Coherence and non-classicality of quantum Markov processes. (arXiv:1709.05267v2 [quant-ph] UPDATED)

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 is able to generate coherences
and to subsequently turn 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.

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