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The Uncertainty Principle as a Deficiency Bound: A Deficiency-Theoretic Perspective on Quantum Incompatibility

Deniz Akdemir

Published Jan 25, 2026• 0 citations
Publisher

Abstract

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 reformulation in terms of experimental deficiency: the Robertson--Schr\"odinger inequality $\det(\Sigma) \geq \hbar^2/4$ yields a lower bound on the Le Cam deficiency between incompatible experiments and any joint simulation. In the Gaussian (LAN) regime, this implies that every joint experiment attempting to simulate both position and momentum must incur strictly positive deficiency, with the Planck constant $\hbar$ setting the information-loss scale. We introduce the \textit{saturation ratio} $\eta = (\hbar^2/4)/\det(\Sigma)$ as a dimensionless diagnostic of experimental optimality, and interpret decoherence as deficiency inflation via increased effective covariance. The framework applies to covariant Gaussian joint measurements (e.g., Arthurs--Kelly-type schemes) and connects quantum uncertainty with classical statistical inference, identifying Gaussian measurements as deficiency-optimal. The resulting quantum--classical ``continuum'' claim is strictly operational: an ordering by saturation loss, not a metaphysical thesis of structural continuity, theory reduction, or ontological continuity.

DOI
10.5281/zenodo.18367259

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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 reformulation in terms of experimental deficiency: the Robertson--Schr"odinger inequality det⁡(Σ)≥ℏ2/4\det(\Sigma) \geq \hbar^2/4det(Σ)≥ℏ2/4 yields a lower bound on the Le Cam deficiency between incompatible experiments and any joint simulation. In the Gaussian (LAN) regime, this implies that every joint experiment attempting to simulate both position and momentum must incur strictly positive deficiency, with the Planck constant ℏ\hbarℏ setting the information-loss scale. We introduce the \textit{saturation ratio} η=(ℏ2/4)/det⁡(Σ)\eta = (\hbar^2/4)/\det(\Sigma)η=(ℏ2/4)/det(Σ) as a dimensionless diagnostic of experimental optimality, and interpret decoherence as deficiency inflation via increased effective covariance. The framework applies to covariant Gaussian joint measurements (e.g., Arthurs--Kelly-type schemes) and connects quantum uncertainty with classical statistical inference, identifying Gaussian measurements as deficiency-optimal. The resulting quantum--classical ``continuum'' claim is strictly operational: an ordering by saturation loss, not a metaphysical thesis of structural continuity, theory reduction, or ontological continuity.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.
Heisenberg’s uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true’ probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.
A universal formulation of the quantum uncertainty regarding quantum indeterminacy, quantum measurement, and its inevitable observer effect is presented with additional focus on the representability of quantum observables over a given state. The operational tangibility of the framework assures that the resultant general relations admit natural operational interpretations and characterisations, and thereby perhaps most importantly, their experimental verifiability. In view of the universal formulation, Heisenberg's original philosophy of the uncertainty principle, most typically exemplified in his famous gamma-ray microscope Gedankenexperiment, is revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. The relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the standard Kennard-Robertson relation, the Arthurs-Kelly-Goodman relations for joint measurement as well as the Ozawa and the Watanabe-Sagawa-Ueda relations for error and error-disturbance.
Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.