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. K. Zachos, D. B. Fairlie, and T. L. Curtright (Editors). Quantum Mechanics in Phase Space: An Overview with Selected Papers, (World Scientific 2005).
 R. W. Spekkens. Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett. 101, 020401 (2008).
 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).
 C. Ferrie and J. Emerson. Framed Hilbert space: hanging the quasi-probability pictures of quantum theory. New J. Phys. 11, 063040 (2009).
 C. Ferrie. Quasi-probability representations of quantum theory with applications to quantum information science. Rep. Prog. Phys. 74, 116001 (2011).
 M. Revzen, P. A. Mello, A. Mann, and L. M. Johansen. Bell's inequality violation with non-negative Wigner functions. Phys. Rev. A 71, 022103 (2005).
 A. Bednorz and W. Belzig. Quasiprobabilistic interpretation of weak measurements in mesoscopic junctions. Phys. Rev. Lett. 105, 106803 (2010).
 A. Bednorz, W. Belzig, and A. Nitzan. Nonclassical time correlation functions in continuous quantum measurement. New J. Phys. 14, 013009 (2012).
 P. P. Hofer and A. A. Clerk. Negative full counting statistics arise from interference effects. Phys. Rev. Lett. 116, 013603 (2016).
 H. Zhu. Quasiprobability representations of quantum mechanics with minimal negativity. Phys. Rev. Lett. 117, 120404 (2016).
 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, C. Ferrie, D. Gross, and J. Emerson. Negative quasi-probability as a resource for quantum computation. New J. Phys. 14, 113011 (2012).
 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).
 L. Mandel. Non-classical states of the electromagnetic field. Phys. Scr. 1986, 34 (1986).
 W. Vogel. Nonclassical correlation properties of radiation fields. Phys. Rev. Lett. 100, 013605 (2008).
 A. A. Clerk, F. Marquardt, and J. G. E. Harris. Quantum measurement of phonon shot noise. Phys. Rev. Lett. 104, 213603 (2010).
 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).
 B. P. Venkatesh, G. Watanabe, and P. Talkner. Quantum fluctuation theorems and power measurements. New J. Phys. 17, 075018 (2015).
 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).
 M. F. Pusey. Anomalous weak values are proofs of contextuality. Phys. Rev. Lett. 113, 200401 (2014).
 J. Dressel and A. N. Jordan. Weak values are universal in von Neumann measurements. Phys. Rev. Lett. 109, 230402 (2012).
 C. Emary, N. Lambert, and F. Nori. Leggett–Garg inequalities. Rep. Prog. Phys. 77, 016001 (2014).
 T. Fritz. Quantum correlations in the temporal Clauser–Horne–Shimony–Holt (CHSH) scenario. New J. Phys. 12, 083055 (2010).
 N. S. Williams and A. N. Jordan. Weak values and the Leggett-Garg inequality in solid-state qubits. Phys. Rev. Lett. 100, 026804 (2008).
 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).
 D. Dasenbrook and C. Flindt. Dynamical scheme for interferometric measurements of full-counting statistics. Phys. Rev. Lett. 117, 146801 (2016).
 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, and Justin Dressel, "Quasiprobability behind the out-of-time-ordered correlator", Physical Review A 97 4, 042105 (2018).
 Gabriele De Chiara, Paolo Solinas, Federico Cerisola, and Augusto J. Roncaglia, Fundamental Theories of Physics 195, 337 (2018) ISBN:978-3-319-99045-3.
 Patrick P. Potts, "Certifying Nonclassical Behavior for Negative Keldysh Quasiprobabilities", Physical Review Letters 122 11, 110401 (2019).
 Harry J. D. Miller, Matteo Scandi, Janet Anders, and Martí Perarnau-Llobet, "Work Fluctuations in Slow Processes: Quantum Signatures and Optimal Control", Physical Review Letters 123 23, 230603 (2019).
 Kang-Da Wu, Elisa Bäumer, Jun-Feng Tang, Karen V. Hovhannisyan, Martí Perarnau-Llobet, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo, "Minimizing Backaction through Entangled Measurements", Physical Review Letters 125 21, 210401 (2020).
 Yu-Xin Wang and A. A. Clerk, "Spectral characterization of non-Gaussian quantum noise: Keldysh approach and application to photon shot noise", Physical Review Research 2 3, 033196 (2020).
 Matteo Lostaglio, "Quantum Fluctuation Theorems, Contextuality, and Work Quasiprobabilities", Physical Review Letters 120 4, 040602 (2018).
 Joonhyun Yeo, "Symmetry and its breaking in a path-integral approach to quantum Brownian motion", Physical Review E 100 6, 062107 (2019).
 Adam Teixidó-Bonfill, Alvaro Ortega, and Eduardo Martín-Martínez, "First law of quantum field thermodynamics", Physical Review A 102 5, 052219 (2020).
 "Leggett-Garg Inequalities for Quantum Fluctuating Work", Entropy 20 3, 200 (2018).
 Amikam Levy and Matteo Lostaglio, "Quasiprobability Distribution for Heat Fluctuations in the Quantum Regime", PRX Quantum 1 1, 010309 (2020).
 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 the Landauer-Büttiker state", arXiv:1706.00561, Physical Review E 96 5, 052124 (2017).
 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).
 J Sperling and W Vogel, "Quasiprobability distributions for quantum-optical coherence and beyond", Physica Scripta 95 3, 034007 (2020).
 Matteo Scandi, Harry J. D. Miller, Janet Anders, and Martí Perarnau-Llobet, "Quantum work statistics close to equilibrium", Physical Review Research 2 2, 023377 (2020).
 F. Krumm, W. Vogel, and J. Sperling, "Time-dependent quantum correlations in phase space", Physical Review A 95 6, 063805 (2017).
 Gaomin Tang, Yanxia Xing, and Jian Wang, "Short-time dynamics of molecular junctions after projective measurement", Physical Review B 96 7, 075417 (2017).
The above citations are from Crossref's cited-by service (last updated successfully 2021-01-24 11:15:45) and SAO/NASA ADS (last updated successfully 2021-01-24 11:15:46). 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.