Multivariable Quantum Statistical Functions

• Multivariable quantum statistical functions generalize classical moment, characteristic, and cumulant generating functions to the noncommutative realm of quantum operators.
• They employ generalized operator orderings to resolve ambiguities in measuring noncommuting observables, thereby recovering moments, cumulants, and correlations through differentiation.
• These functions underpin practical applications in analyzing quantum correlations, phase-space distributions, and measurement protocols such as weak values and quantum central limit theorems.

Multivariable quantum statistical functions generalize the cornerstone statistical tools of classical probability theory—moment-generating functions, characteristic functions, cumulant-generating functions, and related entities—to the noncommutative regime of quantum mechanics. These functions underpin the quantitative analysis of quantum correlations, fluctuations, and higher-order statistical structure in both finite and infinite-dimensional systems, linking operator-based quantum mechanics with phase-space, algebraic, and information-theoretic frameworks.

1. General Definitions and Operator-Ordering Ambiguity

Let $A_1, \ldots, A_n$ be self-adjoint operators on a Hilbert space $\mathcal{H}$, and $\rho$ a density operator. To coherently extend classical statistical functions to multiple noncommuting observables, a generalized operator-ordering function $f^{(N, w)}_A(\theta)$ is defined: $$f^{(N, w)}_A(\theta) = \left[ \sum_{\sigma\in S_n} w(\sigma) \prod_{j=1}^n \exp\left(\theta_{\sigma(j)}/N \cdot A_{\sigma(j)}\right) \right]^N,$$ where $N\in\mathbb{N}$ controls the ordering granularity, $w:S_n\to \mathbb{C}$ is a normalized weight on the symmetric group $S_n$, and $$\theta = (\theta_1, \ldots, \theta_n) \in \mathbb{R}^n$$ parameterizes the multivariate exponential.

Key multivariable quantum statistical functions are:

• Quantum moment-generating function (QMGF): $$M_A(\theta; \rho) = \operatorname{Tr}[f^{(N, w)}_A(\theta)\,\rho]$$
• Quantum characteristic function (QCF): Replace each real exponential with a unitary: $$\tilde{f}^{(N, w)}_A(\theta) = [ \sum_{\sigma} w(\sigma) \prod_{j} \exp(i\theta_{\sigma(j)}/N\, A_{\sigma(j)}) ]^N$$, then $$\chi_A(\theta; \rho) = \operatorname{Tr}[\tilde{f}_A^{(N,w)}(\theta)\,\rho]$$
• Cumulant-generating and second characteristic functions: $K_A(\theta; \rho) = \ln M_A(\theta; \rho)$, $H_A(\theta; \rho) = \ln \chi_A(\theta; \rho)$

All expectation values are taken in the canonical purification $|\Psi\rangle \in \mathcal{H}\otimes\mathcal{H}^*$, with the prescription $\langle X\rangle_\psi = \operatorname{Tr}[X \rho]$ (Emori, 5 Feb 2026).

2. Recovery of Moments, Cumulants, and Correlations

Multivariable quantum statistical functions interpolate all standard moments and cumulants via differentiation:

• Means: $$\left. \frac{dM_A}{d\theta}\right|_0 = \operatorname{Tr}[A \rho]$$
• Variance: For the centered operator $A_0 = A - \langle A\rangle$, $$\left. \frac{d^2M_{A_0}}{d\theta^2}\right|_0 = \operatorname{Tr}[A_0^2 \rho]$$
• Covariance: For the Margenau–Hill (MH) symmetrization (N=1, $w$ symmetric under $(A,B)$ exchange):

$$M^{MH}_{A_0,B_0}(\theta_1,\theta_2) = \frac12 \operatorname{Tr}[e^{\theta_1 A_0}e^{\theta_2 B_0}\rho + e^{\theta_2 B_0}e^{\theta_1 A_0}\rho],$$

the mixed derivative at zero gives the symmetrized covariance:

$$\operatorname{Cov}_\rho(A,B) = \frac12 \operatorname{Tr}[(AB+BA)\rho] - \operatorname{Tr}[A\rho]\operatorname{Tr}[B\rho]$$

• Higher moments: For the Kirkwood–Dirac (KD) ordering (N=1, $w$ at identity), $$M^{KD}_{A_1,\ldots,A_n}(\theta_1,\ldots,\theta_n) = \operatorname{Tr}[e^{\theta_nA_n}\cdots e^{\theta_1A_1}\,\rho]$$, whose $n$-th mixed derivative yields $\operatorname{Tr}[A_1\cdots A_n\rho]$ (Emori, 5 Feb 2026).

Notably, the $n$-point function in KD ordering can be operationally measured as a chain of conditional weak values.

3. Conditional Quantum Statistical Functions and Weak Values

Post-selection on a POVM element $\Pi_m$ yields conditional multivariable QMGF: $$M_A(\theta | \Pi_m, \rho) = \frac{\operatorname{Tr}[\Pi_m e^{\theta A} \rho]}{\operatorname{Tr}[\Pi_m \rho]}$$ The first derivative at $\theta = 0$ gives the complex weak value $A_w$, while the second yields the weak variance. In the multivariable extension, $\Pi_m$ is inserted in the ordering function’s numerator and denominator, generalizing weak measurements to joint distributions and higher moments (Emori, 5 Feb 2026).

4. Operator Orderings, Quasiprobabilities, and Phase Space Functions

Selecting the pair $(N,w)$ recovers important quasiprobabilities:

• Kirkwood–Dirac (N=1, $w$=id): Recovers $$\operatorname{Tr}[e^{\theta_nA_n}\cdots e^{\theta_1A_1}\rho]$$ and the KD joint probability, $$\operatorname{Tr}[P_{A_n}(a_n)\cdots P_{A_1}(a_1)\rho]$$.
• Margenau–Hill: $w$ symmetric.
• Wigner/Weyl-symmetric (N→∞): Yields the symmetrically-ordered exponential $e^{\sum_j \theta_j A_j}$, and direct connection to phase-space Wigner functions: $$\chi^W(\theta;\rho) = \operatorname{Tr}[\exp(i\sum \theta_j A_j)\rho]$$ (Emori, 5 Feb 2026, Calixto et al., 20 Jul 2025, Paul, 2022, Tilma et al., 2011).

Ordering parameters interpolate between standard phase-space distributions ($P$-function, $Q$-function, Wigner, Husimi, etc.), as encoded by Stratonovich–Weyl kernels and operator ordering interpolants (Calixto et al., 20 Jul 2025, Tilma et al., 2011).

5. Extended Theorems and Measurement Decomposition

A quantum extended Bochner’s theorem applies:

• Each quantum characteristic function $\chi_A(\theta;\rho)$ admits an inverse Fourier transform as a tempered distribution on the joint spectrum of the $\{A_j\}$, which is non-negative (classical) if and only if $\chi$ is positive-definite; otherwise, genuine quasiprobability arises.
• The $n$-point derivative of $M^{KD}$ admits expansion as a sum over projectors and a chain of conditional weak values, operationalizing higher correlators in weak measurement protocols.
• Quantum MGFs directly correspond to path-integral generating functionals in field theory, with $K(\theta;\rho)$ paralleling the connected generating functional for quantum field correlators (Emori, 5 Feb 2026).

6. Applications Across Quantum Statistical Mechanics

Multivariable quantum statistical functions underlie several domains:

• Quantum Central Limit Theorems: The multivariate quantum CLT asserts that fluctuations of many-body averages (built from $k$ one-body operators) converge to (possibly complex) Gaussian measures, with the covariance determined by Bogoliubov transformations linearizing the dynamics about mean-field trajectories (Buchholz et al., 2013).
• Quantum Phase Space: For symmetric multi-quDit systems, families of phase-space quasi-distributions $\mathcal{F}^{(s)}_\rho$ (parameter $s$ specifies ordering) reproduce the Wigner, $P$, and $Q$ functions and their marginals are expectation values of observables (Calixto et al., 20 Jul 2025). SU(N)-symmetric generalizations likewise map density matrices to Wigner, $Q$, and $P$ kernels on generalized complex projective phase-spaces (Tilma et al., 2011).
• Quantum Information & Higher-Order Correlations: Multivariable mutual information and higher-order correlation measures (e.g., three-way interaction information $I^3_x$ for three particles) rigorously distinguish quantum-symmetric (bosonic) and antisymmetric (fermionic) structures, and can detect quantum interference or entanglement untraceable to pairwise links (Yépez et al., 2016).

7. Generalizations, Noncommutative Probability, and Beyond

Deformations such as $\mathcal{R}(p,q)$-multivariate distributions extend classical and quantum discrete statistics with noncommutative or quantum-algebraic parameters, linking urn models and stochastic processes to quantum statistical functions, with joint pmfs, probability-generating functions, and explicit covariance formulas given for e.g.\ Pólya and hypergeometric models (Melong, 2022, Melong et al., 2023).

Integrals over phase space (Wigner–Husimi–Toeplitz symbols, classical limit of the grand canonical ensemble) yield phase-space representations for the grand partition function, multi-particle densities, and quantum corrections via commutation and symmetrization functions. Loop/cycle expansions in the symmetrization function efficiently sum quantum exchange and correlation contributions in the thermodynamic limit (Attard, 2018).

The modern framework recasts and unifies classical statistical identities (fluctuation–dissipation, Hellmann–Feynman, response theory, etc.) as specializations of general quantum expectation identities, with multivariate parameter-dependence and full covariance/cumulant structure (Maulén et al., 2024).

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References

• (Emori, 5 Feb 2026) Quantum statistical functions
• (Buchholz et al., 2013) Multivariate Central Limit Theorem in Quantum Dynamics
• (Yépez et al., 2016) Higher-order statistical correlations and mutual information among particles in a quantum well
• (Calixto et al., 20 Jul 2025) Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel
• (Paul, 2022) Husimi, Wigner, T{\"o}plitz, quantum statistics and anticanonical transformations
• (Tilma et al., 2011) SU(N)-symmetric quasi-probability distribution functions
• (Attard, 2018) Quantum Statistical Mechanics in Classical Phase Space
• (Melong, 2022) $\mathcal{R}(p,q)$-multivariate discrete probability distributions
• (Melong et al., 2023) Multi-parameter Fermi-Dirac and Bose-Einstein Stochastic Distributions
• (Maulén et al., 2024) A quantum expectation identity: Applications to statistical mechanics