Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses

TL;DR

This paper investigates supporting hyperplanes for sets of bipartite quantum states with bounded Schmidt number and their witnesses, providing geometric insights and applications to Werner and isotropic states.

Contribution

It introduces a method to find hyperplanes supporting convex sets of states with bounded Schmidt number, linking geometric properties to dual objects and state families.

Findings

Characterization of supporting hyperplanes for Schmidt number sets

Application to Werner and isotropic states

Decomposition of Werner states into product states

Abstract

We consider the compact convex set of all bi-partite states of Schmidt number less than or equal to $k$, together with that of $k$-blockpositive matrices of trace one, which play the roles of Schmidt number witnesses. In this note, we look for hyperplanes which support those convex sets and are perpendicular to a one parameter family through the maximally mixed state. We show that this is equivalent to determining the intervals for the dual objects on the one parameter family. We illustrate our results for the one parameter families including Werner states and isotropic states. Through the discussion, we give a simple decomposition of the separable Werner state into the sum of product states.