Tight Entropic Uncertainty Relations

Abstract

Entropic uncertainty relations $H(A)+H(B)\geqslant γ$ give a nonzero lower bound $γ$ to the sum of the Shannon entropies $H$ of the outcome probabilities of incompatible observables $A$ and $B$. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds $γ$ typically are state independent. Here we provide a state-independent lower bound $γ_s$ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter $s\to 2$, becomes asymptotically tight for all $A,B$. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/

The paper considers state-independent lower bounds on the sum of outcome entropies for two projective measurements with overlap matrix $U_{ji}=\langle b_j|a_i\rangle$. It expresses these bounds using induced operator norms that depend on the full overlap matrix, rather than only its largest entry. For $1<s<2$ and $s'=s/(s-1)$, it establishes $H(A)+H(B)\geq\gamma_s$, where $\gamma_s=-\frac{2s}{2-s}\log_2\|U\|_{s\to s'}$. This is at least the Maassen–Uffink bound $-2\log_2\max_{ij}|U_{ji}|$, and $\gamma_s$ converges to the minimum possible Shannon-entropy sum as $s\to2^-$. The same expression is the exact minimum of $H_{s/2}(A)+H_{s'/2}(B)$ for each $s$. The Shannon bound follows from Riesz–Thorin interpolation and differentiation at the Hilbert-space norm; the Rényi statement follows directly from the induced-norm definition. Numerical evaluation uses nonlinear power iteration with multiple random initializations. The paper explicitly notes that this procedure can converge to local maxima, so finding the global norm requires repeated starts rather than a supplied certificate. It also gives a qubit reduction to a one-variable minimization, with a closed expression in one angular regime and numerical minimization otherwise. The discussion treats pure states and states that the bounds extend to mixed states. An addendum in the abstract states that the main results already appeared on page 6 of a 2018 *Quantum* paper linked there.