ExampleConstruction: Literature-sourcedVerification: 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.

Teleportation branch and Pauli correction

Example

Let $∣ ψ ⟩ = α ∣ 0 ⟩ + β ∣ 1 ⟩$ with $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ . On input $∣ ψ ⟩_{A}$ and shared resource $∣ Φ^{+} ⟩_{B C}$ , compute the four post-gate measurement branches and verify the correction on wire $C$ for each label $(a, b)$ .

Facts & Assumptions

Given: The normalized input, the ordered wires $A, B, C$ , and the teleportation gates, measurement, and correction.

[A1]

$α, β \in C$ and $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ .

[F1]

Teleportation Bell-basis expansion gives the post-gate expansion $\frac{1}{2} \sum_{a, b} ∣ a b ⟩ \otimes X^{b} Z^{a} ∣ ψ ⟩$ .

[F2]

Teleportation outcomes are equiprobable proves that each outcome label has probability $1 / 4$ .

[F3]

The Pauli matrices gives $X ∣ 0 ⟩ = ∣ 1 ⟩$ , $X ∣ 1 ⟩ = ∣ 0 ⟩$ , $Z ∣ 0 ⟩ = ∣ 0 ⟩$ , and $Z ∣ 1 ⟩ = - ∣ 1 ⟩$ .

Verification

1.1givenA1F1algebra

Expanding the input with the Bell resource and applying CNOT $A \to B$ gives $2^{- 1 / 2} (α ∣ 000 ⟩ + α ∣ 011 ⟩ + β ∣ 110 ⟩ + β ∣ 101 ⟩)$ . Applying $H_{A}$ and grouping by $A B$ yields $\frac{1}{2} (∣ 00 ⟩ (α ∣ 0 ⟩ + β ∣ 1 ⟩) + ∣ 01 ⟩ (α ∣ 1 ⟩ + β ∣ 0 ⟩) + ∣ 10 ⟩ (α ∣ 0 ⟩ - β ∣ 1 ⟩) + ∣ 11 ⟩ (α ∣ 1 ⟩ - β ∣ 0 ⟩))$ , which is [F1]'s expansion.

2.1step 1.1A1F2F3algebra

For each label, the receiver vector is respectively $ψ$ , $X ψ$ , $Z ψ$ , and $X Z ψ$ by [F3]. Each has the same norm as $ψ$ , since $X$ swaps its two coordinates and $Z$ changes only the sign of the second. Thus each projected branch has squared norm $\frac{1}{4} ∥ ψ ∥^{2} = 1 / 4$ , agreeing with the probability in [F2].

2.2step 1.1F3algebra

For label $00$ , the correction is $I$ and leaves $α ∣ 0 ⟩ + β ∣ 1 ⟩$ unchanged. For $01$ , it is $X$ , and $X (α ∣ 1 ⟩ + β ∣ 0 ⟩) = α ∣ 0 ⟩ + β ∣ 1 ⟩$ . For $10$ , it is $Z$ , and $Z (α ∣ 0 ⟩ - β ∣ 1 ⟩) = α ∣ 0 ⟩ + β ∣ 1 ⟩$ .

3.1step 2.2F3algebra∎

For label $11$ , the branch is $α ∣ 1 ⟩ - β ∣ 0 ⟩ = X Z ∣ ψ ⟩$ and the correction is $Z X$ . Hence $Z X (X Z ∣ ψ ⟩) = Z (X X) Z ∣ ψ ⟩ = Z^{2} ∣ ψ ⟩ = ∣ ψ ⟩$ using [F3]; this fixed order has no residual phase. Together the four calculations return the input on every branch.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

John Watrous, The Theory of Quantum Information (2018), §6.3.1 (standard reference, not scraped)
Richard Cleve, Quantum Information Processing course notes for QIC 710 / CS 768 / PHYS 767 / CO 681 / AMATH 871 / PMATH 871, version dated October 13, 2026 (standard reference, not scraped)