Improved bounds in the entropic uncertainty and certainty relations for complementary observables

Abstract

The entropic uncertainty relation for sets of $N+1$ complementary observables $\{A_k\}$ in $N$-dimensional Hilbert space, $\sum_kH(A_k)\geq (N+1)\ln[\frac12(N+1)]$, is sharpened to $\sum_kH(A_k)\geq\frac12N\, \ln(\frac12N)+(\frac12 N+1)\!\ln(\frac12N+1)$ for even $N$. A nontrivial upper bound on the entropy sum (entropic certainty relation) is also obtained for not completely mixed states, while a previously given expression for this bound is proved to hold only when $N=2$.