Precursors of non-Markovianity

Open quantum systems provide the framework for describing how a system of interest interacts with its surroundings. The ubiquity of this paradigm has led to a wide variety of techniques to model, characterize, and exploit the manner in which an environment affects a system’s evolution Alicki and Lendi (1987); Weiss (1993); Breuer and Petruccione (2002); Rivas and Huelga (2012). In the context of open quantum systems, typically the system of interest is much smaller than its environment and in this case, if the coupling is weak, the dynamics of the system can be well captured by a memoryless or Markovian evolution.

The failure of such an approximation in the presence of strong coupling, structured reservoirs, or non-negligible system effects on the environment, naturally leads us to explore dynamics with some form of memory. Dynamics of this sort are typically referred to by the catch-all term non-Markovianity and have been the subject of intense activity, evidenced by the development of a range of techniques to simulate and characterize a wide range of non-Markovian dynamics, see e.g. the recent reviews Breuer (2012); Breuer et al. (2016); Rivas et al. (2014); de Vega and Alonso (2017); Li et al. (2018). Indeed beyond being a topic of interest in itself, recent work has shown that non-Markovianity of the dynamics can provide an enhancement in a diverse array of settings and tasks, including quantum-metrology, quantum memories, information processing, and thermodynamic cycles Huelga et al. (2012); Chin et al. (2012); Cimmarusti et al. (2015); Pezzutto et al. (2019); Çakmak et al. (2019); Sakuldee et al. (2018); Dong et al. (2018).

Despite the clear relevance of the field and its success in showing potential applications, understanding the fundamental mechanisms and features that give rise to quantum non-Markovianity remains a difficult task. Indeed the very foundations of the theory are still the object of intense investigations Pollock et al. (2018); Taranto et al. (2019), the reason for which is due to the fact that different approaches to defining non-Markovianity have been devised, whose relationship is currently under study Chruscinski et al. (2011); Wißmann et al. (2015); Breuer et al. (2018); Chruscinski et al. (2018).

In this work we attempt to underpin some fingerprints of a non-Markovian evolution by considering precursors of non-Markovianity, i.e. features of the system and environment state at some time, $s$, that serve as necessary (but not sufficient) indicators that the dynamics can show a non-Markovian behavior at a later time $t\!\char 62\relax\!s$. These quantities can be introduced in the framework of non-Markovianity described as a backflow of information from the environment to the system Breuer et al. (2009, 2016). In this picture, bounds can be introduced that explicitly relate the behavior of a particular indicator of non-Markovianity with the features of the joint system-environment state Laine et al. (2010); Amato et al. (2018). Precursors of non-Markovianity provide necessary conditions for tracing revivals in a suitable distinguishability quantifier back to a combination of established system-environment correlations and changes in the environmental state. A significant limitation of the approach is that it calls for evaluation of features of both the system and the total environment, which is generally too demanding a task, and as such investigations along these lines have rarely been considered Mazzola et al. (2012); Rodríguez-Rosario et al. (2012); Smirne et al. (2013). However, it has been shown that when the environment can be described as a collection of individual subsystems, often only a small subset of the total environmental degrees of freedom plays a significant role in driving the evolution Iles-Smith et al. (2014); Campbell et al. (2018). By relying on the Bures distance and considering collision models of open quantum system dynamics Rau (1963); Brun (2002); Scarani et al. (2002); Ziman and Bužek (2005), where the environment is modelled by an ensemble of individual ancillae with which the system sequentially interacts, we are able to introduce and study a hierarchy of precursors of non-Markovianity for systems in both discrete and continuous variable settings. Our work paves the way for considering the relevance of such precursors in a variety of settings which allow for the systematic introduction of environmental degrees of freedom, e.g. within the framework of reaction coordinate models or chain mappings Hughes et al. (2009); Chin et al. (2010); Martinazzo et al. (2011); Woods et al. (2014); Strasberg et al. (2016); Tamascelli et al. (2018a, b).

The remainder of the paper is organized as follows. In Sec. II we define the precursors of non-Markovianity, and in particular the bound, Eq. (2), that is central to our analysis. In Sec. III we apply our framework to both discrete and continuous variable collision models, and qualitatively compare and contrast the emergence of non-Markovianity in the disparate dimensional systems. Finally, in Sec. IV we draw our conclusions.

In a seminal work, the non-Markovianity of a reduced quantum dynamics has been introduced as an indicator of information backflow from the environment to the system Breuer et al. (2009). In order to detect and estimate this backflow a distance on the state space, namely the so-called trace distance, has been considered. Apart from a normalization factor, this distance, given for any pair of states $\rho$ and $\sigma$ by

corresponds to the natural topology on the state space and has two basic features, which makes it suitable as an estimator of information backflow: (i) it is directly related to a notion of distinguishability among states; (ii) it is a contraction under the action of positive, and in particular completely positive, trace preserving maps. Consequently, it has been possible to consider the evolution in time of the trace distance between two evolved distinct initial states $\rho_{S}^{1}(0)$ and $\rho_{S}^{2}(0)$ as an indicator of non-Markovianity, associated to revivals in time of the corresponding trace distance

for some $t\!\char 62\relax\!s$ and a pair $\rho_{S}^{1}(0)$ and $\rho_{S}^{2}(0)$, as detailed for example in Ref. Breuer et al. (2016). Revivals in the trace distance correspond to revivals in distinguishability, i.e. the capability to ascertain the actual initial state by performing measurements on the system only. The information backflow associated to these revivals has been traced back to the establishment of correlations between system and environment as well as to changes in the state of the environment.

This notion of information backflow can be formalized as follows. Let us identify the total amount of information at time $t$ as the distinguishability of the states of both system and environment

This quantity is a constant and can be naturally written as the sum of two contributions referring to the information that can be obtained by performing local measurements only, namely $\mathcal{I}_{\text{int}}(t)\!=\!D(\rho^{1}_{S}(t),\rho^{2}_{S}(t))$, and to the residual information which can only be accessed measuring also the environment $\mathcal{I}_{\text{ext}}(t)\!=\!D(\rho^{1}_{SE}(t),\rho^{2}_{SE}(t))-D(\rho^{1}_{S}(t),\rho^{2}_{S}(t))$. While their sum is a constant, i.e.

nevertheless revivals in the internal information can take place, such that $\mathcal{I}_{\text{int}}(t)\!\char 62\relax\!\mathcal{I}_{\text{int}}(s)$ for $t\!\char 62\relax\!s$, and they are interpreted as information backflow. This interpretation is substantiated by the following inequality

valid for arbitrary $t\!\geqslant\!s$, thus extending previous seminal work on the study of initial correlations Laine et al. (2010); Amato et al. (2018). It appears that to have a local revival of the trace distance at a time $t$ greater than $s$, that is a positive contribution on the l.h.s., at least one of the contributions on the r.h.s., referring to time $s$, has to be positive. In this respect such quantities act as precursors of non-Markovianity, in that their positivity at time $s$ is a necessary, but not sufficient, condition in order to have a larger trace distance at a later time $t$. Let us further remark that in the case in which the environment is not affected by the system and no correlations are established, corresponding to a perfectly Markovian dynamics, the r.h.s. of the bound is strictly zero, thus forbidding any revival of distinguishability. Evaluation of Eq. (1) is however quite demanding since it calls for the calculation of the trace distance between both system and environment states, where the latter is in general high-dimensional. Indeed, even the evaluation of the trace distance for the system is a difficult task if one considers higher, possibly infinite, dimensional systems. To overcome these difficulties we will consider a different quantifier of the distinguishability among states and build on the structure of the environment in order to obtain alternative bounds, which still provide necessary conditions for the revivals in distinguishability without calling for measurements on both the system and the whole environment.

In order to consider arbitrary dimensional systems it is natural to introduce as a distance on the state space the so-called Bures distance Fuchs and van de Graaf (1999); Vasile et al. (2011)

where $F(\rho,\sigma)=\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$ denotes fidelity which is readily computable for both finite dimensional and Gaussian continuous variable systems Olivares (2012); Banchi et al. (2015). The Bures distance is indeed a metric on the state space, thus satisfying in particular the triangular inequality, and since it is defined in terms of the fidelity, it is a contraction under the action of completely positive trace preserving maps. These properties together with subadditivity with respect to the tensor product allow to reproduce the inequality Eq. (1) in the form

while retaining its physical interpretation. In this formulation we are, however, still bound to the evaluation of the Bures distance involving all system and environment degrees of freedom, which remains a very difficult task.

We now observe that for the case in which the environment exhibits a natural structuring in terms of constituent subunits (ancillae), as is the case in collision models Scarani et al. (2002); Ziman and Bužek (2005), we can construct a hierarchy of environmental marginals by taking the partial trace with respect to a growing number of environmental ancillae. Exploiting the fact that the partial trace is indeed a completely positive trace preserving transformation, supposing the environment to be composed of $N$ ancillae we have the following chain of bounds for the quantifier of correlations

For $k\!=\!0$ we recover the original upper bound $B(\rho_{SE}^{1}(s),\rho_{S}^{1}(s)\otimes\rho_{E}^{1}(s))$, while for $k\!=\!N$ we have the trivial bound $B(\rho_{S}^{1}(s),\rho_{S}^{1}(s))\!=\!0$. Similarly, for the difference in environmental states we have

Thus, it immediately appears that the different terms at the r.h.s. of Eq. (2) can be lower bounded by a whole hierarchy of positive quantities obtained by replacing the environmental states with suitable marginals obtained by tracing out different ancillae. In such a way one can obtain precursors of non-Markovianity which can be more easily evaluated. It is important to stress however that all of these quantities provide lower bounds for the r.h.s., corresponding to quantities which are increasingly easier to evaluate. In this respect, while strict positivity of either of these quantities provides a necessary condition for revivals of Bures distance between distinct pair of initial system states, lower bounding the r.h.s. might lead to a violation of the bound Eq. (2), in the sense that the lower bound for the r.h.s. of Eq. (2) is no longer an upper bound to the l.h.s. On the other hand, as we will show in the collision framework in Sec. III, it turns out that even the simplest approximation can already lead to a very good estimate of Bures distance revivals. Indeed as shown in Sec. III.3, in the collision model framework it appears that already the “minimal approximation” for the environment, treated as being a single ancilla, actually provides quite reasonable bounds, indicating that the contributions arising from the system interacting with this particular ancilla are the most relevant ones for characterizing the dynamics. This is complementary to Ref. Campbell et al. (2018) from which we know that the incoming single ancilla is the only really relevant player in dictating the non-Markovianity of the evolution in the corresponding collision model.

We remark that our choice of the Bures distance over other metrics goes beyond its comparative computational simplicity and applicability to infinite dimensional settings already mentioned. The crucial property that any candidate distinguishability measure must have is to be a contraction under the action of completely positive trace preserving maps. While this condition is met for the Bures and trace distances, other norms, such as the Hilbert-Schmidt norm, fail to satisfy it Wang and Schirmer (2009) and thus make them unsuitable choices. Moreover, in order to explicitly relate the bounds to physically meaningful notions, i.e. the established system-environment correlations or environmental changes, one requires a norm so as to exploit the triangular inequality. This requirement therefore rules out quantifiers such as the relative entropy. For these reasons the Bures distance emerges as the most natural metric for the task at hand.

Let us begin by outlining the basic collision model framework Scarani et al. (2002); Ziman and Bužek (2005); McCloskey and Paternostro (2014); Kretschmer et al. (2016); Çakmak et al. (2017); Campbell et al. (2018); Jin and Yu (2018); Strasberg et al. (2017); Ciccarello (2018) which provides a versatile tool for exploring the emergence of non-Markovianity and, due to their construction, serves as the ideal testbed for studying the precursors of non-Markovianity captured in Eq. (2) by exploiting Eqs. (3) and (4). Following Refs. McCloskey and Paternostro (2014); Campbell et al. (2018), the environment is composed of an array of individual ancillae, $E_{i}$, initially factorized and all with the same initial state. The time evolution is discretized such that at “time-step” $n$ the system, $S$, collides with ancilla, $E_{n}$, after which we retain all correlations established by this system-ancilla (SA) interaction while $E_{n}$ subsequently collides with $E_{n+1}$. At this point we can trace out the degrees of freedom associated with $E_{n}$, after which we begin time-step $n+1$ where the system collides with $E_{n+1}$ and so on. Due to the intra-environment ancilla-ancilla (AA) collisions, i.e. $E_{n}$-$E_{n+1}$, $S$ may already share some correlations with $E_{n+1}$ before they interact McCloskey and Paternostro (2014); Campbell et al. (2018). Furthermore, due to the AA collision the incoming state of the ancilla, $E_{n+1}$, that the system interacts with is not typically its initialized state. Thus, the collision model provides a natural setting to explore non-Markovianity arising due to both the establishment of correlations between system and environment together with changes in the state of the environment, as discussed in Sec. II and captured succinctly by Eq. (2).

As shown in Ref. Campbell et al. (2018), when we restrict to only nearest neighbor AA interactions this corresponds to a memory depth of one and is termed “first-order Markovian” since we need only concern ourselves with keeping track of the system and one additional ancilla to faithfully simulate of the dynamics. Thus, the evolution follows as

where $\rho_{E_{i}}(0)$ denotes the initial ancilla state. We exactly recover the standard evolution according to the Markovian master equation when there are no AA collisions, under the condition that the unitary $\mathcal{U}$ is an energy preserving exchange interaction and all constituents have the same free Hamiltonian terms McCloskey and Paternostro (2014); Pezzutto et al. (2016). Despite the framework is independent of the dimensionality of system and ancilla, most studies restrict to the discrete variable qubit states McCloskey and Paternostro (2014); Kretschmer et al. (2016); Campbell et al. (2018); Çakmak et al. (2017), with only a few exceptions Jin and Yu (2018). Therefore in this work, we are interested in examining any effect that dimensionality may have on the properties of the non-Markovian dynamics, and on the precursors of non-Markovianity, by comparing and contrasting when system and ancilla are discrete variable (DV) qubits with when they are continuous variable (CV) Gaussian states.

When considering the DV case the interaction will be given by the energy preserving partial swap operation

with

We will allow for a generic pure, real initial state of the system, written in the ordered basis $\{\left|0\right\rangle,\left|1\right\rangle\}$,

For the CV setting we will assume that SA collisions are described by a beamsplitter

whereas AA collisions are defined by

We observe that $\mathcal{U}^{\text{CV-AA}}(\theta)=\mathcal{U}^{\text{CV-SA}}(\theta-\pi/2)$: the beam splitter implementing the AA interaction has reflectivity and transmittivity inverted w.r.t. the beam splitter realizing the SA interaction. This corresponds to the shift of a phase factor from the second environmental mode to the first one and allows to avoid a “jagged” behaviour of the covariance matrix phases (see Jin and Yu (2018)). We will consider thermal squeezed initial states for the system with covariance matrices given by

In what follows we will initialize all environmental ancillae in their ground, respectively vacuum, state for both the DV and CV settings and we will fix the SA (AA) interaction strengths, Eqs. (6) and (9), for both the DV and CV cases to be $\theta=0.05\tfrac{\pi}{2}$ ($\theta=0.9\tfrac{\pi}{2}$). The energy preserving nature of the interaction ensures that the steady-state of system is driven to is exactly the initial state of one of the environmental ancillae, i.e. $\rho_{E_{i}}(0)$.

The description of non-Markovianity of quantum dynamics being due to an information backflow between system and environment leads us to identify the establishment of system-environment correlations and changes in the state of the environment as sources of non-Markovian behavior. In the trace distance approach to non-Markovianity, a bound can be introduced to relate revivals of distinguishability to such changes. We have reformulated this bound in terms of the Bures distance, which is based on the quantum state fidelity. This allows to assess a greater range of physical systems since the fidelity is comparatively easier to compute than the trace distance, in particular when one wishes to consider infinite dimensional systems. Revivals in the l.h.s of the bound, indicating periods of non-Markovianity, can be understood in terms of different contributions due to system-environment correlations and changes in the environmental state at an earlier time, thus establishing the r.h.s of the bound as capturing precursors of non-Markovianity. We have shown that the evaluation of these precursors of non-Markovianity can be simplified when the environment can be decomposed into smaller constituent parts, thus leading to a strategy of general applicability. This is possible by considering a hierarchy of lower bounds to the bound in Bures distance revivals set by the overall amount of established correlations and changes in environmental state. Such lower bounds can be considered whenever the environment exhibits a natural partition within its Hilbert space. We applied this framework to a collision model, where we explored both discrete variable (DV) and continuous variable (CV) settings. Exploiting the considered bound we established that the causes of non-Markovianity in such a collision model showed some qualitative differences between the two disparate dimensional settings, in particular while in the DV case the various contributions to non-Markovianity behaved largely the same, we found that the precursors of non-Markovianity exhibited quite a different behavior in the CV setting. While our results were based on zero-temperature environments, we stress that the same features persist for finite temperature environments and for other choices of initial system states. Our results provide a useful tool for studying the causes of a non-Markovian evolution based on readily computable quantities. Furthermore, we believe our analysis is one of the first to comparatively assess the effect that dimensionality can have on the ensuing non-Markovian character of a given evolution.