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.

Teleportation Bell-basis expansion

Statement

For arbitrary $α, β \in C$ , put $∣ ψ ⟩ = α ∣ 0 ⟩ + β ∣ 1 ⟩$ and $V_{A B} = (H_{A} \otimes I_{B C}) {CNOT}_{A \to B}$ , in wire order $A, B, C$ . Then

$V_{A B} (∣ ψ ⟩_{A} \otimes ∣ Φ^{+} ⟩_{B C}) = \frac{1}{2} \sum_{a, b \in {0, 1}} ∣ a b ⟩_{A B} \otimes X_{C}^{b} Z_{C}^{a} ∣ ψ ⟩_{C} .$

No normalization assumption on $∣ ψ ⟩$ is needed for this vector identity.

Facts & Assumptions

Given: $α, β \in C$ , the ordered wires $A, B, C$ , and the operators in the Statement.

[A1]

$∣ ψ ⟩ = α ∣ 0 ⟩ + β ∣ 1 ⟩$ is an arbitrary vector; it need not be normalized.

[F1]

Qubit teleportation procedure specifies initial input $∣ ψ ⟩_{A}$ , shared resource $∣ Φ^{+} ⟩_{B C}$ , wire order $A, B, C$ , CNOT followed by $H_{A}$ , and measurement-record order $(a, b)$ .

[F2]

The Bell vectors defines $∣ Φ^{+} ⟩ = (∣ 00 ⟩ + ∣ 11 ⟩) / \sqrt{2}$ .

[F3]

Exact one-qubit gates and rotations gives $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ .

[F4]

CNOT and Control-Target Order gives ${CNOT}_{1 \to 2} ∣ x, t ⟩ = ∣ x, t \oplus x ⟩$ with the first wire as control.

[F5]

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

[F6]

Tensor products of finitely many linear maps extends the rule $H \otimes I : x \otimes y \mapsto H x \otimes y$ linearly to the three-wire register.

Proof

1.1givenA1F1F2algebra

By [F2], the initial vector is $2^{- 1 / 2} (α ∣ 000 ⟩ + α ∣ 011 ⟩ + β ∣ 100 ⟩ + β ∣ 111 ⟩)$ in wire order $A, B, C$ .

2.1step 1.1F4algebra

Applying [F4] to the four basis vectors gives $2^{- 1 / 2} (α ∣ 000 ⟩ + α ∣ 011 ⟩ + β ∣ 110 ⟩ + β ∣ 101 ⟩)$ after CNOT: the first two terms have control bit $0$ and are fixed, while the last two have control bit $1$ and their target bits flip.

3.1step 2.1F3F6algebra

By [F3] and [F6], $H_{A}$ sends each first-wire $∣ 0 ⟩$ to $(∣ 0 ⟩ + ∣ 1 ⟩) / \sqrt{2}$ and each $∣ 1 ⟩$ to $(∣ 0 ⟩ - ∣ 1 ⟩) / \sqrt{2}$ . Regrouping the resulting eight terms by the first two wires gives $\frac{1}{2} (∣ 00 ⟩ (α ∣ 0 ⟩ + β ∣ 1 ⟩) + ∣ 01 ⟩ (α ∣ 1 ⟩ + β ∣ 0 ⟩) + ∣ 10 ⟩ (α ∣ 0 ⟩ - β ∣ 1 ⟩) + ∣ 11 ⟩ (α ∣ 1 ⟩ - β ∣ 0 ⟩))$ .

4.1A1step 3.1F5algebra∎

The four receiver vectors in step 3.1 are respectively $∣ ψ ⟩$ , $X ∣ ψ ⟩$ , $Z ∣ ψ ⟩$ , and $X Z ∣ ψ ⟩$ by the actions in [F5]. These are exactly $X^{b} Z^{a} ∣ ψ ⟩$ for labels $00, 01, 10, 11$ , so step 3.1 is the sum in the Statement.

Depends on

Used by

Teleportation branch and Pauli correction Example
Teleportation correctness and resources Theorem
Teleportation outcomes are equiprobable Theorem
Teleportation Pauli correction identity Theorem
Teleportation preserves a finite reference Theorem

Dependency tree · two levels

28 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 (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)