LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

Coherent information of a qudit erasure channel

Statement

Let $E_{d, ϵ}$ be the finite-dimensional erasure channel with data dimension $d \geq 1$ and erasure probability $ϵ \in [0, 1]$ , with the complementary dilation in Complement and degradation maps of the erasure channel. For every input state $ρ$ ,

$I_{c} (ρ, E_{d, ϵ}) = (1 - 2 ϵ) S (ρ) .$

Facts & Assumptions

Given: $d \geq 1$ , $ϵ \in [0, 1]$ , and the input state $ρ$ in the Statement.

[F1]

Finite-dimensional quantum erasure channel and Complement and degradation maps of the erasure channel: respectively, the receiver state is $(1 - ϵ) ρ \oplus ϵ ∣ e_{B} ⟩ ⟨ e_{B} ∣$ and the complementary state is $ϵ ρ \oplus (1 - ϵ) ∣ e_{E} ⟩ ⟨ e_{E} ∣$ under the declared data-flag identifications.

[F2]

Coherent information equals output entropy minus environment entropy: $I_{c} (ρ, E) = S (E (ρ)) - S (E^{c} (ρ))$ .

[F3]

Entropy of a finite flagged state: $S (p ρ \oplus (1 - p) ∣ e ⟩ ⟨ e ∣) = h (p) + p S (ρ)$ , including $p = 0, 1$ .

Proof

1.1F1F3givenalgebra

Apply [F3] to the two states in [F1]. Their entropies are $h (ϵ) + (1 - ϵ) S (ρ)$ and $h (1 - ϵ) + ϵ S (ρ)$ , respectively. By the definition of $h$ and the logarithm product and reciprocal laws in Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, $h (1 - ϵ) = h (ϵ)$ , including the endpoint convention $0 \ln 0 = 0$ .

2.1F2F3step 1.1algebracases∎

Subtract the complementary-output entropy from the receiver-output entropy as in [F2]. The equal binary-entropy terms from step 1.1 cancel, leaving $(1 - 2 ϵ) S (ρ)$ . The derivation is valid at $ϵ = 0, 1$ by [F3], so no open-interval restriction is needed.

Depends on

Used by

Exact quantum, private, and entanglement-assisted erasure capacities Theorem

Dependency tree · two levels

32 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, Proposition 8.17 and §8.3.2 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, Proposition 24.7.1 (standard reference, not scraped)