Some Information Theoretic Aspects of De-Sitter Holography

TL;DR

This work analyzes three information-theoretic measures—EE, EoP, and holographic complexity—in the DS/dS holographic framework, using entangling surfaces as great co-dimension one spheres and invoking the surface/state correspondence. It reveals that EE in pure dS exhibits volume-law scaling for central subsystems and can imply a highly degenerate, potentially chaotic energy spectrum, while EoP shows an EWCS Jump with discrete spectra in pure dS and a gapped/continuous spectrum in cutoff dS; CV complexity is ill-defined at half-slice size in pure dS but becomes well-defined under $T\bar T+\Lambda_2$ deformations. The authors reconcile puzzles via the surface/state correspondence, interpret $T\bar T+\cdots$ deformations as operating quantum circuits, and uncover nonlocality signs through boosted strong subadditivity violations. The results point to a consistent information-theoretic picture of dS holography, suggesting that the surface/state framework can extend to general holographic duals and inform the structure of nonlocal field theories in cosmological spacetimes.

Abstract

Built on our observation that entangling surfaces of the boundary field theory are great co-dimension one spheres in the context of DS/dS correspondence, we study some information theoretic quantities of the field theory dual intensively using holographic proposals. We will focus on entanglement entropy (EE), entanglement of purification (EoP) and complexity. Several fundamental observations and analysis are provided. For EE, we focus on its scaling behavior, which indicates the nature of the relevant degrees of freedom. Moreover, we find that EE provides us with important information of the energy spectrum in pure dS and it also leads us to the speculation that the field theory dual is chaotic or non-integrable. For EoP, an interesting phenomenon we call "Entanglement of Purification Jump" is observed according to which we propose two puzzles regarding EoP and EE in the context of dS holography. For complexity, we find that the Complexity=Volume proposal does not provide a well-defined way to compute complexity for pure dS. However, it does provide a well-defined way to compute complexity in the $T\bar{T}+Λ_{2}$ deformed case. At the end, we will use the surface/state correspondence to resolve all the puzzles and hence reach a consistent information theoretic picture of dS holography. Moreover, we will provide evidence for our former proposal that the $T\bar{T}+\cdots$ deformations are operating quantum circuits and study the non-locality of the field theory algebra suggested by the surface/state correspondence.

Figures (12)

• Figure 1: The Scaling Behavior of EE
• Figure 2: Push down the bulk hemisphere onto the plane. The equator (central slice for the whole sphere or dS$_{3}$$\tau=0$ time slice) on the left corresponds to the outer circle on the right.
• Figure 3: Left: If A and B are small then the RT surface for $\text{A}\cup\text{B}$ is $\text{A}\cup\text{B}$ so the entanglement wedge in bulk time slice is empty. There is no neck inside entanglement wedge meaning that EoP is zero. Right: If A and B are large enough then the RT surface of $\text{A}\cup\text{B}$ is its complement $\Sigma$ on the circle so the entanglement wedge is the bulk hemisphere. And the neck is a semi-great circle X stretching through the bulk.
• Figure 4: The cutoff slice is in blue and part of it- subsystems A and B in red. The shaded region is the EW. Left: If A and B are small and not close then the RT surface for $\text{A}\cup\text{B}$ is union of RT surfaces for A and B so the entanglement wedge is disconnected. Hence there is no neck inside the entanglement wedge meaning that EoP is zero. Right: If A and B are large and close enough then the RT surface of $\text{A}\cup\text{B}$ is different from the union of RT surfaces of A and B and the entanglement wedge is connected. The neck is part of a great circle X stretching through the EW.
• Figure 5: The volume in CV proposal, for subsystem $\mathcal{A}$ half as large as the central slice, versus the minimal bulk radial coordinates of RT surfaces.