 Solution

Let $p=|\alpha {|}^2$ and $1-p=|\beta {|}^2$. The first basic measurement has outcome probabilities $p$ and $1-p$, obtained by conjugating the amplitudes in the bras. Conditional on its first outcome, the next computational-basis measurement returns 0 with probability $p$; conditional on its second outcome, that probability is $1-p$. The law of total probability therefore givesConsequently the minimum is $1/2$, attained exactly when $|\alpha |=|\beta |=1/\sqrt{2}$. The relative complex phases do not affect this result. The alternatives after the first measurement form a classical mixture; their amplitudes must not be added coherently as if no measurement occurred.