Quantum Teleportation Protocol: Bell Measurement & Pauli Correction

Unknown State to Teleport

Polar angle θ 90°

Azimuth φ 0°

Apply Pauli correction

Alice and Bob share an entangled pair. Ready.

Bell Measurement

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Outcome probability

25.0%

Runs so far

Result

|ψ⟩ = 1.00|0⟩

Fidelity F = |⟨ψ|φ_B⟩|²

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Avg fidelity (all runs)

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How it works

Alice holds an unknown qubit C in state |ψ⟩ = α|0⟩ + β|1⟩ and shares an entangled Bell pair |Φ⁺⟩ = (|00⟩+|11⟩)/√2 with Bob across qubits A (hers) and B (his). The joint 3-qubit state can be rewritten exactly in the Bell basis of C and A:

|ψ⟩_C ⊗ |Φ+⟩_AB =
  ½[ |Φ+⟩_CA (α|0⟩+β|1⟩)_B
   + |Φ-⟩_CA (α|0⟩-β|1⟩)_B
   + |Ψ+⟩_CA (β|0⟩+α|1⟩)_B
   + |Ψ-⟩_CA (-β|0⟩+α|1⟩)_B ]

Alice performs a Bell-basis measurement on C and A. Every one of the four outcomes occurs with exactly 25% probability, regardless of α and β — this is what makes the classical bits carry no information about |ψ⟩ on their own (no faster-than-light signalling). She sends the 2-bit outcome to Bob over an ordinary classical channel, and he applies the matching correction:

bits 00 (Φ+) → I           (do nothing)
bits 01 (Φ-) → Z           (phase flip)
bits 10 (Ψ+) → X           (bit flip)
bits 11 (Ψ-) → X then Z    (both)

After the correction, Bob's qubit B is in exactly |ψ⟩ = α|0⟩ + β|1⟩ — the original state has been transferred, not copied (the no-cloning theorem forbids copying; measuring C destroys the original). Without the classical bits, Bob's qubit is indistinguishable from a fully mixed state: he cannot recover |ψ⟩ until the bits physically arrive, which is why teleportation cannot send information faster than light.

θ / φ sliders — set the unknown Bloch-sphere state Alice wants to teleport.
Run Teleportation — samples a real random Bell outcome (25% each), animates the classical bits travelling to Bob, and applies (or, if unchecked, withholds) the Pauli correction.
Apply Pauli correction — turn this off to see what happens if Bob ignores the classical bits: his qubit lands on the wrong pole roughly 3 times out of 4.

Alice — qubit to teleport Bob — receiving qubit

State vector (target |ψ⟩) Bloch sphere axes Classical bits in transit

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