Toward a Deficiency-Theoretic Interpretation of Quantum Uncertainty

The uncertainty principle is traditionally viewed as a statement about the epistemic limits of observers or the intrinsic stochasticity of quantum states. We propose an operational framework using the notion of experimental deficiency: the Robertson--Schr"odinger inequality $\det(\Sigma) \geq \hbar^2/4$ implies a strict positivity of the Le Cam deficiency between incompatible experiments and their joint simulation. Specifically, within a suitably specified Gaussian setting, we argue that any joint experiment attempting to simulate both position and momentum measurements must incur a strictly positive deficiency. The obstruction established here is qualitative: it follows from the added noise covariance being non-zero, a fact for which $\hbar$ supplies the physical floor, and we do not derive a quantitative lower bound of the form $\delta \geq f(\hbar)$. We further prove a \emph{quantitative} lower bound: for the Arthurs--Kelly joint measurement on a coherent state, $\delta(\mathcal{E}_{\mathrm{joint}}, \mathcal{E}_{\mathrm{target}}) \geq 2[\Phi(\sqrt{2\ln 2}) - \Phi(\sqrt{\ln 2})] \approx 0.166$, independent of $\hbar$. We introduce the \emph{saturation ratio} $\eta = (\hbar^2/4)/\det(\Sigma)$ as a dimensionless diagnostic of experimental optimality. In this framework, operational classicality implies not necessarily $\hbar \to 0$, but the loss of saturation ($\eta \to 0$) via mechanisms like macroscopic scaling or decoherence. We suggest that decoherence can be interpreted as deficiency inflation. This geometric perspective unifies quantum uncertainty with classical statistical inference, suggesting that Gaussian measurements are natural candidates for deficiency-optimality in the present framework.