Coarse-Graining Holographic States: A Semiclassical Flow in General Spacetimes

Abstract

Motivated by the understanding of holography as realized in tensor networks, we develop a bulk procedure that can be interpreted as generating a sequence of coarse-grained holographic states. The coarse-graining procedure involves identifying degrees of freedom entangled at short distances and disentangling them. This is manifested in the bulk by a flow equation that generates a codimension-1 object, which we refer to as the holographic slice. We generalize the earlier classical construction to include bulk quantum corrections, which naturally involves the generalized entropy as a measure of the number of relevant boundary degrees of freedom. The semiclassical coarse-graining results in a flow that approaches quantum extremal surfaces such as entanglement islands that have appeared in discussions of the black hole information paradox. We also discuss the relation of the present picture to the view that the holographic dictionary works as quantum error correction.

Figures (14)

• Figure 1: The leaf $\sigma$ is split into $\sigma_{\text{int}} \cup \sigma_{\text{ext}}$ such that $\sigma_{\text{int}}$ and $\sigma_{\text{ext}}$ are separated by a small regulating region $\Sigma_{\epsilon}$ on a Cauchy slice $\Sigma$. This induces a division of the Cauchy slice as $\Sigma = \Sigma_{\text{int}} \cup \Sigma_{\epsilon} \cup \Sigma_{\text{ext}}$. We define the location of the holographic screen by requiring that it is marginally quantum trapped/anti-trapped under variations of $\sigma_{\text{int}}$.
• Figure 2: A TN defines a boundary state in the Hilbert space $\mathcal{H}_{\sigma}$ at the outer legs. One can, however, also consider "coarse-grained" states defined at inner layers, e.g. states defined in Hilbert spaces $\mathcal{H}_{\sigma_1}$ and $\mathcal{H}_{\sigma_2}$.
• Figure 3: a) The von Neumann entropy of subregion $A$ is computed by the minimal cut $\gamma_A$ that splits the TN into two parts containing $A$ and $\overline{A}$ respectively. b) By applying a local unitary on $A$, we can find maximally entangled legs across $\gamma_A$, which serve as a bottleneck for the entanglement between $A$ and $\overline{A}$.
• Figure 4: A sequence of coarse-graining steps. At each step, we consider infinitesimal subregions of size $\delta$ ($\rightarrow 0$) and reduce the spacetime region to their respective complementary entanglement wedges.
• Figure 5: Coarse-graining over infinitesimal subregions on $\sigma$ can be performed by considering the intersection of complementary entanglement wedges. This leads to a domain of dependence $R(\sigma)$ which corresponds to a new renormalized leaf $\sigma_1$.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof