Past exam of the laser science and quantum information processing course of the University of Oxford / 2019 / 6 / b

In ZZZ order, the only nonzero amplitudes are those of HHH and VVV. ThusFor an equatorial outcome $a=\pm 1$, use the X and Y eigenvectors in a. For a product basis containing $k$ Y measurements, the bra acting on VVV contributes $(-i{)}^{k}abc/\sqrt{8}$, while that acting on HHH contributes $1/\sqrt{8}$. Hence the amplitude of outcome $(a,b,c)$ in the supplied state isEach of XXY, XYX and YXX has $k=1$, giving $\mathcal{A}=(1-abc)/4$. In each of the three respective product bases,The eigenvector in each position belongs to the observable named in that position. All four outcomes with product $abc=-1$ have probability $1/4$; the four with $abc=+1$ have probability zero. This also specifies all eight probabilities in all three bases, and all single-site equatorial marginals are unbiased.