Quantum symmetries of Hadamard matrices

TL;DR

This paper explores the quantum symmetries and isomorphisms of Hadamard matrices, revealing universal quantum automorphisms for matrices of size at least four and establishing quantum isomorphism among all matrices of the same size.

Contribution

It introduces the concept of quantum automorphisms for Hadamard matrices and proves their existence and universality for matrices of size ≥4, also extending results to associated graphs and quantum spaces.

Findings

Every Hadamard matrix of size ≥4 has quantum symmetries.

All Hadamard matrices of a fixed size are mutually quantum isomorphic.

The results extend to Hadamard graphs and quantum Hadamard matrices.

Abstract

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size $N\ge 4$ has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras.