Global locations of Schmidt number witnesses

TL;DR

This paper classifies the locations of Schmidt number witnesses outside the set of all bipartite states by analyzing the convex geometry of state spaces and their faces, providing conditions for their existence based on subspace properties.

Contribution

It introduces a geometric framework for locating Schmidt number witnesses outside the state set and establishes conditions based on subspace orthogonality and Schmidt rank.

Findings

Locations are classified by faces of the convex state set.

Existence of witnesses depends on orthogonal complement Schmidt ranks.

Schmidt number witnesses of all lower ranks exist once a higher rank witness is found.

Abstract

We investigate global locations of Schmidt number witnesses which are outside of the convex set of all bipartite states. Their locations are classified by interiors of faces of the convex set of all states, by considering the line segments from them to the maximally mixed state. In this way, a nonpositive Hermitian matrix of trace 1 is located outside of one and only one face. Faces of the convex set of all states are classified by subspaces, which are range spaces of states belonging to specific faces. For a given subspace, we show that there exist Schmidt number $k+1$ witnesses outside of the face arising from this subspace if and only if every vector in the orthogonal complement of the subspace has Schmidt rank greater than $k$. Once we have Schmidt number $k+1$ witnesses outside of a face, we also have Schmidt number $2,3,\dots, k$ witnesses outside of the face.