When is a non-Markovian quantum process classical?

More than 90 years after the inception of quantum theory, the question of which traits and phenomena are fundamentally quantum remains under debate. Here, we comprehensively answer this question for temporal processes. Defining classical processes as those that can---in principle---be simulated by means of classical resources only, we fully characterize the set of such processes. Based on this characterization, we show that for non-Markovian processes (i.e., processes with memory), the absence of coherence does not guarantee the classicality of observed phenomena. We then provide a direct connection between classicality and the vanishing of quantum discord between the evolving system and its environment, thus establishing a clear link between discord---a trait that is considered to be exclusively quantum---and the non-classicality of observable phenomena in temporal processes.


I. INTRODUCTION
Coherence is considered one of the fundamental traits that distinguishes quantum from classical mechanics [1][2][3]. Beyond its mathematical deviation from classical theory, it plays an important role in the enhancement of quantum metrology tasks [4,5], constitutes a fundamental requirement for many quantum algorithms [6,7], and has been conjectured to be crucial for the efficiency of biological transport phenomena [8][9][10]. Consequently, the resource theory of coherence [11][12][13][14][15][16][17][18][19] has been of tremendous interest in recent years, and has seen rapid development both on the theoretical as well as the experimental side [20].
Despite such progress and the growing wealth of accompanying evidence that links coherence to non-classical phenomena, the explicit connection between the two remains unclear and subject to active debate [21][22][23][24][25]. Put differently, the mere presence of coherence does not guarantee the existence of effects that cannot be explained on purely classical grounds, and an unambiguous relationship between coherence and non-classicality has not been established yet.
In order to provide such a connection, an operationally meaningful and clear-cut definition of classicality is crucial.
One such possible definition is based on experimentally attainable properties only: the joint probability distributions obtained from sequential measurements of an observable [26]: if these satisfy the so-called Kolmogorov consistency conditions-which are the starting point for the theory of classical stochastic processes [27,28]-then they can, in principle, be explained by a fully classical model and there is therefore nothing inherently quantum about the observed phenomenon. If they do not, then there exists no underlying classical stochastic process that could lead to the observed joint probability distributions, and the corresponding process is considered non-classical. This characterization of classicality is in the spirit of the derivation of Leggett-Garg inequalities, where, instead of classicality, non-invasiveness and macroscopic realism are put to the test [29,30]. Indeed, any set of probability distributions that satisfies the Kolmogorov conditions does not violate the corresponding Leggett-Garg inequalities [31,32].
Following this line of reasoning, in Ref. [33] a one-to-one connection was derived between this notion of classicality and the coherence properties of the dynamics of Markovian (i.e., memoryless) quantum processes: a process is classical iff the corresponding dynamical propagators can never create coherence that can be detected at any later time. Thus, a direct relation between the mathematical notion of coherence and an operationally well-defined and broadly applicable notion of classicality has been established. However, this connection only holds in the memoryless case and does not straightforwardly apply to the non-Markovian scenario, where, amongst others, such propagators cannot be used to compute multi-time statistics [34].
Such more general non-Markovian processes can be described in terms of higher-order quantum maps, so-called quantum combs [35,36].
Recently, this framework has been tailored to the description of open quantum system dynamics [37,38], and has-amongst others-found direct application in the characterization of multi-time memory effects [39,40] and the field of stochastic thermodynamics [41][42][43].
Here, we employ it to extend the results of Ref. [33] to the non-Markovian case. In particular, we provide a clear link between spatial quantum correlations and the non-classicality of observed measurement statistics. Somewhat surprisingly, for the case of general processes-where memory effects play a non-negligible role-the presence of non-classical phenomena is not solely dependent on the creation or detectability of coherence, in stark contrast to the memoryless case. As we will show, the absence of detectable coherence is not necessarily sufficient to enforce classical behaviour in general. Rather, classicality of multi-time statistics is inherently linked to quantum discord-which was originally introduced as a means to distinguish between classical and non-classical spatial correlations [44][45][46]-between the evolving system and its environment. We characterize the complete set of classical processes and derive a concrete relation between the presence and detectability of discord and the non-classicality of observed measurement statistics. This, in turn, allows for a comprehensive categorization of the resources required for the implementation of a non-classical, non-Markovian process, paving the way to a clear-cut understanding of non-classicality on operational grounds. To achieve these results, in Sec. II we first introduce the basic concepts that will be employed throughout this article to examine classicality. In Sec. III, we reiterate and slightly generalize the results of Ref. [33] for the Markovian case, and discuss their breakdown when memory effects are present. This motivates our consideration of the non-Markovian case in Sec. IV, where we fully characterize the set of general classical processes by means of the quantum comb framework. Finally, based on these results, in Sec. V, we establish the direct connection between quantum discord and the classicality of temporal processes. The paper concludes in Sec. VI.

II. GENERAL FRAMEWORK
The overarching aim of this paper is to characterize when a general quantum mechanical process can be considered classical in an operationally consistent manner and identify the structural properties consequently implied. Importantly, our investigation will be operational in the sense that it is based solely on experimentally accessible quantities; as such, it applies to situations where the underlying theory is classical mechanics, quantum mechanics, or some more general theory [47].
Ultimately, any physical theory provides predictions about possible observations-only these can be tested by experiments. That is, any theory must (in principle) provide the correct probabilities for measurement outcomes (or sequences thereof) to occur when a system of interest is experimentally probed.
The fundamental difference of predictions made about these observable quantities by classical physics and quantum or post-quantum theory can then be used as an unambiguous demarcation line between the different theories.
Following Ref. [33], we will thus define our notion of classicality by means of joint probability distributions pertaining to sequences of measurement outcomes, as these are precisely what is obtained when a temporal process is probed.

A. Kolmogorov conditions and classicality
In classical physics, a stochastic process on a set of K times is fully described by a joint probability distribution that yields the probability to measure the realizations {x K , . . . , x 1 } of the random variables {X K , . . . , X 1 } at times {t K , . . . , t 1 }. For example, P 2 (x 2 , t 2 ; x 1 , t 1 ) could describe the probability to obtain outcomes {x 2 , x 1 } when measuring the position of a particle undergoing Brownian motion at times t 2 > t 1 .
In what follows, we will often omit the explicit time label, with the understanding that x j denotes an outcome of a measurement at time t j .
Crucially, in classical physics, joint probability distributions describing a stochastic process for different sets of times satisfy the so-called Kolmogorov conditions [27,28,48,49]: given a joint probability distribution P K for a set of times, all probability distributions for all subsets of times can be obtained by marginalization, that is P n−1 (x n , t n ; . . . ;¨ẍ j , t j ; . . . ; x 1 , t 1 ) (2) = xj P n (x n , t n ; . . . ; x j , t j ; . . . ; x 1 , t 1 ) ∀n ≤ K, ∀j .
Notably, just like the Leggett-Garg inequalities [29][30][31] for temporal correlations, the satisfaction of these requirements is based on the assumptions of realism per se and non-invasive measurements [50]. An experimenter obtaining a family of joint probability distributions that satisfies the Kolmogorov conditions when probing a temporal process at different sets of times would not be able to distinguish said process from a classical one, as every such finite family can be obtained from a-potentially exotic-underlying classical stochastic process.
More generally, the Kolmogorov extension theorem states that if all joint probability distributions for finite subsets of a time interval [0, t] satisfy the consistency conditions of Eq. (2) amongst each other, then there exists an underlying classical stochastic process on said time interval that leads to the observed probability distributions [27,28,48,49]. We therefore define: Def. 1 (K-classical process [33]). Let X be a finite set. A process defined on a set of times T , with |T | = K, that is described by the joint probabilities P n (x n , t n ; . . . ; x 1 , t 1 ), with t n ≥ · · · ≥ t 1 , t i ∈ T , n ≤ K and x i ∈ X , is said to be K-classical if the Kolmogorov consistency conditions of Eq. (2) are satisfied up to n = K.
Throughout this article, we will call a family of joint probabilities on a set of K times (and all subsets thereof) a K-process.
While the above definition of classicality seems intuitive, some comments are in order. First, we choose to define classicality for finite sets of K times, as, in practice, an experimenter can only probe a process at a finite number of times. Second, the classicality of a process according to the above definition depends on the manner in which the system of interest is probed. This is also the case in classical physics: given some underlying classical stochastic process, not every set of measurements that an experimenter might be able to perform on the system of interest will lead to a family of probability distributions that satisfies the above definition of K-classicality. In fact, if performing such measurements might potentially disturb the system (i.e., the measurement is invasive), the Kolmogorov condition fails in general, even if the underlying evolution is classical [50].
For example, suppose that instead of merely measuring the position of a particle at different times when probing a Brownian motion process, an experimenter chooses to displace the particle at each time depending on where it was found. In this case, Eq. (2) would generally fail to hold for the joint probability distributions observed. Consequently, the Kolmogorov consistency conditions in Eq. (2) are in fact a statement of the non-invasiveness of the performed measurements: if they hold true, then not performing a measurement cannot be distinguished (for the given experimental situation) from averaging over their probabilities (i.e., forgetting the outcomes of the measurements performed).
In classical physics one assumes that, in principle, one could measure the system without disturbing it, and that therefore there exists a family of joint probability distributions that can consistently explain all possible outcome probabilities. Such a non-invasive and complete measurement is often referred to as "ideal measurement" in the literature [51].
On the other hand, in quantum mechanics any measurement disturbs some system state and therefore ideal measurements do not exist in general in the strong sense discussed above.
As a consequence, quantum mechanical processes generically do not satisfy Kolmogorov conditions [50,52], a fact that fundamentally distinguishes them from the classical realm.
More generally, the violation of Bell, Kochen-Specker, or Leggett-Garg inequalities, which can be observed in quantum mechanics, precludes the possibility that the measured data was obtained by non-invasive measurements. Particularly, in the case of Leggett-Garg inequalities [29,53], it is precisely the breakdown of Figure 1. Probing a process with projective measurements. At each time tj, the process (depicted in blue) is probed by a projective measurement (depicted in green, respectively) with outcomes xj, where each xj belongs to the same finite set X . If the resulting family of probability distributions Pn (depicted is the cases n ≤ 4) satisfies Kolmogorov consistency conditions, then not performing an experiment at a time tj cannot be distinguished from performing a measurement and averaging over the outcomes. In this case, this experiment cannot be distinguished from a classical one, even though the underlying process might be quantum mechanical.
Kolmogorov conditions that is being probed [33,50], and our above definition of classicality is hence in line with the wider program of determining fundamentally quantum traits of nature.

B. Measurement setup
As mentioned above, the structural properties of families of joint probability distributions depend on the way in which a system of interest is probed. Consequently, before being able to analyze the set of genuinely quantum process, it is crucial to fix the measurements that are used to probe a process at hand. Even though there are no ideal measurements in quantum mechanics, projective measurements share some basic features with the classical ideal measurements discussed above, and are thus a natural choice. In particular, they guarantee repeatability, i.e., that two sequential measurements (without any evolution in between) would give the same value with unit probability, as well as a weaker form of ideality, namely that if an outcome occurs with certainty, then the state of the system before the measurement is not disturbed by the latter [54]. It is therefore natural to start our analysis on the classical reproducibility of quantum processes by focusing on projective measurements; moreover, also following Ref. [33], we will further restrict to the case of orthogonal rank-1 projectors, like, e.g., projective measurements with respect to the eigenbasis of a non-degenerate self-adjoint operator.
In many experimental situations of interest, there is a natural basis to select. For instance, if the dynamics is such that the system dephases to a given basis, the latter provides a natural choice. In other cases it may make sense to choose the basis more arbitrarily (in advance), for instance when analyzing a specific protocol, or attempting to optimize it (see Ref. [55] for more details). Finally, the experimental setup might only allow for measurement of one particular observable, in which case the chosen basis would correspond to the eigenbasis of said observable.
In what follows, we will analyse the classicality of a process based on the joint probability distributions obtained from sequential sharp measurements in a fixed basis {|x } d x=1 -henceforth also called the classical, standard, or computational basis-with the action of a measurement with outcome x on a state ρ given by See Fig. 1 for a graphical representation. This freedom in the considered measurements makes the property of classicality fundamentally contingent on the respective choice of measurement basis. However, this basis dependence is somewhat unsurprising and mirrored by coherence theory [2]. There, the existence of off-diagonal elements m|ρ|n , i.e., coherences, depends on the choice of the basis a quantum state is represented in. As they are considered to be a fundamentally quantum property, it is a natural question to ask how coherences (with respect to the computational basis) and classicality of a process (with respect to the same basis) are interrelated. Importantly, while the existence of coherences cannot be determined by projective measurements in the computational basis alone, the prevalence of non-classical effects can be. Thus, as we shall see below, providing an operationally accessible notion of classicality allows one to link coherence (and, more generally, quantum correlations) in a clear-cut way to experimentally observable deviations from classical physics.

C. Open (quantum) system dynamics and memory effects
The definition of classicality we use (introduced in Ref. [33]) answers the question of whether or not there is a classical stochastic process that can explain the multi-time probabilities obtained by measuring a quantum system at given times in the computational basis. To make our analysis as general as possible, we will consider the possibility that the measured system interacts with a surrounding environment, which can influence the resulting statistics. Explicitly, assuming that, together, system and environment are closed and are described by quantum mechanics, their dynamics between measurements is given by unitary evolutions: U tj+1,tj [η] = U tj+1,tj η U † tj+1,tj , and the resulting joint probability distribution observed reads where η se t1 is the system-environment state at time t 1 , I e signifies the identity channel on the environment and P xj corresponds to a measurement in the computational basis at time t j with outcome x j (see Fig. 2 for a graphical representation.). Naturally, the classicality of the family of joint probability distributions obtained via Eq. (4) crucially depends on the properties of the intermediate evolution and the initial state.
In general, such a multi-time statistics display memory effects, i.e., they are non-Markovian: at any point in time t j , the future statistics does not only depend on the measurement outcome x j at time t j , but also (potentially) on all previous outcomes Indeed, all information about future statistics at t j is contained in the joint state of system and environment, which depends upon the previous measurement outcomes. As this total state cannot be accessed by measurements on the system alone, this dependence on past measurements manifests itself as memory effects on the system level (see Sec. IV for a detailed discussion).
However, under some specific circumstances, such memory effects on the multi-time statistics can be neglected; this is essentially the case when the quantum regression formula (QRF) can be applied [28,[56][57][58]. Under this assumption, the observed statistics can be understood in terms of dynamical propagators that act on the system alone, which, in turn, enables one to directly link the classicality of a process to the properties of said propagators in terms of coherence production and detection. The corresponding result has been obtained in Ref. [33], and we will reiterate and expand it in the coming section. Subsequently, employing quantum combs-the appropriate framework for the description of general, possibly non-Markovian (open quantum system) processes-we comprehensively characterize the set of quantum processes which can be explained classically.

III. COHERENCE AND CLASSICALITY
In this section, we reiterate the main result of Ref. [33] on the connection between coherence and classicality for the memoryless case, generalizing it to the case of a divisible (but not necessarily semigroup [28,59,60]) dynamics. As mentioned above, such a direct connection may be established, because memoryless processes can be understood in terms of propagators that are defined on the system alone, while this property fails to hold in the general, non-Markovian, case.
After introducing an operational notion of Markovianity associated with the multi-time statistics due to sequential measurements of one (non-degenerate) observable, we present a one-to-one connection between the non-classicality of such statistics and the capability of the open system dynamics to generate and detect coherences in the relevant basis. We also clarify the relation between the notion of Markovianity used in this Figure 2. General open quantum process. The initial state of the system is correlated with the environment (depicted by the yellow triangle). Measurements on the system (green boxes) are performed at times t1, t2, . . . . In between, the system and the environment undergo a unitary evolution (blue boxes). The distinction between system and environment is given by the degrees of freedom that the experimenter controls (system) and those that they do not (environment). Figure 3. Markovian process. For a Markovian process, the dynamics in between times (depicted as the blue boxes) can be modeled by mutually independent maps Λt j+1 ,t j (i.e., there is no memory). The measurement statistics are obtained by measuring in the classical basis at times t1, t2, t3, . . . (depicted in green); before the first measurement the system is in the state ρt 1 (depicted in yellow) paper and the QRF, which allows us to straightforwardly recover the main result of Ref. [33]. Finally, we lay out the subtleties that arise when generalizing the framework to allow for memory effects.
A. One-to-one connection in the Markovian case Classically, a process is Markovian (i.e., memoryless), if, for any chosen time t j , the future statistics only depend upon the outcome at time t j , but not on any prior outcomes at t j−1 , t j−2 , · · · ; explicitly, a classical stochastic process is Markovian if its statistics satisfy P(x j |x j−1 , . . . , x 1 ) = P(x j |x j−1 ) ∀j, where P(x j |x j−1 , . . . , x 1 ) is the conditional probability to obtain outcome x j at time t j given that outcomes x j−1 , x j−2 , . . . were measured at earlier times t j−1 , t j−2 , . . . [28]. Extending this definition to general (i.e., not necessarily classical) statistics and taking into account that, in practice, one only deals with systems probed at a finite number of times, we obtain the following definition of K-Markovianity: Def. 2. Let X be a finite set. A process defined on a set of times T , with |T | = K is called K-Markovian if it satisfies: for all ordered tuples of times t k ≥ . . . ≥ t 1 , with t i ∈ T , and x i ∈ X .
Just like our earlier definition of classicality and coherence, the absence of memory effects as defined in Def. 2 is basis dependent: a process that appears Markovian in one basis may appear non-Markovian when probed in a different one. While there exists a basis independent notion of Markovianity in the quantum case [37,38,[61][62][63], the basis dependent one introduced here is best suited for the experimental situation we envision; as such, in what follows, we predominantly understand Markovianity with respect to measurements in the computational basis. We will briefly return to the relation between this basis dependent and the recently developed basis independent notion of Markovianity in the coming section.
To establish a connection between non-classicality of a Markovian process and the coherence properties of the underlying dynamics, we need to introduce the maps that characterize the evolution of the open system. To this end, assume that at an initial time t 0 (with t 0 ≤ t 1 ) the system and environment are in a product state η se t0 = ρ t0 ⊗σ t0 (for some fixed environment state σ t0 ), so that we can define the completely positive and trace preserving (CPTP) dynamical maps {Λ tj ,t0 } of the open system evolution between the initial time and the measurement times t j [28]¿ where tr e denotes the trace over the environmental degrees of freedom. Additionally, let us also assume that the dynamics is divisible [64], i.e, we can define the corresponding propagators {Λ t k ,tj } between any two times via the composition rule and they satisfy the composition law Λ t ,tj = Λ t ,t k • Λ t k ,tj for all times t ≥ t k ≥ t j . Under these assumptions, it is natural to ask, what properties the propagators {Λ t k ,tj } need to satisfy in order for the resulting statistics to be classical. However, Eq. (8) does not yet tell us how to obtain multi-time statistics.
Such a relation is provided by the QRF, which, for example, holds in the weak coupling and the singular coupling limits [65], and constitutes a relation between the definition of Markovian process given by Def. 2 and the corresponding open-system dynamics (see also [62] for an extensive discussion of the QRF and its generalizations). For the case of rank-1 projective measurements (in the computational basis), the QRF states that the multi-time probability distributions in Eq. (4) can be equivalently expressed by Importantly, this means that the full multi-time statistics can be obtained by means of maps that are independent of the respective previous measurement outcomes and which act on the system alone (see Fig. 3 for a graphical representation). It is straightforward to see that satisfaction of the QRF (9) indeed implies Markovian statistics in the sense of Eq. (6) and in particular we have the identity In other words, the action of the propagators on the populations (i.e., the diagonal terms of ρ tj , the state of the system at t j ) can be identified with the conditional probabilities between any two times. Crucially, this is not generally the case, and breaks down if the QRF cannot be applied [66]. More generally, even if the QRF applies, the composition rule on the level of propagators does not imply a composition rule on the level of the resulting measurement statistics, i.e., for a divisible process that satisfies the QRF, we generally have which captures the deviation of quantum Markovian processes from classical ones. As mentioned previously, in order for the resulting process to be classical, not performing a measurement has to be indistinguishable from performing a measurement and averaging over the outcomes. Put differently, for an observer that can only perform measurements in a fixed basis, the process is classical if they cannot detect the invasiveness of measurements in said basis. A measurement at time t j in the classical basis where the measurement outcomes are averaged over can be represented by the completely dephasing map The natural property of the propagators to look at in relation with classicality is thus the following: where I j and Λ j are the identity map and the completely dephasing map at time t j , respectively (see Fig. 4 for a graphical representation), and in the last line of Eq. (13) we used the composition law Λ tj+1,tj−1 = Λ tj+1,tj • Λ tj ,tj−1 ; Eq. (13) is for example satisfied if none of the maps {Λ tj+1,tj } create coherences. More generally, each of the maps in Eq. (13) can in in principle create coherences-as long as these coherences cannot be detected at the next time by means of measurements in the classical basis. Therefore, such a collection of maps satisfying Eq. (13) has been named not-coherence-generating-and-detecting (NCGD) [33]. The precise connection between NCGD and classicality is expressed by the following theorem: Then, the process is also K-classical (Def. 1) if and only if there exist a system state ρ t0 (at a time t 0 ≤ t 1 ) which is diagonal in the computational basis {|x } x∈X and a set of propagators Λ tj ,tj−1 j=1,...,K which are NCGD with respect to {|x } x∈X , such that ρ t0 and Λ tj ,tj−1 j=1,..., Proof. We first show that if a Markovian process can be reproduced by means of NCGD propagators {Λ tj+1,tj } and an initial diagonal state (both properties with respect to the computational basis), then it yields classical statistics. If the statistics is Markovian, then it follows from Eq. (6) that the joint probability distribution on any set of times t K , . . . , t 1 is given by As the process can-by assumption-be reproduced by the maps {Λ tj ,tj−1 } via Eq. (9) then for any time t j we have where we have set Π xj = |x j x j | and the NCGD property was used in the last line. This equation implies Moreover, the (initial) diagonal state ρ t0 guarantees that As a consequence of these two previous relations, the family of joint probability distributions computed via Eq. (14) satisfies Kolmogorov conditions, and is thus classical.
Conversely, if the process is classical and Markovian, Eq. (16) holds. We can then define the maps (18) and the initial diagonal state which also means that we identify the initial time as the time of the first measurement, t 1 = t 0 . The set of maps { Λ tj+1,tj } defined in this way, in conjunction with ρ t0 , reproduces the correct statistics via Eq. (9). As they are diagonal in the computational basis, they form an NCGD set.
Crucially, the connection between classicality and NCGD dynamics is one-to-one; if the obtained Markovian statistics cannot be reproduced by a set of maps that are NCGD, then they are non-classical. Before discussing classicality in the presence of memory effects, it is worth discussing the intuitive meaning of this theorem, and NCGD dynamics in particular.
If the process at hand is Markovian and classical, the maps { Λ tj+1,tj } (as well as the initial state ρ t0 ) introduced in the proof of Thm. 1 define an artificial reduced dynamics of the system, whose propagators correctly reproduce all joint probability distributions for measurements in the (fixed) classical basis via Eq. (9). Note that the actual propagators of the dynamics (i.e., those fixed by the unitary evolution in Eq. (4) via Eqs. (7) and (8)) might differ from the maps Λ tj+1,tj above (and ρ t0 might differ from the actual initial state ρ t0 ); indeed, the fact that they do not coincide is simply a manifestation of the basis dependence of the (sequential) measurement scheme we are focusing on here.
Crucially, a composition rule on the level of the actual propagators does not imply a composition rule on the level of the propagators of the populations. This implication only holds if the propagators of the dynamics are NCGD and the resulting statistics can be computed via Eq. (9), in which case Eq. (13) results in with see Eqs. (18) and (10). This composition law is then-as already mentioned-equivalent to the well-known classical Chapman-Kolmogorov equations which hold for classical Markovian processes; if the measurement statistics of a Markovian process can be reproduced by a set of NCGD maps {Λ tj ,tj−1 }, then it can also be reproduced by the set of maps { Λ tj ,tj−1 }, which act nontrivially only on the populations of the computational basis and satisfies a composition law, thus being classical.
Conversely, as we have seen, if the classical composition rule of Eq. (22) holds for a Markovian process, then there exists a set { Λ tj+1,tj } of propagators (e.g., those defined in Eq. (18)) that are NCGD and correctly reproduce all joint probability distributions for measurements in the (fixed) classical basis.
Thm 1 is a generalisation of the main result of Ref. [33] in two ways. First, it does not impose any restriction on the propagators of the underlying quantum evolution, while in Ref. [33] these were required to form a semigroup, i.e., Λ tj+1,tj = e L(tj+1−tj ) , for some Lindbladian L [59,60].
Second, the definition of Markovianity used here coincides with the standard definition of classical stochastic processes, whereas in Ref. [33] Eq. (9) (for semigroups) was used. Consequently, while the maps {Λ tj+1,tj } cannot be fully probed by measurements in the computational basis alone, the requirement of Eq. (22) can be tested for just by performing sequences of measurements in the classical basis at the relevant times, thus making our theorem fully operational. However, this comes at the cost of dealing with propagators { Λ tj+1,tj } which possibly do not correspond to those of the actual reduced dynamics.
On the other hand, as we show in Appendix A, a one-to-one correspondence between the dynamical propagators Λ tj+1,tj and the non-classicality of the multi-time statistics can be established also in the general (non-semigroup) divisible case, when the QRF applies, provided that one assumes a proper invertibility condition on the restriction of the dynamical maps to the populations of the computational basis. Indeed, this also allows one to recover in a straightforward way the main result of [33] as a corollary by further imposing the semigroup composition law.
Importantly, Thm. 1 comprehensively characterizes the connection between coherences and the classicality of a Markovian process. While it is not necessary that the underlying propagators do not create coherences in order for a Markovian process to be classical, it is necessary and sufficient that coherences-should they be created-cannot be detected at a later point in time by means of measurements in the computational basis. Put differently, the propagators must be such that a classical observer could not decide whether at any point in time an identity map or a completely dephasing map was performed. This requirement is exactly encapsulated in the NCGD property of the propagators.

B. Coherence in the non-Markovian case: preliminary analysis
The above connection between quantum coherence and non-classicality fails to hold in the non-Markovian case. On the one hand, in this case propagators between two times are no longer sufficient to fully characterize the multi-time statistics [67]. On the other hand, even if the state of the system is diagonal in the computational basis at all times, dephasing can still be invasive due to correlations with the environment, making the connection between coherences and the classicality of statistics less straightforward. We will discuss the former problem in the subsequent sections. An explicit ante litteram example of the latter case has already been provided in Ref. [33] (based on a model used in Refs. [61,68,69]), albeit not with an emphasis on the lack of coherence in the system state at all times (even in between the measurements). Here, we reiterate this example, focusing on the absence of coherences in the state of the system. The details of this discussion can be found in Appendices B and C. A simpler, although non-continuous, example for a non-Markovian process that yields non-classical statistics but never displays coherences in the system state is provided in Appendix D.
Example 1. Let the system of interest s consist of a qubit described by ρ s (t) which is coupled to a continuous degree of freedom p of the environment. The global dynamics of system and environment is governed by the unitary evolution U t2,t1 , acting as where {| } =0,1 is the eigenbasis of the system Pauli operatorσ z and φ = ±1.
The initial system-environment state is assumed to be of product form it is straightforward to show that, expressed in the eigenbasis ofσ z , the free open evolution of the state of the system (i.e., without intermediate measurements) is given by where and no coherence w.r.t.σ x will be generated if k(t) is a real function of time (as already noted in Ref. [33]); this is, e.g., the case if f (p) corresponds to a Lorentzian distribution centered around zero, A priori, the fact that there are noσ x -coherences created in the free evolution does not mean that none are created if the system is probed at intermediate times. However, here, noσ x -coherence is generated even when we take into account how the measurements modify the system's state. Specifically, immediately after a measurement in theσ x -basis, yielding outcome ±, is performed at time t 1 , the total state is of product form where ξ (±) (t 1 ) is a state of the environment that depends on the measurement outcome. As we show in Appendix B, any state of the system evolved from the post-measurement state of Eq. (28) according to the described dynamics remains diagonal in the {|± } basis; this also holds true for the state of the system after any sequence of such measurements. Together with the fact that the resulting statistics is non-classical (i.e., it does not satisfy Kolmogorov conditions, as has been shown in Ref. [33]), this constitutes an example of a non-classical process without any coherence with respect to the measured observable ever being generated. Evidently, this behaviour is only possible since the chosen example is non-Markovian [33].
Unlike in the Markovian case, where the absence of coherences trivially leads to classical statistics, when memory effects play a role, it is the coherences of the system state as well as the non-classical correlations between the system and its environment that can lead to non-classical behaviour-in a way which will be specified in the following. Intuitively, while the completely dephasing map leaves the system unchanged if no coherences are created, it does not necessarily leave the overall system-environment state invariant. In detail, in general we can have ∆[ρ tj ] = I[ρ tj ] ∀t j , but not necessarily ∆ ⊗ I e [η se tj ] = I[η se tj ] ∀t j . As we will see, the latter property is sufficient, but not necessary, for the satisfaction of the Kolmogorov conditions. First, though, in order to be able to go beyond the investigation of Markovian processes, and extend the existing connection between classicality and coherences, it is important to introduce quantum combs-a suitable framework to describe general quantum processes [36,37].

IV. NON-MARKOVIAN CLASSICAL PROCESSES
In the previous example, we saw the subtle relation between coherence and classicality in the case of open quantum processes with memory.
There, although no coherence is ever generated on the level of the system with respect to the chosen measurement basis, the system-environment correlations built up throughout the dynamics lead to non-classical statistics.
To develop a more in-depth understanding of the interplay between coherences and classical phenomena, we require a suitable operational framework for approaching such scenarios. We can then employ this framework to comprehensively characterize all quantum processes that display classical statistics.

A. Classicality and processes with memory
The necessity of such a novel framework for the description of quantum processes that display memory effects stems from the breakdown of their modeling in terms of propagators that could be used in the Markovian case; this can already be seen for classical stochastic processes.
Here, a joint probability distribution P K (x K , . . . , x 1 ) fully describes a K-process and can be represented in terms of conditional probabilities as Importantly, all of the above conditional probabilities generally depend upon all preceding measurement results, in contrast to the Markovian case where they only depend on the most recent outcome. Consequently, two-point transition probabilities of the form P(x j |x j−1 ) are not sufficient in general to build up all joint probability distributions and thus do not completely describe the process. Similarly, two-time propagators {Λ tj ,tj−1 } are generally not sufficient to compute multi-time joint probabilities in the quantum case and therefore fail to fully capture the process [66,70].
For classical statistics, the joint probability distribution P K (x K , . . . , x 1 ) contains all information about the K-process, since all distributions for fewer times, as well as all conditional probabilities, can be derived once P K is known. In exactly the same way, a general quantum K-process is fully characterized by the joint probabilities for all possible sequences of K measurements (at times t 1 , . . . , t K ), including non-projective and non-orthogonal ones.
As discussed in the previous section, if the complete system-environment dynamics is known, then all joint probability distributions (on times {t j } K j=1 ) obtained from sequential measurements of the system can be computed via Here, {P xj } correspond to projective measurements in the computational basis, but evidently the same relation can also be used to obtain the correct probabilities when using different probing instruments, e.g., instruments that measure sharply in a different basis or those that perform generalized measurements. More formally, an is a collection of CP maps that add up to a CPTP map [71]. For instance, the instrument corresponding to a measurement in the computational basis is given by J k = {P x k }, and all of its elements add up to the CPTP map x k P x k = ∆ k . Intuitively, each outcome of an instrument corresponds to one of its constituent CP maps, which, in turn, describes how the state of the system changes upon the realization of a specific measurement outcome. With this, the probability to obtain the sequence of outcomes x 1 , . . . , x K , given that the instruments J 1 , . . . , J K were used to probe the system, is given by In what follows, whenever we drop the explicit instrument labels, it is understood that the probabilities were the result of a measurement in the computational basis at each time. The multi-linear functional C K introduced above is known as a quantum comb [36,72] and constitutes the natural generalisation of quantum channels that takes into account the possibility of memory effects [37,38,73,74] (see Fig. 5 for a graphical representation).
It maps any sequence of possible experimental transformations enacted on the system to the corresponding joint probability of their occurrence. In this sense, C K plays exactly the same role that the joint probability distribution P K plays in the classical setting, and thus allows one to decide the classicality of the resulting statistics (see below). For example, for the completely memoryless case, i.e, the case of Markovianity with respect to measurements in any basis, the evolution between any two points in time is described solely by a sequence of independent CPTP maps that act on the system alone [38,75], and we have In general, however, the comb of a K-process does not split in the way above into independent portions of evolution between times. Thus, when analyzing the relation between coherence and classicality, instead of investigating the properties of individual CPTP maps, one must consider those of the multi-time comb C K . The comb C K is an operationally well-defined object that can-just like the joint probability distribution P K -be obtained by means of probing measurements on the system alone through a generalized tomographic scheme [37,76]. Specifically, for its reconstruction, it is not necessary to explicitly know the system-environment dynamics: the comb does not contain direct information about the environment, but solely that of its influence on the multi-time statistics observed from measurements on the system. As such, it encapsulates all that is out of control of the experimenter and thereby clearly separates the underlying process at hand from what can be controlled (i.e., the experimental interventions). An explicit example of the comb formalism is provided in Appendix C, where we rephrase Example 1 in terms of the comb description.
Crucially, the comb framework allows us to consider what it means for a stochastic process with memory to be classical, thereby permitting an extension of the results of Ref. [33] to the non-Markovian case: given the comb C K of a process on times in T , all correct combs describing the process on fewer times T ⊆ T can be deduced by letting C K act on the identity map at the appropriate superfluous times [37,50]. For example, we have (see also Fig. 6) As we have discussed in the previous sections, classicality of a process means that the action of the completely dephasing map cannot be distinguished (by means of measurements in the classical basis) from not performing an operation. With the general method of 'marginalization' given by Eq. (33) at hand, we obtain the following characterization of classical combs: Thm. 2 (K-classical quantum combs). A comb C K on times T , with |T | = K, yields a K-classical process via Eq. (31) iff it satisfies for all subsets T ⊆ T and all possible sequences of outcomes on T \ T .
In slight abuse of notation, here, the argument tj ∈T a j , t k ∈T \T b x k of the comb C K signifies that it acts on the maps a j at times t j ∈ T and on b x k at times t k ∈ T \ T .
Thm. 2 expresses in a concise way that a general process is K-classical iff the action of completely dephasing maps cannot be distinguished from the action of the identity maps. Put differently, Eq. (34) is a requirement of non-invasiveness of the completely dephasing map with respect to measurements in the classical basis.
Proof. The proof of Thm. 2 is thus straightforward: If a comb satisfies Eqs. (34), then the resulting statistics satisfy Kolmogorov conditions. Conversely, any joint probability distribution on a set of times T ⊆ T can either be obtained by direct measurement, or by marginalization of the corresponding distribution on T . The former can be computed via the first line of Eq. (34), the latter via the second one. If the statistics of the process appear classical, then both resulting distributions have to coincide, and Eq. (34) must hold.
In the (basis dependent) Markovian case that we discussed in the previous section, Eq. (34) directly reduces to Eq. (20), the NCGD property on the level of propagators of populations. Thm. 2 therefore provides the proper generalization of the results of Ref. [33] to the non-Markovian case. Nonetheless, its consequences for the structural properties of classical combs, and, in particular, the relation of classicality and coherence remain somewhat opaque in the way Thm. 2 is phrased. In order to address these questions, we now introduce a representation of quantum combs that is favourable for the purposes of our work.

B. Choi-Jamio lkowski representation of general quantum processes
Both the quantum comb describing the K-process at hand and the experimental interventions applied at each time are linear maps (the former being a higher-order multi-linear map). Any such map can be represented in a variety of ways, but the most natural for our present purposes makes use of the Choi-Jamio lkowski isomorphism (CJI) [77,78] between quantum maps and positive Hermitian matrices.
A general quantum map-e.g., one that corresponds to a generalized measurement-at time t k is a CP transformation M x k : B(H i k ) → B(H o k ) that takes linear operators on the (input) Hilbert space H i k onto linear operators on the (output) Hilbert space H o k . Throughout this paper, we will consider the input and output spaces of such maps to be isomorphic (and of finite dimension), and the labels i and o, as well as the time label, are merely introduced for better accounting of the involved spaces. Any such quantum map M x k can be isomorphically mapped onto a positive Hermitian matrix that we will call its Choi state, , by letting it act on one half of an unnormalized maximally-entangled state Φ Through this isomorphism, we have, e.g., the following identifications: Identity Map : Here and throughout this article, we typically denote maps with calligraphic upper-case letters and their Choi state with the corresponding non-calligraphic variant-with the exception of the identity map Eq. (36) and the completely dephasing map Eq. (38). For better orientation, we will continue to denote the respective time at which the maps act by an additional subscript. Analogously, as a quantum comb C K is a multi-linear map it can-in a similar way to Eq. (35)-be mapped onto a positive semi-definite matrix C K [34,36,37]. The action of a quantum comb on a sequence of CP maps {M x K , . . . , M x1 } is then equivalently given by [36] where rT denotes the transposition with respect to the computational basis. Eq. (39) constitutes the Born rule for temporal processes [79], where C K plays the role of a quantum state over time and the Choi states M x K , . . . , M x1 play the role that positive operator-valued measure elements play in the standard Born rule. Concretely, given an instrument sequence J K , . . . , J 1 , by combining Eqs. (31) and (39), the joint probability over the sequence of outcomes x K , . . . , x 1 is given by Through this isomorphism, memory effects of the temporal process correspond directly to structural properties of its Choi state [34,39,40], and analogously, the classicality of a process is reflected in the properties of C K .
Represented in this way, quantum combs and the channels that they generalize have particularly nice properties. Complete positivity and trace preservation for a quantum channel M correspond respectively to M ≥ 0 and satisfaction of tr o [M ] = 1 i . Analogously the Choi state of a quantum comb has to satisfy C K ≥ 0 as well as a hierarchy of trace conditions that fix the causal ordering of events [36], i.e., they ensure that later events cannot influence the statistics of earlier ones.
It is important to note that all K-processes can be represented through the CJI as (unnormalized) quantum states C K . In the converse direction, any operator satisfying these properties admits an underlying open-system description [36,37,72]. Specifically, this means that for every proper comb, there is a (possibly fictitious) environment and a set of system-environment unitaries such that the action of the comb on any sequence of instruments can be written as in Eq. (31). Quantum combs are hence the most general descriptors of open quantum system processes (when the system of interest is probed at fixed times). We will call the respective underlying unitary description that includes the environment the dilation of the comb.
As is the case for quantum channels, any such dilation is non-unique. On the other hand, the comb C K resulting from some underlying evolution is unique, and-just like the joint probability distribution P K in the classical case-constitutes the maximal descriptor of the process on the respective set of times.

C. Structural properties of classical combs
As first step to a structural understanding of classical combs, we rephrase Thm. 2 in terms of Choi states: Thm. 2 (K-classical quantum combs). A comb C K on times T , with |T | = K, yields a K-classical process iff its Choi state satisfies for all subsets T ⊆ T and all possible sequences of outcomes on T \ T .
Using the relations (36) -(38) as well as Eq. (40), it can be seen straightforwardly that this theorem is indeed equivalent to Thm. 2. Importantly, as it is stated in terms of Choi states, Thm. 2 allows one to derive a direct connection between coherences and the classicality of a K-process.
To see how the requirement in Eq. (41) translates to structural constraints on classical combs, first note that any comb that yields the joint probability distribution P K (x K , . . . , x 1 ) when probed in the classical basis can be written as where the term contains the joint probability distribution P K and tr[(P x K ⊗ · · · ⊗ P x1 )χ] = 0 for all x K , . . . , x 1 [76]. Intuitively, C Cl. K corresponds to the part of C K that can be probed by measurements in the classical basis alone, while χ contains all the information about the underlying process that such measurements are blind to. If χ = 0, then C K clearly satisfies the conditions of Eq. (34), as tr[P xj Φ + j ] = tr[P xj D j ] for all x j [80]. Put differently, for χ = 0, the corresponding comb is classical, as it is diagonal in a classical product basis. However, this is not necessary for Eq. (34) to hold; rather, it suffices if χ is such that it does not allow one to distinguish between the action of the identity map and the completely dephasing map. We thus arrive at the following lemma: Lem. 1. Let C K be the comb of a K-process on T , with |T | = K, and let A j := Φ + j −D j . C K yields a K-classical process iff it is of the form where C Cl.
K is obtained from some joint probability distribution P K via Eq. (43) and χ satisfies for all subsets T ⊆ T and T = ∅.
Proof. It is straightforward to see that a comb of the form of Eq. (44) satisfies Eq. (41), whenever χ fulfills Eq. (45), and thus yields K-classical statistics. Conversely, any comb C K on K times can be written as C K = C Cl. K + χ , diagonal in the computational basis classical processes all quantum processes Figure 7. Nested set of processes. Processes that cannot produce coherence and destroy any coherence that is fed in (i.e., their Choi states are diagonal in the computational basis) form a strict subset of processes that appear classical when probed in the computational basis. Both of these sets, as well as the set of all quantum processes are convex.
where C Cl.
K is of the form of Eq. (43) for some P K and tr[(P x K ⊗ · · · ⊗ P x1 )χ] = 0 [76]. When measuring (in the computational basis) at K times, the resulting joint probability distribution is given by P K . As-by assumption-the process is classical, summation over outcomes obtained at any time C K is defined on must yield the same statistics as letting it act on the identity channel at this time. As this has to hold for any collection of times in T , χ has to satisfy the additional requirements given by Eq. (45).
Intuitively, Eq. (45) ensures that the invasiveness of ∆ j cannot be detected at any point in time by means of measurements in the classical basis. Therefore Lem. 1 is equivalent to Thm. 2. However, the latter provides an explicit constraint on the structure of such combs that contain coherences that can be present in the process without making the resulting statistics non-classical.
Indeed, if χ = 0, then the corresponding comb C K is diagonal in the classical product basis and, as such, cannot create coherences and destroys any kind of coherences that could be fed into the process (e.g., by performing coherence creating operations at some time). On the other hand, if χ = 0 and the comb contains off-diagonal terms (with respect to the classical basis), then coherences can be created over the course of the process. However, if χ satisfies Eq. (45), then these coherences-or rather the invasiveness of the completely dephasing map-cannot be detected at any later time by measurements in the classical basis. This understanding of classical non-Markovian combs mirrors the intuition we had built in the Markovian setting for the case of NCGD dynamics. Consequently, Lem. 1 fully characterizes the relation between coherences and the non-classicality of a process (see Fig. 7 for a graphical representation of the different sets of processes we consider).
Unfortunately, in the non-Markovian case, this characterization comes at a price. In order to decide on the K-classicality of a process, it is no longer sufficient to investigate propagators between pairs of times, but rather the full part of the comb C K that is relevant for sequential projective measurements must be known, due to the importance of multi-time effects. However, this behaviour is to be expected, as can already be seen in the case of classical stochastic processes: the full characterization of a non-Markovian process only happens on the level of the full joint probability distribution P K , and not by way of transition probabilities between adjacent times only.
In summary, Thm. 2 and Lem. 1 provide a full characterization of processes that yield classical statistics. For further clarification, and in order to connect such processes to the respective underlying evolution, in the following section, we will discuss some concrete cases of underlying non-Markovian dynamics that lead to classical statistics. Moreover, we will connect the classicality of temporal processes to vanishing quantum discord in the joint state of the system and the environment.

V. DISCORD AND CLASSICALITY
In the Markovian case, classicality of a process could be decided solely in terms of propagators between pairs of times that are defined on the system of interest alone and it was linked to the ability of those maps to create and detect coherences. In particular, the set of dynamics that does not create coherences on the level of the system is contained in the set of maps that lead to classical statistics [33]. As we have seen above, this fails to hold in the non-Markovian case, where even if the state of the system is diagonal in the computational basis at all times, i.e., no coherence on the system level is ever generated, the statistics might not satisfy Kolmogorov conditions.
As soon as memory effects play a non-negligible role, it is both the coherences of the system state, and the correlations between the system and its environment that can lead to non-classical behaviour. In this section, we further investigate the reasons for the detectability of quantum effects in non-Markovian processes and provide a more explicit relation between coherence, correlations and classicality.
To do so, first recall that while the completely dephasing map leaves the system unchanged if the state of the system is classical at all times, it does not necessarily leave the overall system-environment state invariant. Specifically, in this case we have ∆ j [ρ tj ] = I[ρ tj ] ∀t j but not necessarily ∆ j ⊗ I e j [η se tj ] = I se [η se tj ] ∀t j . While the latter is not necessary for the satisfaction of the Kolmogorov conditions, it is sufficient: ti } be sets of probabilities that sum to unity, {Π m j } orthogonal projectors (not necessarily rank-1) on the system that are diagonal in the computational basis, and {ξ m j } states on the environment. If at all times t j ∈ T , with |T | = K, the system-environment state is of the form then the underlying process is K-classical, i.e., it satisfies the Kolmogorov conditions of Eq. (2).
Before we prove this statement, it is insightful to discuss the relation between the concept of classical temporal processes and classical spatial correlations it introduces. States of the form defined in Eq. (46) have vanishing quantum discord [44][45][46]81], i.e., they do not display any genuinely quantum correlations between the system and the environment. For a general zero-discord state, the set {Π m j } in Eq. (46) could be any set of mutually orthogonal projectors, and the correlations between the system and the environment are considered to be classical, since there exists a measurement on the system with perfectly distinguishable outcomes that overall leaves the total state undisturbed [45,46] (see also the proof below).
As we only consider measurements on the system in a fixed basis in our setting, here, vanishing discord at all times does not yet force the resulting statistics to be classical; rather, the discord must vanish in the correct basis, i.e., the one in which the experimenter's measurements act. Consequently, we will call states of the form in Eq. (46) discord-zero with respect to the classical basis. Any system-environment state that cannot be written in this form will be considered to display non-classical correlations in what follows.
While discord is often considered as a basis independent quantity-obtained by a minimization procedure over all possible measurement scenarios [46]-here, and throughout the remainder of this article, we will always consider its basis dependent formulation [44-46, 55, 82]. That is, whenever we consider a state to be of zero discord, we will always implicitly mean that it can be represented as per Eq. (46) with the projectors being diagonal in the classical basis. Importantly, this basis dependence mirrors the basis dependence of coherence, which is also always defined with respect to a fixed classical basis.
Proof. For states of the form in Eq. (46), the completely dephasing map ∆ on the system has the same effect as the 'do-nothing' identity channel I, i.e., Consequently, if the system-environment state is of this form at all times, the resulting statistics satisfy the Kolmogorov conditions.
If a state is of zero discord, it displays neither coherences on the level of the system nor non-classical correlations between the system and the environment, which is, to reiterate, sufficient for the classicality of the resulting process, but not necessary. In this sense, Lem. 2 is a direct extension of the analogous statement in the Markovian case; there, the absence of coherence in the system state at all times is also sufficient but not necessary for the process to be classical.
Put differently, if all of the individual maps making up a Markovian dynamics are maximally incoherent operations (MIO) [2,83], i.e., they map all incoherent states onto incoherent states, then the resulting dynamics satisfies Kolmogorov conditions. However, MIO operations are a strict subset of NCGD maps [33].
While somewhat unsurprising, the above lemma sheds light on the properties that a general non-Markovian dynamics has to satisfy in order to appear classical. For system-environment states that are discord-zero in the computational basis, a measurement in the computational basis is non-invasive, i.e., it leaves the full state unchanged (and not just the system state, as it would be the case if the system state is incoherent at all times).
In particular, this implies that in Example 1, where the state of the system was diagonal at all times, but the resulting statistics non-classical, discord is created throughout the duration of the process. For comprehensiveness, in Appendix E we provide a characterization of non-discord creating processes in terms of their dynamical building blocks.
In general, the absence of discord at all times is not necessary for a process to appear classical. However, what is necessary is that at no time can there be coherences or non-classical system-environment correlations that can be detected by means of measurements in the computational basis at a later time. This mirrors the requirement for classical processes in the Markovian case, where the individual propagators have to be NCGD, i.e., the propagators must be such that they cannot create coherences whose existence can be picked up at a later time by means of classical experimental devices; yet, it is still possible that the individual maps create coherences [33]. NCGD maps are the fundamental building blocks that constitute classical Markovian combs. In what follows, utilizing the connection of classicality and discord discussed above, we will provide a characterization of the building blocks that make up classical non-Markovian processes.

A. Non-Discord-Generating-and-Detecting Dynamics and Classical Processes
In the Markovian case, classicality of a process can be decided on the level of CPTP maps, since in the absence of memory all higher order probability distributions can be obtained from the initial system state ρ t1 and the two-time propagators {Λ tj ,tj−1 }. It suggests itself to employ this intuition in the non-Markovian case, as every non-Markovian process corresponds to a Markovian one if enough additional degrees of freedom are taken into account.
In detail, as we discussed, every non-Markovian process can be dilated to a concatenation of an initial (potentially correlated) system-environment state and unitary total-dynamics [36,37], interspersed by the operations of the experimenter on the system alone that are performed at times {t j } (see Fig. 5 for reference). If the experimenter had access to all the degrees of freedom necessary for the dilation, then the underlying process would appear Markovian, and the results of Ref. [33] could be applied on the system-environment level for the characterization of a classical process. Here, using the Markovian case as a guideline, we aim for a similar characterization of classical processes when only the system degrees of freedom can be accessed.
To compactify notation and simplify later discussions, we can equivalently consider a general open process as a concatenation of CPTP maps that act on both the system and the environment, interspersed by the operations on the system alone. This way of describing general open system dynamics is simply a notational compression of the general case with global unitaries that allows for an easier connection to the Markovian case, but does not lead to a different set of possible combs. In what follows, we will denote these CPTP maps by Γ tj ,tj−1 to clearly distinguish them from the memoryless scenario (where the respective maps act only on the system), so that Eq. (4) generalizes to On this dilated level, the dynamics is Markovianthere are no additional external 'wires' that can carry memory forward-and all higher order joint probability distributions could be built up when the individual CPTP maps {Γ tj ,tj−1 } (and the initial system-environment state) are known. With this, we can define non-discord-generating-and-detecting (NDGD) dynamics: Def.
A graphical representation of this definition can be found in Fig. 8.
Formally, Def. 3 is equivalent to the definition of NCGD dynamics, with the difference that the involved intermediary maps between times are now the system-environment maps, instead of the maps {Λ tj ,tj−1 } acting on the system alone in the Markovian case.
Analogously to the case of NCGD, an NDGD dynamics cannot create discord (with respect to the classical basis) that can be detected at the next time (and, as such, at any later time) by means of classical measurements. Figure 8. NDGD system-environment dynamics. From the perspective of a classical observer, the identity map at any time tj cannot be distinguished from the completely dephasing map. Any discord (with respect to the classical basis) that is present in the system-environment state, and/or created by the system-environment CPTP maps, cannot be detected by a classical observer.
Or, equivalently, an experimenter who can only perform measurements in the classical basis cannot distinguish between a completely dephasing map and an identity map implemented at any time in T . As such, it is the natural extension of NCGD to the non-Markovian case. Indeed, for the memoryless case, Eq. (49) reduces to Eq. (13), and the definitions of NCGD and NDGD coincide. We then have the following theorem: Thm. 3 (NDGD dynamics and classicality). Consider a general, possibly non-Markovian, process on T , with |T | = K, obtained from a system-environment dynamics as in Eq. (48); then the process is K-classical if the initial system-environment state the set {Γ tj ,tj−1 } of maps that corresponds to it are zero discord and NDGD, respectively.
The proof of this theorem is provided in Appendix F. It relies on the fact that measurements in the classical basis commute with the completely dephasing map and proceeds along the same lines as the analogous proof for NCGD dynamics in the Markovian setting provided in Ref. [33]. Importantly, though, it is not a necessity for classical statistics that the corresponding maps are NDGD (see below).
In order to further elucidate the relation of discord and classicality in general quantum stochastic processes, it is insightful to discuss the proximity of Thm. 3 to the corresponding results in Ref. [33] for the Markovian case. Thm. 3 establishes the importance of the role of quantum discord for the classicality of non-Markovian processes. In the memoryless case, it is coherence-or the impossibility of detection thereof-that makes a process classical. Here, this role is played by discord, with the only difference being that instead of describing the process in terms of maps that are solely defined on the system of interest, we are forced to dilate the process to the system-environment space, where it is rendered Markovian. Consequently, the classicality of a process cannot be decided based on the master equation or dynamical maps that describe the evolution of the system alone (as has already been pointed out in Ref. [33]). However, given, e.g., a Hamiltonian that generates the corresponding system-environment dynamics, it can be decided if a process is classical by checking the validity of Eq. (49).
It would be desirable if NDGD dynamics were a sufficient and necessary criterion for the classicality of non-Markovian processes; however, this is not the case. We provide an example of dynamics that is not NDGD, but nevertheless leads to classical dynamics, in Appendix G. NDGD as defined in Eq. (49) is a statement about the entire system-environment dynamics, and holds for any possible input state on the environment. However, by means of projective measurements on the system alone, one only has access to the system part, and the system-environment dynamics cannot be fully probed. Consequently, the criterion of Eq. (49) will, in general, be too strong for a given experimental scenario. Crucially, though, Thm. 3 allows us to understand the role of the discord generated by the system-environment interaction and subsequently detected via projective measurements on the system in establishing non-classical statistics. Nonetheless, even though it is not necessary for the underlying dynamics to be NDGD in order for a non-Markovian process to display classical statistics, for any K-classical process, there always exists a dilation that is NDGD. That is, there exists a set { Γ tj ,tj−1 } of system-environment CPTP maps that are NDGD and a zero-discord initial system-environment state η se t1 that yield the correct classical family of joint probability distributions when probed in the classical basis. Specifically, we have the following theorem: Thm. 4. Let {P n (x n , . . . , x 1 )} define a process on T , with |T | = K, coming from an underlying evolution, fixed by the system-environment maps {Γ tj ,tj−1 } and the initial state η se t1 , according to Eq. (48). The resulting statistics {P n (x n , . . . , x 1 )} is K-classical iff there exists a NDGD evolution given by system-environment maps { Γ tj ,tj−1 } defined on times in T that yields P k (x k , . . . , x 1 ) when probed in the classical basis.
Before we prove this statement, it is important to contrast it with Thm. 1, the analogous result for Markovian processes. There, NCGD propagators of the system dynamics guarantee that the process associated with sequential projective measurements is classical, and classical Markovian processes can be reproduced by a set of NCGD maps (which do not necessarily identify with the actual dynamical propagators). Analogously, here, the NDGD property of the actual system-environment evolution ensures the classicality of the process, while the converse holds for particular dilations, but there can be non-NDGD dilations that nonetheless yield classical statistics.
In both cases the projective measurements in a fixed basis only give a limited amount information about the overall evolution underlying the probed statistics. But, while in the Markovian case the statistics can be traced back to dynamical maps acting on the open system alone, in the more general non-Markovian case it is the whole system-environment evolution which enters into play. As a consequence, only the former case allows one to establish a one-to-one correspondence between classicality and the properties of the actual evolution by enforcing a proper condition on the dynamics, as discussed at the end of Sec. III A.
Proof. As we have already seen in the discussion of Thm. 3, the joint probability distributions obtained from an NDGD dynamics are always classical. We thus only need to prove the opposite direction. Let the underlying system-environment dynamics of the process between times be given by the maps {Γ tj ,tj−1 }. As the process is classical, the set of maps { Γ tj ,tj−1 = ∆ j • Γ tj ,tj−1 • ∆ j−1 } together with an initial state η se t1 = ∆ 1 [η se t1 ], where, again, ∆ k only acts on the system degrees of freedom, yields the same joint probability distributions when probed in the classical basis (see Fig. 9 for reference). The process given by this set { Γ tj ,tj−1 } is NDGD by construction and η se t1 has vanishing discord, which means that for every K-classical process there is an NDGD dilation that reproduces it correctly.
Thm. 3 and Thm. 4 together complete the results of Ref. [33] to the non-Markovian setting, providing an intuitive connection between non-classical spatial correlations (i.e., discord) and completely elucidating the connection between discord and classical processes.

VI. CONCLUSIONS
In this paper, we have provided an operationally motivated definition of general classical stochastic processes and discussed its structural consequences and relation to coherence and non-classical spatial correlations. While we phrased our results predominantly in the language of quantum mechanics, there is-a priori -nothing particularly quantum mechanical about the definition of classicality we introduced. Rather, any process for which the potential invasiveness of performed measurements can be detected by means of said measurements is non-classical, independent of the underlying theory; as such an invasiveness experimentally detectable, this is a fully operational notion. The question of whether or not a process is classical can thus be answered on experimentally accessible grounds and is a priori independent of concepts that the experimenter might not be able to check for, like, e.g., coherences in the system of interest. Nonetheless, our definition allows for the derivation of a direct connection between the classicality of a process and coherences/non-classical correlations that might be present. While this connection is rather straightforward for memoryless processes, it turns out to be more subtle in the non-Markovian case. In general, it is not sufficient for the state of the system to be diagonal in the classical basis at all times for the resulting multi-time statistics to be classical. Rather, it is the interplay of coherences, non-classical system-environment correlations and the underlying dynamics that is of importance, as we have highlighted through a number of examples presented throughout. Using the comb framework-which can encapsulate this complex interplay-for the description of general quantum processes with memory we have provided a comprehensive characterization of quantum processes that yield classical statistics, and derived the structural properties of such processes. In principle, such structural properties can be derived similarly for processes that display classical statistics when probed by means of different measurements, e.g., non-projective and/or non-orthogonal ones.
However, while still enabling the derivation of structural properties, the clear connection between classicality and quantum discord would be lost as soon as sharp measurements in the computational basis are not the probing mechanism of choice anymore. In this paper, orthogonal projections were chosen as the kind of measurements that a priori come closest to the ideal non-invasiveness displayed by classical measurements. More generally, our results could in principle also be extended to post-quantum theories. As the definition of classicality we provided is fully operational, the structure of classical processes in such theories, could be derived in the same vein as we presented in this paper, with coherence and discord being replaced by the analogous properties of the respective theory.
Finally, we investigated the relation between the non-classicality of the statistics observed throughout a process and the quantumness of the prevalent spatial correlations in the underlying dynamics. While the absence of coherence in the state of the system of interest is no longer sufficient in the non-Markovian case to guarantee classicality, the absence of (basis dependent) discord is. This latter fact is intuitive, as the absence of discord at all times means that there are no non-classical system-environment correlations that could influence the multi-time statistics deduced. Specifically, we have shown that the non-Markovian case to some extent mirrors the memoryless one: If the underlying dynamics is NDGD, i.e., any discord that is created at some point in time cannot be detected at a later time, then the process appears classical. While the converse of this statement does not hold, we have further shown that any classical process admits an NDGD dilation.
We have thus provided a comprehensive picture of the interplay between the non-classical resources that are present in the underlying process and the non-classicality of the resulting non-Markovian statistics that can be probed. Together, our results pave the way to the development of a fully-fledged resource theory of classicality that is based solely on experimentally accessible quantities.
In this section we show that the result derived in the main text for the Markovian case (that is, Thm. 1) implies the preceding one in [33]. For the ease of the reader, we restate both results here (changing slightly the terminology of the latter to the one used here).
Thm. 5 (Thm. 2 of [33]). Let {P K (x K , . . . , x 1 )} be the process fixed by the QRF Eq. (9), with respect to a set of propagators forming a CPTP semigroup, i.e., Λ t l ,tj = e L(t l −tj ) for any t l ≥ t j with L a Lindblad generator [59,60], and an initial state ρ t0 . Then the process {P K (x K , . . . , x 1 )} is K-classical (Def. 1) for any ρ t0 diagonal in the computational basis if and only if the family of propagators is NCGD in the sense that While the two theorems are clearly related, there are two relevant differences. The new result is more operational in the sense that the statements only depend on the statistics one obtains by making the measurements in the classical basis at the specified times, whereas the statement in [33] relies on two underlying assumptions on the Markovianity of the quantum dynamics. The first of these assumptions is that the system multi-time statistics satisfy the QRF (Eq. (9)), and the second is that the dynamics forms a semigroup. As we will see below, the second of these assumptions can be relaxed, but the first is crucial if one wants to have the benefit of the statement in [33], which not only relates possible models for the statistics [88], but makes also a statement about how the possibility of modelling a process classically implies that the propagators referred to the actual underlying evolution have to satisfy NGCD. To be able to make this connection between the statistics and the underlying quantum evolution, we need to restrict by assumption the types of evolutions we are considering. For the Markov case, considered here, the natural choice is the QRF (Eq. (9)), as we discussed in the main text that they are closely related.
To prove the connection between the two Theorems, it is useful to consider the following corollary to Thm. 1 of the main text: . , x 1 )} be the process fixed by the QRF Eq. (9), with respect to a set of divisible propagators and an initial state ρ t0 .
Let the classical dynamics of this process be invertible, that is, P K (x j ) = 0 for an initial diagonal state that is full-rank, for any t j < ∞. Then, the process {P K (x K , . . . , x 1 )} is K-classical (Def. 1) for any ρ t0 diagonal in the computational basis if and only if the family of propagators is NCGD, see Eq. (A1).
Since the latter implies Eq. (13) and QRF implies that the process is K-Markovian, for any initial diagonal state in the computational basis K-classicality follows from Thm. 1.
Conversely, let the assumptions hold and the process be K-classical, in particular for an initial diagonal fullrank state. NCGD follows from the equation (for s 3 ≥ s 2 ≥ s 1 in T ) by linearity, since from the assumptions (invertibility of the classical dynamics and taking a diagonal, full-rank initial state) we have that P x1 • Λ s1 [ρ 0 ] = 0 ∀x 1 , s 1 < ∞ (for s 1 , s 2 , s 3 → ∞, Λ si,sj → 1 and NCGD holds trivially).
The only difference between this corollary and Thm. 2 of [33], is that here we have the divisibility of the "full" propagators and invertibility of the classical propagators in the assumptions, while there the dynamics was assumed to be of Lindblad type. This latter assumption is however strictly stronger, as it implies divisibility and that P xj • e Ltj [ρ] = 0 ∀x j , t j < ∞ and for any full-rank ρ, since (finite-dimensional) semigroup evolutions cannot decrease the rank of a state on a finite time [89].
In total we have shown in this section that Thm. 2 of [33] is a corollary of Thm. 1 by using the connection between the QRF and Markovianity and further restricting to the case of Lindblad evolution. Moreover, Corollary 1 shows how, relaxing such restriction and assuming a proper invertibility condition on the classical dynamics, it is possible to establish a one-to-one correspondence between the classicality of a process satisfying the QRF and the NCGD property, where the latter is referred to the propagators of the actual dynamics.
Appendix B: Absence of coherence for the qubit coupled to a continuous degree of freedom In this Appendix, we provide the mathematical details missing in the main text for Example 1. We begin with the expression of the global state at time t 1 , immediately before the first measurement: After a measurement at time t 1 with outcome ±, the state is subsequently given by where we emphasize that we have a tensor product state and have introduced the amplitude ±ρ 01 e i(p+p )t1 ± ρ 10 e −i(p+p )t1 + ρ 11 e −i(p−p )t1 , as well as the normalization factor C Note that nô σ x -coherence is present at this stage.
If we now let the system-environment evolve up to a certain time τ > t 1 , the global state will be where the superscript ± refers to the outcome of the first measurement at time t 1 . The corresponding system state at time τ is then given by tracing out the environmental degrees of freedom, resulting in with Once again, we see that if the initial system state is a convex mixture of |+ and |− and k(t) is real (e.g., a Lorentzian distribution centered at 0) then nô σ x -coherence is present at any time τ . This can be seen because the reduced state can be written as in Eq. (26) for the real α = (±k (±) (τ, t 1 ) + 1)/2. As a side remark, we note that even if the initial state had some coherences w.r.t.σ x , these would have been destroyed after the first measurement at time t 1 and, as long as ρ 01 ∈ R, would not have been 're-generated' by the subsequent evolution.
Indeed, the argument above can be reiterated for the subsequent measurements; for instance, if we consider the global state after the second measurement at time t 2 , we find where s denotes the sequence of + and − obtained in the measurements and sg(s) the sign of the corresponding product. The entire procedure can be iterated, by replacing f 2;t2,t1 (p, p ), so that the state at any subsequent time would remain in the form of Eq. (26), with the off-diagonal elements given by a linear combination with real coefficients of the real function k(t) evaluated at different times. In Appendix C, we will show how Example 1 can be described using a comb representation as introduced in Section IV.
Appendix C: Comb representation of the qubit coupled to a continuous degree of freedom In Appendix B, we showed the absence of coherence in the state of the system at all times for the dynamics of Ex. 1. To do so, we computed the full system-environment dynamics; however, the full knowledge of the system-environment dynamics is not necessary to understand the multi-time probabilities of observables of the system alone. Moreover, the state of the environment is often not experimentally accessible in practice, as it is typically highly complex. Therefore, it is convenient to only describe the influence that the environment has on the multi-time probabilities. Importantly, this influence, and the resulting correct descriptor of the underlying process, can be deduced by probing the system alone.
Such a descriptor can be derived using the concept of quantum combs [36,72], which we briefly reviewed in Section IV. A quantum comb contains all statistical information that can be inferred about the process it describes (on the set of times upon which it is defined). While here we will construct the comb for Ex. 1 by explicitly solving the system-environment dynamics, it is important to note that it could be reconstructed experimentally by means of measurements on the system alone, without any access to or knowledge of the environmental degrees of freedom, through a generalized tomographic scheme [37].
In slight deviation from the notation of the main text, in this appendix, for better orientation, here we explicitly write the labels of the Hilbert spaces a comb acts on, and the times it is defined upon, as sub-and superscripts, respectively.
As described in Example 1, we start with a system-environment state η se (t 0 = 0) = ρ s (t 0 = 0) ⊗ |ϕ e ϕ e | where |ϕ e is fixed. As shown in Fig. 10, the initial system state ρ s (t 0 ) is associated with the Hilbert space with label 1. The channel C t1:t0 (ρ s ) =U t1,t0 ρ s ⊗ |ϕ e ϕ e | (C1) Figure 10. Labeling of Hilbert spaces used for the comb description of Example 1. The grey box contains the comb C t 1 :t 0 2α1 and the red box the comb C t 2 :t 1 4β3α . The comb C t 2 :t 1 :t 0 4β321 corresponds to everything inside the dashed box and consists of the contraction of the two combs C t 1 :t 0 2α1 and C t 2 :t 1 4β3α . maps the initial system state to the full system-environment state at time t 1 directly before the intervention. The corresponding channel in comb description is given by where the superscripts denote the intervention times and the subscripts the Hilbert spaces on which the comb is acting. The object C t1:t0 2α1 above is nothing other than the Choi state associated with the channel. The dynamics from time t 1 to time t 2 is similarly given by the channel applied to the combined system-environment state directly after the first intervention. Again, this channel admits a Choi state description The next step is to eliminate the explicit description of the environment state on Hilbert space α. To do this, we contract the Choi states of the two channels described above using the link product described in Refs. [36,72]. This leaves us with the comb describing the dynamics on both times C t2:t1:t0 4β321 =C t2:t1 4β3α C t1:t0 ds dp f (p)f * (s)e i(φip−φj s)t1 |ii jj| 21 We can also describe the projectors corresponding to the observed measurement outcomes using Choi states, e.g., if we measured in the eigenbasis ofσ x and obtained outcome +, the corresponding Choi state is given by Again, using the link product, we can obtain the unnormalized joint system-environment state directly after the second intervention at time t 2 , conditioned on the initial state of the system ρ s (0) and the interventions M x1 , M x2 (where the superscripts x i refer to the outcomes) as follows For instance, if we observed the outcome + twice, the joint state after the second intervention is given by where we have introduced Since we are mainly interested in the question of whether the obtained measurement statistics can be explained classically, we restrict our attention to the unnormalized state of the system alone, because the probability of obtaining a specific sequence of measurement outcomes is encoded in the trace of the corresponding system state. Therefore we eliminate the description of the environment by tracing over the Hilbert space β, which we can do directly at the level of the comb itself C t2:t1:t0 Following the same procedure as above, we then obtain the system state after the second intervention Similarly, the probability to obtain, e.g., twice the measurement result + is given by P 2 (+, t 2 ; +, t 1 ) = tr ρ (+,+) If we introduce τ n := t n − t n−1 , by way of induction, we find that where we suppressed the subscripts of the combs. As above C tn::t0 , denotes the comb including the outgoing environment and C tn::t0 the comb describing the system alone, see Fig. 11 for a pictorial representation. Therefore, the joint probability distribution for sequences of measurement outcomes is given by P n (x n , t n ; . . . ; x 1 , t 1 ) = tr ρ s (0) T Here, we provide an alternative example of a process where the state of the system is diagonal in the computational basis at all times but does not yield classical statistics. To this end, consider the following circuit (see Fig. 12): Let the initial system-environment state be a maximally entangled two qubit state ϕ + . At t 1 the system alone is thus in a maximally mixed state ρ t1 Between t 1 and t 2 , the system and the environment undergo a CPTP map E t2,t1 (which could-in principle-be dilated to a unitary map [90], but for conciseness, we restrict ourselves to the relevant part of it), that yields output |0 on the system, if system and environment are in the state ϕ + , and |1 otherwise, i.e., when the system-environment state is orthogonal to ϕ + . Consequently, its action can be written as It is easy to check that E t3,t2 is indeed CPTP, and the state of the system at t 2 is a convex mixture of |0 0| and |1 1| for all possible experimental interventions at t 1 ; there are thus no coherences in the state of the system at any of the times {t 1 , t 2 }. However, this process does not satisfy the Kolmogorov condition.
To see this, consider the probabilities for a measurement in the computational basis at t 2 , with no operation performed at t 1 .
In this case, the system-environment state before the action of E t2,t1 is equal to ϕ + , which means that we have ρ t2 = |0 0|. Consequently, a measurement in the computational basis at t 2 yields the probabilities P 1 (0, t 2 ) = 1 and P 1 (1, t 2 ) = 0 . (D2) On the other hand, performing a measurement at t 1 and discarding the outcomes amounts to performing the completely dephasing map ∆ 1 . Immediately after this map, i.e., right before E t2,t1 , the system-environment Figure 12.
Non-classical process that does not display coherences The state of the system is classical, i.e., it does not contain coherences with respect to the classical basis, at every step of the process. The corresponding statistics do not satisfy the Kolmogorov conditions, though. Potential measurements are depicted as green circles. The blue dotted line signifies the comb of the process (see Sec. IV).
Appendix E: Non-discord-creating maps Here, for comprehensiveness, we characterize the set of maps Γ : B(H i s ⊗ H i e ) → B(H o s ⊗ H o e ) that map discord-zero states to discord-zero states, where we mean discord-zero with respect to the classical basis (the generalization to the general case is straightforward). Such system-environment maps form a subset of the NDGD maps of Def. 3 (in the sense that a set of them would satisfy Eq. (49)) and would thus lead to classical statistics on the level of the system. However, for classical statistics, it is not necessary that the underlying maps do not create discord.
To facilitate notation, throughout this Appendix, we will denote discord-zero states as DØ states, and maps that do not create discord as DØ maps. We have the following lemma: where f jk ∈ R. We can choose the basis {ω i j } to consist of the d s rank-1 projectors Π i j in the computational basis and d s (d s − 1) elements Π i s that are orthogonal to these projectors, i.e., such that tr(Π i j Π i s ) = 0 (e.g., one could choose the off-diagonal elements |m n| + |n m| and i(|m n| − |n m|)). With this choice of basis elements Eq. (E2) reads Imposing the requirement that F does not create coherences with respect to the classical basis then yields where p k|j ≥ 0, k p k|j = 1, and τ o r ∈ B(H o s ). Indeed, an F of the form of Eq. (E4) yields an incoherent output state for any incoherent input state ρ cl = ds r=1 q r Π i r ∈ Ξ: Importantly, Eq. (E4) constitutes a decomposition of the form F = F + F ⊥ , where F = j,k p k|j Π o k ⊗ Π i j encapsulates the action of F on incoherent states, and F ⊥ is such that all incoherent states lie in its kernel, i.e., tr(ρF ⊥ ) = 0 for all ρ ∈ Ξ. The fact that F ⊥ does not have to vanish in order for F to be an MIO demonstrates in a transparent way the (well-known) fact that there are MIOs that necessitate coherent resources for their implementation [5,15,17]. As emphasized throughout the main body of this paper, DØ states reduce to incoherent ones when the environment is trivial. Consequently, DØ maps are the natural extension of MIOs, and the proof of Lem. 3 follows similar logic to the above proof for the structural properties of MIOs: Figure 13. Non-NDGD dynamics that leads to classical statistics. The first map Γt 2 ,t 1 (blue transparent box) swaps the system with one half of a maximally entangled state and then discards it. The subsequent CPTP map Γt 3 ,t 2 maps ϕ + and 1/4 onto two different system-environment states with the same reduced system state ρt 3 = 1/2. The final CPTP map Γt 4 ,t 3 is such that it induces a unital dynamics on the system. Consequently, the system state at t2, t3, and t4 is maximally mixed independent of whether the completely dephasing, or the identity map was implemented at t2 and t3. However, the system state is always maximally mixed, independent of whether ∆ 2 or I 2 was implemented at time t 2 . To make the example non-trivial, we add a third dynamics Γ t4,t3 from t 3 to t 4 . We choose Γ t4,t3 such that it induces a unital dynamics on the level of the system, independent of the environment state at t 3 . This happens, e.g., when the corresponding system-environment Hamiltonian is of product form, i.e., H se = H s ⊗ H e , independent of the explicit form of the respective terms [86]. With this final dynamics, the system state at each of the times t 2 , t 3 , and t 4 is maximally mixed, and the resulting statistics satisfy Kolmogorov conditions, i.e., they are classical.