In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. In 1957, Hirschman considered a function f and its Fourier transform g such that where the "≈" indicates convergence in 2, and normalized so that (by Plancherel's theorem), He showed that for any such functions the sum of the Shannon entropies is non-negative, A tighter bound, was conjectured by Hirschman and Everett, proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. The equality holds in the case of Gaussian distributions. Note, however, that the above entropic uncertainty function is distinctly different from the quantum Von Neumann entropy represented in phase space. Sketch of proof The proof of this tight inequality depends on the so-called (q, p)-norm of the Fourier transformation. (Establishing this norm is the most difficult part of the proof.) From this norm, one is able to establish a lower bound on the sum of the (differential) Rényi entropies, , where , which generalize the Shannon entropies. For simplicity, we consider this inequality only in one dimension; the extension to multiple dimensions is straightforward and can be found in the literature cited. Babenko–Beckner inequality The (q, p)-norm of the Fourier transform is defined to be where and In 1961, Babenko found this norm for even integer values of q. Finally, in 1975, using Hermite functions as eigenfunctions of the Fourier transform, Beckner proved that the value of this norm (in one dimension) for all q ≥ 2 is Thus we have the Babenko–Beckner inequality that Rényi entropy bound From this inequality, an expression of the uncertainty principle in terms of the Rényi entropy can be derived. Letting , 2α=p, and 2β=q, so that and 1/2<α<1<β, we have Squaring both sides and taking the logarithm, we get Multiplying both sides by reverses the sense of the inequality, Rearranging terms, finally yields an inequality in terms of the sum of the Rényi entropies, Note that this inequality is symmetric with respect to and : One no longer need assume that ; only