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Statistical Constraints and Measurement Uncertainty for Observables in Quantum Mechanics

Ryosuke Nogami

Published Mar 25, 2026• 0 citations
OpenAlex

Abstract

This dissertation investigates statistical constraints on measurement outcomes in quantum mechanics, addressing two fundamental features that distinguish quantum theory from classical physics: non-reality and uncertainty.We examine these features through two perspectives: observable-dependent statistical constraints on quantum correlations and estimationbased measurement uncertainty in infinite-dimensional quantum systems.In the first part, we study realizability problems in both local realism and quantum theory, focusing on observable-dependent constraints.While previous studies have mainly characterized quantum correlations independently of observables, we derive a necessary and sufficient condition for quantum realizability of correlations in the Clauser-Horne-Shimony-Holt (CHSH) scenario when observables are fixed in advance.Our main result provides an explicit inequality that imposes stronger constraints than previously known observable-independent bounds.We demonstrate that the maximum CHSH inequality violation depends on observable choice and apply our results to Bell's original scenario, revealing that whether Bell's original inequality is violated depends sensitively on observable selection.These findings demonstrate that non-reality does not manifest equally in all measurement configurations.In the second part, we extend the estimation-based formulation of measurement uncertainty by Watanabe, Sagawa, and Ueda to infinite-dimensional quantum systems.The original formulation provided clear physical interpretation but excluded fundamental observables such as position and momentum.Using pseudo-inverse forms of positive-semidefinite sesquilinear forms, we develop classical and quantum estimation theory for statistical models whose parameter space is the set of full-rank quantum states on infinite-dimensional Hilbert spaces.We derive classical and quantum Cramér-Rao inequalities and define estimationbased measurement errors for both bounded and unbounded operators.For each error type, we establish error-error uncertainty relation inequalities stronger than the original Watanabe-Sagawa-Ueda bound due to an improved derivation utilizing the full Cauchy-Schwarz inequality.Our work provides a universal definition of measurement errors with a clear physical interpretation while maintaining mathematical rigor.Our results contribute to quantum foundations by deepening the understanding of how measurement outcome constraints arise from quantum structure and observable properties.The methodological tools introduced-including characteristic functions, quasi-joint probability distributions, symmetry-based reduction techniques, and pseudo-inverse forms-may prove useful for future investigations in quantum foundations and quantum information theory.

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Annalen der Physik•Jul 8, 2025•1 citation
This dissertation investigates statistical constraints on measurement outcomes in quantum mechanics, addressing two fundamental features that distinguish quantum theory from classical physics: non-reality and uncertainty.We examine these features through two perspectives: observable-dependent statistical constraints on quantum correlations and estimationbased measurement uncertainty in infinite-dimensional quantum systems.In the first part, we study realizability problems in both local realism and quantum theory, focusing on observable-dependent constraints.While previous studies have mainly characterized quantum correlations independently of observables, we derive a necessary and sufficient condition for quantum realizability of correlations in the Clauser-Horne-Shimony-Holt (CHSH) scenario when observables are fixed in advance.Our main result provides an explicit inequality that imposes stronger constraints than previously known observable-independent bounds.We demonstrate that the maximum CHSH inequality violation depends on observable choice and apply our results to Bell's original scenario, revealing that whether Bell's original inequality is violated depends sensitively on observable selection.These findings demonstrate that non-reality does not manifest equally in all measurement configurations.In the second part, we extend the estimation-based formulation of measurement uncertainty by Watanabe, Sagawa, and Ueda to infinite-dimensional quantum systems.The original formulation provided clear physical interpretation but excluded fundamental observables such as position and momentum.Using pseudo-inverse forms of positive-semidefinite sesquilinear forms, we develop classical and quantum estimation theory for statistical models whose parameter space is the set of full-rank quantum states on infinite-dimensional Hilbert spaces.We derive classical and quantum Cramér-Rao inequalities and define estimationbased measurement errors for both bounded and unbounded operators.For each error type, we establish error-error uncertainty relation inequalities stronger than the original Watanabe-Sagawa-Ueda bound due to an improved derivation utilizing the full Cauchy-Schwarz inequality.Our work provides a universal definition of measurement errors with a clear physical interpretation while maintaining mathematical rigor.Our results contribute to quantum foundations by deepening the understanding of how measurement outcome constraints arise from quantum structure and observable properties.The methodological tools introduced-including characteristic functions, quasi-joint probability distributions, symmetry-based reduction techniques, and pseudo-inverse forms-may prove useful for future investigations in quantum foundations and quantum information theory.
We establish a necessary and sufficient condition for the existence of a quantum state that reproduces given correlation values in the Clauser--Horne--Shimony--Holt (CHSH) setup for any fixed normalized observables. This result addresses a fundamental question shared by both local realism and quantum mechanics: under what conditions a given set of observed data can be reproduced by a physical model. While previous studies have mainly addressed conditions for correlations achievable without specifying the measurement settings, our result gives a finer characterization by treating the observables as fixed in advance. The resulting quantum condition strengthens previously known constraints, such as Tsirel'son's inequalities and the Tsirel'son--Landau inequality, by characterizing statistical constraints explicitly for each specified set of observables. In particular, we show that our condition applies to Bell's original scenario and reveals that whether Bell's original inequality is violated depends sensitively on the chosen observables. More broadly, this perspective offers new insights into how quantum violations of local realism depend on the measurement settings.
We establish a necessary and sufficient condition for the existence of a quantum state that reproduces given correlation values in the Clauser--Horne--Shimony--Holt setup for any fixed normalized observables. This result addresses a fundamental question shared by both local realism and quantum mechanics: under what conditions a given set of observed data can be reproduced by a physical model. While previous studies have mainly addressed conditions for correlations achievable without specifying the measurement settings, our result gives a finer characterization by treating the observables as fixed in advance. The resulting quantum condition strengthens previously known constraints, such as Tsirel'son's inequalities and the Tsirel'son--Landau inequality, by characterizing statistical constraints explicitly for each specified set of observables. In particular, we show that our condition applies to Bell's original scenario and reveals that whether Bell's original inequality is violated depends sensitively on the chosen observables. More broadly, this perspective offers new insights into how quantum violations of local realism depend on the measurement settings.
A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational tangibility of the framework, the resultant general relations admit natural operational interpretations and characterisations, and are thus also experimentally verifiable. In view of the universal formulation, Heisenberg's philosophy of the uncertainty principle is also revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. In fact, the relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the Arthurs-Kelly-Goodman, Ozawa, and Watanabe-Sagawa-Ueda relations for quantum measurements. The Schr{ö}dinger relation (hence the standard Kennard-Robertson relation as its trivial corollary as well) is also shown to be a special case when the measurement is non-informative.
This paper argues that physical quantities should be reconstructed not as non-contextual intrinsic attributes with measurement-independent determinate values, but as equivalence classes of measurement procedures. Bell–CHSH violations first motivate the problem: they show that Bell-type local causality/factorizability, measurement independence, and the relevant form of outcome determinism or value-definiteness cannot be jointly maintained, without by themselves refuting realism as such. Kochen–Specker contextuality supplies the stronger premise for the sharp-observable case: no global, non-contextual, dispersion-free value assignment to projection-valued observables can preserve the functional relations among compatible observables in Hilbert spaces of dimension at least three. The positive proposal uses algebraic quantum theory to identify physical quantities with classes of measuring procedures that yield the same outcome statistics over a full or tomographically complete state domain. This reconstruction preserves the classical deterministic limit while rejecting the stronger claim that measured quantities are context-independent values possessed independently of measurement. The analysis is restricted to sharp observables and projection-valued measurements; POVMs and unsharp measurements are noted as extensions. The resulting view is realist about an observer-independent world but non-intrinsic about empirical quantities, and it situates Wheeler's "It from Bit" as an interpretive corollary rather than as a foundational premise.
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 Bell nonlocality refers to the nonlocal correlation strength among quantum systems that cannot be duplicated using local hidden variables, and uncertainty relations set the lower bound on the sum or product of variances for incompatible observables within a quantum system. In this work, the study of linking these two fundamental quantum features in terms of statistical characteristics within Bell tests is initiated. Two statistics determine the state‐dependent quantum upper bounds of the Bell values: the expectation values of local observables, and the joint uncertainty that is the product of local uncertainty. Specifically, it is showed that the joint measurement uncertainty determines the upper bounds of covariant Bell inequalities in the quantum region, with more joint measurement uncertainty required for more covariant correlation strengths. Conversely, the covariant Bell value defines the lower bound of joint uncertainty. In addition, given the maximally entangled states, Tsirelson's bounds of Bell inequalities can be achieved when the joint uncertainty reaches the maximum.