LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Entropy of a finite flagged state

Statement

Let $H$ be a nonzero finite-dimensional complex inner-product space, let $τ$ be a density operator on $H$ , let $e$ be a unit vector orthogonal to the data summand in $H \oplus C e$ , and let $p \in [0, 1]$ . With $h (p) : = - p \ln p - (1 - p) \ln (1 - p)$ and $0 \ln 0 : = 0$ , the block-diagonal state

$ω_{p} : = p τ \oplus (1 - p) ∣ e ⟩ ⟨ e ∣$

satisfies $S (ω_{p}) = h (p) + p S (τ)$ .

Facts & Assumptions

Given: The finite-dimensional state, orthogonal flag, and probability $p$ in the Statement.

[F1]

Finite-dimensional von Neumann entropy: entropy is $- \sum_{λ > 0} λ \ln λ$ , and zero-eigenvalue terms contribute zero.

[F2]

Density operator and Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: $τ$ has nonnegative eigenvalues $λ_{j}$ summing to one, with an orthonormal eigenbasis.

[F3]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\ln (x y) = \ln x + \ln y$ for positive $x, y$ , and $\ln 1 = 0$ .

Proof

1.1F1F2F3givenalgebra

The orthogonal summands reduce $ω_{p}$ , so its eigenvalues, including multiplicities, are $p λ_{j}$ on $H$ and $1 - p$ on the flag. For $0 < p < 1$ , [F1]–[F3] give $S (ω_{p}) = - \sum_{j : λ_{j} > 0} p λ_{j} \ln (p λ_{j}) - (1 - p) \ln (1 - p) = h (p) + p S (τ)$ , because $\sum_{j} λ_{j} = 1$ .

2.1F1step 1.1casesalgebra∎

At $p = 0$ , $ω_{p}$ is the pure flag state and both sides equal zero. At $p = 1$ , $ω_{p} = τ$ and both sides equal $S (τ)$ . Thus the formula holds on the closed interval, including its endpoints.

Depends on

Used by

Coherent information of a qudit erasure channel Lemma

Dependency tree · two levels

21 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, §5.2.1 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, §11.1 (standard reference, not scraped)