γ2

γ1

)p1 + p2,

γ2γ1p1

]. (19)

The resource theory then imposes the following family ofconstraints, known as the second laws of thermodynam-ics [4]:

Sα(p(0)‖γ) ≥ Sα(p(tf )‖γ), ∀α ∈ R, (20)

where

Sα(p‖γ) :=sgn(α)

α− 1log

(d∑i=1

pαi γ1−αi

)(21)

is the Renyi relative entropy of order α [24]. One canshow that, for p(tf ) from Eq. (19), all these constraintsare satisfied.3

Nevertheless, one can prove that p(tf ) cannot be ther-modynamically accessed from p(0) by a Markovian ther-mal process. As we shall see in Sec. V, for every Marko-vian thermal process the following generalized entropyproduction equations must hold:

dΣα(t)

dt:= − d

dtSα(p(t)‖γ) ≥ 0, ∀α ∈ R, (22)

which, for α = 1, recover the standard entropy produc-tion inequality from Eq. (8). Now, for d = 2, everydynamical trajectory connecting p(0) with p(tf ) can besimply parametrized as

p(λ) = (1− λ)p(0) + λp(tf ). (23)

In Fig. 2 we plot Σα as a function of α and the trajectoryparameter λ for a particular choice of p(0) and γ. Onecan clearly see in the plotted range of α that Σα at λ = 0is smaller than at λ = 1, but for each α there exists anintermediate point λ∗(α) ∈ (0, 1) at which Σα starts todecrease. Hence, the α-entropy production inequalitiesfrom Eq. (22) are violated at some intermediate time,even if the second laws of Eq. (20) are all satisfied. Thisis why no Markovian thermal process mapping p(0) top(tf ) exists, a fact that cannot be captured by the end-points condition of the resource theory approach. Theconceptual issue is clear: the resource theory approachonly considers discrete transformations and does not in-volve notions such as that of a continuously generatedprocess constantly producing entropy along its path.

Furthermore, it should be clear that even Eq. (22) (astrengthening of the second laws of Ref. [4]) is not entirelysatisfactory. In fact, beyond the simplest cases, one can-not exhaustively check an infinite set of inequalities alongarbitrary trajectories with fixed end-points.

3 This follows immediately from the fact that p(tf ) can be ob-tained from p(0) by applying the stochastic matrix G withG12 = 1, G21 = γ2/γ1. Since Gγ = γ, Eq. (20) follows, seee.g. Sec. II.C.2 of Ref. [1].

2 4 6 80

0.25

0.5

0.75

1

α

λ

−0.3

−0.2

−0.1

0

FIG. 2. Thermodynamic α-entropy for a two-level sys-tem. Value of the relative entropy functional Σα along thetrajectory specified in Eq. (23) connecting p(0) = (0.1, 0.9)with p(tf ) given by Eq. (19). The chosen thermal state isγ = (0.6, 0.4). Note that, for every α, the path connectingλ = 0 to λ = 1, necessarily passes through a region in whichentropy Σα decreases. Hence, no Markovian thermal processcan transform p(0) into p(tf ).

III. PROPOSED HYBRID APPROACH

To address the issues highlighted in the previous sec-tion, we propose to study the set of Markovian ther-mal processes using information theory tools, incorporat-ing from the get-go constraints that are commonplace inmost quantum thermodynamic settings, such as the pres-ence of a large heat bath, weak coupling and Markovian-ity. Our purpose is to leverage the information theorytools to complement the toolkit of the master equationformalism.

The central question investigated in this paper is: whatfinal states ρ(tf ) are accessible from an initial state ρ(0)by means of Markovian thermal processes? When such aprocess exists transforming ρ(0) into ρ(tf ) we will write

ρ(0)MTP7−→ ρ(tf ). (24)

Our main contribution is to find a complete set of condi-tions to answer this question when ρ(tf ) is block-diagonalin the energy basis. Due to property (P2), the problem isreduced to the one involving energy distributions (‘pop-ulations’)

p(0)MTP7−→ p(tf ). (25)

When approached directly, this may look like an ex-tremely complex control problem. As we have seen, it isnot enough to find a trajectory p(t) connecting p(0) top(tf ) involving irreversible entropy production (which initself is a hard task), because such a trajectory does notguarantee that a master equation achieving the desiredtransformation exists. A numerical brute force approachis also unfeasible, since it involves the exploration of avery high dimensional space of control parameters. Ulti-mately, this is related to the fact that characterizing whatdynamics can be realized by a Markovian master equa-tion is an extremely challenging problem even classically:this is known as the embeddability problem [25, 26]. De-spite having been studied for decades in the mathematics

6

literature, general analytic solutions are not known be-yond the simplest d = 2 and d = 3 case [27–30].

Lacking explicit characterizations, we will follow a dif-ferent strategy. It is crucial to highlight that the solutionwill satisfy two desiderata:

(D1) Finite verifiability. One should be able to verify

in a finite number of steps whether p(0)MTP7−→ p(tf )

holds for any given initial and final states.

(D2) Constructability. Whenever p(0)MTP7−→ p(tf )

holds, one should be able to explicitly construct aMarkovian thermal process realizing this transitionthrough a sequence of elementary controls.

These are central requirements for the applicability ofthe framework and in typical resource theory approachesthese are not both satisfied.

IV. CONTINUOUS THERMOMAJORIZATION

Here we introduce the main technical tool to solve theproblem at hand: a generalization of thermomajorization.We start with a summary of well-known results.

A. Recap: majorization and thermomajorization

Majorization is an ubiquitous relation between pairsof vectors that finds applications in fields ranging frommathematics and economy to information theory andquantum physics [31, 32]. Given two probability distri-butions, p and q, we say that p majorizes q, denotedp � q, if

j∑i=1

p↓i ≥j∑i=1

q↓i for j = 1, . . . , d, (26)

where x↓ denotes the vector x sorted in a non-increasingorder. The partial ordering of probability vectors in-duced by majorization can be seen as formalizing themeasure of disorder relative to the uniform distribu-tion η := (1/d, . . . , 1/d): for a fixed dimension d, sharpdistributions majorize all other distributions, and all dis-tributions majorize the uniform distribution η. Further-more, if p � q, then the Shannon entropy of p is smallerthan that of q. The same holds for a whole class of en-tropy functionals known as Schur-concave functions [31](including all Renyi entropies [24]).

However, just like with the entropy production inEq. (8), it is convenient to extend the notion of ma-jorization to thermomajorization [3, 8] (or majorizationrelative to γ, or γ-majorization), so that disorder is mea-sured relative to a generic non-uniform equilibrium dis-tribution γ. To do so, first denote by π(p) the reorderingof {1, . . . , d} that sorts pi/γi in a non-increasing order,

pπi(p)

γπi(p)≥ pπi+1(p)

γπi+1(p)for i = 1, . . . , d− 1. (27)

t

p

qr(t1)�γ r(t2)

FIG. 3. Continuous thermomajorization. The contin-uous thermomajorization relation p Ïγ q holds if and onlyif there is a continuous path of probability distributions r(t)connecting p and q such that r(t1) �γ r(t2) whenever t1 ≤ t2.

This is called the γ-ordering or thermomajorization or-dering of p. Next, we need to introduce the notion ofLorenz curve. The Lorenz curve of p relative to γ (alsocalled thermomajorization curve in Ref. [3]) is a piece-wise linear, concave curve on a plane that connects thepoints l(j) given by

l(j) =

(j∑i=1

γπi(p),

j∑i=1

pπi(p)

)(28)

for j ∈ {1, . . . , d}, where l(0) := (0, 0). Then, p is said tothermomajorize q (relative to γ), denoted p �γ q, whenthe Lorenz curve of p is never below that of q.4 Impor-tantly, in the case of uniform equilibrium distributions,γ = η, thermomajorization reduces to majorization. Fora fixed dimension, the sharp distribution with largestenergy, (0, . . . , 0, 1), thermomajorizes every other distri-bution, and every distribution thermomajorizes γ. Fur-thermore, if p �γ q, then the relative entropy S(p‖γ) islarger than S(q‖γ). The same holds for a general class ofrelative entropy functionals called thermodynamic Schur-concave functions in Ref. [1] (including all α-relative en-tropies Sα(·‖γ) in Eq. (20)).

The crucial property of thermomajorization as a par-tial ordering of probability vectors is that it characterizestransformations under thermal processes [1, 3]:

p(0)TP7−→ p(tf ) ⇔ p(0) �γ p(tf ). (29)

Note that Eq. (29) is satisfied for every Markovian ther-mal process, since these are a subset of thermal processes.However, the problem we pointed out in Sec. II B re-mains: these end-point conditions do not capture the ex-istence of a continuous process generated by a Markovianmaster equation.

4 If we denote the height of the Lorenz curve at point x ∈ [0, 1] byLx(p‖γ), this can also be written as Lxj (p‖γ) ≥ Lxj (q‖γ) for

j ∈ {1, . . . d}, where xj =∑ji=1 γπi(q) [33].

7

B. Continuous thermomajorization

We introduce the following strengthening of thermo-majorization that we illustrate in Fig. 3.

Definition 3 (Continuous thermomajorization). A dis-tribution p continuously thermomajorizes q (or continu-ously majorizes q relative to γ), denoted p Ïγ q, if thereexists a continuous path of probability distributions r(t)for t ∈ [0, tf ) such that

1. r(0) = p,

2. ∀ t1, t2 ∈ [0, tf ) : t1 ≤ t2 ⇒ r(t1) �γ r(t2),

3. r(tf ) = q.

We call r(t) a thermomajorizing trajectory from p to q.

Note that in the particular case of a uniform fixedpoint, γ = η, the above definition corresponds to a con-tinuous version of standard majorization, denoted by Ï

in Ref. [34]. In fact, the notion of continuous majoriza-tion has a decades-long history and appears in a varietyof research fields from thermodynamics and order the-ory [35, 36], through plasma physics [37, 38], to socialsciences [39]. Moreover, this notion was employed andstudied in more detail in Ref. [34], where it was inspiredby a model of heat transport along ideal conducting wiresbetween d objects with different temperatures. Here, weextend these technical considerations to continuous ther-momajorization, which is necessary to capture finite tem-perature thermalizations. We will also highlight the sig-nificance of this notion as the right generalization of theconcept of entropy production.

Our first main result is to show that the notion ofcontinuous thermomajorization correctly encapsulates allthe relevant constraints of Markovian thermal processeson population dynamics.

Theorem 1 (Second law on populations).

p(0)MTP7−→ p(tf ) if and only if

p(0) Ïγ p(tf ). (30)

The proof of the above theorem can be found in Ap-pendix A. As a consequence, continuous thermomajoriza-tion gives a complete (exhaustive) set of constraints forthe evolution of populations in the standard (Markovian)master equations approach to quantum thermodynamics.

It is also worth highlighting that continuous thermo-majorization – which characterizes Markovian processes– and thermomajorization – which characterize generalnon-Markovian processes – coincide when initial and fi-nal states have the same γ-ordering (see Corollary 9 inAppendix A for details):

If π(p) = π(q), p �γ q ⇔ p Ïγ q. (31)

In other words, all the complications with Markovian-ity (or advantages from non-Markovianity) arise from

crossing the boundary between one γ-ordering and an-other. This observation will play a crucial role later, butmore broadly it is noteworthy for the study of the role ofmemory effects in stochastic processes with a given fixedpoint.

V. A COMPLETE SET OF ENTROPYPRODUCTION RELATIONS

We now show how the continuous thermomajorizationcondition of Theorem 1 subsumes (and greatly strength-ens) the standard positive entropy production conditionfrom Eq. (8). Employing Theorem 1, one can translateknown results from the theory of majorization into en-tropic inequalities. In other words, one can constructfamilies of functionals that must be monotonically non-increasing during the Markovian evolution of the systemalong the path ρ(t) with populations p(t). For example,for any well-behaved convex function h : R → R, theh-divergence defined by

Σh(t) = −d∑i=1

γih

(pi(t)

γi

), (32)

must be monotonically non-decreasing

dΣh(t)

dt≥ 0. (33)

Proof. To see that Eq. (33) holds, note that, by The-orem 1, p(0) Ïγ p(tf ). Hence, for any t ∈ [0, tf ) andδ > 0, p(t) �γ p(t + δ). The known results on thermo-majorization (see Theorem 7 in Appendix A) then tell usthat there exists a stochastic matrix T such that

Tp(t) = p(t+ δ), Tγ = γ. (34)

Thus,

Σh(p(t+ δ)) = −d∑i=1

γih

d∑j=1

Tijpj(t)

γi

= −

d∑i=1

γih

d∑j=1

[Tij

γjγi

]pj(t)

γj

≥ −

d∑i,j=1

γi

[Tij

γjγi

]h

(pj(t)

γj

)

≥ −∑j

γjh

(pj(t)

γj

)= Σh(p(t)), (35)

where we used the convexity of h and then the stochastic-ity of T (i.e.,

∑i Tij = 1). We note Eq. (35) could have

also be inferred from Ref. [31], Proposition 14.B.3. Sincethe above holds for every δ > 0, the result follows.

8

For each choice of h, the above qualifies as a valid gen-eralized entropy production inequality. Restrictions onthe thermodynamically admissible paths can be obtainedby studying their level sets within the d-dimensionalprobability simplex (recall Figs. 1a-1b) and constructingthe corresponding “thermodynamic trees”, as detailed inRef. [40] for a special choice of Σh. In the accompanyingpaper [11], we detail how these encompass and strengthenseveral well-known relations in the literature, includingthe standard entropy production relation of Eq. (10),the diagonal entropy production [41], the second laws ofRef. [4], the Tsallis entropies well-known in non-extensivestatistical mechanics and information theory [42–45] andthe ‘vacancy’ [46], which was found to play a crucial rolein low temperature thermodynamics.

As we can see, one can easily generate a huge varietyof entropic inequalities, which helps to see different re-sults as part of a unified framework. At the same time, anatural question arises: Is there a family of entropic con-ditions that implies all others? Our second main result,which follows from Theorem 1, answers this question inthe affirmative and can be interpreted as a sort of ex-haustive H-theorem.

Corollary 2 (Exhaustive H-type theorem).

p(0)MTP7−→ p(tf ) if and only if there exists a con-

tinuous path p(t) for t ∈ [0, tf ] such that Σa(t) ismonotonically increasing in t for all a ∈ [0, 1], where

Σa(t) := −d∑i=1

∣∣∣∣pi(t)− a γiγd∣∣∣∣ . (36)

Proof. First assume that the evolution of populationsp(t) is generated by a Markovian thermal process. Then,from Theorem 1, we know that for every ε > 0 we havep(t) �γ p(t + ε). This is equivalent to (see Chap. 14,Proposition B.4 [31]):∑

i

|pi(t)− aγi| ≥∑i

|pi(t+ ε)− aγi|, ∀a ≥ 0. (37)

Since ε > 0 can be made arbitrarily small, this is thecondition that the functionals

∑i |pi(t)− aγi| are mono-

tonically decreasing in t for every a ≥ 0. These includein particular the monotonicity of the functionals Σa(t).

Conversely, suppose that the evolution of populationsp(t) is such that the functionals Σa(t) are monotoni-cally non-decreasing for all a ∈ [0, 1]. Note that fora > 1 one has aγi/γd > 1 for every i, and hence∑i |pi(t)− a γiγd | =

aγd

independently of the value of p(t).

Thus, we can trivially extend the monotonicity propertyof Σa(t) to all a ≥ 0. In fact, by rescaling a 7→ aγd, weget the equivalent property that

∑i |pi(t)−aγi| is mono-

tonically non-decreasing. As already mentioned above,this is equivalent to p(t) �γ p(t + ε) for every t, ε ≥ 0.Recalling Definition 3, this means that for every t wehave p(0) Ïγ p(t). We conclude using Theorem 1.

Once again, we want to emphasize the “if and only if”in the statement: dΣa(t)/dt ≥ 0 are generalized entropyproduction inequalities which imply all others.

VI. UNIVERSAL THERMODYNAMICCONTROLS

We now change the point of view and consider theequally important question of control. In other words,we start from an initial state p(0) and ask how to de-vise a thermalization process that drives the system toa final target state p(tf ) at some later time tf . FromTheorem 1 we know that every q such that p(0) Ïγ qcan be realized by some choice of controls in the class ofMarkovian master equations of a thermalization process.Such controls, however, may be arbitrarily complex andthe control sequence is unknown. In this section we solvethe first problem by presenting a set of elementary con-trols that are sufficient to perform arbitrary Markovianthermalizations; and in the next section we will solve thesecond problem by presenting an algorithm that returnsthe explicit sequence that is required.

One can reasonably conjecture that a much more re-stricted subclass of physically relevant thermalizationprocesses suffices to grant us the same amount of con-trol as the full set of Markovian thermal processes. Thereader can be reminded of the notion of a universal gateset in quantum computing, where one seeks a minimalset of unitary operations that allows one to approximatearbitrarily well the transformations achievable by arbi-trary unitaries [13]. In the same fashion, we ask hereabout a set of universal thermalization controls.

Following this intuition, Ref. [47] asked whether everytransformation achievable by thermal processes can beachieved by sequentially coupling only two energy levelsof the system to the environment at once, dubbed an‘elementary thermal operation’. Somewhat surprisingly,this question was answered in the negative [47]. In fact,one needs to couple the environment simultaneously to alld energy levels [48], or grant full control of the system’sand an auxiliary thermal qubit’s energy spectra [49].

Remarkably, however, we are not aware that the samequestion was tackled in the standard setup of quantumthermodynamics, where the question is to find controls aspowerful as the most general Markovian master equationof a thermalization process. In this context, a distin-guished candidate for a universal set of thermal controlsis given by two-level partial thermalizations, which we willalso simply call elementary thermalizations. These are aset of thermalizations of both practical and formal inter-est. Each of them acts only on two energy levels (i, j)and is represented by an extremely simple reset Marko-vian master equation

dpidt

=1

τ

(γi

γi + γj(pi + pj)− pi

),

dpjdt

= −dpidt.

9

which describes an exponential relaxation to equilibrium:

pi,j(t) = e−t/τpi,j(0) +Nij(0)(1− e−t/τ )γi,j . (39)

Above, xi,j(t) := (xi(t), xj(t)) and Nij = pi(0) + pj(0).Formally, this can be represented by a matrix equation

pi,j(t) = T i,j(λt)pi,j(0) (40)

with λt = 1− e−t/τ and

T i,j(λ) =

[(1− λ) + λγi

γi+γjλ γiγi+γj

λγj

γi+γj(1− λ) + λγi

γi+γj

]. (41)

These transformations stand out for their formal sim-plicity – they are the stochastic processes with thermalfixed point on two states that can be realized by a Marko-vian master equation (as one can check directly using so-called embeddability conditions [26]). But they also arisenaturally in rather diverse approaches to quantum ther-modynamics [50–52], where they are often used as build-ing blocks for more complex protocols [49, 53, 54]. Here,we prove that elementary thermalizations are a universalset of thermalization controls:

Theorem 3 (Universality of elementary thermaliza-

tions). p(0)MTP7−→ p(tf ) if and only if there exists a finite

sequence of elementary thermalizations such that

p(tf ) = T if ,jf (λf ) . . . T i1,j1(λ1)p(0). (42)

For the proof, see Appendix A. This is a remarkablesimplification of the set of controls required to gener-ate the transformations achievable by the most generalMarkovian thermal process.5 We remark once more thatthis simplification does not hold for thermal processes orthermal operations [47], and so constitutes an importantdifference between the standard and the resource theoryframeworks.6 Our result proves that coupling at oncemore than two system energy levels to the environmentis only required when we want to reproduce effects aris-ing from strong interactions or small environments, butit is not necessary in the Markovian regime.

5 The result also shows that Markovian thermal processes on inco-herent states have the same power as Markovian thermal oper-ations, since every elementary thermalization can be easily seento be a thermal operation as defined in Ref. [23].

6 Notable exceptions are given by infinite temperature limit andsystems with trivial Hamiltonians, when the thermal state is amaximally mixed state and the thermomajorization relation isreplaced by standard majorization. Then, it is known (see, e.g.,Theorem II.1.10 of Ref. [55]) that majorization between p and qis equivalent to the existence of a finite sequence of T -transformsmapping p to q, where the T -transform is a bistochastic matrixacting non-trivially only on two levels of the system.

VII. SECOND LAWS IN THE MARKOVIANREGIME

We are now ready to state the main results of this work.First, we will provide a finite set of necessary and suffi-cient conditions for a given probability distribution p tocontinuously thermomajorize another distribution q and

so, via Theorem 1, for pMTP7−→ q. Second, we will specify a

constructive protocol realizing this transition through asequence of elementary thermalizations. Therefore, ourresults satisfy desiderata (D1) and (D2).

A. Finite set of conditions

In order to state our main result, we will need theconcept of a canonical sequence of γ-orderings, definedas follows.

Definition 4. A sequence of γ-ordering vectors {πk} iscanonical when

1. πk and πk+1 differ only by a transposition of adja-cent elements.

2. Each γ-ordering appears at most once in the se-quence.

We then have the following result.

Theorem 4 (Finite second laws conditions). Given pand q, enumerate all canonical sequences {πk}Nk=1 withπ1 = π(p) and πN = π(q). For each sequence, constructthe state

f :=

N−1∏k=1

T ik,jk(1)p, (43)

where T ik,jk(1) are full elementary thermalizations(Eq. (41) with λ = 1) and the levels ik, jk are the labelsindicating which of the elements of πk and πk+1 differ.Then p Ïγ q if and only if for at least one f

f �γ q, (44)

The proof of the above theorem can be found in Ap-pendix B. Albeit growing quickly with d, this is a finiteset of inequalities and can hence be checked algorithmi-cally. This means that desideratum (D1) is met. In fact,as soon as a sequence satisfying Eq. (44) is found, thesearch ends, so one can also apply heuristic approachesto the search problem.

Let us illustrate Theorem 4 with an explicit examplewith d = 3 and γ = (1/3, 1/3, 1/3), see Fig. 4. In thiscase, the γ-ordering is simply the standard sorting in anon-increasing order and �γ coincides with standard ma-jorization �. Let us choose p(0) with ordering {3, 2, 1}and p(tf ) with ordering {1, 2, 3}. There are then onlytwo canonical sequences from π(p(0)) to π(p(tf )):

{3, 2, 1} → {2, 3, 1} → {2, 1, 3} → {1, 2, 3}, (45a)

{3, 2, 1} → {3, 1, 2} → {1, 3, 2} → {1, 2, 3}. (45b)

10

pT 2,3

T 1,2

T 1,3

T 1,2

T 2,3

T 1,3

(1,0, 0

) (0, 1, 0)

(0, 0, 1)

{1, 2, 3} {2, 1, 3}

{1, 3, 2} {2, 3, 1}

{3, 1, 2} {3, 2, 1}

f1f2

FIG. 4. Verification of continuous thermomajorizationfor d = 3. Simplex representing the state space of all 3-dimensional probability distributions with regions of fixed γ-orderings indicated by {·, ·, ·}. The optimal paths connect-ing a state p(0) with γ-ordering {3, 2, 1} to states f1, f2of γ-ordering {1, 2, 3} are realized by elementary thermal-izations T i,j (indicated by red and blue arrows). The setof states with γ-ordering {1, 2, 3} achievable from p(0) byMarkovian thermal processes is finally obtained as the unionof the set of q thermomajorized by f1 and the set of q ther-momajorized by f2 (for this last construction see, e.g., Ap-pendix E of Ref. [47]). Here the thermal state was chosento be γ = [1/3, 1/3, 1/3], which corresponds to the infinitetemperature limit.

For each sequence we construct a final state:

f1 = T 1,2(1)T 1,3(1)T 2,3(1)p(0), (46a)

f2 = T 2,3(1)T 1,3(1)T 1,2(1)p(0). (46b)

According to Theorem 4, p Ïγ q if and only if eitherf1 �γ p(tf ) or f2 �γ p(tf ), which is simply a set of2(d− 1) = 4 inequalities. Employing Theorem 1, when-ever these inequalities are satisfied there exists a Marko-vian thermal process mapping p(0) to p(tf ).

B. Constructive protocol

Following Theorem 4, suppose we find a sequence ofelementary thermalizations mapping p to a final state fwith the same γ-ordering as the target q and satisfyingf �γ q. Then, one can explicitly construct a sequenceof of M ≤ d−1 elementary thermalizations transformingf into q (see proof of Theorem 12 in Supplemental Ma-terial of Ref. [49]). We thus conclude that a sequence ofMarkovian thermal processes achieving the transforma-tion from p(0) to p(tf ) = q is

q =

M∏s=1

T (fs,f′s)(λs)

N∏k=1

T (ik,jk)(1)p(0), (47)

where the first N elementary thermalizations are ob-tained from a sequence that satisfies Eq. (44), and theconstruction of the remaining ones can be found inRef. [49]. This gives an explicit construction involvinga sequence of N + M ≤ d! + d − 2 elementary thermal-izations achieving any allowed transformation. In otherwords, Theorem 3 is strengthened to a result that alsosatisfies desideratum (D2) of Sec. III:

Corollary 5 (Strengthened universality of elementary

thermalizations). p(0)MTP7−→ p(tf ) if and only if there ex-

ists a finite sequence of elementary thermalizations suchthat

M∏s=1

T fs,f′s(λs)

N∏k=1

T ik,jk(1)p(0) = p(tf ), (48)

where N ≤ d! − 1, M ≤ d − 1 and the sequence can bealgorithmically constructed.

C. Comparison with previous works

We want to emphasize that the two desiderata con-sidered here, (D1)-(D2) in Sec. III, important as theyare to systematically develop and optimize explicit pro-tocols in nonequilibrium thermodynamics, are not typi-cally met by general frameworks. The standard frame-work based on entropy production provides thermody-namic constraints [9], but as discussed in Sec. II B theseare insufficient to characterize the future evolution. Fur-thermore, these entropic relations do not provide tools toconstruct and optimize nonequilibrium thermodynamicprotocols, unless one focuses on special classes of dy-namics, or restricted regimes such as close to equilibriumtransformations and slow driving protocols [56]. Hence,neither of the two desiderata is satisfied. The same holdstrue for prominent results in the resource-theoretic ap-proach to thermodynamics. The second laws constraintsof Ref. [4] are neither finitely checkable, nor they providea way to construct explicit protocols. The same appliesto the “quantum majorization constraints” for thermalprocesses, the main result derived in Ref. [22].

Several other settings satisfy desideratum (D1), butnot (D2). The framework based on correlating catal-ysis [57, 58] provides finitely checkable conditions, butnot explicit protocols. Similarly, the thermomajoriza-tion constraints of Ref. [3] are finitely checkable by linearprogramming, but the construction of explicit operationsachieving the transformations is not known. More pre-cisely, Ref. [59] did provide a set of thermal processes,called β-permutations, whose convex combination real-izes the most general transformation. However, beyondthe case d = 2, it is an open question how these processescan be realized by explicit system-bath interactions. Thesame is true for the work of Ref. [60], which providesGibbs-preserving channels realizing general transforma-tions in the correlating catalyst setting, but not their ex-plicit thermodynamic realization. All these are examples

11

where the desideratum (D2) is not satisfied. Despite no-ticeable progress [61, 62], the difficulty of satisfying bothdesiderata has hindered the application of informationthermodynamics frameworks.

Ref. [49] is an exception in this regard, as it providesconstructive protocols to realize all the transformationsallowed by thermal processes. However, it allows a muchmore extensive control of the system than we consideredhere, including full control over its energy spectrum andthat of a qubit ancilla. These controls are very powerful:as the authors showed, they allow one to achieve everystate transformation possible under thermal processes,including those requiring arbitrary non-Markovian dy-namics. Here, instead, we focused on the characteriza-tion of thermalizations described by Markovian masterequations and involving limited control over the energyspectrum. This paves the way to several applications, asdiscussed in the accompanying paper [11].

VIII. EXPLICIT ALGORITHMICVERIFICATION

Naturally, to verify the complete second laws condi-tions in higher dimensions, one would like to develop anexplicit algorithm that exhaustively verifies the inequal-ities in Theorem 4. Here we propose one such algorithmwith two variants, for which we provide a correspond-ing Mathematica code [10]. The fast version only verifies

whether p(0)MTP7−→ p(tf ) for a fixed initial and final state,

while the slow version constructs the set of all final statesachievable from a given initial state.

Algorithm verifying p Ïγ q

1. Initialize.

(a) Create a set of states Current that initiallycontains only the initial state p.

(b) For each γ-ordering, indexed by k from 1 tod!, create a set Optimal[k]. Initially all setsare empty except for the ones corresponding toγ-orderings of p, which contain only p.

2. Generate optimal states.

(a) Update Current to contain all states achiev-able from old Current via full thermalizationsbetween 2 levels adjacent in the γ-ordering.

(b) [Only for fast version] Remove those elementsof Current that do not thermomajorize q.

(c) Denote by Current[k] all states from Currentwith γ-ordering k. For each k, removefrom Current all those states of Current[k]that are thermomajorized by either another

state from Current[k] or by any state fromOptimal[k].

(d) For each k, remove from Optimal[k] all statesthat are thermomajorized by any state fromCurrent[k], and then add all states fromCurrent[k] to Optimal[k].

(e) Repeat steps (a)-(d) until Current is empty.

3. Verify thermomajorization condition.

(a) [Only for fast version] Verify whether any of thestates from Optimal[k], where k correspondsto γ-ordering of q, thermomajorizes q. If yes,then p Ïγ q; otherwise the relation does nothold.

(b) [Only for slow version] The sets Optimal[k]contain all the information about states contin-uously thermomajorized by p. More precisely,the states with γ-ordering k which are continu-ously thermomajorized by p are those thermo-majorized by Optimal[k].

Let us make a few comments on the above algorithm.First, it is clear that it satisfies desideratum (D2) of finiteverifiability, but it can be easily modified to also satisfydesideratum (D1) of constructability. One simply needsto keep track of the “history” of each state: in step (2a),one should record which elementary thermalization led toa new state. This history should be kept when updatingthe optimal states with current states in step (2d). As aresult, at the end algorithm we will not only have the listof optimal states within each γ-ordering, but we will alsoknow the sequence of elementary thermalizations thatneed to be applied to the initial state to obtain each ofthem.

Second, the possible number of canonical sequencesgrows very fast with the system’s dimension d. Thus,one may try to develop heuristics to distinguish betweenbetter and worse choices of canonical sequences. Usingthe example from Fig. 4, it is intuitively clear that in or-der to go from γ-ordering {1, 2, 3} to γ-ordering {2, 1, 3}one should do it directly rather than following the path{1, 2, 3} → {1, 3, 2} → {3, 1, 2} → {3, 2, 1} → {2, 3, 1} →{2, 1, 3}. Even without any heuristics, we were able torun (on a standard laptop computer) the slow version ofthe algorithm implemented in Mathematica [10] to solvethe d = 6 case in minutes and the d = 7 case in hours.

IX. CONCLUSIONS AND OUTLOOK

In this paper we provided a hybrid framework over-coming current limitations of resource-theoretic and mas-ter equation approaches to quantum thermodynamicsthrough the novel notion of continuous thermomajoriza-tion. Crucially, our approach includes explicit methodsto fully solve the question of the existence of a Markovian

12

thermal process mapping between two non-equilibriumstates and returns a corresponding sequence of elemen-tary controls when these exist. Exhaustive searches arefeasible on a laptop machine up to d = 7 through theMathematica code we provided [10].

While here we tackled the question of describing an al-gorithm which is guaranteed to provide a definite answerto the interconversion problem, in many circumstances itis enough to find a method that is able to construct usefulworking protocols in most cases. A promising directionto probe higher-dimensional systems is then to relax theexhaustive search to a heuristic search. For example, onecan cap the maximum length of a sequence of elemen-tary controls and focus on canonical sequences of (closeto) minimal length. This means looking for paths con-necting the initial distribution p and the target q cross-ing the minimal possible number of distinct thermoma-jorization orderings. This should allow one to constructexplicit protocols and study the thermodynamics of rel-atively high-dimensional systems, which are extremelychallenging to tackle via analytical or numerical meth-ods at the level of generality considered here.

Pushing the achievable dimension up, and combin-ing the current algorithm optimizing the thermaliza-tion stage with alternative methods to optimize unitarystages, will likely open up a range of applications, suchas the optimization of quantum thermodynamic cycles ofheat engines. In the accompanying paper [11] we alreadydiscuss ways of employing the framework developed hereto construct provably optimal thermodynamic protocols.

At the same time, our framework also offers a rigor-ous information-theoretical foundation to the dynamicalviewpoint of quantum thermodynamics. This approachcomplements the master equation toolbox, as we haveseen, for example, with the systematic construction ofgeneralized entropy production inequalities. Another di-rection that should be further explored concerns the roleof quantum coherence in these settings. We provide someinitial remarks in Appendix C, while a solution to thisproblem satisfying both desiderata (D1)-(D2) is still outof reach.

ACKNOWLEDGEMENTS

M.L. thanks A. Levy for useful discussions. K.K. ac-knowledges financial support by the Foundation for Pol-ish Science through TEAM-NET project (contract no.POIR.04.04.00-00-17C1/18-00). M.L. acknowledges fi-nancial support from the the European Union’s MarieSk lodowska-Curie individual Fellowships (H2020-MSCA-IF-2017, GA794842), Spanish MINECO (Severo OchoaSEV-2015-0522 and project QIBEQI FIS2016-80773-P),Fundacio Cellex and Generalitat de Catalunya (CERCAProgramme and SGR 875) and grant EQEC No. 682726.

Appendix A: Proof of Theorem 1 and Theorem 3

To build up towards our final results we will need sev-eral intermediate technical statements. We start by re-calling an important result derived in Ref. [49].

Theorem 6 (Theorem 12, Supplemental Material ofRef. [49]). If p �γ q and π(p) = π(q), there exists a

sequence of elementary thermalizations {T ik,jk(λk)}fk=1such that

T if ,jf (λf ) . . . T i1,j1(λ1)p = q. (A1)

Moreover, f ≤ d− 1.

We will also need the following known result charac-terizing thermomajorization

Theorem 7 (See Refs. [1, 3, 8]). There exists a stochasticmatrix T such that Tp = q and Tγ = γ if and only ifp �γ q.

Next, we link continuous thermomajorization betweentwo distributions with the existence of a sequence of ele-mentary thermalizations bringing one distribution to an-other.

Lemma 8 (Continuous thermomajorization and ele-mentary thermalizations). p Ïγ q if and only if thereexists a finite sequence of elementary thermalizations

{T ik,jk(λk)}fk=1 such that

T if ,jf (λf ) . . . T i1,j1(λ1)p = q. (A2)

Proof. First, assume p Ïγ q. Then, there exists a contin-uous trajectory r(t) with r(0) = p, r(tf ) = q (perhapstf = +∞), and r(t′) �γ r(t′′) for all t′ ≤ t′′. Definet0 = 0, as well as a thermomajorization ordering π1 anda time t1 as follows:

π1 :=π(r(0)), (A3a)

t1 := sup{t|π(r(t)) = π1}. (A3b)

Next, for integer k > 1, define iteratively

πk+1 :=π(r(t+k )), (A4a)

tk+1 := sup{t|π(r(t)) = πk+1}. (A4b)

Clearly tk+1 > tk. Since there are only d! distinct ther-momajorization orderings, ultimately we reach the finalk = f ≤ d!−1, such that πf = π(r(tf )). We now employTheorem 6: for each pair, r(t+k ) and r(tk+1), there existsa sequence of elementary thermalizations such that

r(tk+1) = T ikn ,jkn (λkn) . . . T ik1 ,jk1 (λk1)r(tk), (A5)

with n ≤ d−1. Thus, by sequentially applying the aboveto all k ≤ f we obtain Eq. (A2) with a finite sequence.

Conversely, assume that Eq. (A2) holds. Definer(0) = p and

r(t) = T ik,jk(δ)T ik−1,jk−1(λk−1) . . . T i1,j1(λ1)p,

13

with k and δ ∈ [0, 1] satisfying t = δ +∑k−1i=1 λi. This

defines a continuous path starting at p and terminatingat q. Moreover, using the fact that for any i, j and λ′ ≥ λwe can write T i,j(λ′) = T i,j(µ)T i,j(λ) with µ ∈ [0, 1], wesee that for any t′′ ≥ t′ the distribution r(t′′) is obtainedfrom r(t′) by a finite sequence of elementary thermaliza-tions. Given that elementary thermalizations and theirproducts are stochastic matrices with a fixed point γ,Theorem 7 implies r(t′) �γ r(t′′) for all t′′ ≥ t′. We thusconclude that p Ïγ q.

Note that from the proof above one can conclude thatthe number of elementary thermalizations required for astate transformation is upper-bounded by d!(d− 1), butwe will give a tighter bound later. Also, as a corollaryof Lemma 8 we get that �γ (describing allowed trans-formations under general thermal processes, which em-ploy with memory) and Ïγ (describing allowed trans-formations under Markovian thermal processes) coincidewithin a fixed thermomajorization ordering.

Corollary 9. If π(p) = π(q) and p �γ q then p Ïγ q.Moreover, the thermomajorizing trajectory r(t) connect-ing p to q can be chosen such that for all t ∈ [0, tf ] itbelongs to the same γ-ordering.

Proof. Assuming that p �γ q and π(p) = π(q), Theo-rem 6 tells us then that there exists a sequence of elemen-tary thermalizations mapping p into q. Using Lemma 8,we conclude p Ïγ q. Moreover, the construction of ele-mentary thermalizations presented in the proof of The-orem 6 in Ref. [49] is such that every intermediate statealong the trajectory, r(t), has the same thermomajoriza-tion ordering π(p).

We are now able to discuss Theorem 1 and Theorem 3,which we prove jointly as follows:

Theorem 10. Let ρ(t) by a quantum state with popula-tion vector p(t). The following statements are equivalent:

1. There exists a Markovian thermal process trans-forming ρ(0) with population p(0) into a quantumstate ρ(tf ) with population p(tf ).

2. p(0) Ïγ p(tf ).

3. There exists a finite sequence of elementary ther-malizations such that

T if ,jf (λf ) . . . T i1,j1(λ1)p(0) = p(tf ). (A6)

Proof. Lemma 8 proves the equivalence 2 ⇔ 3. To con-clude we then just prove 3⇒ 1 and 1⇒ 2.

[3⇒ 1]: Given the finite sequence of elementary ther-malizations such that Eq. (A6) holds, we will explicitlyconstruct a time-dependent Lindbladian Lt generating aMarkovian thermal process that maps a state with pop-ulation p(0) to the one with population p(tf ). We defineLt through its action on the basis elements as

〈m|Lt(|n〉〈n′|)|m′〉 = δnn′δmm′T (t)mn− δmnδm′n′ , (A7)

where |x〉 denote the eigenstates of H and T (t) is a d×dstochastic matrix. To see that the above Lt correspondsto a valid Lindbladian, note that it has the form Et − I,where I is the identity channel and Et is the channelthat decoheres in the eigenbasis of H and performs thestochastic map T (t) on the diagonal. Since every channel

can be written as Et(ρ) =∑i LiρL

†i with

∑i L†iLi = I,

Lt has the required form from Eq. (3).Next, let

tk = − log(1− λk), t0 := 0 (A8)

and introduce τk =∑ks=0 tk. Then, choose

T (t) =

γik

γik+γjk

γikγik+γjk

γjkγik+γjk

γjkγik+γjk

⊕ 0\(ik,jk) (A9)

for t ∈ [τk−1, τk] with k = 1, . . . , f , where 0\(ik,jk) de-notes the (d − 2) × (d − 2) matrix of all zeros acting onthe subspace of all energy levels except ik, jk. Now, theequation dρ/dt = Lt(ρt) can be easily solved, and onecan verify that the resulting dynamics implements thesequence of elementary thermalizations from Eq. (A6)on the population vector. We conclude that a Markovianthermal process mapping p(0) into p(tf ) exists.

[1 ⇒ 2] : Given p(0), p(tf ) and a Markovian thermalprocess dynamically evolving the former into the latter,let p(t) be the trajectory followed at the intermediatetimes. By assumption, the process is infinitely divisi-ble, meaning that for every 0 ≤ t′ ≤ t′′ ≤ tf there existsa stochastic matrix T (t′, t′′) such that

T (t′, t′′)p(t′) = p(t′′), T (t′, t′′)γ = γ. (A10)

From Theorem 7, it follows that p(t′) �γ p(t′′). Weconclude that p Ïγ q.

Appendix B: Proof of Theorem 4

We start from introducing the concept of a coarse-grained description of a thermomajorizing trajectoryr(t).

Definition 5 (Coarse-graining). Let r(t) be a thermo-majorizing trajectory from p to q as in Definition 3.Then, a sequence of γ-orderings {πk}Nk=1 is called acoarse-grained description of r(t) if there exists an or-dered set of times {tk}Nk=0,

t0 = 0, tk ≤ tk+1, tN = tf , (B1)

such that the probability vector r(t) belongs to the γ-ordering πk in the interval t ∈ [tk−1, tk] (note that attime tk, r(tk) is associated to both γ-orders πk andπk+1). Moreover, a given coarse-grained description willbe called canonical if the sequence {πk} is canonical ac-cording to Definition 4.

14

Note that since a given state can simultaneously be-long to more than one γ-ordering (this happens whenpk/γk = pl/γl for some k and l), the trajectory r(t) canhave multiple (but finitely many) coarse-grained descrip-tions. However, as we now prove, we may limit our at-tention only to canonical coarse-grainings.

Lemma 11. p Ïγ q if and only if there exists a thermo-majorizing path r(t) connecting p and q with a canonicalcoarse-grained description.

Proof. Assume that p Ïγ q, meaning that there existssome thermomajorizing trajectory r(t) from p to q. Sincer(t) changes continuously in t, so do γ-rescaled entries{ri(t)/γi}di=1. This means that the k-th largest elementamong γ-rescaled entries {ri(t)/γi}di=1 becomes equal tothe (k − 1)-th or (k + 1)-th largest one before (or at thesame time) becoming equal to any other entry. Hence, wecan choose times {tk}Nk=0 (perhaps tk = tk+1 for some k)and assign γ-orderings {πk}Nk=1 such that πk and πk+1

only differ by a transposition of adjacent elements, whichproves the first property of the canonical coarse-graining.

To prove the second one, assume that some γ-orderingπ appears more than once in the coarse-grained descrip-tion of r(t) defined in the previous step. Let us denotethe first time r(t) has the ordering π by t1, and the lasttime it has this ordering by t2. Clearly, r(t1) Ïγ r(t2).Thus, one can introduce a new thermomajorizing pathr′(t) such that it is equal to r(t) for t ∈ [0, t1) andt ∈ (t2, tf ], while for t ∈ [t1, t2] it is the trajectory givenby Corollary 9, connecting r(t1) to r(t2) and lying com-pletely in γ-ordering π. Aa a result, we obtain a thermo-majorizing trajectory r′(t) connecting p and q, and suchthat the γ-ordering π appears only once in the coarse-description. By repeating this for every γ-ordering thatappears more than once in the original coarse-graineddescription of r(t), we end up with a trajectory whosecoarse-grained description satisfies also the second prop-erty of canonical coarse-graining.

Conversely, if there exists a thermomajorizing pathr(t) connecting p and q with whatever coarse-graineddescription, then by definition p Ïγ q.

The next lemma geometrically characterizes theaction of elementary thermalizations on a Lorenzcurve/thermomajorization curve. In words, it shows thatthe effect of T i,j is to decrease the slope of the jth seg-ment of the thermomajorization curve and increase thatof the ith segment till the two are equalized (see Fig. 5and Appendix B.1 of Ref. [49]).

Lemma 12 (Action of elementary thermalization on thethermomajorization curve). Consider a Lorenz curve ofp with elbow points denoted by (xm, ym), see Fig. 5. Forany j > i, denote i = πi(p) and j = πj(p). Then, the

elementary thermalization T i,j(λ) shifts down by an equalamount the y-coordinates (yi, . . . , yj−1). The extremal

map, T i,j(1), equalizes the slopes of the ith and the jth

segments of the curve. Note that a final reordering maybe needed if the thermomajorization ordering is changed.

0.25 0.5 0.75 1

0.25

0.5

0.75

1

i=0

i=1

i=2

i=3i=4

i=5

y1

y2

y3

y4

y5

i=3’

FIG. 5. The defining points of the thermomajorization curveare labelled according to Lemma 12, to visualize the action ofthe two-level partial level thermalization T 3,4(λ). This bringsdown y3 until, at λ = 1, the slopes of the 3rd and 4th segmentare equalized (brown segment connecting 2, 3′ and 4).

Proof. For p′ = T i,j(λ)p we have

p′m =

{pm for m /∈ {i, j},(1− λ)pm + λ

pi+pjγi+γj

γm for m ∈ {i, j}. (B2)

Denote by ym and y′m the y-coordinates of the thermo-majorization curves of p and p′, respectively. Then

y′m =

ym for m < i,

ym − λ (piγj−pjγi)γi+γj

for i ≤ m < j,

ym for m ≥ j.(B3)

This corresponds to shifting down the y-coordinate ofeach point of the thermomajorization curve, startingfrom the ith point to the (j − 1)th point. Setting λ = 1and using yj − yj−1 = pj , one obtains that the slope of

the jth segment is

y′j − y′j−1γj

=pi + pjγi + γj

.

Similarly, the slope of the ith segment is

y′i − y′i−1γi

=pi + pjγi + γj

.

Hence, the two slopes are equalized for λ = 1. Note that,if (and only if) j 6= i + 1, then the thermomajorizationordering will change at some intermediate λ, so that arearrangement of the segments is necessary to sort themaccording to non-increasing slopes.

15

The next lemma identifies states obtained by thermal-izing two adjacent levels in the γ-ordering as the optimalcrossings between one ordering and another.

Lemma 13. Given p, we have

Tπi(p),πi+1(p)(1)p Ïγ q (B4)

for every q satisfying p Ïγ q, π(q) = π(p) and

qπi(p)

γπi(p)=qπi+1(p)

γπi+1(p). (B5)

Proof. Since p Ïγ q implies p �γ q, and furthermorewe have π(q) = π(p), we can use Theorem 6. Its proofshows that there is a sequence of elementary thermaliza-tions such that

q = T ik,jk(λk) · · ·T i1,j1(λ1)p, (B6)

and, furthermore, that all intermediate states have ther-momajorization ordering equal to π(p). Then, we con-clude from Lemma 12 that the thermomajorization curveof q can be obtained from that of p by lowering a set ofits y-coordinates while making the slopes of the i-th and(i + 1)-th segments equal (no re-orderings are involved,so the x-coordinates do not change).

Now let us focus on qi,i+1 := Tπi(p),πi+1(p)(1)p. Usingagain Lemma 12, the thermomajorization curve of thestate qi,i+1 is obtained by lowering the coordinate yi+1

till the slopes of the i-th and (i+1)-th segments are equal,while leaving all other y-coordinates untouched. Cru-cially, note that this is the minimal y-coordinate loweringfor any q that ensures Eq. (B5) is satisfied. From thisproperty and the fact that π(q) = π(qi,i+1), it followsthat qi,i+1 �γ q. That is because, since π(q) = π(qi,i+1)implies that the x-coordinates of the thermomajorizationcurves of q and qi,i+1 coincide, �γ is equivalent to sim-ply comparing the y-coordinates of the respective ther-momajorization curves. Using Corollary 9 we concludeqi,i+1 Ïγ q.

We are now ready to prove the central lemma fromwhich the proof of Theorem 4 follows almost directly.

Lemma 14. Given a canonical sequence {πk}Nk=1 withπ1 = π(p) and πN = π(q), the following two statementsare equivalent:

1. There exists a thermomajorizing trajectory r(t)from p to q with a canonical coarse-grained descrip-tion {πk}Nk=1.

2. The following relation holds:

N−1∏k=1

T ik,jk(1)p �γ q, (B7)

where T ik,jk(1) are full thermalizations between lev-els ik and jk specified by the adjacent elements thatdiffer between πk and πk+1.

Proof. We will prove that the implication holds bothways.

[1 ⇒ 2] Let tk for k ∈ {0, . . . , N} be the times fromDefinition 5, i.e., t0 = 0, tN = tf and the remaining onesdescribing times for which r(t) changes γ-ordering. Bydefinition, there exist indices sk for k ∈ {1, . . . , N − 1}such that πk and πk+1 differ only by a transposition ofthe two adjacent entries, sk and sk + 1. Let us denotethe corresponding energy levels that change order in theγ-ordering by:

ik := πsk(r(tk)), jk := πsk+1(r(tk)). (B8)

By assumption, for k ∈ {0, . . . , N − 1} we haver(tk) Ïγ r(tk+1) and

π(r(tk)) = πk+1, π(r(tk+1)) = πk+1. (B9)

Furthermore, for k ∈ {0, . . . , N − 2}r(tk+1)ik+1

γik+1

=r(tk+1)jk+1

γjk+1

. (B10)

We are then in the conditions to apply Lemma 13 toconclude that for k ∈ {0, . . . , N − 2} we have

T ik+1,jk+1(1)r(tk) Ïγ r(tk+1). (B11)

Applying the above sequentially we arrive at

r(tN−1) Îγ TiN−1,jN−1(1)r(tN−2)

Îγ TiN−1,jN−1(1)T iN−2,jN−2(1)r(tN−3)

Îγ . . .

Îγ

N−1∏k=1

T ik,jk(1)r(t0 = 0). (B12)

Now, recall that r(tN−1) Ïγ r(tN ) = q and thatr(0) = p. Finally, using transitivity and the fact thatrelation Ïγ implies the relation �γ , we conclude that

N−1∏k=1

T ik,jk(1)p �γ q. (B13)

[2⇒ 1] For t ∈ [0, N − 1] define the trajectory

r(t) = T ik,jk(δ)T ik−1,jk−1(λk−1) . . . T i1,j1(λ1)p, (B14)

with k and δ ∈ [0, 1] satisfying t = δ +∑k−1i=1 λi. Note

that the resulting trajectory r(t) has a coarse-graineddescription given by {πk}Nk=1 and that

p Ïγ r(N − 1). (B15)

Now, from the assumption we have that r(N − 1) �γ qand π(r(N − 1)) = π(q). By Corollary 9 we thus havethat r(N − 1) Ïγ q, and that the trajectory connect-ing these two states has a fixed γ-ordering. Hence,p Ïγ r(N−1) Ïγ q and, by transitivity, p Ïγ q. There-fore, we conclude that there exists a thermomajorizingtrajectory from p to q with a canonical coarse-graineddescription given by {πk}Nk=1.

16

We are now ready to present the proof of Theorem 4.

Proof. First, assume that p Ïγ q. Then, fromLemma 11, we know that there exists a thermomajoriz-ing path r(t) connecting p and q with some canonicalcoarse-grained description. Moreover, from Lemma 14we know that such a path with a given coarse-grainedcanonical description exists if and only if Eq. (B7) issatisfied. Thus, if we list all possible canonical coarse-grained paths connecting p and q, then for at least oneof them Eq. (B7) must be satisfied.

Conversely, if Eq. (B7) is satisfied for some canoni-cal coarse-grained path, then from Lemma 14 we havep Ïγ q.

Appendix C: Remarks on fundamental constraintson coherence

In this paper we focused on necessary and sufficientconditions and explicit protocols to generate a given pop-ulation dynamics. What about the characterization ofthe evolution of coherences, i.e., the off-diagonal elementsof the density matrix in the energy basis? Here we pro-vide some general remarks on this notoriously complexissue.

It has been recognized that the thermodynamic evolu-tion of quantum coherence is restricted by symmetry con-siderations [63]. These involve an extension of Noether’stheorem to open quantum system dynamics [64]. Theproperties (P1)-(P2) then allow to construct monoton-ically increasing or decreasing functionals (monotones)which quantify the deterioration of athermality and co-herence during a thermalization process through entropyproduction-like inequalities. The standard entropy pro-duction can be seen as one monotone, to be accompaniedby many more which often have independent operationalor information-theoretical meaning. Let us briefly discusssome notable examples.

First, we have the α-Renyi entropy production rela-tions:

∀α ∈ [0,∞) :dΣα(t)

dt= −dSα(ρ(t)‖γ)

dt≥ 0, (C1)

where γ is the thermal state from Eq. (4) and Sα is thequantum α-Renyi divergence defined by

Sα(ρ‖σ) =

log Tr

(ρασ1−α)

α− 1if α ∈ [0, 1),

log Tr(σ

1−α2α ρσ

1−α2α

)α− 1

if α > 1,

(C2)

whose origin in quantum information lies in quantum hy-pothesis testing [65]. The usual entropy production rela-tion of Eq. (8) is recovered in the limit α→ 1. Next, wehave the α-relative entropy of asymmetry Aα(t) [63, 66],

∀α ∈ [0,∞) : −dAα(t)

dt:= −dSα(ρ(t)‖D(ρ))

dt≥ 0, (C3)

with D denoting the dephasing in the energy basis.Asymmetry is a measure of coherence of the state in thebasis of H. For α = 1 the usual entropy productionrelation can be decomposed as

dΣ(t)

dt=dΣd(t)

dt− dA1(t)

dt, (C4)

where Σd =∑i pi log(pi/γi) is the diagonal relative en-

tropy. Hence, −dA1(t)/dt ≥ 0 can be seen as one of twomonotonically increasing additive terms in the standardentropy production equation, measuring the entropiccontribution due to loss of quantum coherence [63]. Thisconstitutes a refinement of the usual entropy productioninequality, since both Σd(t) and −A1(t) increase in time,not just their sum [41]. The quantity A1(t) operationallyquantifies the coherent contribution in an average workextraction protocol [67], that is the loss to the maximalextractable work due to dephasing of the state. Anotherexample is given by the Fisher information Q(ρ(t)) of theunitary orbit {e−iHδρ(t)eiHδ}δ∈R [68, 69],

− dQ(t)

dt≥ 0, (C5)

where

Q(t) = 2− 2 limδ→0

F 2[ρ(t), e−iHδρ(t)eiHδ)] (C6)

and F (ρ, σ) := Tr(√ρ1/2σρ1/2) is the quantum fidelity.

The quantity Q(t) measures the information that ρ(t)encodes about the phase along the unitary orbit gener-ated by H, i.e. the metrological value of ρ(t) for phaseestimation. Finally, we have the Wigner-Yanase-Dysonskew information [66, 70],

− dIs(t)

dt≥ 0, (C7)

with

Is(t) = −1

2Tr([ρs(t), H][ρ1−s(t), H]

), (C8)

where s ∈ (0, 1). The quantity Is(t) was introduced asa measure of the information contained in measurementsthat do not commute with H [70] (for s = 1/2 one recov-ers the usual Wigner-Yanase skew information). A sys-tematic framework to deal with all these constructionswas developed in Ref. [22]. Leveraging the results ob-tained there, one can formally define a complete set ofmonotones. The drawback is that these form an infinitenumber of extremely involved conditions, and it is notyet clear how these can be simplified.

An alternative approach is to obtain specific com-putable constraints. For example, exploiting proper-ties (P1)-(P2) and the framework of Ref. [71] (see in par-ticular Eq. (22) therein), one can obtain the followingone parameter family of monotones:

dCλ(t)

dt≥ 0, Cλ(t) =

−Z(Ej − Ei)2λe−βEi + e−βEj

|Cij(t)|2, (C9)

17

where Cij(t) is the coherence matrix defined in Eq. (14b)and λ ≥ 0. Note how each coherence element |Cij(t)|2is weighted by a Gibbs factor and the energy differencesquared, neatly combining energetic and coherent con-siderations.

The general problem of Eq. (24) can be explicitlysolved for a single qubit system using the minimal de-

coherence theory of Ref. [72] (see Sec. E3 therein). How-ever, for higher dimensional systems currently there areno tools to obtain a solution to this problem (perhapsup to some approximation) that satisfies both our funda-mental desiderata. We leave this extremely challengingquestion to future work.

[1] M. Lostaglio, An introductory review of the resource the-ory approach to thermodynamics, Rep. Prog. Phys. 82,114001 (2019).

[2] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth,Thermodynamic cost of reliability and low temperatures:tightening Landauer’s principle and the second law, Int.J. Theor. Phys. 39, 2717 (2000).

[3] M. Horodecki and J. Oppenheim, Fundamental limita-tions for quantum and nanoscale thermodynamics, Nat.Commun. 4, 2059 (2013).

[4] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Op-penheim, and S. Wehner, The second laws of quantumthermodynamics, Proc. Natl. Acad. Sci. U.S.A. 112, 3275(2015).

[5] N. H. Y. Ng and M. P. Woods, Resource theory ofquantum thermodynamics: Thermal operations and sec-ond laws, in Thermodynamics in the Quantum Regime(Springer, 2018) pp. 625–650.

[6] S. Vinjanampathy and J. Anders, Quantum thermody-namics, Contemp. Phys. 57, 545 (2016).

[7] R. Kosloff, Quantum thermodynamics: A dynamicalviewpoint, Entropy 15, 2100 (2013).

[8] E. Ruch, R. Schranner, and T. H. Seligman, The mixingdistance, J. Chem. Phys. 69 (1978).

[9] G. T. Landi and M. Paternostro, Irreversible entropy pro-duction, from quantum to classical, Rev. Mod. Phys. 93,035008 (2021).

[10] K. Korzekwa, Continuous thermomajorisation (2021).[11] K. Korzekwa and M. Lostaglio, Optimizing thermaliza-

tions, (In preparation).[12] G. Spaventa, S. F. Huelga, and M. B. Plenio, Non-

Markovianity boosts the efficiency of bio-molecularswitches, arXiv:2103.14534 (2021).

[13] M. A. Nielsen and I. L. Chuang, Quantum computationand quantum information (Cambridge university press,2010).

[14] H.-P. Breuer and F. Petruccione, The theory of openquantum systems (Oxford University Press, 2002).

[15] R. Alicki and R. Kosloff, Introduction to quantum ther-modynamics: History and prospects, in Thermodynamicsin the Quantum Regime: Fundamental Aspects and NewDirections, edited by F. Binder, L. A. Correa, C. Gogolin,J. Anders, and G. Adesso (Springer International Pub-lishing, Cham, 2018) pp. 1–33.

[16] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,Completely positive dynamical semigroups of N-level sys-tems, J. Math. Phys. 17, 821 (1976).

[17] G. Lindblad, On the generators of quantum dynamicalsemigroups, Commun. Math. Phys. 48, 119 (1976).

[18] R. Uzdin, A. Levy, and R. Kosloff, Equivalence of quan-tum heat machines, and quantum-thermodynamic signa-tures, Phys. Rev. X 5, 031044 (2015).

[19] E. T. Campbell, B. M. Terhal, and C. Vuillot, Roadstowards fault-tolerant universal quantum computation,Nature 549, 172 (2017).

[20] J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl,K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walms-ley, R. Uzdin, and E. Poem, Experimental demonstrationof quantum effects in the operation of microscopic heatengines, Phys. Rev. Lett. 122, 110601 (2019).

[21] H. Spohn, Entropy production for quantum dynamicalsemigroups, J. Math. Phys. 19, 1227 (1978).

[22] G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Mar-vian, Quantum majorization and a complete set of en-tropic conditions for quantum thermodynamics, Nat.Commun. 9, 1 (2018).

[23] F. G. S. L. Brandao, M. Horodecki, J. Oppenheim, J. M.Renes, and R. W. Spekkens, Resource theory of quantumstates out of thermal equilibrium, Phys. Rev. Lett. 111,250404 (2013).

[24] A. Renyi, On measures of entropy and information, inFourth Berkeley Symposium on Mathematical Statisticsand Probability (1961) pp. 547–561.

[25] G. Elfving, Zur theorie der Markoffschen ketten, ActaSoc. Sci. Fennicae, n. Ser. A2 8, 1 (1937).

[26] E. B. Davies, Embeddable Markov matrices, Electron. J.Probab. 15, 1474 (2010).

[27] J. F. C. Kingman, The imbedding problem for finiteMarkov chains, Probab. Theory Relat. Fields 1, 14(1962).

[28] J. T. Runnenburg, On Elfving’s problem of imbedding atime-discrete markov chain in a time-continuous one forfinitely many states I, P. K. Ned. Akad. A Math 65, 536(1962).

[29] G. S. Goodman, An intrinsic time for non-stationary fi-nite Markov chains, Probab. Theory Relat. Fields 16, 165(1970).

[30] P. Carette, Characterizations of embeddable 3 × 3stochastic matrices with a negative eigenvalue, New YorkJ. Math 1, 129 (1995).

[31] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities:Theory of Majorization and Its Applications (Springer,2010).

[32] M. A. Nielsen, An introduction to majorization and itsapplications to quantum mechanics, Lecture Notes, De-partment of Physics, University of Queensland, Australia(2002).

[33] A. M. Alhambra, J. Oppenheim, and C. Perry, Fluctuat-ing states: What is the probability of a thermodynamicaltransition?, Phys. Rev. X 6, 041016 (2016).

[34] C. Zylka, A note on the attainability of states by equal-izing processes, Theoret. Chim. Acta 68, 363 (1985).

[35] C. Zylka, On the accessibility of states for systems withdissipative dynamics, Ann. Phys. (Berl.) 502, 268 (1990).

18

[36] P. M. Alberti, B. Crell, A. Uhlmann, and C. Zylka, Orderstructure (majorization) and irreversible processes, Ver-netzte Wissenschaften–Crosslinks in Natural and SocialSciences, PJ Plath, E.-Chr. Hass, eds , 281 (2008).

[37] M. J. Hay, J. Schiff, and N. J. Fisch, Maximal energy ex-traction under discrete diffusive exchange, Phys. Plasmas22, 102108 (2015).

[38] M. Hay, J. Schiff, and N. J. Fisch, On extreme points ofthe diffusion polytope, Physica A 473, 225 (2017).

[39] D. Thon and S. W. Wallace, Dalton transfers, inequalityand altruism, Soc. Choice Welf. 22, 447 (2004).

[40] A. N. Gorban, Thermodynamic tree: The space of admis-sible paths, SIAM J. Appl. Dyn. Syst. 12, 246 (2013).

[41] J. P. Santos, L. C. Celeri, G. T. Landi, and M. Paternos-tro, The role of quantum coherence in non-equilibriumentropy production, npj Quantum Inf. 5, 1 (2019).

[42] C. Tsallis, Possible generalization of boltzmann-gibbsstatistics, J. Stat. Phys. 52, 479 (1988).

[43] S. Abe, Axioms and uniqueness theorem for tsallis en-tropy, Phys. Lett. A 271, 74 (2000).

[44] C. Tsallis, Introduction to nonextensive statistical me-chanics: approaching a complex world (Springer Science& Business Media, 2009).

[45] A. M. Mariz, On the irreversible nature of the tsallis andrenyi entropies, Phys. Lett. A 165, 409 (1992).

[46] H. Wilming and R. Gallego, Third law of thermodynam-ics as a single inequality, Phys. Rev. X 7, 041033 (2017).

[47] M. Lostaglio, A. M. Alhambra, and C. Perry, Elementarythermal operations, Quantum 2, 52 (2018).

[48] P. Mazurek and M. Horodecki, Decomposability and con-vex structure of thermal processes, New J. Phys. 20,053040 (2018).

[49] C. Perry, P. Cwiklinski, J. Anders, M. Horodecki, andJ. Oppenheim, A sufficient set of experimentally imple-mentable thermal operations for small systems, Phys.Rev. X 8, 041049 (2018).

[50] E. B. Davies, Markovian master equations, Commun.Math. Phys. 39, 91 (1974).

[51] W. Roga, M. Fannes, and K. Zyczkowski, Davies mapsfor qubits and qutrits, Rep. Math. Phys. 66, 311 (2010).

[52] V. Scarani, M. Ziman, P. Stelmachovic, N. Gisin, andV. Buzek, Thermalizing quantum machines: Dissipationand entanglement, Phys. Rev. Lett. 88, 097905 (2002).

[53] E. Baumer, M. Perarnau-Llobet, P. Kammerlander,H. Wilming, and R. Renner, Imperfect thermalizationsallow for optimal thermodynamic processes, Quantum 3,153 (2019).

[54] H. J. Miller, M. Scandi, J. Anders, and M. Perarnau-Llobet, Work fluctuations in slow processes: quantumsignatures and optimal control, Phys. Rev. Lett. 123,230603 (2019).

[55] R. Bhatia, Matrix analysis, Vol. 169 (Springer Science &

Business Media, 2013).[56] P. Abiuso, H. JD Miller, M. Perarnau-Llobet, and

M. Scandi, Geometric optimisation of quantum thermo-dynamic processes, Entropy 22, 1076 (2020).

[57] M. Lostaglio, M. P. Muller, and M. Pastena, Stochasticindependence as a resource in small-scale thermodynam-ics, Phys. Rev. Lett. 115, 150402 (2015).

[58] M. P. Muller, Correlating thermal machines and the sec-ond law at the nanoscale, Phys. Rev. X 8, 041051 (2018).

[59] A. M. Alhambra, M. Lostaglio, and C. Perry, Heat-BathAlgorithmic Cooling with optimal thermalization strate-gies, Quantum 3, 188 (2019).

[60] N. Shiraishi and T. Sagawa, Quantum thermodynam-ics of correlated-catalytic state conversion at small scale,Phys. Rev. Lett. 126, 150502 (2021).

[61] P. Lipka-Bartosik and P. Skrzypczyk, All states are uni-versal catalysts in quantum thermodynamics, Phys. Rev.X 11, 011061 (2021).

[62] I. Henao and R. Uzdin, Catalytic transformations withfinite-size environments: applications to cooling andthermometry, Quantum 5, 547 (2021).

[63] M. Lostaglio, D. Jennings, and T. Rudolph, Descriptionof quantum coherence in thermodynamic processes re-quires constraints beyond free energy, Nat. Commun. 6,6383 (2015).

[64] I. Marvian and R. W. Spekkens, Extending Noether’stheorem by quantifying the asymmetry of quantumstates, Nat. Commun. 5, 3821 (2014).

[65] M. Mosonyi and T. Ogawa, Quantum hypothesis test-ing and the operational interpretation of the quantumrenyi relative entropies, Commun. Math. Phys. 334, 1617(2015).

[66] I. Marvian, Symmetry, Asymmetry and Quantum Infor-mation, Ph.D. thesis, University of Waterloo (2012).

[67] E. Baumer, M. Lostaglio, M. Perarnau-Llobet, andR. Sampaio, Fluctuating work in coherent quantum sys-tems: Proposals and limitations, in Thermodynamics inthe Quantum Regime (Springer, 2018) pp. 275–300.

[68] D. Janzing and T. Beth, Quasi-order of clocks and theirsynchronism and quantum bounds for copying timing in-formation, IEEE Trans. Inf. Theory 49, 230 (2003).

[69] H. Kwon, H. Jeong, D. Jennings, B. Yadin, and M. S.Kim, Clock–work trade-off relation for coherence in quan-tum thermodynamics, Phys. Rev. Lett. 120, 150602(2018).

[70] E. Wigner and M. M. Yanase, Information contents ofdistributions, Proc. Natl. Acad. Sci. U.S.A. , 910 (1963).

[71] G. Styliaris and P. Zanardi, Symmetries and mono-tones in markovian quantum dynamics, Quantum 4, 261(2020).

[72] M. Lostaglio, K. Korzekwa, and A. Milne, Markovianevolution of quantum coherence under symmetric dynam-ics, Phys. Rev. A 96, 032109 (2017).