Operational Markov Condition for Quantum Processes

Abstract

We derive a necessary and sufficient condition for a quantum process to be Markovian which coincides with the classical one in the relevant limit. Our condition unifies all previously known definitions for quantum Markov processes by accounting for all potentially detectable memory effects. We then derive a family of measures of non-Markovianity with clear operational interpretations, such as the size of the memory required to simulate a process, or the experimental falsifiability of a Markovian hypothesis.In classical probability theory, a stochastic process is the collection of joint probability distributions of a system's state (described by random variable X) at different times, {P (X k , t k ; X k−1 , t k−1 ; . . . ; X 1 , t 1 ; X 0 , t 0 ) ∀k ∈ N}; to be a valid process, these distributions must additionally satisfy the Kolmogorov consistency conditions [1]. A Markov process is one where the state X k of the system at any time t k only depends conditionally on the state of the system at the previous time step, and not on the remaining history. That is, the conditional probability distributions satisfy P (X k , t k |X k−1 , t k−1 ;. . .; X 0 , t 0 ) = P (X k , t k |X k−1 , t k−1 ) (1) for all k. This simple looking condition has profound implications, leading to a massively simplified description of the stochastic process. The study of such processes forms an entire branch of mathematics, and the evolution of physical systems is frequently approximated to be Markov (when it is not exactly so). This is in part due to the fact that the properties of Markov processes make them easier to manipulate analytically and computationally [2].Implicit in this description of a classical process is the assumption that the value of X j at a given time can be observed without affecting the subsequent evolution. This assumption cannot be valid for quantum processes. In quantum theory, a measurement must be performed to infer the state of system. And the measurement process, in general, must disturb that state. Therefore, unlike its classical counterpart, a generic quantum stochastic process cannot be described without interfering with it [3]. These complications make it challenging to define the process independently of the control operations of the experimenter. From a technical perspective, a serious consequence of this is that joint probability distributions of quantum observables at different times do not satisfy the Kolmogorov conditions [1], and do not constitute stochastic processes in the classical sense.Nevertheless, temporal correlations between observables do play an important role in the dynamics of many open quantum systems, e.g. in the emission spectra of quantum dots [4] and in the vibrational motion of interacting molecular fluids [5]. Quantifying memory effects, and clearly defining * felix.pollock@monash.edu † kavan.modi@monash.edu the boundary between Markovian and non-Markovian quantum processes, represents an important challenge in describing such systems. Attempts at solving this problem tend to take a...

An Introduction to Operational Quantum Dynamics

“…This equation is an analogue of (29) and (34). That is, it is the operator sum (or Kraus) representation for the process tensor, which (like the superchannel) implies that it is CP [74]. It also clearly satisfies the containment property, i.e., T j:k ⊂ T i:l for all i ≤ j ≤ k ≤ l and it is trace preserving in the sense that it maps sequences of CPTP maps to unit trace objects.…”

“…Mathematically, this means that the dynamics is a mapping T k:0 : B(B(H)) ⊗k → B(H), called a process tensor [74], whose action can be written as…”

“…In [20,74] the existence of a generalised Stinespring dilation was proven; a map T k:0 is consistent with se unitary dynamics if it is linear, CP, trace preserving in the sense that it maps sequences of trace preserving control operations to unit trace matrices, and possesses a containment property, T j:k ⊂ T i:l for all i ≤ j ≤ k ≤ l. The latter property is a causality property ensuring that future actions do not affect past dynamics. Conversely, the process tensor can be derived starting from a dilated (unitary) se evolution, as shown in Fig.…”

“…Proving that the dynamics of an open system for a particular initial state is CP amounts to showing that it can be written in terms of a Kraus decomposition [46,4,82,74], and the existence of a minimal Kraus decomposition can be employed to show the existence of generalised Stinespring dilations [19,20].…”

“…Comparable approaches to general quantum stochastic processes were already developed by Lindblad [51] and Accardi et al [1, 2], but have not gained traction with the community of researchers working on open quantum systems. The process tensor framework straightforwardly leads to several important results, most notably an operationally well-defined quantum Markov condition, measures of non-Markovianity (which we will briefly expand on below) [74], and a generalisation of the Kolmogorov extension theorem to general quantum stochastic processes [2].…”