Quantum Teleportation Pauli Feedforward

Worked Example

Let us trace the quantum state through the teleportation protocol for a specific input and measurement outcome.

Assume the input state to teleport is |0> (which corresponds to Polar degrees = 0). The initial state of the three qubits (Alice's input, Alice's Bell half, Bob's Bell half) is:

|ψ>in ⊗ (|00> + |11>) / √2 = |0> ⊗ (|00> + |11>) / √2

Alice applies a CNOT gate from her input qubit to her Bell half, followed by a Hadamard gate on her input qubit. She then measures both of her qubits. Suppose her measurement yields m = 1 (second qubit) and n = 0 (first qubit).

The state collapses such that Bob's qubit is in the state:

Zm Xn |ψ>in = Z1 X0 |0> = Z |0> = |0>

Bob receives the classical bits m = 1 and n = 0. To recover the original state, Bob must apply the inverse correction, which is Xⁿ followed by Z^m (note that X and Z are their own inverses).

1. Since n = 0, no X correction is needed.
2. Since m = 1, a Z correction is needed.
3. Applying Z to Bob's pre-correction state |0> yields Z |0> = |0>.

The original state |0> is successfully recovered with a fidelity of 1.0.

Experiment

Explore the effect of different input states, measurement outcomes, and corrections on the recovered state in the interactive module at https://riano.app/module/quantum-teleportation-pauli-feedforward.

1. Set Polar degrees to 90 and Azimuth degrees to 90. This prepares the input state (|0> + i|1>)/√2.
2. Set Bit m to 1 and Bit n to 1. This simulates Alice measuring both her qubits as 1.
3. Observe Bob pre zero real and other pre-correction amplitudes. Bob's state before correction is X Z (|0> + i|1>)/√2 = X (|0> - i|1>)/√2 = (|1> - i|0>)/√2.
4. Set Correction to "None". The Teleportation fidelity remains low (typically not 1.0) because the state has not been restored.
5. Change Correction to "Both". This applies Z then X to Bob's state. Observe that the post-correction amplitudes exactly match the initial input, and the Teleportation fidelity becomes 1.0.

Limitations

This simulation represents ideal quantum teleportation and relies on certain assumptions.

We assume perfect, noise-free quantum gates and ideal measurements. In a real physical implementation, decoherence, gate errors, and imperfect entanglement would reduce the teleportation fidelity below 1.0. The module also simulates the conditional branches classically rather than implementing a full density matrix evolution. Furthermore, the classical communication channel is assumed to be instantaneous for the purpose of the UI, but in reality, the feedforward is limited by the speed of light, ensuring no faster-than-light communication occurs.

References

The fundamental protocol for quantum teleportation was first proposed in 1993.

• Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., & Wootters, W. K. (1993). Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters, 70(13), 1895–1899. This paper introduces the standard teleportation circuit and the necessity of Pauli feedforward.