Graph Quantum Magic Squares and Free Spectrahedra

TL;DR

This paper introduces a quantum analogue of magic squares based on graphs, demonstrates its failure in the case of the cycle C4 through a counterexample, and shows these structures form compact free spectrahedra.

Contribution

It defines a graph-based quantum magic square concept, provides a counterexample for C4, and characterizes these as free spectrahedra.

Findings

Quantum magic squares fail for C4.

They admit monic linear matrix inequality descriptions.

Form compact free spectrahedra.

Abstract

Recently De les Coves, Drescher and Netzer showed that an analogue of the Birkhoff--von Neumann theorem fails in the quantum setting. Motivated by this and questions arising in the study of quantum automorphisms of graphs, we introduce a graph-based variant of quantum magic squares and show that the analogue already fails for the cycle \(C_4\), via an explicit counterexample. We also show that they admit monic linear matrix inequality descriptions, hence form compact free spectrahedra.