An entropy bound due to symmetries

TL;DR

This work establishes a universal entropy bound for vectors relative to local nets of standard subspaces, showing $S_H(\phi||C) \le S_{\widetilde{U}}(\phi||C)$ where the bound depends only on the anti-unitary symmetry $\widetilde{U}$ and not on the specific net $H$. The authors leverage the Bisognano-Wichmann property, Haag duality, and modular theory to extend the bound to Möbius covariant nets on the circle and to explicit models, notably nets associated with the $U(1)$-current and its derivatives. They derive explicit entropy formulas in 1D and for derivatives, e.g. $S(f||H^{(1)}(B)) = \pi\int_B (1-x^2)f'(x)^2\,dx$ and $S([f]_1||H_{(k)}(B)) = \pi\int_B (1-x^2)f'(x)^2\,dx - \pi k(k-1)\int_B f(x)^2\,dx$, proving nonnegativity and monotonicity in $k$. The paper also develops the entropy-operator framework $\mathcal{E}_H$ and discusses locality, vector classes, and the interplay between conformal nets and duality, including conditions under which Haag duality holds in conformal settings.

Abstract

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 Poincaré group, with the Bisognano-Wichmann property for the wedge modular group. We set an upper bound on the local entropy $S_H(φ|\! | C)$ 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, Möbius 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.

Theorems & Definitions (23)

• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• Proposition 3.4
• Proposition 3.5
• Theorem 3.6
• Corollary 3.7
• Theorem 4.1
• Theorem 5.1
• Proposition 6.1