[Paper Review] Entanglement entropy in de Sitter: no pure states for conformal matter

Dmitry S. Ageev, I. Ya. Aref’eva+3 more|arXiv (Cornell University)|Apr 24, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates entanglement entropy in four-dimensional de Sitter spacetime reduced to two dimensions, focusing on conformal matter. It demonstrates that pure state and complementarity conditions—essential for unitarity—fail universally: entanglement entropy remains bounded away from zero and violates region-complement symmetry, especially under time-dependent Cauchy surfaces. The island formula does not resolve the information paradox in de Sitter, as non-trivial island solutions exist even when entropy does not grow.

ABSTRACT

In this paper, we consider the entanglement entropy of conformal matter for finite and semi-infinite entangling regions, as well as the formation of entanglement islands in four-dimensional de Sitter spacetime partially reduced to two dimensions. We analyze complementarity and pure state condition of entanglement entropy of pure states as a consistency test of the CFT formulas in this geometrical setup, which has been previously used in the literature to study the information paradox in higher-dimensional de Sitter in the context of the island proposal. We consider two different types of Cauchy surfaces in the extended static patch and flat coordinates, correspondingly. For former, we found that entanglement entropy of a pure state is always bounded from below by a constant and never becomes zero, as required by quantum mechanics. In turn, the difference between the entropies for some region and its complement, which should be zero for a pure state, in direct calculations essentially depends on how the boundaries of these regions evolve with time. Regarding the flat coordinates, it is impossible to regularize spacelike infinity in a way that would be compatible with complementarity and pure state condition, as opposed, for instance, to two-sided Schwarzschild black hole. Finally, we discuss the information paradox in de Sitter and show that the island formula does not resolve it, at least in this setup. Namely, we give examples of a region with a time-limited growth of entanglement entropy, for which there is no island solution, and the region, for which entanglement entropy does not grow, but the island solution exists.

Motivation & Objective

• To analyze whether entanglement entropy of conformal matter in de Sitter spacetime satisfies the pure state condition (vanishing entropy) and complementarity (equal entropy for region and complement).
• To investigate the behavior of entanglement entropy on different Cauchy surfaces: constant static time (extended static patch) and constant flat time (flat coordinates).
• To assess the applicability of the island formula in resolving the information paradox in de Sitter spacetime.
• To determine whether regularization of spacelike infinity in flat coordinates can restore pure state and complementarity properties.
• To compare the failure of these quantum information principles in de Sitter with their behavior in two-sided Schwarzschild black holes.

Proposed method

• Analyzing entanglement entropy of free massless Dirac fermions in finite and semi-infinite entangling regions in 2D reduced de Sitter spacetime.
• Using two distinct Cauchy surface foliations: one in extended static coordinates (constant static time) and one in flat coordinates (constant flat time).
• Applying the replica trick and island formula to compute fine-grained entanglement entropy, with attention to UV regularization and boundary conditions.
• Evaluating the time dependence of entanglement entropy and its difference between a region and its complement under different time coordinate choices.
• Performing numerical checks on island configurations for regions with time-limited entropy growth and non-growing entropy.
• Comparing results with the two-sided Schwarzschild black hole case to highlight similarities and differences in the breakdown of pure state and complementarity conditions.

Experimental results

Research questions

• RQ1 Does the entanglement entropy of a pure quantum state in de Sitter spacetime vanish on a Cauchy surface, as required by quantum mechanics?
• RQ2 Can the entanglement entropy of a region and its complement be equal in de Sitter spacetime, satisfying the complementarity condition?
• RQ3 Is it possible to regularize spacelike infinity in flat coordinates such that both pure state and complementarity conditions are satisfied?
• RQ4 Can the island formula resolve the information paradox in de Sitter spacetime, particularly for regions with monotonic entropy growth?
• RQ5 Do non-trivial island solutions exist even when entanglement entropy does not increase over time, undermining the necessity of the island mechanism?

Key findings

• In the extended static patch, entanglement entropy of a Cauchy surface is bounded from below by a non-zero constant, violating the pure state condition.
• The difference between entropies of a region and its complement depends critically on time coordinate choice and can increase with time, invalidating complementarity.
• For flat coordinates, regularization of spacelike infinity leads to divergent entanglement entropy and entropy difference, making it impossible to satisfy pure state and complementarity conditions.
• The island formula fails to resolve the information paradox: for a region with time-limited monotonic entropy growth, no non-trivial island solution exists.
• A non-trivial island solution exists for a region whose entanglement entropy does not grow over time, showing that island existence does not imply entropy increase or information loss.
• The breakdown of pure state and complementarity conditions in de Sitter is more severe than in two-sided Schwarzschild black holes, due to finite spatial coordinates of boundary points in the static patch.