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arXiv:2205.03193 · Communications in Theoretical Physics

Probability density functions of quantum mechanical observable uncertainties

Lin Zhang, Jinping Huang, Jiamei Wang, Shao-Ming Fei

Published Apr 29, 2022• 1 citation
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Abstract

Abstract We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the support of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state-independent uncertainty relations. Our results may be generalized to multiple observable cases in higher dimensional spaces.

DOI
10.1088/1572-9494/ac6b93

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Abstract We study the uncertainties of quantum mechanical observables, quantified by the standard deviation (square root of variance) in Haar-distributed random pure states. We derive analytically the probability density functions (PDFs) of the uncertainties of arbitrary qubit observables. Based on these PDFs, the uncertainty regions of the observables are characterized by the support of the PDFs. The state-independent uncertainty relations are then transformed into the optimization problems over uncertainty regions, which opens a new vista for studying state-independent uncertainty relations. Our results may be generalized to multiple observable cases in higher dimensional spaces.
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We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow for an arbitrary choice of metric for the outcome distance, and the choice of an exponent distinguishing, e.g., absolute and root mean square deviations. The emphasis of this article is on developing a unified treatment, in which one observable takes on values in an arbitrary locally compact Abelian group and the other in the dual group. In all cases, the phase space symmetry implies the equality of measurement and preparation uncertainty bounds. There is also a straightforward method for determining the optimal bounds.
Given two or more non-commuting observables, it is generally not possible to simultaneously assign precise values to each. This quantum mechanical uncertainty principle is widely understood to be encapsulated by some form of uncertainty relation, expressing a trade-off between the standard deviations or other measures of uncertainty of two (or more) observables, resulting from their non-commutativity. Typically, such relations are coarse, and miss important features. It was not until very recently that a broader perspective on quantum uncertainty was envisaged and explored, one that utilises the notion of an uncertainty region. Here we review this new approach, illustrating it with pairs or triples of observables in the case of qubit and qutrit systems. We recall some of the shortcomings of traditional uncertainty relations, and highlight their inability to identify the full uncertainty region. These shortcomings suggest a precautionary note that, we surmise, ought to accompany the presentation of the uncertainty principle in introductory quantum mechanics courses.