Graph Quantum Magic Squares and Free Spectrahedra

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.