LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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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.

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Private information is an average coherent-information difference

Statement

Let $N : A \to B$ be a finite-dimensional channel with chosen complement $N^{c} : A \to E$ , and let $E = {p_{x}, ρ_{x}}_{x \in X}$ be a finite ensemble on $A$ with $\overset{ˉ}{ρ} = \sum_{x} p_{x} ρ_{x}$ . Then

$I_{p} (E, N) = I_{c} (\overset{ˉ}{ρ}, N) - \sum_{x : p_{x} > 0} p_{x} I_{c} (ρ_{x}, N) .$

Facts & Assumptions

Given: The channel, chosen complementary channel, and finite ensemble in the Statement.

[F1]

Private information of a finite channel ensemble: $I_{p} (E, N) = χ ({p_{x}, N (ρ_{x})}) - χ ({p_{x}, N^{c} (ρ_{x})})$ .

[F2]

Finite classical-quantum states and Holevo information: $χ ({p_{x}, σ_{x}}) = S (\sum_{x} p_{x} σ_{x}) - \sum_{x : p_{x} > 0} p_{x} S (σ_{x})$ .

[F3]

Coherent information equals output entropy minus environment entropy: $I_{c} (ρ, N) = S (N (ρ)) - S (N^{c} (ρ))$ for every finite input state and the chosen dilation.

Proof

1.1F1F3givenalgebra

Apply [F3] to $\overset{ˉ}{ρ}$ and to every $ρ_{x}$ with $p_{x} > 0$ . Write $B_{x} = N (ρ_{x})$ , $E_{x} = N^{c} (ρ_{x})$ , $\overset{ˉ}{B} = \sum_{x} p_{x} B_{x}$ , and $\overset{ˉ}{E} = \sum_{x} p_{x} E_{x}$ . The difference of coherent informations becomes $S (\overset{ˉ}{B}) - S (\overset{ˉ}{E}) - \sum_{x} p_{x} (S (B_{x}) - S (E_{x}))$ .

2.1F1F2step 1.1algebra∎

Regrouping the finite sums in step 1.1 gives $[S (\overset{ˉ}{B}) - \sum_{x} p_{x} S (B_{x})] - [S (\overset{ˉ}{E}) - \sum_{x} p_{x} S (E_{x})]$ . Each bracket is the corresponding Holevo information by [F2], so their difference is $I_{p} (E, N)$ by [F1]. Terms with $p_{x} = 0$ are absent in both sums.

Remarks

For a pure-state ensemble every branch has zero coherent information, because a Stinespring isometry maps each pure input to a pure output-environment vector. Thus its private information equals the coherent information of its average input.

Depends on

Used by

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Sources

Igor Devetak, The private classical capacity and quantum capacity of a quantum channel, §3 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, Chapter 23 (standard reference, not scraped)