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Information and Distinguishability of Ensembles of Identical Quantum States

Authors:	Lev?B?Levitin Email author" target="_blank">Tom?Toffoli Email author Zac?Walton

Institution:	(1) Department of Electrical and Computer Engineering, Boston University, Boston, Massachusetts;(2) Department of Electrical and Computer Engineering, Boston University, 8 Saint Mary’s Street, Boston, Massachusetts, 02215

Abstract:	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 (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 $W(N,n,\Omega)=\Omega(\frac{2n}{\pi e})^{\frac{N-1}{2}}$ . The optimal distribution is uniform over the domain in Cartesian coordinates.

Keywords:	information in quantum measurements distinguishability of quantum states number of distinguishable quantum states information criterion for distinguishability
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