Reaching Fleming's dicrimination bound

Abstract

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 $1/n\delta_{A}^{2}$ with $\delta_{A}=|<A>_{1}-<A>_{2}|/(\Delta_{1}A+\Delta_{2}A).$ A larger $\delta_{A}$ results in a smaller upper bound. Yet, according to Fleming, $\delta_{A}$ cannot exceed $\tan\theta$ with $\theta\in(0,\pi/2) $ being the angle between the pure states under consideration. We demonstrate that there exist observables $A$ which reach the bound $\tan\theta$ and we determine all of them.