Reaching Fleming's dicrimination bound

Any rule for identifying a quantum system's state within a set of two non-orthogonal pure states by a single measurement is flawed. It has a non-zero probability of either yielding the wrong result or leaving the query undecided. This also holds if the measurement of an observable $A$ is repeated on a finite sample of $n$ state copies. We formulate a state identification rule for such a sample. This rule's probability of giving the wrong result turns out to be bounded from above by $1/nδ_{A}^{2}$ with $δ_{A}=|<A>_{1}-<A>_{2}|/(Δ_{1}A+Δ_{2}A).$ A larger $δ_{A}$ results in a smaller upper bound. Yet, according to Fleming, $δ_{A}$ cannot exceed $\tanθ$ with $θ\in(0,π/2)$ being the angle between the pure states under consideration. We demonstrate that there exist observables $A$ which reach the bound $\tanθ$ and we determine all of them.

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