TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite quantum conditional mutual information identity

Statement

For every finite-dimensional tripartite state $ρ_{A B C}$ , the entropy-form conditional mutual information defined in Finite quantum mutual, conditional, and conditional mutual information satisfies

$I (A : B ∣ C)_{ρ} = I (A : B C)_{ρ} - I (A : C)_{ρ} .$

This is an algebraic identity; it asserts no non-negativity.

Facts & Assumptions

Given: A finite-dimensional tripartite density operator $ρ_{A B C}$ with the marginal conventions in Finite quantum mutual, conditional, and conditional mutual information.

[F1]

Finite quantum mutual, conditional, and conditional mutual information: $I (A : B ∣ C) = S (A C) + S (B C) - S (C) - S (A B C)$ and $I (A : B) = S (A) + S (B) - S (A B)$ .

Proof

1.1givenF1

By [F1], $I (A : B C)_{ρ} = S (A)_{ρ} + S (B C)_{ρ} - S (A B C)_{ρ}$ .

2.1givenF1step 1.1algebra∎

Also by [F1], $I (A : C)_{ρ} = S (A)_{ρ} + S (C)_{ρ} - S (A C)_{ρ}$ . Subtracting this from step 1.1 cancels $S (A)_{ρ}$ and gives $S (A C)_{ρ} + S (B C)_{ρ} - S (C)_{ρ} - S (A B C)_{ρ}$ , which is $I (A : B ∣ C)_{ρ}$ by [F1]. No inequality or non-negativity premise is used.

Depends on

Finite quantum mutual, conditional, and conditional mutual information

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information, §5.2.2 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §§11.1 and 11.7 (standard reference, not scraped)