Markovian processes in classical and quantum statistical mechanics

Philip Taranto, Thomas J. Elliott, Simon Milz

In classical physics, memoryless dynamics and Markovian statistics are one and the same. This is not true for quantum dynamics, first and foremost because quantum measurements are invasive. Going beyond measurement invasiveness, here we derive a novel distinction between classical and quantum processes, namely the possibility of hidden quantum memory . While Markovian statistics of classical processes can always be reproduced by a memoryless dynamical model, our main result shows that this is not true in quantum mechanics: We first provide an example of quantum non-Markovianity whose manifestation depends on whether or not a previous measurement is performed – an impossible phenomenon for memoryless dynamics; we then strengthen this result by demonstrating statistics that are Markovian independent of how they are probed, but are nonetheless s t i l l incompatible with memoryless quantum dynamics. Thus, we establish the existence of Markovian statistics gathered by probing a quantum process that nevertheless f u n d a m e n t a l l y require memory for their creation.

Masao Nagasawa

Bernard H. Lavenda

Andrea Smirne, Dario Egloff, María García Díaz +2 more

Abstract Although quantum coherence is a basic trait of quantum mechanics, the presence of coherences in the quantum description of a certain phenomenon does not rule out the possibility to give an alternative description of the same phenomenon in purely classical terms. Here, we give definite criteria to determine when and to what extent quantum coherence is equivalent to non-classicality. We prove that a Markovian multi-time statistics obtained from repeated measurements of a non-degenerate observable cannot be traced back to a classical statistics if and only if the dynamics generates coherences and subsequently turns them into populations. Furthermore, we show with simple examples that such connection between quantum coherence and non-classicality is generally absent if the statistics is non-Markovian.

Neil Dowling, Pedro Figueroa–Romero, Felix A. Pollock +2 more

Quantum stochastic processes are shown, within finite time intervals and for finite dimensional time-independent Hamiltonians, to be close to a corresponding stationary equilibrium process. This occurs for multitime observables that are coarse grained in both space and time, and for systems which have many significantly interacting energy eigenstates. This dictates conditions on time scales at when the multitime statistics of any arbitrary non-equilibrium quantum process look equilibrium on average. We show that this leads to the equilibration of general geometric measures of quantum processes, such as the degree of non-Markovianity or classicality. The implications of this result for the emergence of classical stochastic processes from underlying quantum mechanics are also discussed.

Burkhard Kümmerer