Quantum magic squares: dilations and their limitations

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.

v2: 20 pages, very close to published version. v1: 19 pages and 1 Mathematica file attached

References in corpus (4)

• Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
• Factorization and dilation problems for completely positive maps on von Neumann algebras
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Arveson extreme points span free spectrahedra

Cited by in corpus (7)

• Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem
• Genuinely quantum SudoQ and its cardinality
• Magic squares: Latin, Semiclassical and Quantum
• Extreme points of matrix convex sets and their spanning properties
• Matrix Extreme Points and Free extreme points of Free spectrahedra
• Polytope compatibility -- from quantum measurements to magic squares
• Joint measurability meets Birkhoff-von Neumann's theorem