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  "body": "\n## Statement\n\nLet $f \\in L^2(\\R)$ with $\\norm{f}_2 = 1$, and let $\\hat f$ be its Fourier transform on $L^2(\\R)$, normalized so that $\\hat f(\\xi) = \\int_\\R f(x) e^{-2\\pi i x \\xi}\\, dx$ for integrable $f$.\nFor a probability density $p$ on $\\R$, let $H(p) = -\\int_\\R p(x) \\log p(x)\\, dx$ be its differential entropy, with the natural logarithm.\nSuppose that the integrals defining $H(\\abs{f}^2)$ and $H(\\abs{\\hat f}^2)$ converge absolutely.\nThen\n$$H(\\abs{f}^2) + H(\\abs{\\hat f}^2) \\ge \\log \\frac{e}{2}.$$\nEquality holds when $f$ is a Gaussian $f(x) = (2a)^{1/4} e^{-\\pi a x^2}$ with $a \u003e 0$.\n\n## Remarks\n\nBy the [[Plancherel theorem|HPZHRZ0502]], $\\abs{\\hat f}^2$ is also a probability density, so both entropies make sense.\nThe proof differentiates the [[Babenko–Beckner inequality|FFM7Y61KRT]] $\\norm{\\hat f}_q \\le A_p \\norm{f}_p$ with respect to $p$ at $p = 2$.\nAt $p = 2$ both sides are equal to $1$, so the derivative of the difference has a sign, and that sign is the entropy inequality.\nI. I. Hirschman proved a weaker constant and conjectured this one; W. Beckner's sharp constant in the Hausdorff–Young inequality gave the conjecture.\n\nOn $\\R^n$ the bound is $n \\log \\frac{e}{2}$.\n\nThe inequality implies the Heisenberg inequality.\nFor a density $p$ with variance $\\sigma^2$, the entropy satisfies $H(p) \\le \\frac12 \\log(2\\pi e \\sigma^2)$, with equality for normal densities.\nInserting this for $\\abs{f}^2$ and $\\abs{\\hat f}^2$ gives $\\sigma_x^2 \\sigma_\\xi^2 \\ge 1/(16\\pi^2)$.\n\n## Explanations\n\n### L1 human\n\nEntropy measures how spread out a probability distribution is.\nThe theorem says that the spread of $\\abs{f}^2$ and the spread of $\\abs{\\hat f}^2$, measured by entropy, cannot both be small.\nTheir sum is at least the number $\\log(e/2)$, and Gaussians reach this bound.\n\n### L2 human\n\nEntropy is a finer measure of spread than variance: a density can have infinite variance and still finite entropy.\nSo this form of the uncertainty principle is stronger than the Heisenberg inequality, which it implies.\nIt is the model for entropic uncertainty relations in quantum information, where the Fourier transform is replaced by a pair of measurements in two bases.\n",
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