ExampleConstruction: AI-generatedVerification: 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.

Schumacher compression of a diagonal qubit source

Example

For $0 \leq p \leq 1$ , let $ρ_{p} = diag (p, 1 - p)$ be a qubit source state. Its asymptotic quantum compression threshold is $h (p) : = - p \ln p - (1 - p) \ln (1 - p) nats per source qubit,$ with $0 \ln 0 = 0$ . Equivalently the threshold is $h_{2} (p) = h (p) / \ln 2$ transmitted qubits per source qubit.

Facts & Assumptions

Given: The diagonal qubit density operator $ρ_{p}$ , $0 \leq p \leq 1$ .

[F1]

Finite-dimensional von Neumann entropy: $S (ρ) = - \sum_{λ_{j} > 0} λ_{j} \ln λ_{j}$ , with $0 \ln 0 : = 0$ ; the bit entropy is $S_{2} = S / \ln 2$ .

[F2]

Finite-dimensional Schumacher quantum source coding theorem: the optimal reliable compression rate of a finite-dimensional IID source is $S (ρ)$ nats per source letter.

Verification

1.1givenF1casesalgebra

The eigenvalues of $ρ_{p}$ are $p$ and $1 - p$ . Substitution into the entropy definition [F1] gives $S (ρ_{p}) = - p \ln p - (1 - p) \ln (1 - p)$ when $0 < p < 1$ ; at $p = 0$ or $1$ , the zero-term convention gives $S (ρ_{p}) = 0$ .

2.1F1F2step 1.1algebracases∎

Apply the Schumacher theorem [F2] to $ρ_{p}$ . Its threshold is $S (ρ_{p}) = h (p)$ in nats; dividing by $\ln 2$ converts the logarithmic code-dimension rate to bits, hence to qubits sent per source qubit. The endpoint rates are zero, while $p = 1 / 2$ gives $\ln 2$ nats, or one qubit per source qubit.

Depends on

Used by

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Sources

John Watrous, The Theory of Quantum Information, §5.3.2 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, Chapter 18 (standard reference, not scraped)