Quasi-probability distributions for observables in dynamic systems

Patrick P. Hofer

Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland

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.

We develop a general framework to describe the fluctutions of arbitrary non-commuting observables by quasi-probability distributions. Negative values in these distributions imply non-classical behavior in the sense that the system necessarily exhibits coherent superpositions of states corresponding to different measurement outcomes. In addition to giving insight into the behavior of the system, the quasi-probability distribution contains all the information that is necessary to predict the outcomes of any von Neuman type measurement. Popular examples of the introduced quasi-probability distribution are the Wigner function and the full counting statistics. We further connect the developed framework to various time-dependent problems including weak values, fluctuating work, and Leggett-Garg inequalities.

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