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 -