Conditional Quantum Statistical Functions

• Conditional quantum statistical functions are a framework that extends quantum statistics by incorporating pre- and post-selection to handle noncommuting observables.
• They rigorously construct conditional moments and cumulants, linking weak values and quasiprobabilities through precise operator ordering approaches.
• The methods bridge quantum measurement theory and information geometry, enabling analysis of weak variances and anomalous statistical behavior.

Conditional quantum statistical functions form an advanced extension of the quantum statistical function hierarchy. By incorporating pre- and post-selection, or conditioning on measurement outcomes, they unify the structure underlying quantum weak values, weak variances, and the broader landscape of quantum statistics. These functions generalize the conditional expectations and cumulant-generating functions familiar from classical probability theory, but in the context of noncommuting observables and state update rules dictated by quantum measurement theory. Their rigorous construction resolves technical challenges posed by operator noncommutativity and measurement back-action, providing a single coherent framework linking standard moments, quasiprobabilities, and post-selected weak measurement statistics (Emori, 5 Feb 2026).

1. Multivariable Quantum Statistical Functions: Framework and Operator Ordering

General quantum statistical functions (moment-generating, characteristic, cumulant-generating, second characteristic) are defined as expectation values of exponential or polynomial functions of self-adjoint operators $A_1, ..., A_n$ with respect to a purified state $|\Psi\rangle$ encoding the density matrix $\rho$. For $N \in \mathbb{N}$ and a normalized weight $w: S_n \to \mathbb{C}$, the core construction is: $$f_A^{(N,w)}(\theta_1,...,\theta_n) = \left[\sum_{\sigma \in S_n} w(\sigma) \prod_{j=1}^n e^{\frac{\theta_{\sigma(j)}}{N} A_{\sigma(j)}}\right]^N$$ The operator ordering—i.e., the choice of $N$ and $w$—determines which classical or quantum quasiprobability distribution is reproduced upon Laplace or Fourier transform:

• $N=1, w(\mathrm{id})=1$: fully ordered exponentials (Kirkwood–Dirac [KD])
• $N=1$, $w$ symmetric on $\sigma$, $\sigma^{-1}$: Margenau–Hill (MH) symmetrization
• $N \to \infty$: exponential of sum, yielding the Wigner function structure by Lie–Trotter product limit

This encoding of ordering is key for connecting joint statistics of noncommuting observables with their quasiprobability distributions (Emori, 5 Feb 2026).

2. Conditional (Pre- and Post-Selected) Quantum Statistical Functions

A conditional quantum statistical function is defined with respect to a post-selection operation, typically specified by a POVM element $\Pi_m$. The conditional QMGF is given by: $$M_{\vec{A}}(\vec{\theta} | \Pi_m, \rho) = \frac{\langle \Psi | \Pi_m f_{\vec{A}}^{(N,w)}(\vec{\theta}) | \Psi\rangle}{\langle \Psi | \Pi_m | \Psi\rangle}$$ The first derivative with respect to $\theta_i$ at $\theta=0$ yields the (generally complex) weak value,

$$\partial_{\theta_i} M_A(\theta | \Pi_m)\big|_{\theta=0} = \frac{\mathrm{Tr}[\Pi_m A_i \rho]}{\mathrm{Tr}[\Pi_m \rho]} = \langle A_i \rangle_w$$

while higher mixed derivatives generate higher-order conditional (weak) moments and cumulants, such as the weak variance and joint weak covariance for multivariable cases.

Conditional quantum statistical functions thus interpolate between the standard (unconditional) quantum moments and the informationally richer regime of post-selected quantum weak measurement theory. Their output connects to, and generalizes, the spectrum of weak values found experimentally in weak measurement protocols (Emori, 5 Feb 2026).

3. Quasiprobabilities, Multivariable Distributions, and Their Conditioned Versions

Quantum statistical functions encapsulate the structure of a wide variety of quantum quasiprobability distributions via operator ordering choices:

• Kirkwood–Dirac (KD) Distribution: Direct exponentials, Laplace/Fourier transform yields joint sequences $\mathrm{Tr}[\rho P_{A_n}(a_n)...P_{A_1}(a_1)]$
• Margenau–Hill (MH): Real part of KD construction, sampling symmetrized averages
• Wigner Distribution: Full exponential, corresponding to symmetric ordering, with the joint function derived from the Fourier transform of the second characteristic function: $$W(\vec{x}) = \int \frac{d \vec{\theta}}{(2\pi)^n} e^{-i \vec{\theta} \cdot \vec{x}} C^{W}_{\vec{A}}(\vec{\theta})$$ Conditional variants of these distributions correspond to the statistics of sequences of measurement outcomes under both pre- and post-selection, encoding interference and anomalous values absent in classical statistics. This underlies the quantum-structural basis for anomalously large (or negative) conditional means and higher cumulants, a phenomenon thoroughly analyzed for non-commuting variable measurement in both idealized and realistic regimes (Emori, 5 Feb 2026, Franquet et al., 2017).

4. Connections to Weak Measurements and Higher-Order Quantum Moments

The differentiation ladder applied to conditional QMGFs produces the full hierarchy of weak quantum statistical quantities:

• First derivatives: conditional mean/weak value
• Second derivatives: weak variance, weak covariance
• Higher derivatives: weak cumulants, including skewness and kurtosis

The explicit construction ensures that weak values and all higher weak moments—often complex and outside the eigenvalue spectrum—are embedded within the same formalism as unconditional quantum moments. This seamlessly bridges weak measurement theory, quantum information statistics, and the mathematical physics of operator algebras (Emori, 5 Feb 2026).

5. Unified Geometric and Transformation Structure

The entire family of conditional quantum statistical functions interpolates, via operator ordering parameters $(N, w)$, between mixture and exponential orderings. This reflects the dual $e$- and $m$-connections of quantum information geometry. The analytic structure is governed by the extended Bochner theorem: for each (multi-variable) QCF, there exists a tempered (possibly non-positive) distribution as its inverse Fourier transform, with the classical regime characterized by positive-definiteness.

In this geometric picture, conditional functions represent quantum statistical manifolds equipped with pre-selected (initial) and post-selected (final) measurement structures. This opens analytic access to statistical distances, entropic properties, and other invariants under conditional evolution (Emori, 5 Feb 2026).

6. Calculation Example: Bivariate Conditional Functions and Anomalous Statistics

For the $n=2$ case, consider observables $A$ and $B$ with projectors $P_A(a), P_B(b)$. The KD-type conditional QMGF is

$$M^{KD}_{A,B}(\theta_1, \theta_2 | \Pi_m) = \frac{\mathrm{Tr}\left[\Pi_m e^{\theta_2 B} e^{\theta_1 A} \rho\right]}{\mathrm{Tr}[\Pi_m \rho]} = \sum_{a,b} e^{\theta_1 a + \theta_2 b} \frac{\mathrm{Tr}[\Pi_m P_B(b) P_A(a) \rho]}{\mathrm{Tr}[\Pi_m \rho]}$$

First and second derivatives yield all conditional means and covariances, including anomalous weak correlation structures. Such anomaly manifests, for example, as "sudden jumps" or divergent normalized cumulants in short-time, orthogonal pre- and post-selection regimes, as analyzed for weak measurement of two noncommuting observables (Franquet et al., 2017).

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References:

• (Emori, 5 Feb 2026) Quantum statistical functions
• (Franquet et al., 2017) Probability distributions of continuous measurement results for two non-commuting variables and conditioned quantum evolution

Conditional quantum statistical functions thus constitute an encompassing infrastructure for quantum statistics, enabling rigorous, unified treatment of moments, cumulants, weak values, and the full nonclassicality emerging from both operator noncommutativity and quantum measurement conditioning.