Graph Quantum Magic Squares and Free Spectrahedra
Francesca La Piana
TL;DR
This work extends the noncommutative Birkhoff--von Neumann paradigm to graph-structured quantum magic squares, showing that even with graph-imposed commutation constraints the classical convex hull of quantum permutation matrices fails to capture all quantum magic squares, starting already at $n=4$ with $s=2$ on $C_4$. It develops explicit monic linear matrix inequalities that describe both ordinary quantum magic squares and graph-constrained variants, proving that these sets are compact free spectrahedra generated by their Arveson extreme points. For $k$-regular graphs, the authors provide an explicit affine parametrization and a complete LMI description, and they show that graph permutation matrices are Arveson extreme points, indicating a richer extremal structure than the permutation class. The results illuminate the interface between quantum symmetries, noncommutative convexity, and quantum information concepts, and they point to further exploration of extremal structures and connections to graph-based quantum games. Overall, the paper offers a concrete framework to study graph-structured quantum symmetries through free semidefinite descriptions with potential applications in quantum information and operator algebras.
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.