TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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Teleportation Pauli correction identity

Statement

For every qubit vector $∣ ψ ⟩$ and bits $a, b \in {0, 1}$ , the teleportation branch byproduct is $X^{b} Z^{a} ∣ ψ ⟩$ , and the prescribed correction returns it exactly:

$Z^{a} X^{b} X^{b} Z^{a} ∣ ψ ⟩ = ∣ ψ ⟩ .$

The operator order is part of the statement; no Pauli factors are commuted.

Facts & Assumptions

Given: $a, b \in {0, 1}$ and a qubit vector $∣ ψ ⟩$ .

[A1]

$a$ and $b$ are bits.

[F1]

Teleportation Bell-basis expansion gives the branch byproduct $X^{b} Z^{a} ∣ ψ ⟩$ .

[F2]

The Pauli matrices defines $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ and $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ .

Proof

1.1givenA1F2algebra

From the matrices in [F2], direct multiplication gives $X^{2} = I$ and $Z^{2} = I$ . Therefore $X^{b} X^{b} = I$ and $Z^{a} Z^{a} = I$ for the two possible values of each bit.

2.1step 1.1algebra

Keeping the operator order fixed, $Z^{a} X^{b} X^{b} Z^{a} = Z^{a} (X^{b} X^{b}) Z^{a} = Z^{a} I Z^{a} = Z^{a} Z^{a} = I$ by step 1.1. This also covers $a = b = 1$ without using an anticommutation swap.

3.1step 2.1F1algebra∎

Applying the identity in step 2.1 to the byproduct vector from [F1] yields $Z^{a} X^{b} (X^{b} Z^{a} ∣ ψ ⟩) = ∣ ψ ⟩$ for every qubit vector.

Depends on

Used by

Teleportation correctness and resources Theorem
Teleportation preserves a finite reference Theorem

Dependency tree · two levels

8 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)