= Solution
Let $p=|\alpha|^2$, so $|\beta|^2=1-p$. The <projective measurement> has outcome 0 with <probability> $p$ and leaves the state $|0\rangle$; outcome 1 has <probability> $1-p$ and leaves $|1\rangle$. 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 state
$$
\boxed{\rho_B=p|0\rangle\langle0|+(1-p)|1\rangle\langle1|,\qquad
F_B=\langle\psi|\rho_B|\psi\rangle=p^2+(1-p)^2.}
$$
It 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 is
$$
\boxed{\rho_C=I/2,\qquad F_C=1/2.}
$$
More generally, if her prior mean of $|\alpha|^2$ is $q$, her assignment is $\rho_C=\operatorname{diag}(q,1-q)$ 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. <Foundations I: States and Ensembles by John Preskill>[Sections 2.3–2.4, density matrices, ensembles and measurement information]{ref}
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