On the comparison of volumes of quantum states

This paper aims to study the α-volume of K, an arbitrary subset of the set of N ×N density matrices. The α-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the α-volume of K in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the α-volume are also established. We employ our results to obtain bounds for the α-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 [1] for large N in the sense of α-volume. PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.Mn