Quantum processes are fundamentally different from classical ones: therefore, a proper understanding of the classical–quantum boundary, involving the identification of the genuinely quantum features of physical systems—those which cannot be simulated and/or reproduced by any system satisfying the laws of classical physics—is of particular importance (Żurek, 2003; Joos et al., 2004; Schlosshauer, 2005; Horodecki et al., 2009; Modi et al., 2012). Aside from its intrinsic theoretical interest, this boundary also has practical implications for quantum technologies: taking advantage of the underlying quantum nature of the natural world, e.g. quantum correlations (Modi et al., 2012) and coherence (Żurek, 2003; Streltsov et al., 2017), is the key of diverse technological proposals such as quantum teleportation, cryptography or computation (Nielsen and Chuang, 2000). A groundbreaking example is provided by the violation of Bell inequalities by quantum correlations (Bell, 2011); among many other such examples, we may point out quantum states of light corresponding to non-positive values of their Wigner function (or singular Glauber–Sudarshan P-representation) (Mandel and Wolf, 1995; Gardiner and Zoller, 1999).

A thorough analysis of physical systems must involve both their free evolution as well as their behavior under external interventions; this information can be encoded by a multitime statistics of joint probability distributions (correlation functions). In this framework, it was proposed in (Smirne et al., 2019; Milz et al., 2020) to define the classicality of quantum systems in terms of the classicality of their multitime statistics. Let us briefly revise this approach. We shall consider a quantum system living on a Hilbert space H S with finite dimension dim H S = d , interacting with an environment (or bath) associated with a possibly infinite-dimensional Hilbert space H B ; let H be the self-adjoint operator on H S ⊗ H B generating the global dynamics of the system and the environment—the Hamiltonian. ¹ We will denote by U t , t 0 the unitary propagator generated by H. Assuming that, at some initial time t ₀, the global system is in an uncorrelated (product) state ρ t 0 ⊗ ϱ B ∈ B ( H S ⊗ H B ) ≃ B ( H S ) ⊗ B ( H B ) , the map Λ t , t 0 describing the reduced dynamics of the system reads, for any t ≥ t ₀, Λ t , t 0 ρ t 0 = T r B U t , t 0 ρ t 0 ⊗ ϱ B , U t , t 0 = U t , t 0 ⋅ U t , t 0 † , (1) this map being, by construction, completely positive and trace preserving (CPTP). The map is said to be CP-divisible if it satisfies the additional property Λ t 2 , t 0 = Λ t 2 , t 1 Λ t 1 , t 0 and Λ t 2 , t 1 is CPTP for all t ₂ ≥ t ₁ ≥ t ₀.

We shall consider the idealized scenario in which such a system is repeatedly probed via sharp, identical and instantaneous measurements at times t ₁, … , t _n , and evolves freely between each couple of consecutive measurements. This procedure brings about a family of time-continuous joint probability distributions (multitime statistics) defined as follows. Letting { P x } x ⊂ B ( H S ) be a projection-valued measure (PVM) representing the measurement apparatus, and P x = P x ( ⋅ ) P x , we define, for all n ∈ N , P n x n , t n ; x n − 1 , t n − 1 ; … ; x 1 , t 1 = T r P x n ⊗ I B U t n , t n − 1 ⋯ P x 1 ⊗ I B U t 1 , t 0 ρ t 0 ⊗ ϱ B , (2) with I B being the identity map on H B . This is a legitimate family of probability distributions formally analogous to the one associated with a classical stochastic process. However, while in the classical case the Kolmogorov consistency condition, P n − 1 x n , t n ; … ; x j , t j ; … ; x 1 , t 1 = ∑ x j = 0 d − 1 P n x n , t n ; … ; x j , t j ; … ; x 1 , t 1 , (3) holds, the statistics associated with a quantum system as described above will generally violate this condition. In this regard, notice that Eq. 3 essentially means that not performing a measurement at the time t _j is equivalent to performing a measurement at the same time and then forgetting about its outcome—in other words, measurements do not affect the system, which is generally not true in the quantum realm. On the other hand, by the Kolmogorov extension theorem (Kolmogorov, 1933), Eq. 3 is satisfied if and only if there exist some classical stochastic process reproducing the full multitime statistics; in this case the results of the experiment may be consistently explained in the framework of classical physics, whence no genuinely quantum feature is exhibited by the system.

Because of that, following (Smirne et al., 2019; Milz et al., 2020), we shall adopt this definition: the process defined by the global Hamiltonian H, the initial state ρ t 0 ⊗ ϱ B , and the family of projectors { P x } x is.

• N-classical, if Eq. 3 is satisfied for all n = 1, … , N.

Interestingly, a quantum process with classical multitime statistics always satisfies the Leggett–Garg inequalities (LGI) (Leggett and Garg, 1985; Leggett, 2008; Emary et al., 2014), which can be interpreted as a temporal analog of Bell inequalities; a violation of LGI necessarily implies that the underlying process is genuinely quantum.

For an N-classical process, the intrinsically quantum nature of the system will only possibly emerge after N+1 measurements in the prescribed PVM, but will be effectively “hidden” otherwise—that is, N+1 is the minimal number of measurements that must be performed on the system in order to see a genuinely quantum behavior. Most importantly, all these definitions are strictly dependent on both the measurement PVM, the state ρ t 0 in which the system is initially prepared, and the initial state ϱ _B of the environment: even assuming the latter to be fixed, a system exhibiting (N–) classical behavior when measured with a certain apparatus and/or initially prepared in a given state may “reveal” its quantum nature when measured or prepared differently.

A particularly simple scenario is observed when the multitime statistics satisfies the Markov property: P x n , t n | x n − 1 , t n − 1 ; … ; x 1 , t 1 = P x n , t n | x n − 1 , t n − 1 , (4) with P ( ⋅ | ⋅ ) corresponding to the conditional probability. In this special case, the whole multitime statistics of the process can be entirely reconstructed by the 1-time distribution P 1 ( x 1 , t 1 ) and the conditional probability P ( x 2 , t 2 | x 1 , t 1 ) , whence the full (infinite) hierarchy of consistency conditions (3) ends up reducing to the Chapman–Kolmogorov equations, which can be written as P 1 x 2 , t 2 = ∑ x = 0 d − 1 P 2 x 2 , t 2 ; x , t , t 2 ≥ t ≥ t 0 ; (5) P 2 x 2 , t 2 ; x 1 , t 1 = ∑ x = 0 d − 1 P 3 x 2 , t 2 ; x , t ; x 1 , t 1 , t 2 ≥ t ≥ t 1 ≥ t 0 , (6) thus reducing to only two consistency conditions; in particular, in the Markovian case, 3-classicality is sufficient for (hence equivalent to) classicality. It is also worth noting that the validity of the Markov property is ensured whenever the regression formula holds (Lax, 1963; Gardiner and Zoller, 1999), that is, when the full multitime statistics (2) induced by the system can be expressed in terms of the reduced dynamics alone: P n x n , t n ; x n − 1 , t n − 1 ; … ; x 1 , t 1 = T r P x n Λ t n , t n − 1 ⋯ P x 1 Λ t 1 , t 0 ρ t 0 ; (7) conversely, if the statistics is Markovian and Eq. 7 holds for n = 1, 2, then Eq. 7 holds for all n.

While adopting in this paper these mathematical definitions of Markovianity and classicality, it should be stressed that both physical concepts have been variously identified with diverse, generally inequivalent, mathematical properties. We refer to the reviews (Breuer et al., 2016; Li et al., 2019a; Li et al., 2019b) for a general discussion about quantum non-Markovianity; various definitions of quantum Markovianity and the intricate relations between them are presented in (Rivas et al., 2014; Li et al., 2018), while we refer to (de Vega and Alonso, 2017) for an overview of various approaches to describe the dynamics of non-Markovian open systems, and (Chruściński, 2022) for the mathematical and physical properties of non-Markovian dynamical maps.

The definition adopted in this paper is closely related to the mathematical formulation of quantum Markov stochastic processes proposed in (Davies and Lewis, 1970; Lindblad, 1975; Accardi et al., 1982) (cf. also the review (Milz and Modi, 2021)) and closely related to the recent approach to quantum Markovianity proposed in (Pollock et al., 2018a; Pollock et al., 2018b; Milz et al., 2019), where the Markovianity of the corresponding process is characterized in terms of the factorization of the so-called quantum process tensor of the underlying quantum system, this property in turn being equivalent to the validity of quantum regression (Li et al., 2018). For a discussion of Markovianity based on the quantum regression formula see also (Guarnieri et al., 2014; Lo Gullo et al., 2014); exact results on the validity of quantum regression in simple models were found in (Lonigro and Chruściński, 2022a; Lonigro and Chruściński, 2022b; Chruściński et al., 2022).

Similarly, many other approaches to the problem of classicality of quantum systems can be found in the literature. For example, in (Chen et al., 2018) the quantum evolution of an open system represented by a dynamical map Λ _t,s is said to be classical it can be simulated by an ensemble of Hamiltonians {p _k , H _k } on H S , with { p k } k being a probability distribution, such that, for any system state ρ t 0 , one has Λ t , t 0 ρ t 0 = ∑ k p k U t , t 0 k ρ t 0 U t , t 0 k † , (8) with U t , t 0 k = e − i ( t − t 0 ) H k : in this case, the (non-unitary) evolution of the system can be reproduced via a purely classical averaging procedure over distinct unitary evolutions. The above representation is often applied for disordered quantum systems described by Hamiltonian ensembles (Gneiting et al., 2016; Kropf et al., 2016; Gneiting and Nori, 2017a; Gneiting and Nori, 2017b). Note that such a concept of classicality immediately implies that the dynamics has to be unital, i.e. Λ t , t 0 ( 1 ) = 1 for all t ≥ t ₀ (cf. also (Burgarth et al., 2017)). With this approach, in (Chen et al., 2019) the (non) classicality of pure dephasing processes is analyzed for qubit systems, and a corresponding non-classicality measure is proposed.

In this paper we shall investigate the multitime statistics for a quantum system undergoing a pure dephasing evolution, focusing on the existence of particular choices of the initial state ρ t 0 and the measurement basis for which classicality (possibly up to a finite number of measurements) is achieved, thus effectively “hiding” its quantum nature. This problem was recently analyzed in (Sakuldee and Cywiński, 2022), albeit from a slightly different perspective: the authors consider the scenario in which the system, after each measurement, is reset to the initial state ρ t 0 . We shall return to this point in Section 5. Here we will first find non-trivial dephasing processes that are 2-classical, and then focus on the Markovian scenario, where necessary and sufficient conditions for the classicality (at any order) of a dephasing process are found. Interestingly, excepting particular cases, classicality can be achieved when the dephasing basis and the measurement one are either fully compatible or fully incompatible—that is, in the latter case, mutually unbiased bases (MUBs) (Wootters and Fields, 1989) (cf. also the review (Durt et al., 2010)), possibly hinting at a deeper relation between the two concepts.

The paper is organized as follows. In Section 2 we provide a brief introduction about dephasing processes, in particular recalling a simple characterization of Markovian dephasing processes. In Section 3 we show that a system initially prepared in a state which is diagonal in the dephasing basis, when measured with a proper class of unbiased bases with respect to the dephasing one, is 2-classical but generally not 3-classical; an analogous result is then shown to hold, under some conditions on the environment, for an arbitrary measurement basis if the initial state is the maximally mixed one. In Section 4 we finally focus on the Markovian scenario and find conditions under which, in this case, the system is in fact classical when measured with each of these unbiased bases. Final considerations are outlined in Section 5.

Reprising the notation of Section 1, we shall consider global Hamiltonians on H S ⊗ H B in the following form. Given a PVM { E j } j ⊂ H S on the Hilbert space of the system, define H = ∑ j E j ⊗ H j , (9) with { H j } j being a family of self-adjoint operators on H B , whence H is self-adjoint itself. Suppose that, at the initial time t ₀, the global state of the system and the environment is a product state ρ t 0 ⊗ ϱ B , and define. U t , s j , ℓ = U t , s j ⋅ U t , s ℓ † , (10) E j , ℓ = E j ⋅ E ℓ , (11) with U t , s j being the unitary propagator associated with H _j . The map Λ t , t 0 describing the reduced dynamics of the system is then given by Λ t , t 0 ρ t 0 = ∑ j , ℓ T r U t , t 0 j , ℓ ϱ B E j , ℓ ρ t 0 , (12) thus being dependent on a matrix-valued function which we shall refer to as the dephasing matrix of the process, crucially depending on the interplay between the block Hamiltonians { H j } j and the environment state ϱ _B. This is essentially a slight generalization of a typical dephasing channel, since at this stage we do not assume the projectors E _j to be rank-one.

The multitime statistics describing repeated measurements with a given PVM { P x } x can be readily computed: for every n, the joint probability distribution reads P n x n , t n ; … ; x 1 , t 1 = ∑ j n , l n ⋯ ∑ j 1 , l 1 T r P x n E j n , ℓ n ⋯ P x 1 E j 1 , ℓ 1 ρ t 0 T r U t n , t n − 1 j n , ℓ n ⋯ U t 1 , t 0 j 1 , ℓ 1 ϱ B , (13) again with P x = P x ( ⋅ ) P x . Each term of the sum decomposes into the product of a term containing all information about the preparation–measurement process, and another encoding all information about the environment. We may refer to the latter as the dephasing tensor. In particular, by a direct check one obtains the following result:

Proposition 2.1

(Lonigro and Chruściński, 2022b). The dephasing system satisfies the regression equality (2) if and only if T r U t n , t n − 1 j n , ℓ n ⋯ U t 1 , t 0 j 1 , ℓ 1 ϱ B = ∏ k = 1 n T r U t k , t k − 1 j k , ℓ k ϱ B , (14) with U t , s j , ℓ as in Eq. 10 .

While this claim is proven in (Lonigro and Chruściński, 2022b) for a standard dephasing channel (i.e., all projectors E _j being rank-one), its extension to the general case is immediate. Eq. 14 means that, for dephasing processes, the validity of the regression formula—and hence Markovianity—reduces to a factorization property of the n-time dephasing tensor in terms of 2-time tensors, i.e. dephasing matrices. This condition is clearly independent of the particular preparation–measurement protocol: the Markovianity of any dephasing process only depends on the initial environment state ϱ _B and the Hamiltonian H, without any dependence on either the initial system state ρ t 0 nor the particular choice of measurement PVM. In this sense, Proposition 2.1 provides a complete characterization of (non) Markovianity in dephasing processes. As we will see, the situation is much more involved for classicality, in which both the initial state ρ t 0 and the measurement PVM play a fundamental role.

The hierarchy of conditions (14) has two important consequences. First of all, it contains the following condition T r U t 2 , t 0 j , ℓ ϱ B = T r U t 2 , t 1 j , ℓ ϱ B T r U t 1 , t 0 j , ℓ ϱ B , (15) which, for time-independent Hamiltonians, implies, for some ϵ j ℓ ∈ R and γ _jℓ ≥ 0, T r U t , s j , ℓ ϱ B = e − i ϵ j ℓ + 1 2 γ j ℓ t − s , t ≥ s ≥ t 0 , (16) and clearly means that the reduced dynamics induced by H must satisfy the semigroup property, whence Λ _t,s must be a GKLS semigroup (Gorini et al., 1976; Lindblad, 1976). However, keep in mind that this is only a necessary condition: counterexamples of dephasing semigroups blatantly violating regression can be easily constructed. However, while very restrictive, this condition is satisfied in (at least) one case of physical interest: by choosing H as the dephasing-type spin–boson model with flat form factor, and ϱ _B as the vacuum state of the boson field, the couple (H, ϱ _B) does indeed satisfy the full hierarchy (14), as shown in (Lonigro and Chruściński, 2022b).

Furthermore, Eq. 14 cannot hold in the commutative case—that is, when the Hamiltonians { H j } j form a commutative family ² —unless the dephasing is trivial, that is, all elements of the dephasing matrix have unit modulus, T r U t 2 , t 1 j , ℓ ϱ B 2 = 1 , (17) for all values of the parameters: the reduced dynamics induced on ρ t 0 (cf. Eq. 12) only involves its off-diagonal elements acquiring a time-dependent phase term, their modulus being unchanged (Lonigro and Chruściński, 2022b).

To conclude, for future convenience let us list here some elementary properties of the dephasing matrix and tensor.

Proposition 2.2

For every n and every value of the indices, we have T r U t n , t n − 1 j n , j n U t n − 1 , t n − 2 j n − 1 , ℓ n − 1 ⋯ U t 1 , t 0 j 1 , ℓ 1 ϱ B = T r U t n − 1 , t n − 2 j n − 1 , ℓ n − 1 ⋯ U t 1 , t 0 j 1 , ℓ 1 ϱ B ; (18) besides, in the case in which the dephasing is either Markovian or generated by commuting Hamiltonians, for every k = 1, … , n we have T r U t n , t n − 1 j n , ℓ n ⋯ U t k + 1 , t k j k + 1 , ℓ k + 1 U t k , t k − 1 j k , j k U t k − 1 , t k − 2 j k − 1 , ℓ k − 1 ⋯ U t 1 , t 0 j 1 , ℓ 1 ϱ B = T r U t n , t n − 1 j n , ℓ n ⋯ U t k + 1 , t k j k + 1 , ℓ k + 1 U t k − 1 , t k − 2 j k − 1 , ℓ k − 1 ⋯ U t 1 , t 0 j 1 , ℓ 1 ϱ B . (19)

That is: in the general case, all entries of the n-time dephasing tensor characterized by the nth couple of indices (j _n , ℓ _n ) having equal values can be evaluated by substituting the unitary map U t n , t n − 1 ( j n , j n ) with the identity map; furthermore, if either Eq. 14 or the block Hamiltonians form a commuting family, the same holds for every other pair (j _k , ℓ _k ). As we will see, the latter property will greatly simplify the calculation of multitime probabilities in the particular cases we are interested in.

We shall now search for particular preparation–measurement protocols which yield a classical or N-classical process. This will ultimately depend on the interplay between the two PVMs that enter the process:

• the dephasing PVM { E j } j in the definition (9) of the global Hamiltonian;

but in general may—and will—also depend non-trivially on the properties of the environment.

Clearly, if the two PVMs coincide, then the process is indeed classical—and shows a trivial dependence on time. Indeed, in this case we must set P x n = E x n , x n in Eq. 13, whence P n x n , t n ; … ; x 1 , t 1 = δ x n , x n − 1 ⋯ δ x 2 , x 1 T r P x 1 ρ t 0 , (20) thus obtaining an elementary process, independent of all times t ₁, … , t _n , which is clearly classical: this is an obvious consequence of the fact that the measurement do not interfere at all with the dephasing. More generally, the same happens if the two PVMs are commuting. Clearly, a genuinely time-dependent (and generally non-classical) process can only be obtained when the two PVMs do not commute, that is, when the measurement is not fully compatible with the dephasing.

We may wonder whether there exist preparation–measurement protocols which yield time-dependent dephasing processes exhibiting a classical behavior, that is, satisfying the consistency condition 3) possibly up to some finite order, despite the fact that the system is measured with some PVM not fully compatible with its “natural” one. We shall start in this section by analyzing some of these configurations which ensure 2-classicality, and then proceed in Section 4 to investigate the Markovian case, where classicality at any order can be in fact achieved.