Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 61 / 4 / i

Represent any universally quantum device, including its auxiliaries and arbitrary final readout, by a positive operator-valued measure $\{E_i\}$ on the input. Perfect identification would requireSince a positive operator has a positive square root, zero expectation implies $E_i|{\varphi }_j⟩=0$ for $j\ne i$. Also $I-E_i\ge 0$ and its expectation in ${\varphi }_i$ is zero, so $E_i|{\varphi }_i⟩=|{\varphi }_i⟩$. ConsequentlyThis proves that perfect discrimination of pure states requires orthogonality. Any nonzero overlap contradicts perfect identification, independently of whether the device disturbs its input.

Equivalently, a unitary operator coupling the input to a readout apparatus must preserve inner products. Distinct perfectly readable records have zero inner product, so their corresponding input vectors must already be orthogonal. Nonorthogonal candidate states cannot be identified with certainty in one use. Probabilistic unambiguous quantum state discrimination can have an inconclusive outcome, which is excluded by the guaranteed-identification requirement.