= Solution
Bob knows the initial amplitudes and the <quantum measurement>[measurement] basis but not its result, so he averages the two conditional states:
$$
\boxed{\rho_B=|\alpha|^2|0\rangle\langle0|+|\beta|^2|1\rangle\langle1|
=\begin{pmatrix}|\alpha|^2&0\\0&|\beta|^2\end{pmatrix}.}
$$
The initial <matrix> had off-diagonal entries $\alpha\beta^*$ and $\alpha^*\beta$, which the unread <quantum measurement>[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.
Back to article page
