Information and Distinguishability of Ensembles of Identical Quantum States

Abstract

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.