The quantum entropic uncertainty principle is more restrictive than the Heisenberg uncertainty principle. From the inverse logarithmic Sobolev inequalities H x ≤ ≤ 1 2 log ⁡ ⁡ ( 2 e π π σ σ x 2 / x 0 2 ) , H p ≤ ≤ 1 2 log ⁡ ⁡ ( 2 e π π σ σ p 2 / p 0 2 ) , (equivalently, from the fact that normal distributions maximize the entropy of all such with a given variance), it readily follows that this entropic uncertainty principle is stronger than the one based on standard deviations, because σ σ x σ σ p ≥ ≥ ℏ ℏ 2 exp ⁡ ⁡ ( H x + H p − − log ⁡ ⁡ ( e h 2 x 0 p 0 ) ) ≥ ≥ ℏ ℏ 2 .