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We develop a general framework to investigate fluctuations of non-commutingobservables. To this end, we consider the Keldysh quasi-probabilitydistribution (KQPD). This distribution provides a measurement-independentdescription of the observables of interest and their time-evolution.Nevertheless, positive probability distributions for measurement outcomes canbe obtained from the KQPD by taking into account the effect of measurementback-action and imprecision. Negativity in the KQPD can be linked to aninterference effect and acts as an indicator for non-classical behavior.Notable examples of the KQPD are the Wigner function and the full countingstatistics, both of which have been used extensively to describe systems in theabsence as well as in the presence of a measurement apparatus. Here we discussthe KQPD and its moments in detail and connect it to various time-dependentproblems including weak values, fluctuating work, and Leggett-Garginequalities. Our results are illustrated using the simple example of twosubsequent, non-commuting spin measurements.

Quasi-probability distributions for observables in dynamic systems