Entropic uncertainty

The (q, p)-norm of the Fourier transform is defined to be[6]

${\displaystyle \|{\mathcal {F}}\|_{q,p}=\sup _{f\in L^{p}(\mathbb {R} )}{\frac {\|{\mathcal {F}}f\|_{q}}{\|f\|_{p}}},}$ where ${\displaystyle 1<p\leq 2~,}$   and ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.}$

In 1961, Babenko[7] found this norm for even integer values of q. Finally, in 1975, using Hermite functions as eigenfunctions of the Fourier transform, Beckner[3] proved that the value of this norm (in one dimension) for all q ≥ 2 is

${\displaystyle \|{\mathcal {F}}\|_{q,p}={\sqrt {p^{1/p}/q^{1/q}}}.}$

Thus we have the Babenko–Beckner inequality that

${\displaystyle \|{\mathcal {F}}f\|_{q}\leq \left(p^{1/p}/q^{1/q}\right)^{1/2}\|f\|_{p}.}$

From this inequality, an expression of the uncertainty principle in terms of the Rényi entropy can be derived.[6][8]

Letting ${\displaystyle g={\mathcal {F}}f}$, 2α=p, and 2β=q, so that 1/α + 1/β = 2 and 1/2<α<1<β, we have

${\displaystyle \left(\int _{\mathbb {R} }|g(y)|^{2\beta }\,dy\right)^{1/2\beta }\leq {\frac {(2\alpha )^{1/4\alpha }}{(2\beta )^{1/4\beta }}}\left(\int _{\mathbb {R} }|f(x)|^{2\alpha }\,dx\right)^{1/2\alpha }.}$

Squaring both sides and taking the logarithm, we get

${\displaystyle {\frac {1}{\beta }}\log \left(\int _{\mathbb {R} }|g(y)|^{2\beta }\,dy\right)\leq {\frac {1}{2}}\log {\frac {(2\alpha )^{1/\alpha }}{(2\beta )^{1/\beta }}}+{\frac {1}{\alpha }}\log \left(\int _{\mathbb {R} }|f(x)|^{2\alpha }\,dx\right).}$

Multiplying both sides by

${\displaystyle {\frac {\beta }{1-\beta }}=-{\frac {\alpha }{1-\alpha }}}$

reverses the sense of the inequality,

${\displaystyle {\frac {1}{1-\beta }}\log \left(\int _{\mathbb {R} }|g(y)|^{2\beta }\,dy\right)\geq {\frac {\alpha }{2(\alpha -1)}}\log {\frac {(2\alpha )^{1/\alpha }}{(2\beta )^{1/\beta }}}-{\frac {1}{1-\alpha }}\log \left(\int _{\mathbb {R} }|f(x)|^{2\alpha }\,dx\right)~.}$

Rearranging terms, finally yields an inequality in terms of the sum of the Rényi entropies,

${\displaystyle {\frac {1}{1-\alpha }}\log \left(\int _{\mathbb {R} }|f(x)|^{2\alpha }\,dx\right)+{\frac {1}{1-\beta }}\log \left(\int _{\mathbb {R} }|g(y)|^{2\beta }\,dy\right)\geq {\frac {\alpha }{2(\alpha -1)}}\log {\frac {(2\alpha )^{1/\alpha }}{(2\beta )^{1/\beta }}};}$

${\displaystyle H_{\alpha }(|f|^{2})+H_{\beta }(|g|^{2})\geq {\frac {1}{2}}\left({\frac {\log \alpha }{\alpha -1}}+{\frac {\log \beta }{\beta -1}}\right)-\log 2~.}$

Note that this inequality is symmetric with respect to α and β: One no longer need assume that α<β; only that they are positive and not both one, and that 1/α + 1/β = 2. To see this symmetry, simply exchange the rôles of i and −i in the Fourier transform.

Taking the limit of this last inequality as α, β → 1 yields the less general Shannon entropy inequality,

${\displaystyle H(|f|^{2})+H(|g|^{2})\geq \log {\frac {e}{2}},\quad {\textrm {where}}\quad g(y)\approx \int _{\mathbb {R} }e^{-2\pi ixy}f(x)\,dx~,}$

valid for any base of logarithm, as long as we choose an appropriate unit of information, bit, nat, etc.

The constant will be different, though, for a different normalization of the Fourier transform, (such as is usually used in physics, with normalizations chosen so that ħ=1 ), i.e.,

${\displaystyle H(|f|^{2})+H(|g|^{2})\geq \log(\pi e)\quad {\textrm {for}}\quad g(y)\approx {\frac {1}{\sqrt {2\pi }}}\int _{\mathbb {R} }e^{-ixy}f(x)\,dx~.}$

In this case, the dilation of the Fourier transform absolute squared by a factor of 2π simply adds log(2π) to its entropy.