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).
 Gaomin Tang, Yanxia Xing, and Jian Wang, "Short-time dynamics of molecular junctions after projective measurement", Physical Review B 96 7, 075417 (2017).
 Camille Aron, Giulio Biroli, and Leticia Cugliandolo, "(Non) equilibrium dynamics: a (broken) symmetry of the Keldysh generating functional", SciPost Physics 4 1, 008 (2018).
 Mihail Mintchev, Luca Santoni, and Paul Sorba, "Quantum Fluctuations of Entropy Production for Fermionic Systems in Landauer-Buttiker State", Physical Review E 96 5, 052124 arXiv:1706.00561 (2017).
 F. Krumm, W. Vogel, and J. Sperling, "Time-dependent quantum correlations in phase space", Physical Review A 95 6, 063805 (2017).
 Nicole Yunger Halpern, Brian Swingle, and 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).
 Patrick P. Potts, "Certifying Nonclassical Behavior for Negative Keldysh Quasiprobabilities", Physical Review Letters 122 11, 110401 (2019).
 "Leggett-Garg Inequalities for Quantum Fluctuating Work", Entropy 20 3, 200 (2018).
 Kang-Da Wu, Yuan Yuan, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Martí Perarnau-Llobet, "Experimentally reducing the quantum measurement back action in work distributions by a collective measurement", Science Advances 5 3, eaav4944 (2019).
The above citations are from Crossref's cited-by service (last updated 2019-03-20 09:20:01) and SAO/NASA ADS (last updated 2019-03-20 09:20:02). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.