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Information and Distinguishability of Ensembles of Identical Quantum States

Lev B. Levitin·Tommaso Toffoli·Zac D. Walton
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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 $W(α_1,α_2,n)=|α_1-α_2|\sqrt{\frac{2n}{πe}}$, where $(α_1,α_2)$ 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 $W(N,n,Ω)=Ω(\frac{2n}{πe})^{\frac{N-1}{2}}$. 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 $W(α_1,α_2,n)=|α_1-α_2|\sqrt{\frac{2n}{πe}}$, where $(α_1,α_2)$ 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 $W(N,n,Ω)=Ω(\frac{2n}{πe})^{\frac{N-1}{2}}$. The optimal distribution is uniform over the domain in Cartesian coordinates.

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Lev B. Levitin, Tommaso Toffoli, Zac D. Walton. 2001-12-12. Information and Distinguishability of Ensembles of Identical Quantum States. https://arxiv.org/abs/quant-ph/0112075

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