Quasi-probability distributions for observables in dynamic systems
Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
|Published:||2017-10-12, volume 1, page 32|
|Citation:||Quantum 1, 32 (2017).|
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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.
► BibTeX data
 E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932).
 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).
 W. K. Wootters. A Wigner-function formulation of finite-state quantum mechanics. Ann. Phys. 176, 1 (1987).
 K. S. Gibbons, M. J. Hoffman, and W. K. Wootters. Discrete phase space based on finite fields. Phys. Rev. A 70, 062101 (2004).
 K. E. Cahill and R. J. Glauber. Density operators and quasiprobability distributions. Phys. Rev. 177, 1882 (1969).
 D. F. Walls and G. J. Milburn. Quantum Optics, (Springer 1994).
 N. Lütkenhaus and S. M. Barnett. Nonclassical effects in phase space. Phys. Rev. A 51, 3340 (1995).
 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).
 L. S. Levitov, H. Lee, and G. B. Lesovik. Electron counting statistics and coherent states of electric current. J. Math. Phys. 37, 4845 (1996).
 Y. V. Nazarov (Editor). Quantum Noise in Mesoscopic Physics, (Springer 2003).
 Y. V. Nazarov and M. Kindermann. Full counting statistics of a general quantum mechanical variable. Eur. Phys. J. B 35, 413 (2003).
 A. Bednorz and W. Belzig. Quasiprobabilistic interpretation of weak measurements in mesoscopic junctions. Phys. Rev. Lett. 105, 106803 (2010).
 A. A. Clerk. Full counting statistics of energy fluctuations in a driven quantum resonator. Phys. Rev. A 84, 043824 (2011).
 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).
 J. von Neumann. Mathematische Grundlagen der Quantenmechanik, (Springer 1932).
 S. Stenholm. Simultaneous measurement of conjugate variables. Ann. Phys. 218, 233 (1992).
 A. Kamenev. Field Theory of Non-Equilibrium Systems, (Cambridge University Press 2011).
 J. Schwinger. Brownian motion of a quantum oscillator. J. Math. Phys. 2, 407 (1961).
 A. Altland and B. D. Simons. Condensed Matter Field Theory, (Cambridge University Press 2010).
 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).
 H. Carmichael. An Open Systems Approach to Quantum Optics, (Springer 1991).
 A. Kenfack and K. Życzkowski. Negativity of the Wigner function as an indicator of non-classicality. J. Opt. B 6, 396 (2004).
 A. J. Leggett and A. Garg. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett. 54, 857 (1985).
 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).
 M. Howard, J. Wallman, V. Veitch, and J. Emerson. Contextuality supplies the `magic' for quantum computation. Nature 510, 351 (2014).
 U. M. Titulaer and R. J. Glauber. Correlation functions for coherent fields. Phys. Rev. 140, B676 (1965).
 L. Mandel. Non-classical states of the electromagnetic field. Phys. Scr. 1986, 34 (1986).
 J. Sperling and W. Vogel. Representation of entanglement by negative quasiprobabilities. Phys. Rev. A 79, 042337 (2009).
 W. Vogel. Nonclassical correlation properties of radiation fields. Phys. Rev. Lett. 100, 013605 (2008).
 F. Krumm, J. Sperling, and W. Vogel. Multitime correlation functions in nonclassical stochastic processes. Phys. Rev. A 93, 063843 (2016).
 F. Krumm, W. Vogel, and J. Sperling. Time-dependent quantum correlations in phase space. Phys. Rev. A 95, 063805 (2017).
 A. A. Clerk, F. Marquardt, and J. G. E. Harris. Quantum measurement of phonon shot noise. Phys. Rev. Lett. 104, 213603 (2010).
 A. N. Korotkov. Continuous quantum measurement of a double dot. Phys. Rev. B 60, 5737 (1999).
 A. N. Korotkov. Output spectrum of a detector measuring quantum oscillations. Phys. Rev. B 63, 085312 (2001).
 A. Chantasri, J. Dressel, and A. N. Jordan. Action principle for continuous quantum measurement. Phys. Rev. A 88, 042110 (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).
 A. Chantasri and A. N. Jordan. Stochastic path-integral formalism for continuous quantum measurement. Phys. Rev. A 92, 032125 (2015).
 A. Di Lorenzo. Strong correspondence principle for joint measurement of conjugate observables. Phys. Rev. A 83, 042104 (2011).
 K. Husimi. Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn. 22, 264 (1940).
 U. Leonhardt and H. Paul. Measuring the quantum state of light. Prog. Quant. Electron. 19, 89 (1995).
 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).
 W. Belzig and Y. V. Nazarov. Full counting statistics of electron transfer between superconductors. Phys. Rev. Lett. 87, 197006 (2001).
 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).
 P. Solinas and S. Gasparinetti. Full distribution of work done on a quantum system for arbitrary initial states. Phys. Rev. E 92, 042150 (2015).
 P. Solinas and S. Gasparinetti. Probing quantum interference effects in the work distribution. Phys. Rev. A 94, 052103 (2016).
 P. Talkner, E. Lutz, and P. Hänggi. Fluctuation theorems: Work is not an observable. Phys. Rev. E 75, 050102 (2007).
 P. Talkner and P. Hänggi. Aspects of quantum work. Phys. Rev. E 93, 022131 (2016).
 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).
 J. Dressel. Weak values as interference phenomena. Phys. Rev. A 91, 032116 (2015).
 M. F. Pusey. Anomalous weak values are proofs of contextuality. Phys. Rev. Lett. 113, 200401 (2014).
 D. Bohm. A suggested interpretation of the quantum theory in terms of "hidden" variables. I. Phys. Rev. 85, 166 (1952).
 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).
 R. Ruskov, A. N. Korotkov, and A. Mizel. Signatures of quantum behavior in single-qubit weak measurements. Phys. Rev. Lett. 96, 200404 (2006).
 A. N. Jordan, A. N. Korotkov, and M. Büttiker. Leggett-Garg inequality with a kicked quantum pump. Phys. Rev. Lett. 97, 026805 (2006).
 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).
 H. J. Briegel, D. E. Browne, W. Dur, R. Raussendorf, and M. Van den Nest. Measurement-based quantum computation. Nat. Phys. 5, 19 (2009).
 Timo Kerremans, Peter Samuelsson, and Patrick Potts, "Probabilistically violating the first law of thermodynamics in a quantum heat engine", SciPost Physics 12 5, 168 (2022).
 Nicole Yunger Halpern, Brian Swingle, and Justin Dressel, "Quasiprobability behind the out-of-time-ordered correlator", Physical Review A 97 4, 042105 (2018).
 Patrick P Potts, Alex Arash Sand Kalaee, and Andreas Wacker, "A thermodynamically consistent Markovian master equation beyond the secular approximation", New Journal of Physics 23 12, 123013 (2021).
 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).
 Shayan Majidy, Jonathan J. Halliwell, and Raymond Laflamme, "Detecting violations of macrorealism when the original Leggett-Garg inequalities are satisfied", Physical Review A 103 6, 062212 (2021).
 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).
 Ahana Chakraborty and Rajdeep Sensarma, "Nonequilibrium Dynamics of Renyi Entropy for Bosonic Many-Particle Systems", Physical Review Letters 127 20, 200603 (2021).
 Adam Teixidó-Bonfill, Alvaro Ortega, and Eduardo Martín-Martínez, "First law of quantum field thermodynamics", Physical Review A 102 5, 052219 (2020).
 Harry Miller and Janet Anders, "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).
 Kazuhisa Ogawa, Natsuki Abe, Hirokazu Kobayashi, and Akihisa Tomita, "Complex counterpart of variance in quantum measurements for pre- and postselected systems", Physical Review Research 3 3, 033077 (2021).
 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", 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).
 Devashish Pandey, Rui Sampaio, Tapio Ala-Nissila, Guillermo Albareda, and Xavier Oriols, "Identifying weak values with intrinsic dynamical properties in modal theories", Physical Review A 103 5, 052219 (2021).
 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).
 Mihail Mintchev, Luca Santoni, and Paul Sorba, "Quantum Fluctuations of Entropy Production for Fermionic Systems in Landauer-Buttiker State", arXiv:1706.00561, (2017).
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