Statistical Constraints and Measurement Uncertainty for Observables in Quantum Mechanics

Ryosuke Nogami

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.

PEKKA J. LAHTI

Stephen D. Bartlett, Terry Rudolph, Robert W. Spekkens

How would the world appear to us if its ontology was that of classical mechanics but every agent faced a restriction on how much they could come to know about the classical state? We show that in most respects it would appear to us as quantum. The statistical theory of classical mechanics, which specifies how probability distributions over phase space evolve under Hamiltonian evolution and under measurements, is typically called Liouville mechanics, so the theory we explore here is Liouville mechanics with an epistemic restriction. The particular epistemic restriction we posit as our foundational postulate specifies two constraints. The first constraint is a classical analog of Heisenberg's uncertainty principle; the second-order moments of position and momentum defined by the phase-space distribution that characterizes an agent's knowledge are required to satisfy the same constraints as are satisfied by the moments of position and momentum observables for a quantum state. The second constraint is that the distribution should have maximal entropy for the given moments. Starting from this postulate, we derive the allowed preparations, measurements, and transformations and demonstrate that they are isomorphic to those allowed in Gaussian quantum mechanics and generate the same experimental statistics. We argue that this reconstruction of Gaussian quantum mechanics constitutes additional evidence in favor of a research program wherein quantum states are interpreted as states of incomplete knowledge and that the phenomena that do not arise in Gaussian quantum mechanics provide the best clues for how one might reconstruct the full quantum theory.

Stanley Gudder

Jean-Paul Marchand

Deepak Mohan

Abstract Uncertainty-like precision tradeoffs appear across domains that share no common physical substrate: quantum observables, time-frequency representation, statistical learning, control systems, and adaptive cognition. These are typically treated as domain-specific curiosities. This paper argues that they are instances of a single structural class: state-space admissibility constraints on bounded, dynamic systems. We identify the primitive condition from which this class emerges — when observables are constraint-coupled projections of an evolving state trajectory, their simultaneous deterministic fixation becomes structurally forbidden. This is not a consequence of measurement disturbance. It is a consequence of the requirement that the system persist. We derive this result from persistence-theoretic first principles, demonstrate its instantiation across domains, and distinguish two contributing mechanisms: structural uncertainty arising from constraint geometry, and interactional uncertainty arising from scale-dependent probe coupling. We further show that physical constants and invariants are more accurately described as high-solvency attractors — asymptotically stable but bounded by nonzero epistemic uncertainty. The paper positions these results as a category-level abstraction. No domain-specific theory is replaced; each remains intact as a specialisation within the structural class identified here. Keywords: persistence, constraint coupling, generalized uncertainty, state-space admissibility, bounded dynamical systems, measurement theory, scale-dependent observation