Comments on the entanglement spectrum of de Sitter space

Tom Banks, Patrick Draper

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number135
JournalJournal of High Energy Physics
Issue number1
StatePublished - Jan 2023


  • Black Holes
  • Models of Quantum Gravity

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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