How often is a random quantum state k-entangled?

TL;DR

This paper analyzes the volume ratios of nested sets of k-positive maps in quantum state space, providing asymptotically tight bounds and insights into the structure of k-entangled states in bipartite systems.

Contribution

It derives asymptotically tight bounds for the volumes of k-positive maps, revealing the small size of (k+1)-positive maps within k-positive maps.

Findings

Inner set of (k+1)-positive maps is a small fraction of k-positive maps

Provides asymptotically tight volume bounds for these sets

Results relate to the volume of k-entangled states in bipartite systems

Abstract

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.