Information and Distinguishability of Ensembles of Identical Quantum States

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 (α1, α2, n) = |α1 − α2| √ 2n πe , 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,Ω) = Ω( 2n πe ) N−1 2 . The optimal distribution is uniform over the domain in Cartesian coordinates.