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We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is $$W(\alpha_1,\alpha_2,n)=|\alpha_1-\alpha_2|{\sqrt{\frac{2n}{\pi e}}}$$, where (α1,α2) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is $$W(N,n,\Omega)=\Omega(\frac{2n}{\pi e})^{\frac{N-1}{2}}$$. The optimal distribution is uniform over the domain in Cartesian coordinates.

Information and Distinguishability of Ensembles of Identical Quantum States