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Bell States, Teleportation, and Dense Coding — Examples

1 · Prerequisites

2 · Summary

Objective

This companion expands the teleportation circuit for a general normalized qubit, lists each measured branch, and checks the Pauli correction in the declared phase-bit/flip-bit order, including the two-bit case where operator order matters.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

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.