ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Heralded success and the zero-success branch

Example

Let $ρ$ be a density operator and let $p \in [0, 1]$ . Define the two outcome maps $I_{s} (X) = p X$ and $I_{f} (X) = (1 - p) X$ . The success probability is $p$ ; if $p > 0$ , the conditional success state is $ρ$ . At $p = 0$ , the subnormalized success output is zero and there is no normalized conditional state.

Facts & Assumptions

Given: A finite-dimensional density operator $ρ$ , $p \in [0, 1]$ , and the two-outcome instrument specified above.

[F1]

Finite instruments and conditional states: an instrument is a finite family of completely positive outcome maps whose sum is trace-preserving; a conditional state is $I_{s} (ρ) / p_{s}$ only when $p_{s} = tr I_{s} (ρ) > 0$ , and is $⊥$ when $p_{s} = 0$ .

[F2]

Heralded success and conditional verification contract: success probability, conditional output guarantee, any unconditional guarantee including failure, and verification/output cost are separate contract fields.

[F3]

Completely positive maps: complete positivity requires positivity after extension by every finite-dimensional identity map.

[F4]

Density operator: $ρ$ is positive semidefinite and has trace one.

Verification

1.1givenF1F3F4algebra

For every finite reference space, the extensions of $I_{s}$ and $I_{f}$ are multiplication by the nonnegative scalars $p$ and $1 - p$ , so they preserve positive operators by [F3]. For every operator $X$ , $tr (I_{s} (X)) + tr (I_{f} (X)) = p tr X + (1 - p) tr X = tr X$ . Thus the maps form an instrument by [F1], and $p_{s} = tr (p ρ) = p$ by [F4].

2.1step 1.1F1F2cases∎

If $p > 0$ , then $I_{s} (ρ) / p_{s} = p ρ / p = ρ$ . If $p = 0$ , both numerator and success probability are zero, so [F1] assigns $⊥$ rather than a density operator. The unconditional failure probability is $1 - p$ , including $1$ at $p = 0$ ; it is reported separately from the conditional-state guarantee, as required by [F2].

Depends on

Used by

Nothing in the library uses this result yet.

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 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)