Quasi-probability distributions for observables in dynamic systems

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.

Experimentally reducing the quantum measurement back action in work distributions by a collective measurement

“…Describing work fluctuations in genuinely coherent processes remains a subtle and open question in quantum thermodynamics, although relevant progress has been achieved recently [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][38][39][40]. Here we report the first experimental observation of work distributions, or, more precisely of transition probabilities, using an implementation based on a CM scheme [18].…”

Certifying Nonclassical Behavior for Negative Keldysh Quasiprobabilities

“…where we consider α = β = γ = δ = 1/ √ 2 in the main text. The KQPD can then be written as [19] P c (Σ 1 , Σ 2 ) = 1 (2π) 2 dλ 1 dλ 2 e iλ1Σ1+iλ2Σ2 Tr e −i λ 2 2σ 2 e −i( λ 1 2 +γ1)σ1 |+ +|e −i( λ 1 2 −γ1)σ1 e −i λ 2 2σ 2 = σ1=0,±1 σ2=±1 P(σ 1 , σ 2 )δ(Σ 1 − σ 1 )δ(Σ 2 − σ 2 ),…”

“…In quantum theory, such scenarios are well described by von Neumann type measurements [15][16][17][18], where observables of interest are coupled to detectors which are subsequently measured projectively. The probability distribution describing the measurement outcomes have a natural description in terms of a quasiprobability distribution that we abbreviate with KQPD due to its reminiscence of the Keldysh path-integral formulation [19,20]. The KQPD depends on the observables of interest and can reduce to the Wigner function [21] or the full counting statistics [22].…”

“…The KQPD depends on the observables of interest and can reduce to the Wigner function [21] or the full counting statistics [22]. Other applications include quantum thermodynamics [23][24][25][26][27][28], quantum optics [29], generalized Wigner functions [30], weak values [19] (see also [31][32][33]), and non-equilibrium phenomena in quantum systems [20]. Importantly, the KQPD can become negative, indicating non-classical behavior [18,29,[34][35][36][37][38][39].…”

“…After the interaction, a projective measurement of the detectors is performed to complete the measurement of the system observables j . The measured distribution reads [19] (see also [46,47])…”