While magic squares have intrigued mathematicians for over two thousand years, they are more than just interesting puzzles. A paper by De las Cuevas et al. puts this ancient piece of mathematics in conversation with modern physics. The authors show a famous theorem about magic squares cannot be generalized to their quantum cousins.

N-by-N magic squares where each row and column sum to 1 can represent transition probabilities between N states. A subset of magic squares, known as permutation matrices, have rows and columns which contain a single entry of 1 with the rest 0.

“The famous Birkhoff-von Neumann Theorem now just states that every magic square can be decomposed into permutation matrices,” said author Tom Drescher. “In other words, we can also say that the permutation matrices have an extreme stance among all magic squares and, more importantly, that every magic square with an extreme stance must be a permutation matrix.”

In quantum physics, the transition probabilities between states can be described by matrices. The authors define quantum magic squares, where each entry is a matrix, and each row and column sum to the identity. Quantum permutation matrices can be constructed by analogy.

Using the theory of matrix-convex sets, an extension of the geometric notion of convexity, the group shows there are extreme points in the set of quantum magic squares that are not always quantum permutation matrices, disproving the quantum analogue of the Birkhoff-von Neumann Theorem.

Matrix convexity isn’t widely used in the physics community, said Drescher. “I hope the paper is a contribution to making the framework of matrix-convex sets more popular to a wider range of people.”

**Source:** “Quantum magic squares: Dilations and their limitations,” by Gemma De las Cuevas, Tom Drescher, and Tim Netzer, *Journal of Mathematical Physics* (2020). The article can be accessed at https://doi.org/10.1063/5.0022344.