An entropy bound due to symmetries

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.