ExampleConstruction: Literature-sourcedVerification: 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.

Trace distance of orthogonal pure states

Example

Let $H = C^{2}$ with orthonormal computational basis $∣ 0 ⟩, ∣ 1 ⟩$ , and set $ρ = ∣ 0 ⟩ ⟨ 0 ∣$ and $σ = ∣ 1 ⟩ ⟨ 1 ∣$ . Their trace distance is maximal, and the computational-basis projective measurement distinguishes them with total variation one.

Facts & Assumptions

Given: The two orthogonal computational-basis unit vectors in $C^{2}$ .

[F1]

The rank-one operators associated with orthonormal unit vectors are rank-one orthogonal projections (Pure state and ray, Ray-to-projector is well-defined and bijective).

[F2]

$D (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}$ (Trace Distance).

[F3]

A PVM with effects $E_{a}$ has outcome law $p_{a} = tr (ρ E_{a})$ (Finite POVMs and their outcome law).

Verification

1.1givenF1F2

In the basis $∣ 0 ⟩, ∣ 1 ⟩$ , $ρ - σ = diag (1, - 1)$ , whose singular values are $1, 1$ . Thus $∥ ρ - σ ∥_{1} = 2$ and [F2] gives $D (ρ, σ) = 1$ .

2.1givenF1F3step 1.1algebra∎

The computational-basis PVM has effects $E_{0} = ∣ 0 ⟩ ⟨ 0 ∣$ and $E_{1} = ∣ 1 ⟩ ⟨ 1 ∣$ . By [F3], the outcome laws are $(1, 0)$ on $ρ$ and $(0, 1)$ on $σ$ , with total variation $\frac{1}{2} (∣ 1 - 0 ∣ + ∣ 0 - 1 ∣) = 1$ . This equals the trace distance from step 1.1.

Depends on

Used by

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Dependency tree · two levels

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Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)