Considering the simultaneous measurement of non-commuting observables, we define a geometric measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. The rationality of our geometric measure is demonstrated and the application of it is presented. The connection between our measure and Heisenberg's uncertainty principle is discussed as well. Our work deepens the understanding of quantum non-commuting measurement.
Considering the simultaneous measurement of non-commuting observables, we define a geometric measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. The rationality of our geometric measure is demonstrated and the application of it is presented. The connection between our measure and Heisenberg's uncertainty principle is discussed as well. Our work deepens the understanding of quantum non-commuting measurement.
Simultaneous measurement of non-commuting observables deepens the understanding of Heisenberg’s uncertainty principle and has significant applications to quantum feedback control theory. Here we use techniques from machine learning to define a measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. Both the measurement strength effect and the angle effect between two observables have been constructed by using the K-means clustering method to reflect the defined non-commutativity measure. The rationality of the measure is demonstrated and its application in obtaining the measurement limitation that physically realizable is presented. Our work helps to advance the understanding of quantum non-commuting measurement and quantum control theory.
Quantum uncertainty, famously encapsulated by Heisenberg's principle, is a cornerstone of quantum mechanics, traditionally understood through the algebraic non-commutativity of observable operators. This paper proposes a novel framework: the commutator geometry of wavefunctions, aiming to unify diverse manifestations of quantum uncertainty within a single, intrinsic geometric description of the quantum state space. By establishing a direct link between the algebraic structure of commutators and the differential geometry of projective Hilbert space, we demonstrate how the inherent non-commutativity of observables fundamentally sculpts the geometry of quantum states. This approach allows for the derivation of geometric measures of uncertainty that intrinsically capture the limitations on simultaneous measurability. We show how this geometric formulation not only recovers existing uncertainty relations, such as the Robertson-Schroedinger inequality, but also provides a more profound, unified interpretation of quantum uncertainty as a manifestation of curvature and metric properties within the space of wavefunctions. The methodology involves constructing a commutator-induced metric and curvature tensor on the quantum state manifold, revealing how the degree of non-commutativity directly dictates the "fuzziness" or "spread" in observable values. Illustrative examples, including position-momentum and angular momentum, are used to demonstrate the efficacy and intuitive appeal of this geometric perspective. The implications of this unified geometric understanding extend to foundational issues in quantum mechanics, potentially offering new insights into quantum gravity, quantum information theory, and the very nature of quantum reality, where uncertainty is not merely a statistical limitation but an intrinsic geometric feature of the universe.
Abstract The existence of incompatible measurements, i.e. measurements which cannot be performed simultaneously on a single copy of a quantum state, constitutes an important distinction between quantum mechanics and classical theories. While incompatibility might at first glance seem like an obstacle, it turns to be a necessary ingredient to achieve the so-called quantum advantage in various operational tasks like random access codes or key distribution. To improve our understanding of how to quantify incompatibility of quantum measurements, we define and explore a family of incompatibility measures based on non-commutativity. We investigate some basic properties of these measures, we show that they satisfy some natural information-processing requirements and we fully characterize the pairs which achieve the highest incompatibility (in a fixed dimension). We also consider the behavior of our measures under different types of compositions. Finally, to link our new measures to existing results, we relate them to a robustness-based incompatibility measure and two operational scenarios: random access codes and entropic uncertainty relations.
refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrodinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.
Quantum uncertainty lies at the heart of quantum mechanics, fundamentally distinguishing it from classical physics. Traditionally quantified through variance-based measures, these approaches often rely on the expectation values of observables, providing a macroscopic statistical view rather than an intrinsic characterization of the wavefunction's inherent indeterminacy. This paper introduces a novel commutator-induced metric designed to directly quantify quantum uncertainty within the wavefunction itself, independent of specific measurement outcomes or a pair of conjugate variables. By leveraging the fundamental non-commutativity of quantum operators, this metric provides a more fundamental and intrinsic measure of the delocalization and inherent indeterminacy encoded in the quantum state. We define this metric based on the expectation value of an operator derived from the self-commutator of the wavefunction's defining operators, mapping the non-commutativity inherent in the Hilbert space onto a scalar value that reflects the degree of quantum uncertainty. This approach offers a new perspective on interpreting the uncertainty principle, providing a generalized framework that extends beyond canonical conjugate pairs and potentially offering insights into the structure of quantum information. The paper details the theoretical construction of this metric, explores its mathematical properties, and discusses its potential applications in quantum information theory, quantum measurement, and the foundational understanding of quantum mechanics.