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.
 C. Ferrie and J. Emerson. Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations. J. Phys. A: Math. Theor. 41, 352001 (2008).
 E. Arthurs and J. L. Kelly. B.S.T.J. briefs: On the simultaneous measurement of a pair of conjugate observables. Bell Syst. Tech. J. 44, 725 (1965).
 V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson. Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation. New J. Phys. 15, 013037 (2013).
 S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan, K. W. Murch, and I. Siddiqi. Mapping the optimal route between two quantum states. Nature 511, 570 (2014).
 S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche. Reconstruction of non-classical cavity field states with snapshots of their decoherence. Nature 455, 510 (2008).
 M. Perarnau-Llobet, E. Bäumer, K. V. Hovhannisyan, M. Huber, and A. Acin. No-go theorem for the characterization of work fluctuations in coherent quantum systems. Phys. Rev. Lett. 118, 070601 (2017).
 M. Esposito, U. Harbola, and S. Mukamel. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665 (2009).
 T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra. Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system. Phys. Rev. Lett. 113, 140601 (2014).
 Y. Aharonov, D. Z. Albert, and L. Vaidman. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988).
 J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo. Partial-measurement backaction and nonclassical weak values in a superconducting circuit. Phys. Rev. Lett. 111, 090506 (2013).
 A. V. Lebedev, G. B. Lesovik, and G. Blatter. Optimal noninvasive measurement of full counting statistics by a single qubit. Phys. Rev. B 93, 115140 (2016).
 Nicole Yunger Halpern, Brian Swingle, Justin Dressel, "Quasiprobability behind the out-of-time-ordered correlator", Physical Review A 97 4, 042105 (2018).
 Matteo Lostaglio, "Quantum Fluctuation Theorems, Contextuality, and Work Quasiprobabilities", Physical Review Letters 120 4, 040602 (2018).
 "Leggett-Garg Inequalities for Quantum Fluctuating Work", Entropy 20 3, 200 (2018).
 Camille Aron, Giulio Biroli, Leticia Cugliandolo, "(Non) equilibrium dynamics: a (broken) symmetry of the Keldysh generating functional", SciPost Physics 4 1, 008 (2018).
 Mihail Mintchev, Luca Santoni, Paul Sorba, "Quantum fluctuations of entropy production for fermionic systems in the Landauer-Büttiker state", Physical Review E 96 5, 052124 (2017).
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