TY - JOUR

T1 - Comments on the entanglement spectrum of de Sitter space

AU - Banks, Tom

AU - Draper, Patrick

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023/1

Y1 - 2023/1

N2 - We argue that the Schwarzschild-de Sitter black hole entropy formula does not imply that the entanglement spectrum of the vacuum density matrix of de Sitter space is flat. Specifically, we show that the expectation value of a random projection operator of dimension d ≫ 1, on a Hilbert space of dimension D ≫ d and in a density matrix ρ = e–K with strictly positive spectrum, is dD(1+o(1d)), independent of the spectrum of the density matrix. In addition, for a suitable class of spectra the asymptotic estimates Tr (ρK) ~ ln D – o(1) and Tr [ρ(K – 〈K〉)2] = a〈K〉 are compatible for any order one constant a. We discuss a simple family of matrix models and projections that can replicate such modular Hamiltonians and the SdS entropy formula.

AB - We argue that the Schwarzschild-de Sitter black hole entropy formula does not imply that the entanglement spectrum of the vacuum density matrix of de Sitter space is flat. Specifically, we show that the expectation value of a random projection operator of dimension d ≫ 1, on a Hilbert space of dimension D ≫ d and in a density matrix ρ = e–K with strictly positive spectrum, is dD(1+o(1d)), independent of the spectrum of the density matrix. In addition, for a suitable class of spectra the asymptotic estimates Tr (ρK) ~ ln D – o(1) and Tr [ρ(K – 〈K〉)2] = a〈K〉 are compatible for any order one constant a. We discuss a simple family of matrix models and projections that can replicate such modular Hamiltonians and the SdS entropy formula.

KW - Black Holes

KW - Models of Quantum Gravity

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U2 - 10.1007/JHEP01(2023)135

DO - 10.1007/JHEP01(2023)135

M3 - Article

AN - SCOPUS:85146772012

SN - 1126-6708

VL - 2023

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 1

M1 - 135

ER -