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

Alice conditions on her observed computational-basis result. For outcome $0$, occurring with probability $|\alpha {|}^2$, her post-measurement density matrix is $|0⟩⟨0|$; for outcome $1$, occurring with probability $|\beta {|}^2$, it is $|1⟩⟨1|$. Explicitly,These are the conditional Lüders rule updates for the two rank-one projectors. They assume the corresponding outcome has nonzero probability and Alice knows which one occurred.

Bob knows the initial amplitudes and the measurement basis but not its result, so he averages the two conditional states:The initial matrix had off-diagonal entries $\alpha {\beta }^{\ast }$ and ${\alpha }^{\ast }\beta$, which the unread measurement removes. Alice's description is conditioned on an available record; Bob's is the nonselective projective measurement mixture. Averaging Alice's records gives exactly Bob's predictions, and communicating the result lets him make the same conditional update. This is a difference of conditioning information, not two inconsistent physical state assignments.