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Entanglement constant for conformal families

Pawel Caputa, Alvaro Veliz-Osorio

TL;DR

The paper shows that in 1+1D Virasoro CFTs, exciting a state with any operator from a given conformal family increases the second Renyi entanglement entropy by a constant that is identical for all descendants of the primary. For rational CFTs this constant equals the logarithm of the primary’s quantum dimension, linking entanglement to the operator’s fusion data. Through explicit calculations for primaries and descendants up to level 2, it argues a general mechanism: the constant arises from universal conformal data and persists across the entire conformal family. The work provides a unifying entanglement signature of conformal families and suggests deep connections to left-right entanglement and topological aspects of quantum field theories.

Abstract

We show that in 1+1 dimensional conformal field theories, exciting a state with a local operator increases the Renyi entanglement entropies by a constant which is the same for every member of the conformal family. Hence, it is an intrinsic parameter that characterises local operators from the perspective of quantum entanglement. In rational conformal field theories this constant corresponds to the logarithm of the quantum dimension of the primary operator. We provide several detailed examples for the second Renyi entropies and a general derivation.

Entanglement constant for conformal families

TL;DR

The paper shows that in 1+1D Virasoro CFTs, exciting a state with any operator from a given conformal family increases the second Renyi entanglement entropy by a constant that is identical for all descendants of the primary. For rational CFTs this constant equals the logarithm of the primary’s quantum dimension, linking entanglement to the operator’s fusion data. Through explicit calculations for primaries and descendants up to level 2, it argues a general mechanism: the constant arises from universal conformal data and persists across the entire conformal family. The work provides a unifying entanglement signature of conformal families and suggests deep connections to left-right entanglement and topological aspects of quantum field theories.

Abstract

We show that in 1+1 dimensional conformal field theories, exciting a state with a local operator increases the Renyi entanglement entropies by a constant which is the same for every member of the conformal family. Hence, it is an intrinsic parameter that characterises local operators from the perspective of quantum entanglement. In rational conformal field theories this constant corresponds to the logarithm of the quantum dimension of the primary operator. We provide several detailed examples for the second Renyi entropies and a general derivation.

Paper Structure

This paper contains 15 sections, 122 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Replica trick for locally excited states
  3. Example: EPR-primary
  4. Descendants examples
  5. Energy-momentum tensor
  6. Derivative of a primary
  7. Both derivatives
  8. Descendants at level 2
  9. General derivation
  10. Conclusions and discussion
  11. Correlators of the EPR operators
  12. Transformation of descendants
  13. Correlators at level 1
  14. Correlators at level 2
  15. Comments on large c and holography

Figures (4)

  • Figure 1: Finite value of the Rényi entropies as function of $a$ for different replica numbers $n$ in a state excited by operator \ref{['eq:EPR primary']}
  • Figure 2: Evolution of the energy density for three different times. Plot for $l=0$ and $\epsilon=1$.
  • Figure 3: Evolution of the energy density for three different times in the state excited by the descendant \ref{['Desa']}. For comparison, the dashed lines show the evolution in the state excited by a primary. Plot for $l=0$ and $\epsilon=1$.
  • Figure 4: Evolution of the energy density for three different times in the state excited by $\bar{\partial}\partial O$. The dashed lines show the evolution in the state excited by a primary. Plot for $l=0$ and $\epsilon=1$.