  Let $H$ be a local net of real Hilbert subspaces of a complex Hilbert space
on the family of double cones of the spacetime $\mathbb{R}^{d+1}$, covariant
with respect to a positive energy, unitary representation $U$ of the Poincar\'e
group, with the Bisognano-Wichmann property for the wedge modular group. We set
an upper bound on the local entropy $S_H(\phi|\! | C)$ of a vector in a region
$C$ that depends only on $U$ and the PCT anti-unitary canonically associated
with $H$. A similar result holds for local, M\"obius covariant nets of standard
subspaces on the circle. We compute the entropy increase and illustrate this
bound for the nets associated with the $U(1)$-current derivatives.
