Quantum statistical functions
Haruki Emori
TL;DR
The work establishes a unified framework for quantum statistical functions by defining expectation-value-based quantum analogues of the classical MGF, CF, CGF, and second CF on the purified state, thereby reproducing standard quantum moments and enabling conditional statistics via pre- and post-selection. By introducing a multivariable operator-ordering function, it links these quantum functions to Kirkwood-Dirac, Margenau-Hill, and Wigner quasiprobabilities, and demonstrates a robust extension to weak values through conditional moments. The authors also reveal a Golden-Thompson hierarchy that interpolates between ordering schemes and connect the formalism to information-geometric structures, including e- and m-connections, providing a rigorous mathematical basis via an extended Bochner theorem. They further apply the framework to quantum parameter estimation, introducing the quantum method of moments (QMM) and quantum generalized method of moments (QGMM) and illustrate them with a transverse-field Ising model, highlighting practical advantages for efficiently inferring Hamiltonian parameters from quantum data.
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.