Quantum magic squares: dilations and their limitations

TL;DR

This paper investigates whether all quantum magic squares can be represented as convex combinations of quantum permutation matrices, revealing limitations in their dilation properties and classifying special cases.

Contribution

It demonstrates that not all quantum magic squares dilate to quantum permutation matrices and classifies those that do with commuting entries.

Findings

Not all quantum magic squares dilate to quantum permutation matrices.

Classified quantum magic squares that dilate to permutation matrices with commuting entries.

Established a lower bound on the diameter of the set of dilatable quantum magic squares.

Abstract

Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff--von Neumann Theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries, and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.