Quantum statistical functions

TL;DR

This paper develops a comprehensive framework for quantum statistical functions, unifying classical statistical tools with quantum mechanics, including weak values and quasiprobability distributions, to deepen understanding of quantum statistical properties.

Contribution

It introduces a unified mathematical framework for quantum statistical functions that incorporates nonclassical features and connects standard quantum statistics with quasiprobability distributions.

Findings

Reproduces quantum expectation values, variance, and covariance through differentiation.

Defines conditional quantum statistical functions yielding weak values.

Links multivariable quantum functions to known quasiprobability distributions.

Abstract

Statistical functions such as the moment-generating function, characteristic function, cumulant-generating function, and second characteristic function are cornerstone tools in classical statistics and probability theory. They provide a powerful means to analyze the statistical properties of a system and find applications in diverse fields, including statistical physics and field theory. While these functions are ubiquitous in classical theory, a quantum counterpart has remained elusive due to the fundamental hurdle of noncommutativity of operators. The lack of such a framework has obscured the deep connections between standard statistical measures and the non-classical features of quantum mechanics. Here, we establish a comprehensive framework for quantum statistical functions that transcends these limitations, naturally unifying the disparate languages of standard quantum statistics, quasiprobability distributions, and weak values. We show that these functions, defined as expectation values with respect to the purified state, naturally reproduce fundamental quantum statistical quantities like expectation values, variance, and covariance upon differentiation. Crucially, by extending this framework to include the concepts of pre- and post-selection, we define conditional quantum statistical functions that uniquely yield weak values and weak variance. We further demonstrate that multivariable quantum statistical functions, when defined with specific operator orderings, correspond to well-known quasiprobability distributions. Our framework provides a cohesive mathematical structure that not only reproduces standard quantum statistical measures but also incorporates nonclassical features of quantum mechanics, thus laying the foundation for a deeper understanding of quantum statistics.