Von Neumann Entropy in QFT


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.

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Our collaboration took place, in particular, during the CERN workshop on “Advances in Quantum Field Theory” in March–April 2019 and the program “Operator Algebras and Quantum Physics” at the Simons Center for Geometry and Physics at Stony Brook in June 2019. We are grateful to both institutions for the invitations. R.L. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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Correspondence to Roberto Longo.

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R. Longo: Supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM

F. Xu: Supported in part by NSF Grant DMS-1764157.

Communicated by Y. Kawahigashi

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Longo, R., Xu, F. Von Neumann Entropy in QFT. Commun. Math. Phys. 381, 1031–1054 (2021). https://doi.org/10.1007/s00220-020-03702-7

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