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In classical stochastic theory, the joint probability distributions of a stochastic process obey by definition the Kolmogorov consistency conditions. Interpreting such a process as a sequence of physical measurements with probabilistic outcomes, these conditions reflect that the measurements do not alter the state of the underlying physical system. Prominently, this assumption has to be abandoned in the context of quantum mechanics, yet there are also classical processes in which measurements influence the measured system. Here, conditions that characterize uniquely classical processes that are probed by a reasonable class of such invasive measurements are derived. We then analyze under what circumstances such classical processes can simulate the statistics arising from quantum processes associated with informationally-complete measurements. It is expected that this investigation will help build a bridge between two fundamental traits of non-classicality, namely, coherence and contextuality.
One often assumes that one can completely measure classical systems without altering them. In real scenarios, however, this may not always be possible: measuring the position of a small particle can randomly change its momentum and subsequent evolution. Here, the exact conditions are derived, which characterize when a statistics can be simulated classically, but assuming invasive measurements. It is shown that not all quantum experiments can be simulated in such a way.image
