Almost Markovian processes from closed dynamics

Pedro Figueroa-Romero; Kavan Modi; Felix A. Pollock

doi:10.22331/q-2019-04-30-136

Almost Markovian processes from closed dynamics

Pedro Figueroa-Romero, Kavan Modi, and Felix A. Pollock

School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia

Published:	2019-04-30, volume 3, page 136
Eprint:	arXiv:1802.10344v5
Doi:	https://doi.org/10.22331/q-2019-04-30-136
Citation:	Quantum 3, 136 (2019).

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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 $\textit{almost}$ 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.

Featured image: In the space of all processes $\Upsilon$, the probability that the non-Markovianity $\mathcal{N}$ is larger than $\mathcal{B}_k\geq\mathbb{E}[\mathcal{N}]$ by some $\epsilon > 0$ decreases exponentially in $\epsilon^2$. When the environment dimension is much larger than that of the system, $\mathcal{B}_k$ becomes small and quantum processes concentrate around the Markovian ones $\Upsilon^{\scriptstyle{(\mathrm{M})}}$.

► BibTeX data

@article{FigueroaRomero2019almostmarkovian,
  doi = {10.22331/q-2019-04-30-136},
  url = {https://doi.org/10.22331/q-2019-04-30-136},
  title = {Almost {M}arkovian processes from closed dynamics},
  author = {Figueroa-Romero, Pedro and Modi, Kavan and Pollock, Felix A.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {136},
  month = apr,
  year = {2019}
}

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