= Solution
Alice sends the $m$ conjugate-coded <qubits> through the noiseless <quantum channel>. Bob independently chooses a uniformly random basis bit $y_i'$ for each qubit and performs a <projective measurement> in the computational basis when $y_i'=0$ or the Hadamard basis when $y_i'=1$.
After all measurements, Alice and Bob publicly reveal $Y$ and $Y'$ but keep the data bits secret. They discard every position with $y_i\ne y_i'$ and retain the measured bits at positions where the bases agree. In the ideal channel those retained bits equal Alice's $x_i$ with certainty. Each pair of bases agrees with probability $1/2$, so the shared secret key has expected length $m/2$.
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