Almost Markovian processes from closed dynamics

Abstract

It is common, when dealing with quantum processes involving a subsystem of a much larger composite closed system, to treat them as effectively memory-less (Markovian). While open systems theory tells us that non-Markovian processes should be the norm, the ubiquity of Markovian processes is undeniable. Here, without resorting to the Born-Markov assumption of weak coupling or making any approximations, we formally prove that processes are close to Markovian ones, when the subsystem is sufficiently small compared to the remainder of the composite, with a probability that tends to unity exponentially in the size of the latter. We also show that, for a fixed global system size, it may not be possible to neglect non-Markovian effects when the process is allowed to continue for long enough. However, detecting non-Markovianity for such processes would usually require non-trivial entangling resources. Our results have foundational importance, as they give birth to almost Markovian processes from composite closed dynamics, and to obtain them we introduce a new notion of equilibration that is far stronger than the conventional one and show that this stronger equilibration is attained.

Quantifying non-Markovianity via conditional mutual information

“…Finally, we remark that in many circumstances the non-Markovianity will be small if the environment becomes very large. For example, in [24], it is shown by using random unitaries that almost all open quantum processes will concentrate on the Markov case, when the environment is large enough. This almost Markovian phenomenon can be intuitively understood from the perspective of local propagation of information (cf.…”

Resource theories of multi-time processes: A window into quantum non-Markovianity

“…More importantly, the process tensor enables the systematic exploration of the rich structure of quantum non-Markovian memory [38][39][40][41][42][43][44][45][46][47][48][49]. The formalism has also led to a pathway to generalise the theory of stochastic thermodynamics to quantum mechanics [50,[50][51][52][53][54].…”

Randomized benchmarking for non-Markovian noise

“…where tr j:i means trace over all except between steps i to j, or otherwise placing relevant bounds on N for Schatten-norm measures, as done in Ref. [51,52] to study some statistical properties of non-Markovian processes. Here, we will care about quantifying how non-Markovian an RB experiment is, which will boil down to quantifying how distinguishable a non-Markovian ASF is from a sensible Markovian counterpart.…”