Bob receives a two-qubit state guaranteed to be one of the four Bell states on qubits…
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
Bob receives a two-qubit state guaranteed to be one of the four Bell states ∣βjk⟩=(ZjXk⊗I)∣Φ+⟩ on qubits 1 and 2, with ∣Φ+⟩=21(∣00⟩+∣11⟩) (Chapter 4's Bell-state notation). He wants to learn k without doing a full Bell measurement, which would also reveal, and disturb, j. He introduces an ancilla qubit 3, prepared in ∣+⟩ (Chapter 1), and applies the two-qubit gate C(Z) with qubit 3 as control and qubit 1 as target, immediately followed by C(Z) with qubit 3 as control and qubit 2 as target. He then measures qubit 3 in the X basis, that is he applies H to qubit 3 and measures in the computational basis.
((a)) Write the 4×4 matrix of C(Z), control first and target second, in the computational basis ∣00⟩,∣01⟩,∣10⟩,∣11⟩, and check it is unchanged if the roles of control and target are swapped.
((b)) The state ∣βjk⟩ is an eigenstate of Z1Z2 with eigenvalue (−1)k, since Z1X1=−X1Z1 and Z1Z2∣Φ+⟩=∣Φ+⟩. Show that the two controlled-Z gates together act on the ancilla and the Bell pair as a controlled version of the joint operator Z1Z2, derive by phase kickback the resulting state of qubit 3, and hence show that measuring qubit 3 in the X basis reveals k with certainty while leaving qubits 1 and 2 in the state ∣βjk⟩, undisturbed.
((c)) Bob now wants to apply an X gate to qubit 1, conditioned on the qubit-3 outcome, so as to convert the pair to one with k=0. Use the deferred measurement principle to redraw the protocol so that qubit 3 is measured only at the very end of the whole circuit, and state why the final joint statistics of qubits 1 and 2 are unchanged by this reordering.