Three faces of the Bures form (type III)

Status: proved — a complete structured proof.

Area: Foundations: CMI as a Markov defect (Notes 1-3) · Card type: theorem · Knowledge-base id: bures-three-faces

Statement

For a von Neumann algebra M with cyclic separating vector Omega and any a in H with Im<Omega,a> = 0, putting phi(K) = -2 Im<a, K Omega>: (1/4) g_B(a) = (1/2) ||(1-J)(1+Delta)^{-1/2} a||^2 = dist(a, closure of M'_sa Omega)^2 = sup over K in M_sa of [phi(K) - Var_omega(K)], with no domain hypothesis on a.

How to verify

rigor/lemma_second_variation.tex, Theorem 4.4 (thm:three); imported verbatim in rigor/uhlmann_upper_bound.tex (thm:three-faces) and rigor/universality_theorem_B.tex (thm:three).

Evidence

• rigor/lemma_second_variation.tex — document, at thm:three

Related claims

None recorded.

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-B1.

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.

Documentation map · All claims by area · What is open · Symbols · Definitions