# Holographic Interpretation of Shannon Entropy of Coherence of Quantum Pure States

## TL;DR

This paper proposes a holographic interpretation of the Shannon entropy of quantum coherence in pure states within conformal field theory, linking it to geometric quantities in the AdS/CFT correspondence.

## Contribution

It introduces a conjectured differential geometric formula for the Shannon entropy of coherence, connecting it to holographic complexity and action in the bulk space.

## Key findings

- Conjectured a geometric formula relating Shannon entropy to holographic complexity and action.
- Provided a holographic interpretation of quantum coherence entropy in AdS/CFT context.
- Suggested a new way to define bulk qubit model actions at thermal and momentum equilibrium.

## Abstract

For a quantum pure state in conformal field theory, we generate the Shannon entropy of its coherence, that is, the von Neumann entropy obtained by introducing quantum measurement errors. We give a holographic interpretation of this Shannon entropy, based on Swingle's interpretation of anti-de Sitter space/conformal field theory (AdS/CFT) correspondence in the context of AdS$_3$/CFT$_2$. As a result of this interpretation, we conjecture a differential geometrical formula for the Shannon entropy of the coherence of a quantum pure or purified state in CFT$_2$ at thermal and momentum equilibrium as the sum of the holographic complexity and the abbreviated action, divided by $\pi\hbar$, in the bulk domain enclosed by the Ryu--Takayanagi curve. This result offers a definition of the action of a bulk model of qubits dual to the boundary CFT$_2$ at this equilibrium.