Math 2.0

Harmonic analysis on Euclidean spaces Uncertainty principles Theorem

Entropic uncertainty principle

Builds on Fourier transformLp spaceBabenko–Beckner inequalityPlancherel theorem

Statement

Let f∈L2(ℝ)f \in L^2(\R) with ∥f∥2=1\norm{f}_2 = 1, and let f^\hat f be its Fourier transform on L2(ℝ)L^2(\R), normalized so that f^(ξ)=∫ℝf(x)e−2πixξ dx\hat f(\xi) = \int_\R f(x) e^{-2\pi i x \xi}\, dx for integrable ff. For a probability density pp on ℝ\R, let H(p)=−∫ℝp(x)log⁡p(x) dxH(p) = -\int_\R p(x) \log p(x)\, dx be its differential entropy, with the natural logarithm. Suppose that the integrals defining H(∣f∣2)H(\abs{f}^2) and H(∣f^∣2)H(\abs{\hat f}^2) converge absolutely. Then H(∣f∣2)+H(∣f^∣2)≥log⁡e2.H(\abs{f}^2) + H(\abs{\hat f}^2) \ge \log \frac{e}{2}. Equality holds when ff is a Gaussian f(x)=(2a)1/4e−πax2f(x) = (2a)^{1/4} e^{-\pi a x^2} with a>0a > 0.

Remarks

By the Plancherel theorem, ∣f^∣2\abs{\hat f}^2 is also a probability density, so both entropies make sense. The proof differentiates the Babenko–Beckner inequality ∥f^∥q≤Ap∥f∥p\norm{\hat f}_q \le A_p \norm{f}_p with respect to pp at p=2p = 2. At p=2p = 2 both sides are equal to 11, so the derivative of the difference has a sign, and that sign is the entropy inequality. I. I. Hirschman proved a weaker constant and conjectured this one; W. Beckner's sharp constant in the Hausdorff–Young inequality gave the conjecture.

On ℝn\R^n the bound is nlog⁡e2n \log \frac{e}{2}.

The inequality implies the Heisenberg inequality. For a density pp with variance σ2\sigma^2, the entropy satisfies H(p)≤12log⁡(2πeσ2)H(p) \le \frac12 \log(2\pi e \sigma^2), with equality for normal densities. Inserting this for ∣f∣2\abs{f}^2 and ∣f^∣2\abs{\hat f}^2 gives σx2σξ2≥1/(16π2)\sigma_x^2 \sigma_\xi^2 \ge 1/(16\pi^2).

Going deeper

For a student

Entropy is a finer measure of spread than variance: a density can have infinite variance and still finite entropy. So this form of the uncertainty principle is stronger than the Heisenberg inequality, which it implies. It is the model for entropic uncertainty relations in quantum information, where the Fourier transform is replaced by a pair of measurements in two bases.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Almost everywhere, concept, no evidence levelAlmost everywhereconceptFourier transform, concept, no evidence levelFourier transformconceptHilbert space, concept, no evidence levelHilbert spaceconceptIntegrable function, concept, no evidence levelIntegrable functionconceptIntegral of a non-negative function, concept, no evidence levelIntegral of a non-neg…conceptLebesgue integral, concept, no evidence levelLebesgue integralconceptLp space, concept, no evidence levelLp spaceconceptMeasurable function, concept, no evidence levelMeasurable functionconceptOperator norm, concept, no evidence levelOperator normconceptSequence, definition, no evidence levelSequencedefinitionSigma-finite measure, concept, no evidence levelSigma-finite measureconceptUnitary operator, concept, no evidence levelUnitary operatorconceptPlancherel theorem, statement, no evidence levelPlancherel theoremstatementMonotone convergence theorem, statement, no evidence levelMonotone convergence …statementRiesz–Thorin theorem, statement, no evidence levelRiesz–Thorin theoremstatementReal sequence, definition, no evidence levelReal sequencedefinitionHausdorff–Young inequality, statement, no evidence levelHausdorff–Young inequ…statementBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit inferior, definition, no evidence levelLimit inferiordefinitionFatou's lemma, statement, no evidence levelFatou's lemmastatementDominated convergence theorem, statement, no evidence levelDominated convergence…statementDifferentiation and multiplication under the Fourier transform, statement, no evidence levelDifferentiation and m…statementFourier transform of the Gaussian, statement, no evidence levelFourier transform of …statementBabenko–Beckner inequality, statement, no evidence levelBabenko–Beckner inequ…statementEntropic uncertainty principle, statement, no evidence levelEntropic uncertainty …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Almost everywhere concept
  2. Fourier transform concept
  3. Hilbert space concept
  4. Integrable function concept
  5. Integral of a non-negative function concept
  6. Lebesgue integral concept
  7. Lp space concept
  8. Measurable function concept
  9. Operator norm concept
  10. Sequence definition
  11. Sigma-finite measure concept
  12. Unitary operator concept
  13. Plancherel theorem statement
  14. Monotone convergence theorem statement
  15. Riesz–Thorin theorem statement
  16. Real sequence definition
  17. Hausdorff–Young inequality statement
  18. Bounded sequence definition
  19. Limit inferior definition
  20. Fatou's lemma statement
  21. Dominated convergence theorem statement
  22. Differentiation and multiplication under the Fourier transform statement
  23. Fourier transform of the Gaussian statement
  24. Babenko–Beckner inequality statement
  25. Entropic uncertainty principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Entropic_uncertainty
About this page
TagCBRTFS0RVC
URIm2://CBRTFS0RVC
Kindstatement, theorem
WritingW0 stub
MSC42A38, 94A17, 81S07
StatusThis copy of the site has no build data. Run m2 status CBRTFS0RVC.
AboutFourier transform
UsesFourier transform, Lp space, Babenko–Beckner inequality, Plancherel theorem
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