Tutte Colloquium - Michael Brennan

Title: Hadamard matrices, quantum groups, and quantum games.

Abstract: A Hadamard matrix is a square matrix of complex numbers whose entries have modulus 1, and whose columns are pairwise orthogonal.  Hadamard matrices appear all over mathematics and its applications, including computer science, statistics, and quantum physics. In this talk, I will give a brief introduction to Hadamard matrices with a particular focus on Hadamard matrices whose entries are powers of a fixed root of unity (a.k.a. Butson Hadamard matrices).  Butson matrices are interesting combinatorial objects, and can be thought of as generalizations of the Fourier transform matrix associated to a finite abelian group.  For a general Hadamard matrix, I will explain how one can associate to it two natural "group-like" objects, called quantum groups (or Hopf algebras).   The first type of quantum group associated to a Hadamard matrix can be thought of as a generalization of the abelian group associated to a Fourier matrix, and the second type of quantum group can be thought of as the "quantum automorphism group" of the Hadamard matrix.  This latter quantum group is particularly interesting as it can be used to define a notion of quantum equivalence of Hadamard matrices, and it turns out that many classically inequivalent Hadamard matrices can be quantum equivalent.  Time permitting, I'll interpret quantum equivalence of Hadamard matrices in terms of a certain non-local game involving Hadamard matrices that can be played perfectly with the aid of quantum entanglement.  This is joint work with Daniel Gromada, Roberto Hernandez-Palomares, and Nicky Priebe.