Quantum advantage in simulating stochastic processes

We investigate the problem of simulating classical stochastic processes through quantum dynamics, and present three scenarios where memory/time quantum advantages arise. First, by introducing and analysing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory. Second, by extending the notion of space-time cost of a stochastic process $P$ to the quantum domain, we prove an exponential advantage of the quantum cost of simulating $P$ over the classical cost. Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to an advantage in cooling protocols. To achieve this last point, we develop the notion of continuous thermo-majorisation, which strengthens the so-called"second laws"by including the typical constraint that the dynamics is (effectively) Markovian.


I. INTRODUCTION
What tasks can we perform more efficiently by employing quantum properties of nature? And what are the quantum resources powering them? These are the central questions that need to be answered not only to develop novel quantum technologies, but also to deepen our understanding of the foundations of physics. Over the last few decades, these questions were successfully examined in the context of cryptography [1], computing [2], simulations [3] and sensing [4], proving that the quantum features of nature can indeed be harnessed to our benefit.
More recently, an area of active theoretical and experimental interest focused on the memory advantages offered by quantum mechanics for the simulation of stochastic processes in the setting of classical causal models [5][6][7][8]. An experimentally accessible and relevant measure of such an advantage is the dimensionality of the memory required for the simulation [8,9]. These dimensional advantages have been identified experimentally (a qubit system has been used to simulate a stochastic process that classically requires three bits [8]), and theoretically for a certain class of Poisson processes [9].
Here we take a complementary approach starting from the following simple observation: although all fundamental interactions are memoryless, the basic informationprocessing primitives (such as the bit-flip operation) cannot be performed classically in a time-continuous fashion without employing a memory [10]. We show that this picture changes dramatically if instead we consider memoryless quantum dynamics. This is due to quantum coher-ence, arising from the superposition principle, which can effectively act as an internal memory of the system during the evolution. In this work we identify three aspects of potential quantum advantage in simulating stochastic processes.
First, in Sec. II, we investigate the possibility to simulate classical processes requiring memory using quantum memoryless dynamics. More precisely, we compare the sets of all stochastic processes that can be generated by time-continuous memoryless dynamics in the classical and quantum domains. We prove that the latter set is strictly larger than the former one, i.e., that there exist stochastic processes that classically require memory to be implemented, but can be realised by memoryless quantum dynamics. As an example, consider a random walk on a cyclic graph with three sites, where the walker can either move clockwise, anti-clockwise, or stay in place. As we present in Fig. 1, only a small orange subset of such walks can arise from a continuous classical evolution that does not employ memory (note that, differently from other investigations [11], we do not put any restriction on the classical dynamics beyond the fact that it is memoryless). However, if we allow for continuous memoryless quantum evolution, all stochastic processes in the much larger blue set can be achieved. Besides this particular class, in this work we provide general constructions for whole families of stochastic processes for any finitedimensional systems that require memory classically, but can be implemented quantumly in a memoryless fashion.
Second, in Sec. III, we go beyond the simple distinction between stochastic processes that can or cannot be simulated without memory, and take a more quantitative approach, thus investigating quantum memory advantages. To this end, we employ the recent formalism of Ref. [10], that allows one to quantify the classical spacetime cost of a given stochastic process, i.e., the minimal amount of memory and time-steps needed to classically The vertices of the triangle correspond to deterministic processes (S: stay, C: move clockwise, A: move anti-clockwise) for a random walker moving between three states. Points inside the triangle correspond to probabilistic mixtures (convex combinations) of these three deterministic processes, e.g., the centre of the triangle corresponds to the maximally mixing dynamics (with S, C and A each happening with probability 1/3). The orange petal-shaped region contains all stochastic processes that can arise from time-continuous memoryless classical dynamics. For time-continuous memoryless quantum dynamics this set is enlarged by the remaining shaded region in blue. For details see Sec. II C and, in particular, Fig. 4.
implement a given process. We extend this approach to the quantum domain in order to analyse the quantum space-time cost. An illustrative example is given by the bit-flip process presented in Fig. 2, which in the classical setting requires either one memory state and three timesteps, or two memory states and two time-steps. However, if one allows for quantum evolution, such a bit-flip can be performed in a continuous and memoryless fashion through a simple unitary evolution exp(iσ x t) with σ x denoting the Pauli x operator. More generally, the authors of Ref. [10] have characterised the space-time cost for the family of {0, 1}-valued stochastic processes (i.e., all discrete functions). Their bound shows an unavoidable classical trade-off between the number of memory states m and the number of time-steps τ needed to realise a given stochastic process on N systems of dimension d. Crucially, a typical process requires exponential resources, meaning that either m or τ is exponential in N . In this paper we prove that in the quantum regime all such processes can be simulated with zero memory states and in at most two time-steps. That is, most processes can be simulated quantumly with exponentially less memory states and/or time-steps than the best possible classical implementation. Third, in Sec. IV we study memory advantages in Each time-step is composed of a continuous memoryless dynamics that does not affect one of the states, and maps the remaining two to one of them. (b) In the quantum regime, a bit flip can be performed without any memory, simply by a time-continuous unitary process exp(iσxt) that continuously connects the identity operation at time t = 0 with the bit flip, represented by Pauli x operator σx, at time t = π/2. During the process, the information about the initial state of the system is preserved in quantum coherence.
control by comparing classical and quantum continuous memoryless dynamics in terms of the set of accessible final states. We assume a fixed point of the evolution is given, which is a realistic physical constraint in dissipative processes and typically, but not necessarily, coincides with the thermal Gibbs state. A standard example is given by a thermalisation of the system to the environmental temperature. Here, our contribution is twofold. Firstly, in Sec. IV B we characterise the set of all states accessible from a given state by classical memoryless dynamics. In order to achieve this, we employ the results on Markov majorization arising from the studies on continuous c-processes and w-processes [12], and extend them to the thermodynamic setting. The upshot is a more stringent form of the "second laws conditions" introduced in Ref. [13]: the constraints derived here are stronger due to the inclusion of the realistic assumption that the dynamics is generated by a Markovian master equation. Note that these results bridge the gap between stochastic thermodynamics [14], where Markovian master equations form the basic underlying assumption, and the previously disconnected resource-theoretical framework [15]. Secondly, in Sec. IV C we show how quantum memoryless dynamics allows one to access a larger set of final states compared to what is possible through clas- 3. Markovian cooling of a qubit. Classical memoryless processes can only cool the initial state ρ of a twodimensional system to the thermal state γ at the environmental temperature (path along the solid line arrow). Quantum memoryless dynamics allows one to cool the system below that, all the way to state ρ with the lowest temperature achievable by classical processes with memory (path along the dotted line arrow). For details see Sec. IV C and, in particular, Fig. 9.
sical memoryless dynamics. This is most evident in the case of maximally mixed fixed points (corresponding to the environment in the infinite temperature limit), since every transformation that is classically possible with arbitrary amounts of memory can be realised in a memoryless fashion in the quantum domain. For general fixed points, we prove that an analogous result holds for systems of dimension d = 2, and argue that the set of accessible states is strictly larger in the quantum regime than in the classical one for all d. Since it is known that memory effects enhance cooling [16,17], a direct consequence of our results is that quantumly it is possible to bring the two-dimensional system below the environmental temperature without employing memory effects, something that is impossible classically (see Fig. 3).

A. Classical embeddability
Given a discrete state space, {1, . . . , d}, the state of a finite-dimensional classical system is described by a probability distribution p over these states. A stochastic matrix or process P is a matrix P i|j of transition probabilities, which describes the evolution of the system from one state p to another P p.
A stochastic matrix P is embeddable if it can be generated by a continuous Markov process [18]. This can be understood as a control problem involving a master equation. Namely, introducing a rate matrix or generator L as a matrix with finite entries satisfying a continuous one-parameter family L(t) of rate matrices generates a family of stochastic processes P (t) satisfying d dt The aim of the control L(t) is to realize a target stochastic process P at some final time t f as P = P (t f ). If this is possible for some choice of L(t), then P is embeddable; and if there exists a time-independent generator L such that P = e Lt f , then we say that P can be embedded by a time-homogeneous Markov process. A final technical comment is that we also consider the case t f = ∞ to be embeddable (in Ref. [10] this case was referred to as limitembeddable). Then, P cannot be generated in any finite time, but can be approximated arbitrarily well. This is the case, e.g., with the bit erasure process: 0 → 0, 1 → 0 [10]. The question of which stochastic matrices P are embeddable is a challenging open problem that has been extensively investigated for decades [18][19][20][21][22][23]. The full characterization does not go beyond 2×2 and 3×3 stochastic matrices, however various necessary conditions have been found. In particular, in Ref. [22] it was proven that every embeddable stochastic matrix P satisfies the following inequalities: The condition det P ≥ 0 is in fact also known to be sufficient in dimension d = 2 [20], and a time-independent rate matrix L can then be found.

B. Quantum embeddability
A state of a finite-dimensional quantum system is given by a density operator ρ, i.e. a positive semi-definite operator with trace one that acts on a d-dimensional Hilbert space H d . A general evolution of a density matrix is described by a quantum channel E, which is a completely positive trace-preserving map from the space of density matrices to itself. Now, focussing on the computational basis {|k } d k=1 of H d , suppose we input the quantum state ρ p = k p k |k k|, apply the channel E and measure the resulting state E(ρ p ) in the computational basis. The measurement outcomes will be distributed according to P p, where In this way, the preparation of ρ p , followed by a channel E and the computational basis measurement, simulates the action of a stochastic process P on the classical state p. We say that a stochastic matrix P is quantumembeddable if it can be simulated by a quantum process as in Eq. (5), with E a Markovian quantum channel, i.e. a channel that can be generated by a continuous quantum Markov process [24]. Despite the difference in jargon between the two communities, Markovianity for channels is the quantum analogue of the classical notion of embeddability; it can also be understood as a control problem, but this time involving a quantum master equation. More precisely, the rate matrix L is replaced by a Lindbladian [25,26], which is a superoperator L acting on density operators and satisfying with the first term describing unitary evolution and the remaining ones encoding the dissipative dynamics, e.g., due to the interaction with an external environment.
Here H is a Hermitian operator, [A, B] := AB − BA denotes a commutator, Φ is a completely positive superoperator, Φ * denotes the dual of Φ under the Hilbert-Schmidt scalar product, and {A, B} := AB + BA stands for the anticommutator. In analogy with Eq. (3), a continuous one-parameter family of Lindbladians L(t) generates a family of quantum channels E(t) satisfying where I denotes the identity channel. A quantum channel E is Markovian [24] if E = E(t f ) for some choice of the Linbladian L(t) and t f (perhaps t f = +∞). Any given Markovian channel E gives a stochastic process P through Eq. (5). The aim of the control L(t) is to achieve a target stochastic matrix P after some time t f . More formally we introduce the following definition.
Definition 1 (Quantum-embeddable stochastic matrix). A stochastic matrix P is quantum-embeddable if where E is a Markovian quantum channel.

C. Quantum advantage
One can easily see that all (classically) embeddable stochastic processes are also quantum-embeddable: given a classical generator L one chooses the CP map Φ defining the Lindbladian L in Eq. (6) to be However, the converse is not true. There exist many stochastic matrices P which can be generated by a quantum, but not a classical Markov process. The simplest example is given by a non-trivial permutation Π, satisfying Clearly, Eq. (4) is violated and hence Π is not embeddable. However, noting that every unitary U is quantum embeddable (by choosing the Lindbladian with no dissipative part and H such that U = exp(iHt f ), and that a permutation matrix Π is unitary, we conclude that every permutation Π is quantum-embeddable. This also proves that neither of the two conditions in Eq. (4) are necessary for quantum-embeddability. More generally, a larger class of stochastic matrices that are quantum-embeddable is given by the set of unistochastic matrices [27,28]. These are defined as all stochastic matrices P satisfying for some unitary matrix U , and the argument for quantum embeddability is analogous to the one given for permutation matrices. The set of unistochastic matrices includes permutations, but also other (classically) nonembeddable stochastic matrices. As an example consider a bistochastic matrix P P = 1/3 2/3 2/3 1/3 .
Since, in dimension d = 2, every bistochastic matrix is unistochastic, P is quantum-embeddable. At the same time it fails to satisfy Eq. (4), and thus is not (classically) embeddable.
Beyond these examples we prove a simple general result that allows one to find larger families of quantum embeddable stochastic matrices.
Lemma 1 (Monoid property). The set of quantumembeddable stochastic matrices contains identity and is closed under composition, i.e., if P and Q are quantumembeddable, then also P Q is.
Proof. First, identity is obviously quantum-embeddable as it arises from a trivial Lindbladian L = 0. Now, note that the composition of Markovian quantum channels gives a Markovian quantum channel. Next, notice that a completely dephasing map is a Markovian quantum channel. Finally, that the composition E = E P • D • E Q , with E P and E Q being quantum channels describing the quantum embeddings of P and Q, is a Markovian quantum channel which quantum-embeds the stochastic process described by P Q.
Let us now discuss the consequences of Lemma 1 with increasing generality. We start with the following corollary for dimension d = 2.  Proof. A general 2×2 stochastic matrix P can be written as If det P ≥ 0, then P is embeddable and hence quantum embeddable. Otherwise, if det P < 0, we can write P = ΠP with Π the denoting the non-trivial 2×2 permutation and P being a stochastic matrix with det P ≥ 0.
Since P can be written as a composition of two quantum embeddable maps, by Lemma 1 it is also quantum embeddable.
For d ≥ 3 comparing quantum and classical embeddability becomes complicated due to the lack of a complete characterisation of classical embeddability. One can, however, focus on certain subclasses of stochastic processes that are better understood. For example, for the family of 3 × 3 circulant stochastic matrices, defined by the necessary and sufficient conditions for (classical) embeddability are known. Denoting the eigenvalues of P by λ k = r k e iθ k with θ k ∈ [−π, π], these are given by [29] ∀k : r k ≤ exp [−θ k tan(π/3)] .
We illustrate the set of classically embeddable circulant matrices by a green region in parameter space [a, b] in Fig. 4. On the other hand, due to Lemma 1, quantum embeddable circulant stochastic matrices also include permutations of P , i.e., ΠP with Π denoting circulant 3 × 3 permutation matrices. In fact, this set contains not only permutations of P but also compositions of P with any unistochastic matrix; however, numerical verification suggests that this does not further expand the investigated set. As a result, the set of quantum embeddable stochastic matrices in the parameter space [a, b] contains not only the region corresponding to classically embeddable matrices, but also its two copies (corresponding to two permutations), which we illustrate in blue in Fig. 4. Moreover, all unistochastic circulant matrices (which are fully characterised by the "chain-links" conditions from Ref. [28]) are also quantum embeddable. The resulting region is also plotted in Fig. 4 in orange. We thus clearly see that the set of quantum-embeddable stochastic circulant matrices is much larger than the classically embeddable one, since it contains the union of green, blue and orange regions. However, we do not expect that all such matrices are quantum embeddable, with the case a = b = 1/2 being the least likely to arise from quantum Markovian dynamics.
For general dimension d one can generate families of quantum-embeddable matrices using Lemma 1 in an analogous way, by composing classically embeddable matrices with unistochastic ones. Moreover, employing Corollary 2 we note that the set of quantum embeddable stochastic matrices also includes all matrices P that can be written as products of elementary stochastic matrices P ei (also known as pinching matrices), i.e., where P 2 is a general 2×2 stochastic matrix, I d−2 denotes identity on the remaining states, and Π i is an arbitrary permutation. Notably, for d ≥ 4 this contains matrices that are not unistochastic [30] and hence cannot be reduced to the examples above. In conclusion, quantum embeddings allow one to achieve many stochastic processes, which necessarily require memory from a classical standpoint.

D. Discussion
From a physical perspective, it is now natural to ask: why the set of quantum embeddable stochastic matrices is strictly larger than the set of classically embeddable ones? To address this question, let us first consider the simple example of classically non-embeddable permutation matrices, where ⊕ here denotes addition modulo d. A direct calculation shows that Π m = e iHm with Hamiltonian and We thus see that the continuous and memoryless Hamiltonian evolution creates a superposition of classical states |n on the way between identity and Π m . The intuitive picture that emerges is that the quantum superposition between classical states created during the evolution effectively acts as a memory. For example, when we perform a rotation of the Bloch sphere around the y axis, we can implement a bit flip sending |0 to |1 and vice versa, but the path the state follows (going through |+ if the initial state was |0 , and through |− if the initial state was |1 ) will preserve the memory about the initial state. At the same time, a classical memoryless process moving (1, 0) towards (0, 1) and (0, 1) towards (1, 0) cannot proceed beyond the point at which the two trajectories meet.
Our results can be naturally connected to a result by Montina [31], who proved that Markovian hidden variable models reproducing quantum mechanical predictions necessarily require a number of continuous variables that grows linearly with the Hilbert space dimension, and hence exponentially with the system size. Intuitively, here we are showing that this "excess baggage" [32] can be exploited to simulate memory effects. In what follows we provide a quantification of the advantage beyond the embeddable/non-embeddable dichotomy. We will see that quantum theory allows for exponential advantages in the simulation of stochastic processes by memoryless dynamics.

III. SPACE-TIME COST OF A STOCHASTIC PROCESS
In this section we first recall a recently introduced framework for the quantification of the space and time costs of simulating a stochastic process by memoryless dynamics [10]. We then extend it to the quantum domain and prove an exponential quantum advantage in the corresponding costs.

A. Classical space-time cost
Let P be a non-embeddable stochastic matrix acting on d so-called visible states. We then want to ask: how many additional memory states m does one need to add, in order to implement P by a classical Markov process? Formally, one looks for an embeddable stochastic matrix Q acting on d + m states whose restriction to the first d rows and columns is identical to P . When this happens, Q is said to implement P with m memory states. In fact, given any d-dimensional distribution p, if we take the d + m dimensional distribution q = (p, 0 . . . 0), then Qq = (P p, 0, . . . 0). Following Ref. [10] we now have the following.
As a technical comment we note that the above definition can be extended to situations in which visible and memory states are not disjoint, e.g., when the visible states on which P acts are logical states defined by a coarse graining of the states on which Q acts. Since this does not change any of the result presented here, we refer to Ref. [10] for further details and adopt the simpler definition given here.
Once we find a matrix Q that implements P , the next question is: what is the number of time-steps required to realise Q? The notion of a time-step is meant to capture the number of independent controls that are needed to achieve Q. A natural definition would be that the number of time-steps necessary to realise an embeddable stochastic matrix Q is the minimum number n such that where L (1) , . . . L (n) are time-independent generators, i.e. each L (k) is a control applied for some time t k . However, this definition would assign an infinite cost to practically feasible protocols in which controls are switched on and off in a continuous fashion. To overcome this issue, note that by Levy's lemma [33], a crucial property is that at each step k the set of non-zero transition probabilities of e L (k) t is the same for all t > 0. One can interpret timesteps as the number of times the set of "blocked" transitions changes, since each change in principle requires raising or lowering some infinite energy barrier [10]. Hence, the definition of a time-step can be extended to allow for time-dependent controls L (k) (t) as follows [10].
can be chosen such that the set of non-zero transition probabilities of P (t) is the same for all t ∈ (0, t f ).
Putting all this together we obtain the notion of time cost from Ref. [10]: Definition 4 (Time cost). The time cost C time (P, m) of a d×d stochastic matrix P , while allowing for m memory states, is the minimum number τ of one-step stochastic matrices The framework presented above allows one to quantify the memory and time costs of implementing a given stochastic process by classical master equations. We now introduce a natural extension of the above to the quantum domain.
The quantum time cost of a stochastic matrix, in analogy with the classical counterpart, admits more or less restrictive definitions. Since we want to prove a quantum advantage, it is sufficient to adopt a stricter definition (i.e., give less power in the quantum domain) with time independent generators only.
Definition 6 (Quantum time cost). The quantum time cost Q time (P, m) of a d × d stochastic matrix P , while allowing for m memory states, is the minimum τ such that there exist time-independent Lindbladians L 1 , . . . L τ on a (d + m) × (d + m) dimensional Hilbert space and implements P .
The central question in the classical setting is to find C space (P ) and then characterise C time (P, m) for m ≥ C space (P ). The main result of Ref. [10] was to solve this problem for stochastic matrices P that are {0, 1}-valued or, in other words, represent a function f over the set of states {1, . . . , d}. How do these results compare with what can be done quantum mechanically? In the next section we give a protocol realising every {0, 1}-valued stochastic matrix that scales much better (typically, exponentially better) than the corresponding minimal classical cost.  The optimal trade-off between space cost and time cost of implementing stochastic matrices for a system of s = 32 bits, i.e., with dimension d = 2 32 (plotted in log-log scale). Solid coloured curves correspond to optimal trade-offs for classically implementing exemplary {0, 1}-valued stochastic matrices described by functions f1(i) = i ⊕ 1 (addition modulo d) and f2(i) = min{i + 2 s/2 , 2 s − 1}, as analysed in Ref. [10]. Dashed black curve corresponds to optimal trade-offs for quantumly implementing any {0, 1}-valued stochastic matrix, thus illustrating a quantum advantage.
Theorem 3 (Classical cost of a function [10]). The time cost of a {0, 1}-valued stochastic matrix P f described by a function f is given by where b f (m) = 0 or 1 and · is the ceiling function.
Suppose that the state space is given by all bit strings of length s, so that d = 2 s . Theorem 3 shows that, , then P f is expensive to simulate by memoryless dynamics unless the number of fixed points is , an exponential number of memory states are required to have an efficient simulation in the number of time-steps. Conversely, one needs an exponential number of time-steps to have an efficient simulation for a fixed number of memory states. One of the examples discussed in Ref. [10] is that of f 1 (i) = i ⊕ 1 (addition modulo d), which may be interpreted as keeping track of a clock in a digital computer. From Theorem 3 we see that one has C time (P f1 , m) ≥ 2 s /m with m the number of memory states introduced (see Fig. 5). However, as we discussed already above, any permutation is quantum embeddable by a unitary, and hence Q time (P f1 , 0) = 1. The existence of this (exponential) advantage is generalised by the following result.
Theorem 4 (Quantum cost of a function). For any m ≥ 0 and any function f we have Q time (P f , m) ≤ 2.
The explicit proof is given in Appendix B, but it is based on the simple fact that every function can be realised quantumly by a unitary process realising a permutation followed by a classical master equation achieving an idempotent function f I in a single time-step. Hence, one can achieve every function quantumly using zero memory states and only two time-steps, an exponential advantage as compared to the classical case. This result, illustrated in Fig. 5, is a quantitative evidence of the power of superposition to act as an effective memory.

A. Accessibility regions
In this section, we change the focus from processes to states. We will investigate whether a given state transformation can be realised by either a classical or a quantum master equation. In other words, given an input distribution p, is it possible to get a given final state q through an embeddable (or quantum embeddable) stochastic matrix P ? Of course, given full control it is always possible to choose a master equation with q being the unique fixed point. However, more realistically, a fixed point of the evolution is constrained rather than being arbitrary -and typically corresponds to the thermal Gibbs distribution γ with Here, E k are the energy levels of the system interacting with an external environment at inverse temperature β. Hence, suppose some (full-rank) fixed point γ is given (which may or may not be the thermal state of the system). We then introduce the following two definitions.
Definition 7 (Classical accessibility). A distribution q is accessible from p by a classical stochastic process with a fixed point γ if there exists a stochastic matrix P , such that P p = q and P γ = γ. We denote the set of all q accessible from p given γ by C Mem γ (p).
Definition 8 (Classical memoryless accessibility). A distribution q is accessible from p by a classical master equation with a fixed point γ if there exists a continuous one-parameter family L(t) of rate matrices generating a family of stochastic matrices P (t), such that P (t f )p = q and L(t)γ = 0 for all t ∈ [0, t f ). We denote the set of all q accessible from p given γ by C γ (p).
The former definition encapsulates the set of inputoutput relations achievable by means of general processes with a fixed point γ, while the latter captures the subset achievable without exploiting memory effects, i.e. by Markovian master equations. These definitions naturally generalise to quantum dynamics. Denote by ρ p the density matrix diagonal in the computational basis with entries given by the probability distribution p: ρ p = k p k |k k|. We then have the following.
Definition 9 (Quantum accessibility). A distribution q is accessible from p by quantum dynamics with a fixed point γ if there exists a quantum channel E, such that E(ρ p ) = ρ q and E(ρ γ ) = ρ γ . We denote the set of all q accessible from p given γ by Q Mem γ (p).
Definition 10 (Quantum memoryless accessibility). A distribution q is accessible from p by a quantum master equation with a fixed point γ if there exists a continuous one-parameter family of Lindbladians L(t) generating a family of quantum channels E(t), such that E(t f )[ρ p ] = ρ q and L(t)[ρ γ ] = 0 for all t ∈ [0, t f ). We denote the set of all q accessible from p given γ by Q γ (p).
Note that in the above definitions the requirements L(t)γ = 0 and L(t)[ρ γ ] = 0 for all times ensure that Thus, by characterising the difference between the sets C Mem γ (p) and C γ (p), one can capture the state transformations that can be achieved only through controls exploiting memory effects. That is, all states q ∈ C Mem γ (p), but not in C γ (p), can only be achieved from p via a transformation that employs memory. Analogous statements hold for Q Mem γ (p) and Q γ (p). In this section we will study relations between all four accessibility regions. Our main result will be that C γ (p) ⊂ Q γ (p) in the particular case of a uniform fixed point η := (1/d, . . . , 1/d) and for general fixed points for a qubit system. This signals a quantum advantage, i.e., some transitions that classically require memory can be achieved through memoryless quantum dynamics. However, to prove the existence of such an advantage, we will first derive the necessary and sufficient conditions for a given q to belong to C γ (p), i.e., we will show when q is accessible from p by means of a (classical) Markovian master equation with a fixed point γ. We believe this is a result of independent interest, especially for the field of quantum thermodynamics (note that the main result of Ref. [13] reduces to answering the question: is q in C Mem γ (p)?). In particular, we will show that: 1. The accessibility region C γ (p) is characterised by a geometric criterion corresponding to a continuous version of the thermo-majorisation relation described in Ref. [13].
2. Partially thermalising pairs of energy levels by coupling them to a fixed thermal environment is sufficient to achieve any state transformation allowed by general master equations with a given fixed point. We will start by first reviewing the known results on the accessible region C Mem γ (p). Consider a system in a classical state p, that is distributed over the allowed energy states according to p. Then, C Mem γ (p) describes the set of classical states that can be obtained from p by thermal operations, i.e., energy-preserving couplings of the system with arbitrary thermal baths at temperature fixed by the choice of γ [13,15]. This scenario was analysed in Refs. [13,34], and it was proven there that C Mem γ (p) is fully specified by the notion of thermo-majorisation (also known as majorisation relative to γ [35]).
To explain the structure of C Mem γ (p), let us then briefly remind the definition of thermo-majorisation. Given a fixed point γ and an arbitrary distribution p, let π(p) denote a vector describing a permutation of {1, . . . , d} such that We will refer to π(p) as to the thermo-majorisation ordering of p, and it simply orders the γ-rescaled version of p in a non-increasing order. The thermo-majorisation curve of p is defined as the piecewise linear curve in R 2 obtained by joining the points for k ∈ {0, . . . d}, with k = 0 corresponding to the point (0,0). The points at which the thermo-majorisation curve changes slope are called elbow points. Then, p is said to thermo-majorise q, which we denote p γ q, when the thermo-majorisation curve of p lies above that of q (we do not mean strictly above). Importantly, in the case of a uniform fixed point, γ = η := (1/d, . . . 1/d), thermomajorisation coincides with the fundamental relation of majorisation, usually denoted by and defined by where p ↓ denotes a non-increasing reordering of p. Now, we are ready to state the full characterisation of C Mem γ (p).
Transforming p into q ∈ C Mem γ (p) may require control over memory effects that would manifest themselves as information back-flows from the environment. As a simple example of this phenomenon, consider a twodimensional system with an energy gap E and the Gibbs state γ as in Eq. (24). Then, one can straightforwardly verify that (1/2, 1/2) γ 1 − e −βE /2, e −βE /2 . However, any continuous trajectory r(t) connecting these two distributions has to pass through γ. This means that the (non-equilibrium) free energy of the system, will first decrease all the way to its minimal value, and then increase again, thus signalling an information backflow from the thermal environment. It is obvious that such phenomenon does not occur when dissipation is welldescribed by a Markovian master equation, as is the case in many typical thermalisation scenarios. Hence, we now proceed to extending the approach of Ref. [13] to include the constraint of Markovian interactions with the environment, and this way characterise C γ (p). Using the language of Ref. [13]: we will describe when a given state transformation can be realised by a Markovian thermal operation.

Region Cγ(p): classical memoryless transformations
We start by introducing the relation that will take the place of thermo-majorisation. The definition requires the existence of a continuous, oriented path connecting the two distributions, such that any point along the trajectory is thermo-majorised by those coming before.
Definition 11 (Markovian thermo-majorisation). We say that a distribution p Markov thermo-majorises q, denoted p Ï γ q, if there exists a continuous path of probability distributions r(t) for t ∈ [0, t f ) such that 1. r(0) = p, Note that in the particular case of a uniform fixed point, γ = η, the above definition corresponds to a continuous version of standard majorisation, denoted by Ï in Ref. [12]. In fact, the notion of continuous majorisation has a decades-long history and appears in a variety of research fields from thermodynamics and order theory [36,37], through plasma physics [38,39], to social sciences [40]. Moreover, the study in Ref. [12], inspired by a model of heat transport along ideal conducting wires between d objects with different temperatures, characterised the set C η . Therefore, the results we present here form a non-uniform generalisation of these previous studies.
The next step is to recall the definition of a family of operations known as two-level partial thermalisations, which will play the role of a sufficient set of controls. These are given by stochastic matrices that partially thermalise any two given energy levels. Definition 12 (Two-level partial thermalisation). A two-level partial thermalisation is a stochastic matrix T (λ) = T i,j (λ) ⊕ 1 \i,j , with 1 \i,j the identity over all levels not equal to i, j and with G i,j the full thermalisation for energy levels i, j: Before we proceed further, let us note that partial level thermalisations describe transformations of both practical and formal interest. They are a standard toy model for Markovian thermalisation processes that naturally emerge from collision models [41], and describe the set of population dynamics resulting from the weak interaction of a two-level system with a large bath [42,43]. Moreover, T i,j (λ) corresponds to dynamics generated during time t by the following master equations with λ = 1 − e −t/τ . These are used as building blocks for more complex protocols [44], e.g., in the context of work extraction [45] and slow driving [46]. Furthermore, it is not difficult to show using Eq. (4) that two-level partial thermalisations form exactly the set of embeddable stochastic processes with a fixed point γ acting on two energy levels.
We now have all the tools to state our result. The following theorem, the proof of which can be found in Appendix C, fully characterises the accessibility region C γ (p).

A distribution q is obtained from p by a sequence
of at most d! + d − 2 two-level partial thermalisations between adjacent elbow points of the (current) thermo-majorisation curve. In fact, the first d! − 1 partial thermalisations can be chosen to be full thermalisations (with λ = 1).
We will now discuss two important consequences of the above theorem. First, let us address the universality of the set of two-level partial thermalisations as controls for arbitrary Markovian master equations with a fixed point γ (equivalence 1 ⇔ 3 in the theorem above). It means that if a state is accessible via some general Markovian master equation, it is also accessible via a sequence of two-level partial thermalisations. Although it may seem to be an intuitive result, one should recall that in the general case with memory, i.e. the set C Mem γ (p), one cannot obtain all state transformations by elementary operations on two levels [41]. In fact, one then needs to couple the environment simultaneously to all d energy levels [47]. This also proves that C γ (p) forms a proper subset of C Mem γ (p). The fact that this decomposition into two-level partial thermalisations exists for Markovian thermal processes is a crucial simplification. As an application, recall that the authors of Ref. [44] presented a sufficient set of "experimentally implementable" controls allowing one to access every state in C Mem γ (p). These controls included: two-level partial thermalisations, arbitrary control over the set of energy levels, and full control of a thermal qubit ancilla. Here, we showed that two-level partial thermalisations alone already generate the full set C γ (p).
Second, let us comment on some simple but noteworthy consequences of the equivalence 1 ⇔ 2 in the theorem above. Condition 2 allows one to use the known results on d-majorisation [35] to construct a family of functionals that must be monotonically non-increasing during the Markovian evolution of the system along the path r(t).

More precisely, for any convex function
must be monotonically non-increasing along any admissible path: df h (t) dt ≤ 0 for all t and convex h.
Note that for h(x) = (log(x) − log Z)/β and γ as in Eq. (24), one recovers the standard non-equilibrium free energy from Eq. (28): More generally, h(x) = sign(α) α−1 x α − log Z β yield the α-free energies of the "second laws" derived in Ref. [48]. These play the role of Lyapunov functions for the dynamical system, hence restrictions to the admissible paths can be obtained by studying their level sets and constructing the corresponding thermodynamic trees [49]. In this way one clearly obtains conditions much more stringent than those presented in Ref. [48], which only prescribe the endpoint conditions f h (p) ≥ f h (q). These stronger second laws will be relevant in many circumstances, since often one can describe thermalisation processes by means of Markovian master equations.
These considerations are particularly powerful if we now include condition 3 of the theorem. In geometric terms condition 3 provides a finite set of conditions to determine whether an admissible path connecting p with q exists. These conditions generalise the thermomajorisation relation described in Ref. [13] to the Markovian scenario and can be formulated as follows. An admissible path connecting p with q exists if and only if the thermo-majorisation curve of p can be transformed by a sequence of at most d! − 1 lowerings of the elbow points (each straightening two adjacent segments) into a curve with • the same thermo-majorisation ordering as that of q; • and lying all above the curve of q.
An illustration of these conditions, which generalise thermo-majorisation to the Markovian setting, is illustrated in Fig. 6.

C. Quantum advantage
So far, we have reminded the reader how the classical accessibility region C Mem γ (p) is characterised in terms of thermo-majorisation [13] and obtained a characterisation of the memoryless accessibility region C γ (p) in terms of a Markov thermo-majorisation condition. We also observed that C γ (p) ⊂ C Mem γ (p). A natural question that arises then is whether quantum dynamics provides any advantage, i.e., whether the sets Q γ (p) and Q Mem γ (p) are larger than their classical counterparts.
It is straightforward to prove that without the memoryless constraint there will be no quantum advantage. In other words, we have q ∈ Q Mem γ (p) if and only if q ∈ C Mem γ (p). The "if" part is obvious, as the set of all quantum channels with a fixed point ρ γ contains as a subset the set of classical stochastic processes with the same fixed point. Now, assume there exists q ∈ Q Mem γ (p). This mean that there exists a channel E such that E(ρ p ) = ρ q and E(ρ γ ) = ρ γ . Then we can construct a stochastic process P with matrix elements P i|j given by i| E (|j j|) |i . Matrix P is stochastic because E is positive and trace-preserving. Furthermore, it satisfies P p = q and P γ = γ. Therefore, q ∈ C Mem γ (p). However, as we will now prove, a quantum advantage is exhibited by Q γ (p) ⊃ C γ (p), i.e., there are states classically accessible only with memory that can be achieved by quantum memoryless dynamics. First, in the case of a uniform fixed point, going from classical to quantum memoryless dynamics allows one to achieve the maximal quantum advantage: all transformations involving memory can be realised quantum mechanically with no memory.
Proof. Assume that q ∈ C Mem η (p), which by Theorem 5 means p q. We will show that q can be achieved from p by a composition of two quantum embeddable processes (so, according to Lemma 1, by a quantum-embeddable process). First, note that every permutation is quantumembeddable, as discussed in Sec. II C. Thus, one can rearrange p into p with p p and sorted in the same way as q. By transitivity of majorisation we have p q and then, by Corollary 10 from Appendix C, we obtain p Ï q. Now, Theorem 6 implies that there exists a classically (and so, quantum) embeddable stochastic matrix with a fixed point η mapping p to q. Thus, q ∈ Q η (p).
Conversely, if q ∈ Q η (p), there is a unital quantum channel E mapping ρ p to ρ q . This implies that the eigenvalues of ρ p majorise those of ρ q [50,51], which means p q and thus, by Theorem 5, q ∈ C Mem η (p).
Next, we analyse the quantum advantage in the scenario with a non-uniform fixed point γ. The first step here would be to characterise the set of diagonal quantum states achievable from a given state ρ p via Markovian quantum master equations with a given fixed point ρ γ . However, even without the constraint that the channel is generated by a master equation, finding a simple characterisation of the set of accessible states for d > 2 has remained an open problem for decades [52]. Therefore, here we will focus on the simplest non-trivial case of a qubit system, where such problem has been fully solved [52][53][54]. We will numerically show that in d = 2 one also achieves a maximal quantum advantage.

Result 1 (Numerics). For
The above result shows that for a two-dimensional classical system all thermodynamic state transformations involving memory can be realised quantum mechanically by a Markov master equation. This showcases that the advantage of Theorem 7 is not limited to the special case of a uniform fixed point. Superposition can substitute memory in the control of classical systems at every finite temperature.
Before we formally prove the result let us discuss some consequences. Any classical Markovian master equation with a fixed point γ evolves p along the path p(t) that can never go "on the other side of the fixed point" (recall Fig. 3): memory is required for that to happen. Instead, the corresponding quantum Markovian master equations access all states achievable under general stochastic maps with fixed point γ. Creation of quantum coherence is crucial since it opens new pathways that "go around" the fixed state. What is surprising is that, in d = 2, the creation of coherence in a Markovian dissipative process can replace all memory effects. Even the "β-swap", the classical process with a thermal fixed point which requires the largest free energy back-flow and achieves the farthest accessible state on the other side of the fixed point γ, can be approximated arbitrarily well by a quantum Markovian master equation with a thermal fixed point. Hence, the optimal heat bath algorithmic cooling protocol derived in Ref. [16], which requires β-swaps and hence classical control over memory effects, can be realised by a Markovian master equation with a thermal fixed point. We expect this phenomenon to be relevant also for higher dimensional systems, since it is based on the following general behaviour. Suppose for simplicity that there are no degeneracies in the Bohr spectrum of the system, i.e., the allowed energy differences, {E i − E j } i =j , for the studied system are all distinct. Given a quantum Markovian evolution ρ(t), decompose the state as where r(t) is the population in the energy basis and C(t) are the off-diagonal terms ("coherence") at time t. Any classical Markov evolution with a thermal fixed point requires d dt F (r(t)) ≤ 0 and C(t) = 0 at all t ≥ 0. Any quantum Markovian dynamics requires d dt F Q (ρ(t)) ≤ 0 for all t ≥ 0, where F Q is the quantum non-equilibrium free energy:

F (r(t))
A(ρ(t)) β t t * FIG. 7. Free energy stored in coherence. Under a Markovian master equation with a thermal fixed point, the quantum free energy F Q is monotonically decreasing in time. Since F Q = F + A/β, part of the classical component F can be stored in the coherent component A at times t ≤ t * and recovered later. At time t = t * the population is thermal, so the classical free energy is at a minimum, but A = 0 so F Q is above the minimum. For t ≥ t * part of this coherent free energy is converted back into classical free energy. Hence, the latter undergoes a backflow which classically would require memory effects.
with H S denoting the Hamiltonian of the system and S(ρ) = −Tr (ρ log ρ) being the von Neumann entropy. Recall that F Q (ρ) can be additively decomposed into two non negative components [15]: The first is the (classical) non-equilibrium free energy and the second is a quantum component (called "asymmetry"), which measures the coherent contribution to F Q [55]. At t = 0 we have r(0) = p and C(0) = 0, which implies A(ρ(0)) = 0. Hence, both classical and quantum free energies for the initial state are equal to F (p). However, a Markovian quantum evolution can store some free energy in coherence at times t ≥ 0, since only the sum of the classical and quantum components of the free energy must monotonically decrease in time. This way, at time t * when r(t * ) = γ, classically one is stuck in a free energy minimum F (γ) and cannot proceed further. But quantum mechanically one can have C(t * ) = 0 and hence F Q (ρ(t * )) > F (γ). A Markovian quantum dynamics can hence access other states at t > t * by converting back some of quantum component of the free energy into classical free energy. This allows one to achieve the required backflow in the classical component of the free energy, see Fig. 7. Storing free energy in coherence is of course a nontrivial task (it requires the aid of an external source of coherence [55]), however here we showed that for a single qubit it can be done with a Markovian master equation.
In order to prove Result 1, we present an explicit construction and numerical evidence for an even stronger result.
Result 2 (Numerics). Every qubit state accessible via a qubit channel with given fixed point can be achieved by a qubit Markovian master equation with the same fixed point.
Let us start by recalling the result of Ref. [52], where the authors provided necessary and sufficient conditions for the existence of a qubit channel E satisfying: for any two pairs of qubit density matrices (ρ, ρ ) and (σ, σ ). Moreover, whenever such a channel exists, the authors provided a construction of the Kraus operators of E. Setting σ = σ = ρ γ one obtains a characterisation of all states accessibile from ρ through arbitrary channels with a given fixed point ρ γ (we choose a basis in which the fixed point is diagonal). In Ref. [54] the continuous set of conditions presented in Ref. [52] was reduced to just two inequalities: These are best understood through the standard Bloch sphere parametrisation of the states involved. Recall that a general qubit state can be written as where σ denotes the vector of Pauli matrices (σ x , σ y , σ z ), while r ρ is a 3-dimensional real vector which uniquely represents ρ as a point inside a unit Bloch ball in R 3 . We parametrise the initial, final and fixed point as follows: r ρ = (x, y, z), r ρ = (x , y , z ), r γ = (0, 0, ζ). (42) Unitary rotations about the z axis leave ρ γ unchanged. By performing such rotations before and after the channel E, without loss of generality we can set x ≥ 0, x ≥ 0 and y = y = 0. The monotones R ± from Eq. (40) are then defined as [54]: where with analogous (primed) definitions for ρ . The two inequalities from Eq. (40) can be then used to find extremal states accessible from ρ via qubit channels with fixed point ρ γ . As shown in Fig. 8, these are given by: • States with a constant R + lying on a circle (c 0 , R 0 ) if z ≥ z, where • States with a constant R − lying on a circle (c 1 , R 1 ) if z < z, where Here, the parameters for initial and fixed states are chosen to be x = 1/2, z = 0 and ζ = 1/4.
The crucial observation we make here is as follows. Consider the case z ≥ z. Divide the extremal path into n parts by choosing states ρ 0 , . . . , ρ n along the (c 0 , R 0 ) circle with ρ 0 = ρ. Since Eq. (40) is satisfied, for each i ∈ {0, . . . , n − 1} there exists E i with E i (ρ i ) = ρ i+1 and E i (ρ γ ) = ρ γ . Similar considerations hold for z < z considering the (c 1 , R 1 ) circle. This suggests that there indeed exists a continuous Markov evolution that evolves the state along the extremal path.
To construct a time-dependent Lindbladian that evolves the state along the extremal path (say, the one with z ≥ z) we fix some arbitrarily small ∆ > 0 and find the state ρ 1 on the extremal path with z = z + ∆. Using the construction of Ref. [52] we obtain an explicit form for the quantum channel E 0 mapping ρ 0 to ρ 1 , while preserving ρ γ . Next, we define the Lindbladian L 0 = E 0 − I and evolve the state according to e L0 , obtainingρ 1 := e L0 ρ 0 . We then repeat the same procedure, but instead of ρ 0 we start withρ i for i > 0. In this way we construct a whole set of Lindbladians L i . The procedure ends when Eq. (40) is no more satisfied for z = z + ∆. Due to the extremely complicated form of the Kraus operators describing the channels E i (and hence L i ), instead of their explicit expressions we provide their construction in Appendix D.
We have thus constructed a quantum Markovian evolution i e Li passing through the pointsρ i . Numerical investigations show that this Markovian dynamics evolves ρ 0 = ρ approximately along the extremal circle (c 0 , R 0 ) (or (c 1 , R 1 ) for ∆ < 0), with the approximation improving as ∆ → 0. We illustrate these results for particular choices of initial and fixed states in Fig. 9, and note that this is a strong evidence that Result 2 holds.

V. OUTLOOK
The central task of this paper was to prove that quantum dynamics offers memory improvements over the classical stochastic evolution in a variety of settings, from the more computational ones to the more physical ones. The unifying notion is that of the underlying Markovian master equation and the advantages one gains with quantum controls as compared to the classical ones. The driving force behind these advantages is the superposition principle, which provides a wider arena for memoryless evolutions to unfold. While Holevo's theorem [2] prevents us from retrieving more than n bits of information from n qubits, we see that in many other respects superpositions can take over the role played by a classical memory. This is most clearly captured by the notion of quantum embeddable stochastic processes introduced here: processes which do not require any memory quantum mechanically but they do classically. We found several classes of such processes, but the full characterisation is left as a big open problem for future research. It may be especially hard taking into account that the classical version of the problem is still unsolved for d > 3, however recent progress on accessibility of quantum channels via the Lindblad semigroup [56] is promising. Moreover, one may still hope for a partial characterisation, e.g., it would be of particular interest to identify the outer limits to the quantum advantage by means of necessary conditions for quantum embeddability.
We have also proved the quantum advantage in terms of memory and time-step cost of implementing a given stochastic process. By means of computational basis input states and measurements, a quantum Markovian master equation on a (d+m Q )-dimensional quantum system realises a d×d stochastic process P in τ Q time-steps. We compared (m Q , τ Q ) with the minimal (m C , τ C ) required in order to simulate the same P through any classical Markovian master equation. When P is deterministic, i.e., it is a function over a discrete state space, we saw that typically one has an unbounded gap between (m Q , τ Q ) and (m C , τ C ).
From a technological perspective, our investigations lead to the question whether these in-principle simulation advantages translate into practical ones. There is a long history, dating back to Landauer, of associating a cost to erasure [57], due to the unavoidable dissipation of entropy required for the implementation of such processes. Notably, these costs are common to classical and quantum scenarios. On the other hand, we showed here that the memory and time-step costs of computations under memoryless quantum dynamics are typically much lower than the corresponding classical costs. If these savings make up for the challenge of controlling quantum, rather than classical, degrees of freedom is an intriguing open question.
Our results also open several new directions in the realm of quantum thermodynamics. Our proof that qubit systems can be quantum mechanically cooled below the environmental temperature using neither memory effects nor ancillas (something that is impossible classically) suggests that we should look for realistic Markovian master equations in which this phenomenon can be observed. Moreover, our study of memoryless thermalisations has shown that the standard partial thermalisation model provides the same amount of control as the most general classical Markovian master equation with a thermal fixed point. Furthermore, it also gave necessary and sufficient conditions for a thermodynamic transformation to occur through a Markovian process. This strengthens the so-called "second laws" of Ref. [13], enforcing on them the often realistic constraint of Markovianity. It also relates them to previously disconnected thermodynamic frameworks (such as stochastic thermodynamics) which take Markovianity as a basic underlying assumption. Thus, our results should find applications in addressing the questions of thermodynamic control, and could bring the abstract resource-theoretic approach to quantum thermodynamics closer to realistic setups. 2018/MAB/5) and through TEAM-NET project (contract no. POIR.04.04.00-00-17C1/18-00 Proof. Given a function f : Z d → Z d let us denote the size of the image of f by r = |img(f )|. Next, we denote the elements of img(f ) by {y k } r k=1 , and the remaining elements belonging to Z d \ img(f ) by {y k } d k=r+1 . Moreover, for each y k with k ≤ r, i.e., for each of the r elements of the image of f , let us denote the corresponding pre-image as follows: Note that the sets {x k j } d k j=1 are disjoint and that their union is the full set Z d . Now, we will construct a permutation function f π and an idempotent function f I , both mapping Z d to Z d and such that First, f π is defined by where the convention is that 0 l=1 ≡ 0. Then, introducing n s := r + s−1 l=1 (d l − 1), the idempotent map f I is given by f I (y k ) = y k for k ≤ r, y s for k ∈ {n s + 1, n s + d s − 1}.
With the above definitions, it is a straightforward calculation to show that Eq. (B2) holds. Finally, we need to show that there exist timeindependent Lindbladians L π and L I that generate P fπ and P f I , i.e., {0, 1}-valued stochastic matrices realising functions f π and f I , respectively. This way, by Definition 6, we will prove that any function f can be realised quantumly without the use of memory and in at most 2 time-steps. First, since P fπ is a permutation, its generator L π exists and is simply given by the commutator with the Hamiltonian (see discussion in Sec. II C). Now, in the case of f I , notice that it is a function sending r disjoint sets Y k of size d k , to a single element y k of the given set Y k . This mapping can be easily realised for t f → ∞ by a classical generator L (so also by the corresponding quantum Lindbladian) given by L y k |y l =    −1 for k = l and l > r, 1 for k = f I (l) and l > r, 0 otherwise.

(B6)
Appendix C: Proof of Theorem 6 Before we present the proof of Theorem 6, we first build up several partial results. We start by recalling an important result derived in Ref. [44].
Theorem 8 (Theorem 12, Supplementary Material of Ref. [44]). If p γ q and π(p) = π(q), there exists a sequence of two-level partial thermalizations Moreover, all intermediate states have the same thermomajorisation order and f ≤ d − 1.
Next, we link continuous thermo-majorisation between two distributions with the existence of a sequence of twolevel partial thermalisations bringing one distribution to another.
Lemma 9 (Continuous thermo-majorisation and two-level partial thermalizations). We have p Ï γ q if and only if there exists a finite sequence of two-level partial thermalizations {T i k ,j k (λ k )} f k=1 such that Proof. First, assume p Ï γ q. Then, there exists a continuous trajectory s(t) with s(0) = p, s(t f ) = q (perhaps t f = +∞), and s(t ) γ s(t ) for all t ≤ t . Define t 0 = 0 as well as the thermo-majorisation ordering π 1 and a time t 1 as follows: π 1 :=π(s(0)), (C3) t 1 := sup{t|π(s(t)) = π 1 }. (C4) Next, for k ≥ 1 define iteratively Clearly t k+1 > t k . Since there are only d! distinct thermo-majorisation orderings, ultimately we reach the final k = f ≤ d! − 2, such that π f = π(s(t f )). We will now employ Theorem 8: for each pair, s(t k ) and s(t k+1 ), there exists a sequence of two-level partial thermalizations such that with n ≤ d − 1. Thus, by sequentially applying the above to all k ≤ f we obtain Eq. (C2).
Conversely, assume that Eq. (C2) holds. Define s(0) = p and This defines a continuous path starting at p and terminating at q. Moreover, using the fact that for λ ≥ λ we can write T i,j (λ ) = T i,j (µ)T i,j (λ) with µ ∈ [0, 1], we see that for any t ≥ t the distribution s(t ) is obtained from s(t ) by a finite sequence of partial level thermalisations. As partial level thermalisations are stochastic matrices with a fixed point γ, Theorem 5 implies s(t ) γ s(t ) for all t ≥ t . We thus conclude that p Ï γ q.
Note that from the proof above one can conclude that the number of two-level partial thermalizations required for a state transformation is upper-bounded by (d! − 1)(d − 1), but we will give a tighter bound later. Also, as a corollary of Lemma 9 we get that γ (describing allowed transformations with memory) and Ï γ (describing allowed transformations without memory) coincide within a fixed thermo-majorisation ordering.
Proof. The implication p Ï γ q ⇒ p γ q holds trivially. Thus, assume that p γ q and π(p) = π(q). Theorem 8 tells us then that there exists a sequence of two-level partial thermalisations mapping p into q. Using Lemma 9, we conclude p Ï γ q.
The next lemma geometrically characterises the action of a two-level partial thermalisation on a thermomajorisation curve (discussed in Sec. IIIA of Ref. [44]). In words, it shows that the effect of T i,j is to decrease the slope of the j th segment of the thermo-majorisation curve and increase that of the i th segment till the two are equalised. Lemma 11 (Action of two-level partial thermalisation on the thermo-majorisation curve). Given a thermomajorisation curve, let i label its defining points (sorted from left to right) and y i denote the corresponding ycoordinates, see Fig. 10. Given i, j with i < j, let T i,j (λ) be a two-level partial thermalisation (order the labels within the constant slope segments so that points i and j are as close as possible). The action of T i,j (λ) shifts down by an equal amount the y-coordinates (y i , . . . , y j−1 ). The extremal map, T i,j (1) = G i,j , equalises the slopes of the i th and the j th segment of the curve. Note that a final reordering may be needed if the thermo-majorisation ordering is changed.
Proof. For p = T i,j (λ)p we have This corresponds to shifting down the y-coordinate of each point of the thermo-majorization curve, starting from the i th point to the (j − 1) th point. Setting λ = 1 one obtains that the slope of the j th segment is Similarly, the slope of the i th segment is Hence, the two slopes are equalised for λ = 1. Note that, if j = i + 1, then the thermo-majorisation ordering will change at some intermediate λ, so that a rearrangement of the segments is necessary to sort them according to non-increasing slopes.
Let us now split the probability simplex into closed subsets with a fixed thermo-majorisation ordering: R π = {p|π(p) = π}, with π a permutation of {1, . . . , d}. Some of these sets overlap at some of the boundaries defined as: ∂R h1,h2 π = {p ∈ R π and p h1 /γ h1 = p h2 /γ h2 }. (C9) The next lemma shows the optimal way of crossing from one subset R π to an adjacent R π via a Markovian stochastic process with a fixed point γ. More precisely, for p ∈ R π it identifies G h1,h2 p as the "best" state at the boundary ∂R h1,h2 p to q. Since (4) gives a sequence of at most (d! − 1) twolevel partial thermalisations mapping p top, we conclude that there is a sequence of at most (d! + d − 2) two-level partial thermalisations mapping p to q.
(3) ⇒ (1) It follows by definition of C γ and the fact that partial level thermalisations are embeddable stochastic matrices, and are thus generated by a master equation dynamics with L(t)γ = 0 for all t.
(1) ⇒ (2) Given p and q, let s(t) be the trajectory defined by the master equation described in the definition of C γ , i.e. s(t) = P (t)p. By assumption, the process is infinitely divisible, meaning that for every 0 ≤ t ≤ t ≤ t f there exists a stochastic matrix T (t , t ) such that T (t , t )s(t ) = s(t ), T (t , t )γ = γ. (C10) From Theorem 5, it follows that s(t ) γ s(t ). Since s(0) = p, s(t f ) = q, it follows that p Ï γ q.

Appendix D: Extremal path
Consider the initial qubit state ρ described by the Bloch vector (x, 0, z), and a fixed state ρ γ with the Bloch vector (0, 0, ζ). Here, we will show how to construct quantum channels E 0 and E 1 with a fixed point ρ γ , and evolving ρ along the extremal circles (c 0 , R 0 ) and (c 1 , R 1 ), as derived in Ref. [54] and described in Sec. IV C. More precisely, for a given ∆ > 0 we look for E 0 that evolves ρ to ρ with Similarly, for a given ∆ > 0 we look for E 1 that evolves ρ to ρ with Note that in both cases there is a maximal value of ∆ for x to stay real, and we assume that ∆ is below that maximal value (otherwise the map we are looking for does not exist). Below, we will explain how to construct Kraus operators {A i , B i , C i } for E i , so that The construction is based on the general construction for channels mapping between pairs of qubit states provided by Alberti and Uhlmann in Ref. [52]. The first step is to define the following projectors: The above four projectors are used to define four unitary matrices, with i ∈ {0, 1} and analogous primed definition for U i . These are then employed to define four rotated fixed states Γ i := U i ρ γ U † i and analogously for Γ i . Let us parametrise these states as follows: These eight parameters are then used to calculate the following eight new parameters: These, in turn, allow us to introduce the following operators: which after unitary rotations yield the final Kraus operators we are looking for: [1] Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden, "Quantum cryptography," Rev. Mod.