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IEEE Signal Processing Letters

Sharper NNN-D Heisenberg's Uncertainty Principle

Zhichao Zhang, Xi-Ya Shi, An-Yang Wu, Dong Li

Published Jan 1, 2021• 26 citations
Publisher

Abstract

A sharper uncertainty inequality which exhibits a lower bound tighter than that in the classical N-dimensional Heisenberg's uncertainty principle is obtained. The condition that reaches an equality relation of the uncertainty inequality is deduced. Example and simulations are carried out to illustrate that the newly derived uncertainty principle is truly sharper than the classical one. The new proposal's applications in time-frequency analysis, optical signal processing and physical optics propagation are also given.

DOI
10.1109/lsp.2021.3101114

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A sharper uncertainty inequality which exhibits a lower bound tighter than that in the classical N-dimensional Heisenberg's uncertainty principle is obtained. The condition that reaches an equality relation of the uncertainty inequality is deduced. Example and simulations are carried out to illustrate that the newly derived uncertainty principle is truly sharper than the classical one. The new proposal's applications in time-frequency analysis, optical signal processing and physical optics propagation are also given.
A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.
In Fourier analysis and quantum mechanics, the Heisenberg uncertainty principle stands as a fundamental result. Its classical version provides a product lower bound for the variances of position and frequency. More recently, for vector-valued functions, Dang, Deng and Qian introduced an improved inequality featuring an extra covariance term derived from the current density. The present work performs a one-dimensional numerical comparison of the two lower bounds using four test functions: a pure Gaussian, an asymmetric sum of two Gaussians, a linear chirp, and a vector-valued signal with spatially separated orthogonal components. For the Gaussian, the covariance term vanishes up to machine precision, making the two bounds identical. For the asymmetric sum, the refined bound exceeds the classical one by about three percent. The chirp signal yields a dramatically larger refined bound, approximately sixteen times above the classical one. For the vector-valued case with each component a real Gaussian, the refined bound is about eight times higher. These numerical outcomes confirm the validity of the refined inequality and highlight the sensitivity of the covariance term to phase structure and vectorial coupling.
The position-momentum uncertainty-like inequality based on moments of arbitrary order for ddd-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the R\'enyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in ddd-dimensional physics: the hydrogenic and oscillator-like systems.
New uncertainty relations in quantum mechanics are derived. They express restrictions imposed by quantum theory on probability distributions of canonically conjugate variables in terms of corresponding information entropies. The Heisenberg uncertainty relation follows from those inequalities and so does the Gross-Nelson inequality. The purpose of this paper is to derive a new, stronger version of the Heisenberg uncertainty relation in wave mechanics. This new uncertainty relation has a simple interpretation in terms of information theory. It is also closely related to newly discovered logarithmic Sobolev inequalities. The new uncertainty relation has the form (for wave functions normalized to unity)
The phenomenon in the essence of classical uncertainty principles is well known since the thirties of the last century. We introduce a new phenomenon which is in the essence of a new notion that we introduce: "Generalized Uncertainty Principles". We show the relation between classical uncertainty principles and generalized uncertainty principles. We generalized "Landau-Pollak-Slepian" uncertainty principle. Our generalization relates the following two quantities and two scaling parameters: 1) The weighted time spreading ∫−∞∞∣f(x)∣2w1(x)dx\int_{-\infty}^\infty |f(x)|^2w_1(x)dx∫−∞∞​∣f(x)∣2w1​(x)dx, (w1(x)w_1(x)w1​(x) is a non-negative function). 2) The weighted frequency spreading ∫−∞∞∣f^(ω)∣2w2(ω)dω\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω∫−∞∞​∣f^​(ω)∣2w2​(ω)dω. 3) The time weight scale aaa, w1a(x)=w1(xa−1){w_1}_a(x)=w_1(xa^{-1})w1​a​(x)=w1​(xa−1) and 4) The frequency weight scale bbb, w2b(ω)=w2(ωb−1){w_2}_b(ω)=w_2(ωb^{-1})w2​b​(ω)=w2​(ωb−1). "Generalized Uncertainty Principle" is an inequality that summarizes the constraints on the relations between the two spreading quantities and two scaling parameters. For any two reasonable weights w1(x)w_1(x)w1​(x) and w2(ω)w_2(ω)w2​(ω), we introduced a three dimensional set in R3R^3R3 that is in the essence of many uncertainty principles. The set is called "possibility body". We showed that classical uncertainty principles (such as the Heiseneberg-Pauli-Weyl uncertainty principle) stem from lower bounds for different functions defined on the possibility body. We investigated qualitative properties of general uncertainty principles and possibility bodies. Using this approach we derived new (quantitative) uncertainty principles for Landau-Pollak-Slepian weights. We found the general uncertainty principles related to homogeneous weights, w1(x)=w2(x)=xkw_1(x)=w_2(x)=x^kw1​(x)=w2​(x)=xk, k∈Nk\in Nk∈N, up to a constant.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: <||x||^2> <||u||^2> ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy un- der statistical moments' constraints ( <||x||^a> and <||u||^b> ). Our result improves a relation proposed very recently by Dehesa et al. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems.