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arXiv:2312.15544

Heisenberg uncertainty principle and its analogues in higher dimension: via Wigdersons' method

Yiyu Tang

Published Dec 24, 2023• 0 citations
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Abstract

The following question was proposed by Avi Wigderson and Yuval Wigderson: Is it possible to use the method in their paper(The uncertainty principle: variations on a theme) to prove Heisenberg uncertainty principle in higher dimension R^d, and get the correct dependence of the constant on d? We answer this question affirmatively, and also prove some generalizations of Heisenberg uncertainty principle in R^d via Wigdersons' method.

DOI
10.48550/arxiv.2312.15544

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The following question was proposed by Avi Wigderson and Yuval Wigderson: Is it possible to use the method in their paper(The uncertainty principle: variations on a theme) to prove Heisenberg uncertainty principle in higher dimension R^d, and get the correct dependence of the constant on d? We answer this question affirmatively, and also prove some generalizations of Heisenberg uncertainty principle in R^d via Wigdersons' method.
All SpS_pSp​ notions of quantum expansion are equivalent
In a recent work Li, Qiao, Wigderson, Wigderson and Zhang introduced notions of quantum expansion based on SpS_pSp​ norms and posed as an open question if they were all equivalent. We give an affirmative answer to this question.
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.