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arXiv:1809.07249 · Quantum Reports

The Measure Aspect of Quantum Uncertainty, of Entanglement, and the Associated Entropies

Ivan Horváth A

Published Sep 12, 2021• 11 citations
arXivHTMLPDF

Abstract

Indeterminacy associated with the probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here, we express it as an effective amount (measure) of distinct outcomes instead. The resulting μ-uncertainties are described by the effective number theory whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive μ-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-like characteristics, the μ-entropies, convey how many degrees of freedom are effectively involved in a given measurement process. In order to construct quantum μ-entropies, we are led to quantum effective numbers designed to count independent, mutually orthogonal states effectively comprising a density matrix. This concept is basis-independent and leads to a measure-based characterization of entanglement.

DOI
10.3390/quantum3030035

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The information-theoretic formulation of quantum measurement uncertainty \nrelations (MURs), based on the notion of relative entropy between measurement \nprobabilities, is extended to the set of all the spin components for a generic \nspin $s$. For a physical class of approximate joint measurements of the spin \ncomponents, we define the device information loss as the maximum loss of \ninformation per observable occurring in approximating the ideal incompatible \ncomponents with the joint measurement at hand. By optimizing on the measuring \ndevice, we define the notion of minimum information loss. By using this \nnotions, we show how to give a significant formulation of state independent \nMURs in the case of infinitely many target observables. The same construction \nworks as well for finitely many observables, and we study the related MURs for \ntwo and three orthogonal spin components. The minimum information loss plays \nalso the role of measure of incompatibility and in this respect it allows us to \ncompare quantitatively the incompatibility of various sets of spin observables, \nwith different number of involved components and different values of $s$.

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Indeterminacy associated with the probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here, we express it as an effective amount (measure) of distinct outcomes instead. The resulting μ-uncertainties are described by the effective number theory whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive μ-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-like characteristics, the μ-entropies, convey how many degrees of freedom are effectively involved in a given measurement process. In order to construct quantum μ-entropies, we are led to quantum effective numbers designed to count independent, mutually orthogonal states effectively comprising a density matrix. This concept is basis-independent and leads to a measure-based characterization of entanglement.
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.
Uncertainty is a fundamental characteristic of quantum system. The degree of uncertainty of an observable has long been investigated by the standard deviation of the observable. In recent years, however, by analyzing some special examples, researchers have found that the Shannon entropy of the measurement outcomes of an observable is more suitable to quantify its uncertainty. Formally, Shannon entropy is a special limit of a more general Rényi entropy. In this paper, we discuss the problem of how to predict the measurement outcome of an observable by the existing measurement results of the observable, and how to quantitatively describe the uncertainty of the observable from the perspective of the repeatable probability of the measurement results of this observable in an unknown state. We will argue that if the same observable of different systems in the same state is repeatedly and independently measured many times, then the probability of obtaining an identical measurement result is a decaying function of the number of measurements of obtaining the same result, and the decay rate of the repeatable probability for obtaining the same measurement results and the repeatable number of measurements can represent the degree of uncertainty of the observable in this state. It means that the greater the uncertainty of an observable, the faster the repeatable probability decays with the number of repeatable measurements; conversely, the smaller the uncertainty, the slower the repeatable probability decays with the number of repeatable measurements. This observation enables us to give the Shannon entropy and the Rényi entropy of an observable uniformly by the functional relation between the repeatable probability and the number of repeatable measurements. We show that the Shannon entropy and the Rényi entropy can be formally regarded as the “decay index” of the repeatable probability with the number of repeatable measurements. In this way we also define a generalized Rényi entropy by the repeatable probability for consecutively observing identical results of an observable, and therefore we give a proof of the Maassen-Uffink type entropic uncertainty relation by using this generalized Rényi entropy. This method of defining entropy shows that entropic uncertainty relation is a quantitative limitation for the decay rate of the total probability for obtaining identical measurement results when we simultaneously measure two observables many times.
Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring noncommuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabeling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in M. H. Partovi, Phys. Rev. A 84, 052117 (2011). Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.
Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constrains on the accuracy of two non-commuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation of coherence between two measurement non-commuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1l_1l1​ norm of coherence.
Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R\`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R\`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R\`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. log2log 2log2 for a single qubit.
The information-theoretic formulation of quantum measurement uncertainty \nrelations (MURs), based on the notion of relative entropy between measurement \nprobabilities, is extended to the set of all the spin components for a generic \nspin sss. For a physical class of approximate joint measurements of the spin \ncomponents, we define the device information loss as the maximum loss of \ninformation per observable occurring in approximating the ideal incompatible \ncomponents with the joint measurement at hand. By optimizing on the measuring \ndevice, we define the notion of minimum information loss. By using this \nnotions, we show how to give a significant formulation of state independent \nMURs in the case of infinitely many target observables. The same construction \nworks as well for finitely many observables, and we study the related MURs for \ntwo and three orthogonal spin components. The minimum information loss plays \nalso the role of measure of incompatibility and in this respect it allows us to \ncompare quantitatively the incompatibility of various sets of spin observables, \nwith different number of involved components and different values of sss.