DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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.

Symmetric and antisymmetric swap-test measurement

Definition

Let $H$ be a finite-dimensional complex inner-product space. On $H \otimes H$ define the swap $F$ by $F (u \otimes v) = v \otimes u$ and linear extension. Define the symmetric and antisymmetric effects $P_{+} = (I + F) / 2, P_{-} = (I - F) / 2.$ The “equal” outcome corresponds to $P_{+}$ and “unequal” to $P_{-}$ . When $H \neq {0}$ , the following theorem proves well-definedness as a two-outcome projective measurement and calculates its pure-product law. For $H = {0}$ the operators are still defined, but no density input or measurement interpretation under the cited nonzero-space convention is asserted.

Depends on

Used by

Simultaneous-message equality by quantum fingerprints Algorithm
Fingerprint swap-test error and communication cost Theorem
Swap-test measurement and pure-state outcome law Theorem

Dependency tree · two levels

32 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

Harry Buhrman, Richard Cleve, John Watrous, Ronald de Wolf, Quantum Fingerprinting (standard reference, not scraped)
John Watrous, The Theory of Quantum Information (standard reference, not scraped)