Classical Uncertainty Relations

• Classical uncertainty relations are fundamental statistical constraints that limit the simultaneous precision of conjugate observables in classical systems.
• They are derived using moment formalisms, dimensionless scaling, and information-theoretic methods to reveal deep structural parallels with quantum uncertainty.
• Extensions include stochastic, thermodynamic, and optical contexts where techniques like the Cramér–Rao inequality and geometric bounds refine traditional uncertainty measures.

Classical uncertainty relations are fundamental constraints satisfied by statistical ensembles in classical physics, restricting the simultaneous predictability of conjugate observables such as position and momentum. These relations originate purely from the statistical, probabilistic, and geometric architecture of classical phase space or stochastic processes and exhibit deep structural parallels with their quantum analogs—despite the absence of operator non-commutativity or Planck-scale corrections. Modern research has placed them within a hierarchy of inequalities for moments, entropic measures, optimal transport, and stochastic thermodynamics, and has clarified their correspondence to quantum uncertainty through rigorous mathematical and information-theoretic frameworks.

1. Statistical Moment Formalism and Canonical Second-Order Bound

Given a normalized classical phase space probability density $\rho(q,p)$, one defines the statistical moments: $$C^{a,b} := \iint (p-p_0)^a (q-q_0)^b\, \rho(q,p)\,dq\,dp,$$ where $p_0, q_0$ are the means. The covariance matrix

$$\Sigma = \begin{pmatrix} C^{2,0} & C^{1,1}\ C^{1,1} & C^{0,2} \end{pmatrix}$$

is positive semidefinite ($\Sigma \succeq 0$), yielding the classical uncertainty relation

$$(\Delta p)^2 (\Delta q)^2 \geq [\mathrm{Cov}(p,q)]^2,$$

where $\Delta p^2 = C^{2,0}$, $\Delta q^2 = C^{0,2}$, and $\mathrm{Cov}(p,q) = C^{1,1}$ (Brizuela, 2014). For uncorrelated distributions, this reduces to

$\Delta p\,\Delta q \geq 0,$

expressing the impossibility of perfect simultaneous sharpness.

Higher-order generalizations use the Cauchy–Schwarz inequality for functions in $L^2(\rho)$. For all nonnegative integers $a,b,c,d$: $[C^{a+c,\,b+d}]^2 \leq C^{2a,\,2b}\,C^{2c,\,2d},$ encapsulating an infinite hierarchy of moment-based uncertainty-type inequalities. The full set is equivalent to the statement that the infinite “moment matrix” $M_{ij}=C^{i+j,0}$ is positive semidefinite and that all its finite principal minors are nonnegative (Brizuela, 2014).

2. Classical–Quantum Correspondence: Dimensionless Scaling and the Limit

Uncertainty bounds in the classical ensemble formalism admit algebraic identities with the quantum analogs when observables are scaled by the system’s intrinsic action scale, typically via dimensionless variables. For a classical microcanonical ensemble of energy $E$ in a potential $V(x)$, after introducing dimensionless variables $X = x/A$, $P = p/\sqrt{2mE}$ (with $A$ the amplitude), all central moments of $X$ and $P$ coincide with those of the corresponding quantum stationary state. Specifically, for the harmonic oscillator: $$\Delta X = \Delta P = 1/\sqrt{2} \implies \Delta X\,\Delta P = 1/2,$$ identical to the quantum result for any eigenstate—demonstrating the equivalence of static uncertainties after scaling (Devi et al., 2011, Gattus et al., 2021). At large quantum numbers (classical limit), the classical and quantum uncertainties coincide, with quantum corrections vanishing as $\hbar \to 0$ and/or $n \to \infty$.

3. Classical Information Theory, Entropic, and Fisher–Cramér–Rao Formulations

The classical uncertainty relation can be derived information-theoretically for measurement of two observables, using the Cramér–Rao inequality and statistical estimation theory. For any unbiased estimator of parameters $\theta = (\theta_1, ..., \theta_N)$ associated to a probability density $f(x;\theta)$, the covariance matrix of the estimators satisfies

$$\Sigma \geq J^{-1}, \qquad J_{ij} = \mathbb{E} \left[ \frac{\partial\ln f}{\partial\theta_i} \frac{\partial\ln f}{\partial\theta_j}\right],$$

where $J$ is the Fisher information matrix (Kurihara, 2012). For conjugate variables related by a Sturm–Liouville operator, boundary conditions or prior information can reduce the minimal product below the standard Kennard–Robertson bound.

Entropic uncertainty relations, when restricted to the classical regime (mixed states diagonal in the observable basis), reduce to lower bounds governed entirely by Shannon/von Neumann entropy, i.e., the classical component

$C(O,\rho) = S(\rho),$

where $S(\rho) = -\sum p_k \ln p_k$ dominates (Korzekwa et al., 2014).

4. Probabilistic, Stochastic, and Thermodynamic Classical Uncertainty

In classical stochastic dynamics, F\"urth’s (1933) uncertainty relation gives a universal lower bound for Markov processes or diffusions. For any observable $f$ of a diffusion process, the variance product is bounded as

$$\mathrm{Var}[f(\xi_t, t)]\; \mathrm{Var}[\mathfrak{d} f(\xi_t, t)] \geq \frac{1}{4}\left|\mathbb{E}\left[\langle \nabla f, A \nabla f \rangle \right]\right|^2,$$

with $A$ the diffusion tensor and $\mathfrak{d}f$ the time-symmetric osmotic derivative (Muratore-Ginanneschi et al., 2023). This extends to Markov jump processes, providing trade-off bounds between the variance of cumulative observables (currents) and the total entropy production or dynamical activity: $$\frac{\mathrm{Var}[J]}{\langle J\rangle^2} \geq \frac{2}{\Sigma_\tau},\quad \frac{\mathrm{Var}[J]}{\langle J\rangle^2} \geq \frac{1}{\mathcal{A}_\tau},$$ representing thermodynamic and kinetic uncertainty relations (TUR, KUR), which constrain precision in non-equilibrium phenomena (Kwon et al., 6 Dec 2024).

5. Classical Uncertainty in Optical, Information-Theoretic, and Relational Contexts

For optical fields, the classical Gabor time–frequency bound asserts

$\Delta t\,\Delta \omega \geq \frac{1}{2}$

for all classical (separable) wavepackets, with the two-photon product obeying

$\Delta\tau\,\Delta\Omega \geq 1,$

and quantum corrections subside as intensity increases, recovering the classical bound in the strong-field (large $\bar N$) limit (Willemann et al., 10 Dec 2025).

In finite-dimensional measurement scenarios, “classical” (preparation noncontextual) models satisfy linear predictability tradeoff bounds, e.g.,

$P_X + P_Z \leq 1$

for binary-outcome measurements, with quantum theory achieving strictly concave (circular) trade-offs beyond the classical polytope—thus functional differences in uncertainty relations serve as contextuality witnesses (Catani et al., 2022).

Classical information-locking and data-hiding schemes rely on explicit uncertainty relations for outcome distributions: families of random unitary matrices can be constructed so that, for any initial state, the distribution over measurement outcomes in a local subsystem remains $\varepsilon$-close to uniform in classical fidelity, an optimal metric for quantifying uncertainty and hiding (Adamczak, 2016).

6. Boundary Conditions, Geometry, and Nontrivial Classical Generalizations

Boundary conditions or geometric constraints (finite domains, curvature) can modify classical uncertainty products. For classical systems governed by Sturm–Liouville operators, the minimal product of standard deviations for a pair of dual observables is

$\Delta\xi\,\Delta\lambda \geq \tilde h/2$

where $\tilde h$ is the system-dependent scale (not a fundamental constant). Restrictive boundaries generate information gain and can reduce the uncertainty product below the standard free value. This is observed, e.g., in the measurable reduction of position–energy disturbance products for electrons confined in molecular cages (Kurihara, 2012). In geometric generalizations, the extended uncertainty principle (EUP) on curved spaces reads

$$\Delta p \cdot \rho \gtrsim \hbar\,\pi\,\left[1 - \frac{R}{12\pi^2}\rho^2 + \ldots\right],$$

with $R$ the Ricci scalar curvature and $\rho$ the position uncertainty (Wagner, 2022).

7. Connection to Quantum Uncertainty and Epistemic Restrictions

Classical uncertainty relations arise from the fundamental statistical and geometric properties of classical phase space distributions and are unrelated to noncommutativity of observables. Quantum uncertainty generalizes the same structural constraints and introduces $\hbar$-dependent corrections arising from operator noncommutativity. For instance, in quantum mechanics, the covariance determinant acquires a strict quantum lower bound: $G^{2,0} G^{0,2} - (G^{1,1})^2 \geq (\hbar/2)^2,$ whereas for any classical distribution the determinant is only constrained to be nonnegative (Brizuela, 2014). The passage to quantum theory is marked by the impossibility of jointly vanishing all fluctuations due to epistemic restrictions—an irreducible coupling of statistical moments, visible even in classical phase space when one imposes a smearing by the uncertainty principle (Budiyono et al., 2021, Radozycki, 2016).

Aspect	Classical Uncertainty	Quantum Correction (if any)
Moment bound (2nd order)	$$(\Delta p)^2 (\Delta q)^2 \geq [\mathrm{Cov}(p,q)]^2$$	$+ (\hbar/2)^2$
Entropic bound	$H_O(\rho) = S(\rho)$ when $\rho$ diagonal	$+ Q(O,\rho)$ coherence term
Stochastic process	F\"urth’s martingale-type bound	TUR/KUR with quantum unravelings
Dimensional scale	System action scale	$\hbar$ as universal scale

The classical–quantum correspondence is realized via scaling and structural isomorphism of moment or entropy inequalities, with quantum effects vanishing in the limit $\hbar \to 0$ or large quantum numbers.

• (Brizuela, 2014) Statistical moments and generalized uncertainty relations
• (Devi et al., 2011) Classical–quantum correspondence for uncertainty
• (Gattus et al., 2021) Dimensional analysis of uncertainty
• (Kurihara, 2012) Classical information theory and uncertainty
• (Korzekwa et al., 2014) Quantum and classical entropic uncertainty
• (Catani et al., 2022) Operational and noncontextual classical tradeoffs
• (Muratore-Ginanneschi et al., 2023) F\"urth’s stochastic uncertainty and entropy production
• (Kwon et al., 6 Dec 2024) TUR/KUR and Markov stochastic uncertainty
• (Willemann et al., 10 Dec 2025) Classical bounds for optical uncertainty
• (Radozycki, 2016) “Uncertainty-smearing” of classical distributions
• (Budiyono et al., 2021) Epistemic restriction and classical hidden variable models
• (Wagner, 2022) Geometric and curvature-induced generalizations

Classical uncertainty relations, therefore, are not merely analogs but foundational statistical constraints inherent to any bona fide probability ensemble. They regulate the mutual sharpness of marginal distributions, entropic measures, and dynamical observables, and their tightness or modification reflects the interplay of geometry, dynamical structure, and (in the quantum regime) noncommutativity or epistemic restriction.