On the comparison of volumes of quantum states

TL;DR

This paper investigates the $ ext{ extalpha}$-volume of subsets of quantum states, providing bounds and estimates that relate it to known volumes, and applies these to important classes like separable and PPT states.

Contribution

It introduces two-sided estimates for the $ extalpha$-volume in terms of Hilbert-Schmidt and Bures volumes, advancing understanding of quantum state space geometry.

Findings

Derived bounds for $ ext{ extalpha}$-volume of quantum state subsets.

Established relations between $ ext{ extalpha}$-volume, Hilbert-Schmidt, and Bures volumes.

Provided asymptotic results for large $N$ relevant to quantum information theory.

Abstract

This paper aims to study the $\a$-volume of $\cK$, an arbitrary subset of the set of $N\times N$ 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 $N$ in the sense of $\a$-volume. \vskip 3mm PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.Mn