Graph Quantum Magic Squares and 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.