  We develop a general framework to investigate fluctuations of non-commuting
observables. To this end, we consider the Keldysh quasi-probability
distribution (KQPD). This distribution provides a measurement-independent
description of the observables of interest and their time-evolution.
Nevertheless, positive probability distributions for measurement outcomes can
be obtained from the KQPD by taking into account the effect of measurement
back-action and imprecision. Negativity in the KQPD can be linked to an
interference effect and acts as an indicator for non-classical behavior.
Notable examples of the KQPD are the Wigner function and the full counting
statistics, both of which have been used extensively to describe systems in the
absence as well as in the presence of a measurement apparatus. Here we discuss
the KQPD and its moments in detail and connect it to various time-dependent
problems including weak values, fluctuating work, and Leggett-Garg
inequalities. Our results are illustrated using the simple example of two
subsequent, non-commuting spin measurements.
