  We consider a fixed quantum measurement performed over $n$ identical copies
of quantum states. Using a rigorous notion of distinguishability 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
$(\alpha_1,\alpha_2)$ is the angle interval from which the states are chosen.
In the general case of an $N$-dimensional Hilbert space and an area $\Omega$ 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.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
$(\alpha_1,\alpha_2)$ is the angle interval from which the states are chosen.
In the general case of an $N$-dimensional Hilbert space and an area $\Omega$ 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.
