Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 5 / b / ii / Solution

Arrange the amplitudes of $|{\varphi }_1⟩$ as the matrixThe second state has $C_2=C_{1}X$, where $X$ swaps the $A_2$ basis states. If Bob receives $A_1$, his two reduced density matrices areand are identical. No measurement on $A_1$ contains any information about the shared state.

If Bob instead receives $A_2$,They differ because $a^2+c^2<b^2+d^2$. Bob can therefore distinguish them with better-than-random success. For equal priors,so the optimal success probability is $\frac{1}{2}(1+D)$. It is generally below one, so a single copy does not permit certain identification.