Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 4 / b

Apply a Hadamard gate before a Z-basis measurement: $H|+⟩=|0⟩$ and $H|-⟩=|1⟩$. The result probabilities areFor measurement statistics the network is $H$ followed by $M_Z$. If the measured qubit survives and should be left in the original X eigenstate, append a second $H$ after the nondestructive $M_Z$. The resulting instrument has Kraus operators $H{P}_{0}H=|+⟩⟨+|$ and $H{P}_{1}H=|-⟩⟨-|$, not Z-eigenstate outputs.

Bob must discard both results. An unread X measurement gives ${\rho }_X=p_{+}|+⟩⟨+|+p_{-}|-⟩⟨-|$. Each of these two X eigenstates gives Z outcomes with probabilities $1/2,1/2$. The final unread Z measurement therefore givesEquivalently, in the Bloch-vector description the first measurement retains only the X component, and the second retains only the Z component, leaving none. The final state is independent of the original $\alpha ,\beta$.