Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

TL;DR

This paper analyzes the geometric structure and volume ratios of various classes of quantum maps, revealing that most high-dimensional positive maps are not completely positive or decomposable, with implications for quantum state properties.

Contribution

It provides explicit volume bounds for nested sets of quantum maps and demonstrates that generic positive maps in high dimensions are typically not completely positive or decomposable.

Findings

Most positive maps become non-decomposable as dimension increases

Explicit volume bounds for sets of quantum maps are derived

High-dimensional positive maps are generally not completely positive

Abstract

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.