 Solution

Alice makes a two-outcome local measurement in quantum mechanics with Kraus operatorsSince $0<\tan \alpha /\tan \beta <1$, these are valid and satisfy $K_{\mathrm{s}}^{†}{K}_{\mathrm{s}}+K_{\mathrm{f}}^{†}{K}_{\mathrm{f}}=I$. Alice sends Bob the success or failure outcome. The successful unnormalized branch isTherefore the normalized success state is exactly the target andBob needs no quantum correction in this known Schmidt basis. The failure branch is proportional to $|00⟩$, a product state, and has probability $1-s/t$. The successful branch saturates the previous bound: $p_{\mathrm{succ}}{E}_2(\varphi )=E_2(\psi )$ and failure contributes zero. This establishes optimal stochastic conversion of a two-qubit pure state.