Quantum Statistics Framework Unlocks Hidden Links Between Classical and Non-Classical Mechanics

Researchers have long sought to extend classical statistical functions, including moment-generating, characteristic, cumulant-generating, and second characteristic functions, to the realm of quantum mechanics, a pursuit hindered by the noncommutativity of quantum operators. Haruki Emori from Hokkaido University and RIKEN, along with colleagues, now present a comprehensive framework addressing this challenge and unifying standard statistical measures with non-classical mechanical features. Their work establishes a novel approach defining these functions as expectation values of purified states, naturally reproducing expectation values, variance, and covariance through differentiation. Significantly, the team demonstrate that incorporating pre- and post-selection concepts yields conditional statistical functions directly linked to weak values and variances, while multivariable functions correspond to established quasiprobability distributions, paving the way for a more profound understanding of quantum statistics.

Cornerstone tools in classical statistics and probability theory provide a powerful means to analyse 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. We establish a comprehensive framework for quantum statistical functions that transcends these limitations, naturally unifying the disparate.

Derivation of conditional quantum statistics and quasiprobability relationships

Scientists are investigating the languages of standard quantum statistics, quasiprobability distributions, and weak values. 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, researchers define conditional quantum statistical functions that uniquely yield weak values and weak variance. Furthermore, they demonstrate that multivariable quantum statistical functions, when defined with specific operator orderings, correspond to well-known quasiprobability distributions.

This 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. In classical statistics and probability theory, statistical functions such as the moment-generating function, characteristic function, cumulant-generating function, and second characteristic function play a central role in characterizing the statistical properties of a system under consideration.

The moment-generating function provides a compact way to encode all the moments of a probability distribution, while the characteristic function, being its Fourier transform, is always well-defined, even when the moments do not exist. Their logarithmic counterparts, the cumulant-generating and second characteristic functions, are particularly useful for analyzing the independence and additivity of random variables.

These tools are indispensable not only in statistics but also in physics, finding applications in statistical mechanics for establishing quantum fluctuation theorems and full counting statistics, as well as in quantum field theory for providing a complete description of correlation functions. Despite the profound success of these concepts in classical statistics and probability theory, a unified framework for such statistical functions in quantum counterparts has been lacking except for specific applications.

Quantum mechanics introduces two fundamental features that complicate this analogy: the noncommu-tative nature of operators and the intrinsic probabilistic nature of measurement outcomes. While the expectation value of an observable is a well-defined statistical quantity, a systematic framework that generates all higher-order moments and cumulants in a manner analogous to their classical counterparts remains to be fully developed.

This work bridges this fundamental gap by constructing a complete hierarchy of quantum statistical functions. Their approach does not merely mimic classical definitions but reveals an intrinsic geometric structure governing quantum features. Their approach departs from a direct analogy with classical probability distributions and instead leverages the generalized operator ordering function, which provides a rich structure of quantum quasiprobability distributions.

By defining their quantum statistical functions as expectation values of these functions with respect to a purified state, they establish a robust and comprehensive framework. They report three key advances that reshape our understanding of quantum statistics. First, they define a set of four quantum statistical functions: the quantum moment-generating function, characteristic function, cumulant-generating function, and second characteristic function.

They demonstrate that these functions, when differentiated, correctly reproduce standard quantum statistical quantities such as expectation values, variance, and covariance. Second, they show that the multivariable versions of these functions, when defined with specific operator orderings, are intimately connected to well-known quasiprobability distributions like the Kirkwood-Dirac (KD), Margenau-Hill (MH), and Wigner distributions.

This link highlights the role of noncommutativity of operators in shaping the statistical properties of quantum systems. Third, they extend their framework to the regime of pre- and post-selection to address the statistical nature of weak values. Since their inception, weak values have garnered significant attention regarding their physical interpretation and anomalous properties.

Their framework offers a definitive understanding: weak values are the precise quantum analogues of conditional expectations derived from a conditional moment-generating function. With these advances, their framework provides a unified mathematical structure that not only reproduces fundamental statistical quantities of quantum mechanics but also incorporates the non-classical aspects of quasiprobability distributions.

By generalizing the original Bochner’s theorem relating characteristic functions to probability measures, they establish a rigorous criterion that demarcates classical statistics from quantum statistics. This theorem explicitly links the non-positive definite nature of quantum characteristic functions to the emergence of quasiprobability distributions within the framework of tempered distributions, thereby providing a firm mathematical basis for observing non-classicality.

Therefore, their framework presented here lays the foundation for a deeper understanding of quantum statistics and could inspire new experimental protocols for characterizing quantum states and processes. The paper is organized as follows. In Sec.

II, they review the foundational concepts required for their framework, including classical statistical functions, the path integral generating functional, the canonical purification of quantum states, and the matrix geometric mean. Section III presents the core of their theoretical framework. They introduce the definitions of single-variable and multivariable quantum statistical functions, and extend them to the conditional regime, using the generalized operator ordering function.

Here, they also establish the connection to quasiprobability distributions such as the KD, MH, and Wigner distributions, and provide the extended Bochner’s theorem. In Sec. IV, they apply their framework to quantum estimation, introducing the quantum method of moments (QMM) and quantum generalized method of moments (QGMM) for efficient parameter estimation.

Section V compares their approach with other frameworks, specifically those based on the matrix geometric mean and information geometry, to highlight the unique advantages of their framework. Finally, they summarize their Results and discuss future perspectives in Sec. VI.

Detailed mathematical proofs and the connection to discrete Wigner functions are provided in the Appendices. To establish the foundation for their framework of quantum statistical functions, they first review essential concepts. They begin with a detailed overview of statistical functions for classical systems, which serve as the conceptual blueprint for their quantum analogues.

Next, they discuss the generating functional in quantum field theory, highlighting the powerful idea of a single master function that encodes all of a system’s correlation functions. They then provide a thorough description of state purification, a key mathematical tool in quantum mechanics that is central to their definitions.

Finally, they briefly introduce a technique of the matrix geometric mean used in the definition of other quantum statistical functions. In classical statistics and probability theory, the statistical properties of a random variable X are completely characterized by its probability distribution. Statistical functions provide a powerful alternative representation of this information, often in a more analytically tractable form.

They define the four primary functions that form the basis of their analogy. Definition 1 (Classical moment-generating function). For a real-valued random variable X, the moment-generating function (MGF) MX(θ) is defined as the expectation value of exp(θX): MX(θ) := Ex(eθX). where θ ∈R is a real parameter and Ex(· · · ) stands for an expectation value.

The MGF is so named because it generates the moments mn of the distribution through differentiation. The MGF is equivalent to the Laplace transform of the probability density function of X. By performing a Taylor series expansion of the exponential function around θ = 0, they yield the moments as coefficients: MX(θ) = 1 + Σn=1∞ mn n. θn.

The n-th moment is therefore given by mn = dn dθn MX(θ) θ=0. While moments are intuitive, cumulants provide a deeper insight into the properties of a distribution, such as additivity for independent random variables. Cumulants are generated by the logarithm of the MGF.

Definition 3 (Classical cumulant-generating function). The cumulant-generating function (CGF) KX(θ) is defined as KX(θ) := log MX(θ). The Taylor series expansion of the CGF yields the cumulants kn as coefficients: KX(θ) = Σn=1∞ kn n. θn.

The n-th cumulant is therefore given by kn = dn dθn KX(θ) θ=0. The first cumulant k1 is the mean, and the second cumulant k2 is the variance. Higher-order cumulants describe more subtle statistical properties and are related to, but distinct from, the higher-order central moments.

Analogously, the logarithm of the characteristic function generates the same cumulants. Definition 4 (Classical second characteristic function). The second characteristic function HX(θ) is defined as HX(θ) := log CX(θ).

The cumulants are obtained by differentiation: kn = i−n dn dθn HX(θ) θ=0. These four functions provide a complete and powerful toolkit for the analysis of statistical properties of classical systems, a toolkit they aim to construct for quantum systems. In quantum field theory, the concept of a generating function is elevated to that of a generating functional, a master object from which all physical quantities of a theory can be derived.

The path integral formalism, pioneered by Feynman, provides the natural language for this concept. The central quantity is the transition amplitude, or propagator, which is expressed as a sum over all possible paths of a system. For a particle moving from (xi, ti) to (xf, tf), this is given by K(xf, tf; xi, ti) = Z D[x(t)]e i ħS[x(t)]. where S[x(t)] = Z tf ti L(x(t), x(t), t)dt is the classical action and the integral is a functional integral over all paths x(t) connecting the endpoints.

Quantum statistical functions from expectation values and differentiation

Statistical functions, including the moment-generating function, characteristic function, cumulant-generating function, and second characteristic function, form essential tools in classical statistics and probability theory. These functions provide a powerful means to analyze statistical properties and find applications in fields like statistical physics.

A corresponding framework has remained elusive due to the noncommutativity of quantum operators, obscuring connections between standard statistical measures and non-classical mechanics. This work establishes a comprehensive framework for quantum statistical functions, unifying the languages of standard statistics, quasiprobability distributions, and weak values.

Specifically, these functions, defined as expectation values with respect to a purified state, reproduce fundamental statistical quantities upon differentiation. Expectation values, variance, and covariance are all naturally generated through this process. By extending the framework to include pre- and post-selection, conditional statistical functions are defined, uniquely yielding weak values and weak variance.

Multivariable statistical functions, when defined with specific operator orderings, correspond to well-known quasiprobability distributions such as the KD, MH, and Wigner distributions, and provide the extended Bochner’s theorem. The research introduces the quantum method of moments (QMM) and quantum generalized method of moments (QGMM) for efficient parameter estimation.

These methods leverage the newly defined statistical functions to provide a robust approach to quantum estimation problems. A comparison with frameworks based on the matrix geometric mean and information geometry highlights the unique advantages of this approach. The framework provides a cohesive mathematical structure that reproduces standard statistical measures while incorporating nonclassical features of mechanics, laying the foundation for a deeper understanding of quantum statistics.

Classical moment-generating functions are defined as the expectation value of exp(θX), where θ is a real parameter and X is a real-valued random variable. Differentiating the moment-generating function with respect to θ yields the moments of the distribution, with the n-th moment given by dn dθn MX(θ) evaluated at θ = 0.

The first moment represents the mean, while the second central moment defines the variance, providing insights into the distribution’s shape. The characteristic function, defined as the expectation value of exp(iθX), converges for all real θ and is related to the moment-generating function through an analytic continuation.

Cumulants, generated by the logarithm of the moment-generating function, offer a deeper understanding of distribution properties, particularly additivity for independent random variables. The n-th cumulant is obtained by differentiating the cumulant-generating function with respect to θ and evaluating the result at θ = 0.

The second characteristic function, defined as the logarithm of the characteristic function, also generates the same cumulants through differentiation. These four functions collectively provide a complete toolkit for analyzing the statistical properties of classical systems, serving as the conceptual blueprint for the quantum analogues developed in this study.

Quantum statistical unification via generalised operator orderings and Bochner’s theorem

Scientists have established a unified framework for statistical functions in quantum mechanics, resolving ambiguities regarding operator ordering that have long existed in the field. This framework introduces quantum versions of classical statistical tools, including moment-generating, characteristic, cumulant-generating, and second characteristic functions, by utilising a generalized operator ordering function and a generalized unitary operator ordering function.

Through differentiation, these functions reproduce standard quantum statistical quantities such as expectation value, variance, and covariance, effectively bridging the gap between classical and quantum statistical descriptions. Furthermore, the research incorporates weak values and weak variance as intrinsic statistical moments within pre- and post-selected systems, rather than treating them as anomalous quantities.

An extended Bochner’s theorem rigorously underpins this unification, identifying the violation of positive definiteness in the characteristic function as a clear mathematical indicator of non-classical behaviour. The connection between multivariable quantum statistical functions and established quasiprobability distributions underscores the framework’s ability to capture the unique statistical properties of noncommuting observables.

The authors acknowledge a limitation in that the framework’s full potential requires further exploration in complex systems and experimental verification. Future research may focus on applying this toolkit to emerging technologies in quantum information processing, potentially offering new insights and capabilities. This work clarifies foundational issues concerning quasiprobability distributions and provides a versatile set of tools for analysing quantum systems, ultimately laying the groundwork for a deeper understanding of quantum statistics.