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arXiv:0807.2190

Generalized uncertainty principles

Ronny Machluf

Published Jul 14, 2008• 4 citations
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Abstract

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 $\int_{-\infty}^\infty |f(x)|^2w_1(x)dx$, ($w_1(x)$ is a non-negative function). 2) The weighted frequency spreading $\int_{-\infty}^\infty |\hat{f}(ω)|^2w_2(ω)dω$. 3) The time weight scale $a$, ${w_1}_a(x)=w_1(xa^{-1})$ and 4) The frequency weight scale $b$, ${w_2}_b(ω)=w_2(ωb^{-1})$. "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 $w_1(x)$ and $w_2(ω)$, we introduced a three dimensional set in $R^3$ 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, $w_1(x)=w_2(x)=x^k$, $k\in N$, up to a constant.

DOI
10.48550/arxiv.0807.2190

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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.
Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.
Historically, the element of uncertainty in quantum mechanics has been expressed through mathematical identities called uncertainty relations, a great many of which continue to be discovered. These relations use diverse measures to quantify uncertainty (and joint uncertainty). In this paper we use operational information-theoretic principles to identify the common essence of all such measures, thereby defining measure-independent notions of uncertainty and joint uncertainty. We find that most existing entropic uncertainty relations use measures of joint uncertainty that yield themselves to a small class of operational interpretations. Our notion relaxes this restriction, revealing previously unexplored joint uncertainty measures. To illustrate the utility of our formalism, we derive an uncertainty relation based on one such new measure. We also use our formalism to gain insight into the conditions under which measure-independent uncertainty relations can be found.
Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.
Standard-deviation and entropic formulations of uncertainty principle capture the spread of the probability distribution but say little about the probability itself contained in a small region. We introduce the confidence uncertainty Δcx(θx)Δ^{c}x(θ_x)Δcx(θx​) as the minimal Lebesgue measure of the support set in which the particle is found with probability at least θxθ_xθx​, and the companion interval confidence uncertainty ΔIx(θx)Δ^{I}x(θ_x)ΔIx(θx​) which restricts the support to a single interval. We prove two complementary uncertainty inequalities. (i) For θx+θp≤1θ_x+θ_p\le 1θx​+θp​≤1 both confidence uncertainties can be made arbitrarily small simultaneously, so that no nontrivial product bound holds; in particular, position and momentum can be jointly localised with probability at least~50%50\%50%. (ii) For θx+θp>1θ_x+θ_p>1θx​+θp​>1 a lower bound holds: combining Lenard's projection inequality with the Donoho--Stark operator-norm bound we obtain Δcx Δcp≥2πℏ(θxθp−(1−θx)(1−θp)) ⁣2Δ^{c}x\,Δ^{c}p\geq 2π\hbar\bigl(\sqrt{θ_xθ_p}-\sqrt{(1-θ_x)(1-θ_p)}\bigr)^{\!2}ΔcxΔcp≥2πℏ(θx​θp​​−(1−θx​)(1−θp​)​)2, and for the interval version we obtain the sharp implicit Landau--Pollak bound ΔIx ΔIp≥4ℏ λ0−1 ⁣((θxθp−(1−θx)(1−θp))2)Δ^{I}x\,Δ^{I}p\geq 4\hbar\,λ_{0}^{-1}\!\bigl((\sqrt{θ_xθ_p}-\sqrt{(1-θ_x)(1-θ_p)})^{2}\bigr)ΔIxΔIp≥4ℏλ0−1​((θx​θp​​−(1−θx​)(1−θp​)​)2), where λ0(c)λ_{0}(c)λ0​(c) is the largest prolate-spheroidal eigenvalue. We support the analytical bounds with numerical evaluation of λ0(c)λ_{0}(c)λ0​(c), provide closed-form small-ccc and large-ccc asymptotics, compute the optimal Slepian-superposition states that saturate the interval bound, and compare the resulting product against the variance Heisenberg--Kennard, the Białynicki-Birula--Mycielski entropic, and the Donoho--Stark concentration bounds. The unified picture provides a complete phase diagram on (θx,θp)∈[0,1]2(θ_x,θ_p)\in[0,1]^{2}(θx​,θp​)∈[0,1]2.