Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S03 / 2 / b

z

-spin measurement. The Born rule gives

p (z + ∣ ψ_{+}) = p (z + ∣ ψ_{-}) = p (z - ∣ ψ_{+}) = p (z - ∣ ψ_{-}) = \frac{1}{2} .

(1)

Thus this measurement cannot distinguish the two preparations, even by repeating the same measurement on many identically prepared copies: it discards their relative-phase difference in this basis.

Solution

 0  0

A horizontal Stern-Gerlach spin measurement measures

S_{x}

. The inputs are its orthogonal eigenstates, so

p (x + ∣ ψ_{+}) = 1, p (x + ∣ ψ_{-}) = 0.

(1)

The output beam identifies the input with certainty in a single ideal measurement.

The negative-outcome probabilities are complementary. Both outcomes can occur for both inputs, so one result cannot distinguish them perfectly. Repeated measurements on independently prepared copies distinguish their different statistics with arbitrarily high confidence. Reversing the diagonal direction exchanges the two numbers. This separates single-copy certainty from statistical distinguishability.