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On singular limits of relative entropies

Feng Xu

TL;DR

The work advances the understanding of singular limits of relative entropies in conformal nets by removing prior constraints (strong additivity and finite index) and establishing a general relationship between entropy limits and the index [A:B]. By analyzing Connes spatial derivatives, modular theory, and subnet structure, it provides a unified framework showing that the limit of S(ω, ω·E_n) equals ln[A:B] under broad conditions, and proves a pivotal auxiliary result asserting divergence in certain limiting configurations. The approach splits into irreducible and infinite-index cases and also handles non-irreducible situations via coset subnets, with implications for the mathematical structure of entanglement in conformal field theory. These results connect entropy, subfactor index, and subnet theory in a way that is broadly applicable to split Möbius nets and their Virasoro subnets.

Abstract

In this paper we generalize a key result relating singular limits of certain relative entropies with index in the setting of conformal nets, which has played an important role recently in the mathematical theory of relative entropies in the context of Conformal Field Theory.

On singular limits of relative entropies

TL;DR

The work advances the understanding of singular limits of relative entropies in conformal nets by removing prior constraints (strong additivity and finite index) and establishing a general relationship between entropy limits and the index [A:B]. By analyzing Connes spatial derivatives, modular theory, and subnet structure, it provides a unified framework showing that the limit of S(ω, ω·E_n) equals ln[A:B] under broad conditions, and proves a pivotal auxiliary result asserting divergence in certain limiting configurations. The approach splits into irreducible and infinite-index cases and also handles non-irreducible situations via coset subnets, with implications for the mathematical structure of entanglement in conformal field theory. These results connect entropy, subfactor index, and subnet theory in a way that is broadly applicable to split Möbius nets and their Virasoro subnets.

Abstract

In this paper we generalize a key result relating singular limits of certain relative entropies with index in the setting of conformal nets, which has played an important role recently in the mathematical theory of relative entropies in the context of Conformal Field Theory.

Paper Structure

This paper contains 9 sections, 10 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spatial derivatives, relative entropy and index theory for general subfactors
  4. A result on relative entropy
  5. Nets and subnets
  6. Nets
  7. Subnets
  8. Singular limits
  9. Proof of Th. \ref{['thm2']}

Key Result

Theorem 2.1

(1) Let $M$ be a von Neumann algebra and $M_1$ a von Neumann subalgebra of M. Assume that there exists a faithful normal conditional expectation $E$ of $M$onto $M_1$. If $\psi$ and $\omega$ are states of $M_1$ and $M$, respectively, then $S(\omega, \psi\cdot E) = S(\omega\!\restriction\! M_1, \psi)

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Lemma 2.9
  • Theorem 3.1
  • ...and 3 more