Von Neumann Entropy in QFT

Abstract

In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones $O \subset {\widetilde{O}}$ of the spacetime, where the closure of O is contained in ${\widetilde{O}}$. Given a QFT net ${\mathcal {A}}$ of local von Neumann algebras ${\mathcal {A}}(O)$, we consider the von Neumann entropy $S_{\mathcal {A}}(O, {\widetilde{O}})$ of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras ${\mathcal {A}}(O)\subset {\mathcal {A}}({\widetilde{O}})$ (split property). We show that this canonical entanglement entropy $S_{\mathcal {A}}(O, {\widetilde{O}})$ is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy ${\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})$, the infimum of the vacuum von Neumann entropy of ${\mathcal {F}}$, where ${\mathcal {F}}$ here runs over all the intermediate, discrete type I von Neumann algebras. We prove that ${\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})$ is finite for the local chiral conformal net generated by finitely many commuting U(1)-currents.