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.

• The paper completes the classification of order-six complex Hadamard matrices by proving that every equivalence class has a finite 3×3-corner witness and is recovered by a branch-complete dilation algorithm.
• Its division-free exact-algebraic method handles vanishing denominators and infinite fibers, establishing the disjoint decomposition into Karlsson’s three-parameter family, Tao’s isolated matrix, and the remaining algebraically reconstructed classes.
• The geometric analysis shows that regular reconstructions use one quadratic and one cubic in each direction, while only Tao’s matrix and one explicit Karlsson matrix lack product-regular frames, supporting applications to six-mode interferometry and mutually unbiased bases.

Complex Hadamard matrices of order six have been the last unresolved case in the classification program for small orders: orders two, three, and five contain only the Fourier matrix, order four a single one-parameter family, while order six is the first dimension where several inequivalent continuous families coexist with an isolated solution. The paper under review completes this classification (2608.18053). Its central result is that every equivalence class of order-six complex Hadamard matrices admits a dephased representative with a finite $3\times3$ corner witness, so that a branch-complete refinement of Szöllősi's dilation algorithm enumerates all classes exactly. As corollaries, the authors prove Szöllősi's Conjecture 4.2 — that every class outside Karlsson's three-parameter family and Tao's isolated matrix is recovered algebraically from a suitable corner — and they characterize the geometry of the reconstruction over a four-phase seed torus.

Background and problem structure

A complex Hadamard matrix $H\in\mathbb T^{n\times n}$ satisfies $HH^\dagger=nI_n$, and equivalence is generated by independent row/column permutations and diagonal unitary phasings. Every class contains a unique dephased representative for fixed row and column ordering. In order six, two named sectors anchor the classification: the Karlsson sector $\mathcal K_6^{(3)}$, consisting exactly of the $H_2$-reducible matrices (those equivalent to one containing a $2\times2$ Hadamard submatrix), which Karlsson showed coincides with his complete three-real-parameter family; and the Tao sector $\mathcal T_6$, the singleton orbit of the cubic-root matrix $S_6^{(0)}$ introduced by Tao in disproving Fuglede's conjecture in dimensions five and higher.

Szöllősi's dilation method starts from a dephased $3\times3$ corner $E(a,b,c,d)$ with four free phases, solves fixed-Gram constraints for the adjacent blocks $H\in\mathbb T^{n\times n}$0 and $H\in\mathbb T^{n\times n}$1, and forces the fourth block by orthogonality as $H\in\mathbb T^{n\times n}$2. His fixed-corner completeness theorem holds when the normalized invertible candidate sets ("side fibers") are finite and nonempty. Two gaps separated this from a full classification: published formulas use divisions whose denominators can vanish on exceptional branches, and some corners admit infinitely many candidates, so fixed-corner completeness does not guarantee that every matrix possesses such a corner. The paper closes both gaps.

The branch-complete dilation procedure

The authors replace Szöllősi's generic companion-function quotients with division-free elimination on the uncancelled fixed-Gram equations, retaining all solutions on vanishing-denominator branches. Exceptional cases are handled through exact real-algebraic pipelines: real-and-imaginary formulation with unit-circle constraints, Gröbner-basis or rational-univariate elimination, exact root isolation, and direct verification against the parent system. Soundness is elementary: any retained pair $H\in\mathbb T^{n\times n}$3 yields unitarity of the assembled matrix via the block Gram identities, and retention requires all nine entries of the forced block to be unimodular.

The exhaustiveness argument proceeds by a global routing over the $H\in\mathbb T^{n\times n}$4 positional corners. The key local input is an infinite-fiber trichotomy: if the normalized fixed-Gram fiber of an invertible $H\in\mathbb T^{n\times n}$5 phase matrix is infinite, then either its Gram matrix equals $H\in\mathbb T^{n\times n}$6 (a Fourier block), the real part of the cyclic cubic Gram invariant $H\in\mathbb T^{n\times n}$7 is strictly negative, or the matrix contains a $H\in\mathbb T^{n\times n}$8 Hadamard submatrix. The proof uses Haagerup's trick to derive polynomial eliminants and carefully treats the common-root branch that premature cancellation would destroy.

Complementary blocks reverse the sign of the cubic invariant, since off-diagonal Gram entries satisfy $H\in\mathbb T^{n\times n}$9 on complementary triples. A corner-routing proposition then shows that outside the Karlsson sector, either all four blocks of some partition are order-three Hadamard matrices, or a permutation and dephasing produce a finite-corner witness. The Fourier-block case is resolved by reducing to a normal form with $HH^\dagger=nI_n$0 and solving the remaining autocorrelation conditions, which force membership in the Karlsson or Tao sectors. Finally, separate propositions supply witnesses inside both named sectors: every Karlsson class admits a finite-corner witness (with the affine-Fourier boundary covered by six explicit corners certified through 245 pairwise resultants and Bernstein-basis positivity of a residual polynomial), and the leading corner of Tao's matrix is itself a witness. Together these establish:

$HH^\dagger=nI_n$1

i.e., the total output of the branch-complete procedure equals the entire class space. This proves Szöllősi's Conjecture 4.2 in the form $HH^\dagger=nI_n$2 with pairwise disjoint sectors.

Geometry of the four-phase reconstruction

Beyond enumeration, the paper describes how each class is reconstructed from the four seed phases $HH^\dagger=nI_n$3. On a product-regular locus — defined by eleven nonvanishing guards including block determinants, leading sextic coefficients, companion determinants, coordinate-cubic discriminants, and companion resultants — the reconstruction reduces to solving one quadratic and one cubic equation in each of the horizontal and vertical directions: a self-inversive product quadratic determines the product $HH^\dagger=nI_n$4, a self-inversive cubic $HH^\dagger=nI_n$5 gives the coordinates, and a companion quotient recovers the paired phases.

The physical domain is characterized sharply: a product-regular lift is physical if and only if the normalized product discriminant satisfies $HH^\dagger=nI_n$6. For $HH^\dagger=nI_n$7 there are two distinct physical sheets, which coalesce at $HH^\dagger=nI_n$8; horizontal and vertical roots cannot be chosen independently but are paired by an affine matching. Notably, the generic double cover is proven nonsplit: the residual discriminant is not a square in $HH^\dagger=nI_n$9, certified by a specialization to an octic with simple irreducible factors.

A strong global statement bounds the reach of these regular charts. Writing $\mathcal K_6^{(3)}$0 for the classes admitting a product-regular frame, the paper proves

$\mathcal K_6^{(3)}$1

where $\mathcal K_6^{(3)}$2 is a single explicitly displayed Karlsson matrix. The proof combines a counting argument — absence of regular frames would force at least 100 positive-dependent blocks while the exact incidence bound permits at most 80 — with exhaustive exact enumerations: 14,400 ordered frames checked for $\mathcal K_6^{(3)}$3 and for Tao (in $\mathcal K_6^{(3)}$4), and a 49-case dispatcher analysis. Outside these two exceptions, every class also carries at least 200 distinct finite-corner witnesses among the 400 positional corners.

Formal verification and computation

The classification argument is accompanied by a Lean 4 formalization whose public theorem derives the two-sided equivalence between being Hadamard and belonging to $\mathcal K_6^{(3)}$5, keeping the two published structural inputs (Karlsson's $\mathcal K_6^{(3)}$6 characterization and Szöllősi's cubic-root criterion) as explicit hypotheses. The repository reports no sorry, admit, or additional axioms beyond the standard foundational trio. The post-classification geometry relies instead on separately verified exact certificates (resultant computations, Bernstein subdivision, Eisenstein-integer enumerations) indexed against individual claims; these certificate calculations are not Lean-formalized, a boundary the authors state plainly. The acknowledgments disclose substantial interactive use of LLM assistants in proof exploration and code development, with the authors asserting responsibility for all retained material.

Limitations and open questions

Several boundaries are conceded explicitly. The ramification locus $\mathcal K_6^{(3)}$7 is shown to be disjoint from Tao and $\mathcal K_6^{(3)}$8 by containment, but its possible intersection with the remaining Karlsson classes is not determined. The nonsplitting result concerns rational selection over the original seed field only; nonrationality of the compactified cover after arbitrary birational change of coordinates remains open. The numerical diagnostics at the representative ramification seed (a Cayley seed involving the real root of a degree-seven polynomial) certify separation from known constructions at that point only, not on a neighborhood. Whether analogous finite-corner arguments extend to complex Hadamards of higher orders is left as an explicit question. The classification does not settle the existence question for mutually unbiased bases in $\mathcal K_6^{(3)}$9: it removes the need to search over an unclassified set of Hadamards, but joint compatibility conditions across shared bases must still be imposed directly on the classified space.

Conclusion

This work resolves the order-six complex Hadamard classification by proving that every class possesses a finite-corner witness and that a division-free, branch-complete dilation procedure is both sound and exhaustive. It thereby establishes Szöllősi's conjecture, identifies the exact exceptions to a universal quadratic–cubic reconstruction (Tao's matrix and one explicit Karlsson matrix), and provides a finite-chart parametrization of the class space with a sharply characterized physical domain. The combination of exact symbolic computation, careful treatment of degenerate branches, and machine-checked verification supplies a rigorous foundation for subsequent work on six-mode interferometry, dual-unitary circuits, and the mutually unbiased bases problem in dimension six.