cft-cmi

CMI as the data-processing defect of tracing out C (Note 2)

Status: proved — a complete structured proof.

Area: Foundations: CMI as a Markov defect (Notes 1-3) · Card type: result · Knowledge-base id: cmi-markov-defect-interpretation

Statement

In finite dimensions I(A:C|B) = D(rho_ABC || rho_A x rho_BC) - D(rho_AB || rho_A x rho_B), the defect of the data-processing inequality for Tr_C (eq:cmi-dpi-defect); equality (Markov state) is equivalent to exact Petz recovery and to the Hayden-Jozsa-Petz-Winter decomposition (literature). In type III, for an inclusion N subset M and FAITHFUL NORMAL states rho, sigma on M with D_M(rho || sigma) < infinity (Note 2, l. 304 and 311-315), vanishing of the data-processing defect delta_{N subset M}(rho, sigma) = D_M(rho||sigma) - D_N(rho|N || sigma|N) is equivalent, by Petz's sufficiency theorem, to exact recovery by a normal unital 2-positive map alpha: M -> N with rho|N o alpha = rho and sigma|N o alpha = sigma (eq:petz-vn-equivalence), and, for faithful states, to the Connes-cocycle criterion [D rho : D sigma]_t in N for all real t (eq:connes-criterion, l. 317-321); without D_M(rho || sigma) < infinity the defect is infinity - infinity and the equivalence is not established. This is the reading that makes the recovery error a quantitative companion of the CMI.

How to verify

cmi_as_quantum_markov_defect.tex eq:cmi-dpi-defect (l. 229-236), Petz/HJPW equivalences (l. 270-296, literature), type-III hypotheses and eq:petz-vn-equivalence (l. 304-315), eq:connes-criterion (l. 317-321); rigor/check_notes_1_2.tex.

Evidence

Related claims

None recorded.

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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