# Holographic Shannon Entropy: The Outer Entropy of Entanglement Wedges

## TL;DR

This paper introduces holographic Shannon entropy, a geometrical quantity in AdS/CFT that resembles classical Shannon entropy and is related to the outer entropy of entanglement wedges, obeying key entropy inequalities.

## Contribution

The paper defines holographic Shannon entropy, linking it to outer entropy of entanglement wedges and proving its key properties like monotonicity and inequalities.

## Key findings

- Holographic Shannon entropy is divergent but can be regulated.
- It obeys monotonicity and all known balanced inequalities.
- In static cases, it matches classical Shannon entropy of boundary variables.

## Abstract

We introduce a simple geometrical construction similar to covariant holographic entanglement entropy but with the addition of a new term proportional to boundary region volume. This new procedure has properties strongly resembling classical Shannon entropy of probability distributions rather than von Neumann entropy, so we call the quantity holographic Shannon entropy. The holographic Shannon entropy of a region $A$ is divergent in AdS/CFT, but upon regulation, it appears to be equal to the outer entropy of the entanglement wedge of $A$ with the entanglement wedge of the complement of $A$ held fixed. The construction is unambiguous when applied to compact surfaces with convex shape in general spacetimes obeying the null curvature condition (of which AdS with a regulator is a special case). In this context we prove that holographic Shannon entropy obeys monotonicity, a key property of Shannon entropy, as well as all known balanced inequalities of dynamical holographic entanglement entropy. In the static case, we explain why there must exist some classical probability distribution on random variables locally distributed on the boundary with the property that the Shannon entropies of all marginals are exactly reproduced by the holographic Shannon entropy formula.