CounterexampleConstruction: 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.

Normalized states can have infinite quantum relative entropy

Statement refuted

Every pair of density operators on the same finite-dimensional space has finite quantum relative entropy.

Facts & Assumptions

Given: A two-dimensional register $H = C^{2}$ with computational orthonormal basis $e_{0}, e_{1}$ .

[F1]

Finite-dimensional quantum relative entropy: $D (ρ ∥ σ) = + \infty$ whenever $supp (ρ) ⊈ supp (σ)$ .

[F2]

Density operator: a density operator is a non-negative endomorphism of trace one.

[F3]

Finite registers and their ordered computational bases: $e_{0}, e_{1}$ are orthonormal.

[F4]

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: an orthogonal rank-one projection is non-negative.

Counterexample

1.1givenF2F3F4construct

Define $P_{j} x : = ⟨ x, e_{j} ⟩ e_{j}$ for $j = 0, 1$ , and set $ρ = P_{0}$ , $σ = P_{1}$ . Each is an orthogonal rank-one projection, hence non-negative by [F4]; in the basis $e_{0}, e_{1}$ each has trace one, so both are density operators by [F2].

2.1givenF1F3step 1.1algebra∎

The range/support of $P_{0}$ is $C e_{0}$ , while the support of $P_{1}$ is $C e_{1}$ ; these are orthogonal by [F3]. Thus $supp (ρ) ⊈ supp (σ)$ , and [F1] gives $D (ρ ∥ σ) = + \infty$ . This contradicts the asserted finiteness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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, arXiv:1106.1445v8, §11.8 (standard reference, not scraped)