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.

10 pages, 1 figure, IQSA 2001