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Relative entropy of excited states in two dimensional conformal field theories

Gábor Sárosi, Tomonori Ugajin

TL;DR

The paper derives the leading small-interval expansion of the relative entropy between excited states in two-dimensional conformal field theories, showing that for equal conformal weights the leading term is controlled by differences in OPE coefficients to light primaries, and that the relative entropy is proportional to the trace square distance in this limit. When the states have different weights, the leading term is universal if there are no lighter primaries than the stress tensor; otherwise, a relevant primary can dominate. The authors verify the general formulas with explicit calculations for generalized free fields and the critical Ising model, and discuss implications for black hole microstate distinguishability in holography and the validity of bulk effective field theory near horizons. The work connects replica-trick methods to concrete CFT data, providing a framework to quantify how distinguishable reduced density matrices of excited states are under local measurements.

Abstract

We study the relative entropy and the trace square distance, both of which measure the distance between reduced density matrices of two excited states in two dimensional conformal field theories. We find a general formula for the relative entropy between two primary states with the same conformal dimension in the limit of a single small interval and find that in this case the relative entropy is proportional to the trace square distance. We check our general formulae by calculating the relative entropy between two generalized free fields and the trace square distance between the spin and disorder operators of the critical Ising model. We also give the leading term of the relative entropy in the small interval expansion when the two operators have different conformal dimensions. This turns out to be universal when the CFT has no primaires lighter than the stress tensor. The result reproduces the previously known special cases.

Relative entropy of excited states in two dimensional conformal field theories

TL;DR

The paper derives the leading small-interval expansion of the relative entropy between excited states in two-dimensional conformal field theories, showing that for equal conformal weights the leading term is controlled by differences in OPE coefficients to light primaries, and that the relative entropy is proportional to the trace square distance in this limit. When the states have different weights, the leading term is universal if there are no lighter primaries than the stress tensor; otherwise, a relevant primary can dominate. The authors verify the general formulas with explicit calculations for generalized free fields and the critical Ising model, and discuss implications for black hole microstate distinguishability in holography and the validity of bulk effective field theory near horizons. The work connects replica-trick methods to concrete CFT data, providing a framework to quantify how distinguishable reduced density matrices of excited states are under local measurements.

Abstract

We study the relative entropy and the trace square distance, both of which measure the distance between reduced density matrices of two excited states in two dimensional conformal field theories. We find a general formula for the relative entropy between two primary states with the same conformal dimension in the limit of a single small interval and find that in this case the relative entropy is proportional to the trace square distance. We check our general formulae by calculating the relative entropy between two generalized free fields and the trace square distance between the spin and disorder operators of the critical Ising model. We also give the leading term of the relative entropy in the small interval expansion when the two operators have different conformal dimensions. This turns out to be universal when the CFT has no primaires lighter than the stress tensor. The result reproduces the previously known special cases.

Paper Structure

This paper contains 17 sections, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the basic definitions
  3. Reduced density matrices of excited states
  4. Relative entropy
  5. Trace square distance
  6. $S(\rho_{V}|| \rho_{W})$ and $T(\rho_{V}|| \rho_{W})$ in the small interval limit
  7. Contributions of the vacuum
  8. Small subsystem expansion
  9. Small interval limit for operators with equal weight
  10. Leading universal contribution when $h_V \neq h_W$
  11. Generalized free fields
  12. Relative entropy
  13. Trace square distance
  14. Critical Ising model
  15. Conclusions
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Illustration of the relevant correlation functions for the 3-sheeted manifold $\Sigma_3$. The left figure corresponds to $\text{Tr}\rho_V^3$ while the right one to $\text{Tr}\rho_V \rho_W^2$. The cyclic $Z_3$ replica symmetry becomes the cyclic symmetry of the trace.