Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 2 / b

The joint Born rule uses the tensor product of the two outcome bras. Its amplitude isThe other source component vanishes on contraction with $⟨0|$ on A. ThereforeThis is a joint probability: $\mathrm{Pr}(A=0)=16/25$, while $\mathrm{Pr}(B={\theta }_0\mid A=0)={\cos }^2\theta$. Dropping the first factor would give only the conditional probability.

Here $⟨+y|=(⟨0|-i⟨1|)/\sqrt{2}$. Keeping that complex conjugation, the joint amplitude isHence $\mathrm{Pr}(A=+,B=+y)=0$. The cancellation is destructive quantum interference between the two source components, not the absence of either local outcome. Each local outcome individually has probability $1/2$, since each reduced density matrix is $I/2$. Omitting the conjugation in the second bra would turn the cancellation into reinforcement and give the wrong answer.