Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 2 / c

Measure the first qutrit in the computational basis and the second in the orthonormal basis $\{|{\varphi }_0⟩,|{\varphi }_1⟩,|{\varphi }_2⟩\}$ from Question 1(a), with those labels as outcomes. For first-party outcome 0, the unnormalized conditional vector of the second party is $|0⟩/\sqrt{3}$, so this outcome occurs with probability $1/3$ and leaves the normalized state $|0⟩$.

The second-party Born rule probabilities are then $|⟨{\varphi }_0|0⟩{|}^2=1/2$, $|⟨{\varphi }_1|0⟩{|}^2=0$, and $|⟨{\varphi }_2|0⟩{|}^2=1/2$. HenceThe relevant joint probabilities are $1/6,0,1/6$, each divided by $\mathrm{Pr}(A=0)=1/3$. This distinguishes the conditional statement from an unconditional outcome probability. More generally, conjugate-state steering from a maximally entangled pair explains how the first local measurement determines the second party's conditional vector; the basis choice here makes that vector especially simple.