Abstract
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.
Highlights
Quasi-probability distributions such as the Wigner function have been an important tool in quantum mechanics since the early days of the theory [1]
The Wigner function lies at the heart of the phase-space representation of quantum mechanics which is tightly connected to classical statistical mechanics and provides an alternative route to quantization [2]
We introduce two well-known examples of the Keldysh quasi-probability distribution (KQPD): the Wigner function and the Full counting statistics (FCS)
Summary
Quasi-probability distributions such as the Wigner function have been an important tool in quantum mechanics since the early days of the theory [1]. To classical statistical mechanics, the fluctuations in quantum mechanics can be encoded in quasi -probability distributions These distributions can take on negative values which is a consequence of the non-commutativity of operators, reflecting the fact that a measurement inevitably disturbs all subsequent measurements of observables which do not commute with the first one. Negative quasi-probability distributions reflect the impossibility of accessing observables without disturbing them, indicating a non-classical behavior of the system which requires to take into account the action of the measurement apparatus to predict its outputs. The goal of the present paper is to introduce a general quasi-probability distribution that can be tailored to the problem at hand and provides a general framework for investigating fluctuations of non-commuting observables.
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