On the comparison of volumes of quantum states
arXiv:1003.0727 · doi:10.1088/1751-8113/43/31/315301
Abstract
This paper aims to study the $\a$-volume of $\cK$, an arbitrary subset of the set of density matrices. The $\a$-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the $\a$-volume of $\cK$ in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the $\a$-volume are also established. We employ our results to obtain bounds for the $\a$-volume of the sets of separable quantum states and of states with positive partial transpose (PPT). Hence, our asymptotic results provide answers for questions listed on page 9 in \cite{K. Zyczkowski1998} for large in the sense of $\a$-volume. \vskip 3mm PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.Mn
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