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In the context of an Open Quantum System (OQS) i.e. a quantum system coupled to a quantum environment, what is really meant by information backflow from the environment to the system? I'm newly exposed to this field and learning basic standard stuff related to the formalism. Looking forward to a simple explanation.

More details I came across this term in an ICTP video lecture on open quantum systems by Ines de Vega (here, around $4.40$mins). The phrase 'information backflow' was used as one of the qualifiers that characterize non-Markovian dynamics. My impression is that if we start with the system in a pure state at time $t=0$, the interaction will establish entanglement between the system and the bath. Therefore, at $t>0$ the reduced density matrix for the system will not satisfy $\rho^2_S(t)=\rho_S(t)$, even though initially $\rho^2_S(0)=\rho_S(0)$. Is this referred to as information outflow? Then, what's backflow?

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  • $\begingroup$ Do you have a reasonably thorough reference talking about this? -- Other than that, it sounds like on the one hand information goes from the system into the bath, but on the other hand if the bath is not markovian, quite obviously information from the bath should also be transferred to the system. If this is information which previously went from the system to the bath, this is something I would call "backflow". $\endgroup$ – Norbert Schuch Aug 5 at 19:28
  • $\begingroup$ @NorbertSchuch To be honest, I am not much knowledgable in this field. In an ICTP video lecture on open quantum systems, I heard the phrase 'information backflow' as a qualifier of non-Markovian dynamics. My impression is that if we start with the system in a pure state at time $t=0$, the interaction will establish entanglement between the system and the bath. Therefore, at $t>0$ the reduced density matrix for the system will not satisfy $\rho^2_S(t)=\rho_S(t)$, even though initially $\rho^2_S(0)=\rho_S(0)$. Is this referred to as information outflow? Then, what's backflow? $\endgroup$ – SRS Aug 6 at 5:42
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    $\begingroup$ Your comment above contains more useful information about where you heard this phrase and in what context than your question (but it would still be good if you gave an explicit reference for where you heard the phrase you're asking about). Please edit all relevant information into your question. $\endgroup$ – ACuriousMind Aug 6 at 14:36
  • $\begingroup$ Edited! Thanks! $\endgroup$ – SRS Aug 6 at 14:48
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There are various definitions of what "information" means in this context and multiple measures have been introduced, some based on entropies and some based on other quantifiers.

One way to think about it that maybe makes the concept a bit clearer is that in a Markovian system, the reduced quantum state at time $t$ becomes more and more distinguishable from the initial state as time goes on. In a non-Markovian system, you can have a revival of indistinguishability.

In terms of literature, I would have a look at https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.012120 (arxiv: https://arxiv.org/abs/1804.06445) and references therein. There, a trace distance of the density matrices is used as. the distinguishability measure, but other measures are also discuussed. It is also pointed out that this information backflow can be neatly related to what happens in classical non-Markovian stochastic process.

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