ExampleConstruction: Literature-sourcedVerification: 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.

Entropy of a diagonal qubit state

Example

For $p \in [0, 1]$ , let $ρ_{p}$ be the qubit state with matrix $diag (p, 1 - p)$ . Its entropy is $S (ρ_{p}) = - p \ln p - (1 - p) \ln (1 - p)$ nats, with zero terms interpreted as $0 \ln 0 = 0$ . In bits it is $h_{2} (p) : = - p \log_{2} p - (1 - p) \log_{2} (1 - p)$ , where $\log_{2} u : = \ln u / \ln 2$ , with the same endpoint convention. Thus pure basis states have entropy $0$ , and the maximally mixed qubit has entropy $\ln 2$ nats, or one bit.

Facts & Assumptions

Given: $p \in [0, 1]$ , the diagonal density operator $ρ_{p}$ , and the entropy definition.

[F1]

Finite-dimensional von Neumann entropy: entropy is the negative sum of $λ \ln λ$ over positive eigenvalues, and entropy in bits divides by $\ln 2$ .

[F2]

A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S: a non-negative diagonal operator has the displayed diagonal entries as its eigenvalues.

[F3]

Density operator: a density operator is non-negative and has trace one.

[F4]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\ln (1 / 2) = \ln 1 - \ln 2 = - \ln 2$ .

Verification

1.1givenF1F2F3algebra

The diagonal entries $p, 1 - p$ are nonnegative and sum to one, so $ρ_{p}$ is a density operator by [F3]. By [F2] its eigenvalues are exactly $p, 1 - p$ . Applying [F1] gives $S (ρ_{p}) = - p \ln p - (1 - p) \ln (1 - p)$ , with a zero-eigenvalue term omitted.

2.1givenF1F4step 1.1algebracases∎

If $p = 0$ or $p = 1$ , the only positive eigenvalue is $1$ , so $S = 0$ . If $p = 1 / 2$ , [F4] gives $S = - 2 (\frac{1}{2} \ln \frac{1}{2}) = \ln 2$ , and [F1] gives $S_{2} = 1$ . For general $p$ , the stated definition of $\log_{2}$ gives $S_{2} = h_{2} (p)$ .

Depends on

Used by

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Sources

John Watrous, The Theory of Quantum Information, §5.2.1 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, arXiv:1106.1445v8, §11.1 (standard reference, not scraped)