Annales Henri Poincaré

, Volume 20, Issue 7, pp 2353–2375 | Cite as

Relative Entropy Close to the Edge

  • Stefan HollandsEmail author


We show that the relative entropy between the reduced density matrix of the vacuum state in some region A and that of an excited state created by a unitary operator localized at a small distance \(\ell \) of a boundary point p is insensitive to the global shape of A, up to a small correction. This correction tends to zero as \(\ell /R\) tends to zero, where R is a measure of the curvature of \(\partial A\) at p, but at a rate necessarily slower than \(\sim \sqrt{\ell /R}\) (in any dimension). Our arguments are mathematically rigorous and only use model independent, basic assumptions about quantum field theory such as locality and Poincare invariance.



It is a pleasure to thank Centro Atomico Balseiro, Bariloche, Argentina, for hospitality during my visit in March 2018, as well as the Simons Foundation for financially supporting that visit. I have greatly benefited from discussions with H. Casini, M. Huerta, F. Otto, and D. Pontello. I am grateful to the Max-Planck Society for supporting the collaboration between MPI-MiS and Leipzig U., grant Proj. Bez. M.FE.A.MATN0003.


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Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany

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