Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 3 / a / Solution

Let $p=|\alpha {|}^2$, so $|\beta {|}^2=1-p$. The projective measurement has outcome 0 with probability $p$ and leaves the state $|0⟩$; outcome 1 has probability $1-p$ and leaves $|1⟩$. Alice knows her actual result, so her conditional description is a pure state. Her fidelity of quantum states with the initial state is $p$ after outcome 0 or $1-p$ after outcome 1. Its outcome average is $p^2+(1-p{)}^2$.

Bob knows the preparation but not the outcome, so his density matrix is the nonselective postmeasurement stateIt is mixed for $0<p<1$ and pure only at the endpoints. The relative phase has disappeared because the classical outcome record is unavailable to him.

The undisclosed qubit measurement outcome with an unspecified preparation prior requires care: Charlene has no specified prior over preparations. Consequently the information given does not determine a unique posterior probability for the two outcomes. Under a symmetric ignorance or maximum-entropy prior, her best assignment isMore generally, if her prior mean of $|\alpha {|}^2$ is $q$, her assignment is ${\rho }_C=\mathrm{diag}(q,1-q)$ and its fidelity of quantum states with the actual initial state is $pq+(1-p)(1-q)$. Knowing that a measurement occurred guarantees diagonality in its basis, but does not by itself prove $q=1/2$. Different descriptions here reflect different available records, rather than different physical outcomes. ^[ref]