A Complete Classification of Complex Hadamard Matrices of Order Six

Abstract

Complex Hadamard matrices encode perfectly balanced unitary transformations. They underlie mutually unbiased quantum measurements and multiphoton interferometry. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We first prove that every such matrix can be constructed from an initial dephased $3 \times 3$ corner by a finite, branch-complete procedure. This supplies the global step missing from Szöllősi's dilation method and proves his conjecture: up to standard equivalence, every class outside Karlsson's three-parameter family and Tao's isolated matrix is recovered algebraically from a suitable corner. We then describe the geometry of the reconstruction from four initial phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.