cft-cmi

Longo-Xu CMI of adjacent intervals for the free fermion net A_r, its finite-index subnets and extensions

Status: proved — a complete structured proof.

Area: Foundations: CMI as a Markov defect (Notes 1-3) · Card type: theorem · Knowledge-base id: longo-xu-cmi

Statement

Let the chiral net be the free fermion net A_r (central charge c = r), a finite-index subnet of it, or one of the finite-index extensions treated after Longo-Xu Thm 4.2, and assume strong additivity. For adjacent intervals A = (-a,0), B = (0,b), C = (b,b+c) (here c is the length of C) in the vacuum, the conditional mutual information (Longo-Xu generalized mutual information, equivalently the gap limit) is finite and I(A:C|B) = (r/6) log[(a+b)(b+c)/(b(a+b+c))] = (r/6) log(1+z) = -(r/6) log eta (adjacent_interval_cmi_longo_xu.tex eq:abc-formula; Note 2 eq:longoxu-limit; paper 1 thm:LX). Read as the quantum-Markov defect of the chain A-B-C; the Petz recovery error is the natural quantitative companion. Scope (Note 1 scope box, l. 327-330; paper 1 rem:LX-scope): the rigorous conclusion covers only this free-fermion class; the formula with r replaced by the central charge for every rational conformal net is Longo-Xu's Remark 4.3, a conjecture (card longo-xu-cmi-general-c), and a full 2D CFT needs a separate left-right analysis.

How to verify

adjacent_interval_cmi_longo_xu.tex eq:abc-formula (l. 254) and scope box (l. 327-330); cmi_as_quantum_markov_defect.tex eq:Ieps-free, eq:longoxu-limit (l. 372-386, 'For the free r-fermion net'); paper1/sec_setting.tex thm:LX (l. 77-87) and rem:LX-scope (l. 101-107); Longo-Xu Def. 3.17, Thm 3.18, Thm 4.1-4.2 (refs/LongoXu_1712.07283.pdf).

Evidence

Related claims

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B3.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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