Classicality, Markovianity and local detailed balance from pure state dynamics

Abstract

When describing the effective dynamics of an observable in a many-body system, the repeated randomness assumption, which states that the system returns in a short time to a maximum entropy state, is a crucial hypothesis to guarantee that the effective dynamics is classical, Markovian and obeys local detailed balance. While the latter behaviour is frequently observed in naturally occurring processes, the repeated randomness assumption is in blatant contradiction to the microscopic reversibility of the system. Here, we show that the use of the repeated randomness assumption can be justified in the description of the effective dynamics of an observable that is both slow and coarse, two properties we will define rigorously. Then, our derivation will invoke essentially only the eigenstate thermalization hypothesis and typicality arguments. While the assumption of a slow observable is subtle, as it provides only a necessary but not sufficient condition, it also offers a unifying perspective applicable to, e.g., open systems as well as collective observables of many-body systems. All our ideas are numerically verified by studying density waves in spin chains.

Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace

“…A further related result is given by Strasberg et al [28], who consider repeated measurements at 0 < t 1 < t 2 < • • • < t r < T of all P ν 's and argue that the probability distribution of the outcomes is essentially indistinguishable from the joint distribution of X t 1 , . .…”

Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach

“…3) is about how to prove classicality by extending recent research results [7,8,15]. Therein it has been observed that isolated many-body systems can behave classical even in presence of large amount of coherences and that non-integrability and chaos might be the key to understand this behaviour.…”

“…Classicality as considered here therefore has a clear operational meaning, which was also used in Ref. [7][8][9][10][11][12][13][14][15]: a process is classical if (at least in principle) a classical stochastic process can be used to generate the same measurement statistics. The idea of defining classicality in this way is rooted in a "black-box-mentality": there might be some very expensive quantum computer in front of you, but if the available measurement statistics can be simulated, or emulated, with a classical stochastic processes, then the measurement statistics alone do not allow you to draw the conclusion that there is anything quantum going on in the computer.…”

“…In particular, Ref. [15] argued that this is a generic effect for a large class of observables (specified in greater detail later on) and estimated that deviations from classical behaviour are exponentially small in the system size. Here, we confirm this estimate and provide an alternative derivation of it in Sec.…”

“…The first contribution is to give a focused overview over three different approaches to the question what needs to be proved. These are the decoherence approach in open quantum systems (OQS) [1][2][3][4], the consistent/decoherent histories formalism [5,6], and a recent approach based on the notion of Kolmogorov consistency [7][8][9][10][11][12][13][14][15]. The second contribution emphasizes the crucial role played by the rigorous definition of (multi-time) quantum Markovianity [16][17][18] (for introductions see Refs.…”