TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Fingerprint states are normalized and encode agreement as overlap

Statement

For every $n \geq 1$ , every positive-length binary code map $E : {0, 1}^{n} \to {0, 1}^{N}$ with $N \geq 1$ , and all $x, y \in {0, 1}^{n}$ , $∣ h_{x} ⟩$ has norm one, uses $⌈ \log_{2} N ⌉ + 1$ qubits, and $⟨ h_{x} ∣ h_{y} ⟩ = N^{- 1} ∣ {i < N : E_{i} (x) = E_{i} (y)} ∣$ .

Facts & Assumptions

Given: Integers $n, N \geq 1$ , a binary code map $E : {0, 1}^{n} \to {0, 1}^{N}$ , and inputs $x, y \in {0, 1}^{n}$ .

[F1]

Code-based quantum fingerprint: $∣ h_{x} ⟩ = N^{- 1 / 2} \sum_{i = 0}^{N - 1} ∣ i ⟩ ∣ E_{i} (x) ⟩$ in an index register of $⌈ \log_{2} N ⌉$ qubits and one data qubit.

[F2]

The computational basis is orthonormal: The computational basis vectors of a finite register are orthonormal.

[F3]

The product basis is orthonormal and tensor dimensions multiply: The ordered tensor-product basis is orthonormal and has dimension equal to the product of the factor dimensions.

Proof

technique · direct

1.1givenF1F2F3algebra

The $N$ summands in [F1] are distinct orthonormal basis vectors by [F2]–[F3], even when code bits repeat, because their index labels differ. Therefore $∥ h_{x} ∥^{2} = N^{- 1} \sum_{i < N} 1 = 1$ . Its ambient register has $m + 1 = ⌈ \log_{2} N ⌉ + 1$ qubits by [F1].

2.1step 1.1F1F2F3algebra∎

Expanding the inner product using [F1] and orthonormality gives $⟨ h_{x} ∣ h_{y} ⟩ = \frac{1}{N} \sum_{i, j < N} ⟨ i ∣ j ⟩ ⟨ E_{i} (x) ∣ E_{j} (y) ⟩ = \frac{1}{N} \sum_{i < N} 1_{E_{i} (x) = E_{i} (y)} .$ The last sum counts agreeing code coordinates, proving the overlap identity.

Depends on

Used by

Simultaneous-message equality by quantum fingerprints Algorithm
Finite length-eight fingerprint overlap Example
Fingerprint swap-test error and communication cost Theorem

Cited to discharge well-definedness by Code-based quantum fingerprint.

Dependency tree · two levels

12 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)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)