Bekenstein's bound for wave packets

Let $B$ be a spatial region of width $2R$ and $\Phi$ a Klein-Gordon wave packet localized in $B$ at time zero. We show the inequality $S \leq 2\pi R E$; here, $S$ is the entropy of $\Phi$ contained in a region $B$, and $E$ is the energy content of $\Phi$ within $B$. We consider a wider setting and formulate a variational problem aimed at minimizing our bound when $\Phi$ is not localized in $B$. Our inequality holds in more generality in the framework of local, Poincar\'e covariant nets of standard subspaces and is related to the Bekenstein inequality. We point out a general bound that is compatible with the recent numerical computations by Bostelmann, Cadamuro, and Minz concerning the one-particle modular Hamiltonian of a scalar massive quantum Klein-Gordon field. We also provide a version of the entropy balance and ant formulas for wave packets.