Quantum uncertainty is the cornerstone of quantum mechanics which underlies many counterintuitive nonclassical phenomena. Recent studies remarkably showed that it also fundamentally limits nonclassical correlation, and crucially, a deviation from its exact form may lead to a violation of the second law of thermodynamics. Are there deep and natural principles which uniquely determine its form? Here we work within a general epistemic framework for a class of nonclassical theories, introducing an epistemic restriction to an otherwise classical theory, so that the distributions of positions are irreducibly parametrized by the underlying momentum fields. It was recently shown that the mathematics of quantum mechanics formally arises within an operational scheme, wherein an agent makes a specific estimation of the momentum given information on the positions and the experimental settings. Moreover, quantum uncertainty can be traced back to the ``specific'' choice of estimator and the associated estimation error. In the present work, we show that a plausible inferential-causality principle of estimation independence, which requires that the estimation of momentum of one system must be independent of the position of another system independently prepared of the first, singles out the specific forms of the estimator, and especially the estimation error up to its strength given by a global-nonseparable random variable on the order of Planck constant.
One of the advantages of a reconstruction of quantum mechanics based on transparent physical axioms is that it may offer insight to naturally generalize quantum mechanics by relaxing the axioms. Here, we discuss possible extensions of quantum mechanics within a general epistemic framework based on an operational scheme of estimation of momentum given information on the conjugate positions under epistemic restriction. The epistemic restriction is parameterized by a global-nonseparable random variable on the order of Planck constant, an ontic extension to the separable classical phase-space variables. Within the estimation scheme, the canonical quantum laws are reconstructed for a specific estimator and estimation error. In the present work, keeping the Born's quadratic law intact, we construct a class of nonlinear variants of Schr\"odinger equation and generalized Heisenberg uncertainty principle within the estimation scheme by assuming a more general class of estimation errors. The nonlinearity of the Schr\"odinger equation and the deviation from the Heisenberg uncertainty principle thus have a common transparent operational origin in terms of generalizations of estimation errors. We then argue that a broad class of nonlinearities and deviations from the Heisenberg uncertainty principle arise from estimation errors violating a plausible inferential-causality principle of estimation independence which is respected by the standard quantum mechanics. This result therefore constrains possible extensions of quantum mechanics and suggests directions to generalize quantum mechanics which comply with the principle of estimation independence.
Consider a statistical model with an epistemic restriction such that, unlike in classical mechanics, the allowed distribution of positions is fundamentally restricted by the form of an underlying momentum field. Assume an agent (observer) who wishes to estimate the momentum field given information on the conjugate positions. We discuss a classically consistent, weakly unbiased, best estimation scheme for the momentum field (by minimizing the mean-squared error) based on which the abstract mathematical rules of quantum mechanics can be derived. The results suggest that a quantum wave function is not an objective agent-independent attribute of reality but represents the agent's best estimations of the momentum, given the positions, under epistemic restriction. Quantum uncertainty and complementarity between momentum and position find their epistemic origin from the trade-off between the mean-squared errors of simultaneous estimations of momentum field and mean position, with the Gaussian wave function representing the simultaneous efficient estimations, achieving the Cram\'er-Rao bounds of the associated mean-squared errors. We then argue that unitary time evolution and wave-function collapse in measurement are normative rules for an agent to update her estimations given information on the experimental settings.
What does it take for real-deterministic
c-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic
c-valued variables out of the weak values obtained via a nonperturbing weak measurement of quantum operators with a postselection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to convert the incompatibility between the quantum operator and the basis into the fluctuation of an ``error term,'' and then superimpose it onto the real part of the weak value. We show that this class of ``
c-valued physical quantities'' provides a real-deterministic contextual hidden-variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the
c-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two
c-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty principle is captured by a specific irreducible epistemic restriction between the two
c-valued physical quantities, foreign in classical mechanics, constraining the allowed form of their joint distribution. We then suggest an epistemic interpretation of the two terms decomposing the
c-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.
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.
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an "ontic extension" defines a nonseparable (global) random variable that generates physical correlations, and an "epistemic restriction" constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck's constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.
Abstract Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an “ontic extension” defines a nonseparable (global) random variable that generates physical correlations, and an “epistemic restriction” constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck’s constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.