Qubits 1 and 2 are prepared in the Bell state (Chapter 4)
PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · The quantum circuit model and quantum algorithms
Quantum Information and Technology, Ch The quantum circuit model
Problem
Qubits 1 and 2 are prepared in the Bell state ∣Φ+⟩12=21(∣00⟩+∣11⟩) (Chapter 4). An ancilla qubit 3 is prepared in ∣+⟩ (Chapter 1). Recall that Rn^(θ)=e−iθn^⋅σ/2 for any axis n^, so Rn^(2π)=−I on a single qubit. The gate C(Rz^(2π))3→1, with qubit 3 as control and qubit 1 as target, is applied.
((a)) Write the 4×4 matrix of C(Rz^(2π)), control first, qubit 3, and target second, qubit 1, in the computational basis ∣00⟩,∣01⟩,∣10⟩,∣11⟩.
((b)) Because Rz^(2π)=−I, every state of qubit 1, indeed every state of the pair of qubits 1 and 2 treating that pair as the target, is an eigenstate of it with eigenvalue −1, whether or not qubit 1 is entangled with qubit 2. Use phase kickback to find the exact state of the ancilla and the Bell pair after C(Rz^(2π))3→1 is applied to ∣+⟩3⊗∣Φ+⟩12, and state which Bloch-sphere fact from Chapter 1 this result makes observable.
((c)) The protocol now measures qubit 3 in the X basis, an outcome certain from part (b), and uses the classical outcome to decide whether to apply a Z gate to qubit 2 before Bob measures qubit 2 in the computational basis. Use the deferred measurement principle to redraw this as a circuit in which qubit 3 is measured only at the end, and explain why Bob's measurement statistics on qubit 2 are the same either way.