Schmidt balls around the identity

TL;DR

This paper introduces the concepts of Schmidt robustness and random Schmidt robustness to quantify entanglement resilience, providing bounds, a distillability criterion, and conjectures on Schmidt number balls and witnesses.

Contribution

It presents the novel notions of Schmidt robustness and random Schmidt robustness, analyzes bounds for pure states, and proposes conjectures on Schmidt number balls and optimal witnesses.

Findings

Upper bounds enable a simple distillability criterion

Analysis of bounds for pure states

Conjectures on Schmidt number balls and witnesses

Abstract

Robustness measures as introduced by Vidal and Tarrach [PRA, 59, 141-155] quantify the extent to which entangled states remain entangled under mixing. Analogously, we introduce here the Schmidt robustness and the random Schmidt robustness. The latter notion is closely related to the construction of Schmidt balls around the identity. We analyse the situation for pure states and provide non-trivial upper and lower bounds. Upper bounds to the random Schmidt-2 robustness allow us to construct a particularly simple distillability criterion. We present two conjectures, the first one is related to the radius of inner balls around the identity in the convex set of Schmidt number n-states. We also conjecture a class of optimal Schmidt witnesses for pure states.