A Complete Classification of Complex Hadamard Matrices of Order Six

• URL: http://arxiv.org/abs/2608.18053v1
• Date: Tue, 18 Aug 2026 17:46:58 GMT
• Title: A Complete Classification of Complex Hadamard Matrices of Order Six
• Abstract summary: Complex Hadamard matrices encode perfectly balanced unitary transformations.<n>Their classification is complete through order five, but order six has remained open for decades.<n>Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• GPTQ-2D: Cubic-Time Two-Sided Adaptive Rounding [54.10836630612739]
  GPTQ, or equivalently Babai's nearest plane algorithm, rounds a real matrix to integers under a quadratic metric.<n>We present GPTQ-2D, which produces the identical rounded matrix in cubic time.
  arXiv  Detail & Related papers  (2026-07-29T15:34:00Z)
• Generalized Numerical Construction of MUBs: A Group Theoretical Investigation [0.0]
  Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory.<n>In non-prime-power dimensions, the existence of maximal sets of MUBs remains an open question.
  arXiv  Detail & Related papers  (2026-04-05T16:11:00Z)
• Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices [7.799711162530712]
  We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal.<n>One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples.
  arXiv  Detail & Related papers  (2026-02-26T19:14:16Z)
• High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces [48.465738895704455]
  We construct path matrices for decomposition of Cartesian tensors up to rank $n=9$ with reduced and affordable complexity.<n>Our method avoids the RREF algorithm and maintains a fully analytical derivation of each ICT decomposition matrix.<n>We extend our result to the arbitrary tensor product and direct sum spaces, enabling free design between different spaces while keeping symmetry.
  arXiv  Detail & Related papers  (2024-12-24T08:25:38Z)
• Two-Unitary Complex Hadamard Matrices of Order $36$ [0.0]
  A family of two-unitary complex Hadamard matrices (CHM) stemming from a particular matrix, of size $36$ is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all $36$ officers.
  arXiv  Detail & Related papers  (2024-01-03T11:10:00Z)
• Generalized unistochastic matrices [0.4604003661048266]
  We measure a class of bistochastic matrices generalizing unistochastic matrices. We show that the generalized unistochastic matrices is the whole Birkhoff polytope.
  arXiv  Detail & Related papers  (2023-10-05T10:21:54Z)
• Mutually-orthogonal unitary and orthogonal matrices [6.9607365816307]
  We show that the minimum and maximum numbers of an unextendible maximally entangled bases within a real two-qutrit system are three and four, respectively. As an application in quantum information theory, we show that the minimum and maximum numbers of an unextendible maximally entangled bases within a real two-qutrit system are three and four, respectively.
  arXiv  Detail & Related papers  (2023-09-20T08:20:57Z)
• Generalization Bounds for Inductive Matrix Completion in Low-noise Settings [46.82705757568271]
  We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption. We obtain for the first time generalization bounds with the following three properties.
  arXiv  Detail & Related papers  (2022-12-16T08:30:41Z)
• Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation [64.49871502193477]
  We propose a novel semi-supervised subspace clustering method, which is able to simultaneously augment the initial supervisory information and construct a discriminative affinity matrix. Comprehensive experimental results on six commonly-used benchmark datasets demonstrate the superiority of our method over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-05-21T01:47:17Z)
• Relative Pose from SIFT Features [50.81749304115036]
  We derive a new linear constraint relating the unknown elements of the fundamental matrix and the orientation and scale. The proposed constraint is tested on a number of problems in a synthetic environment and on publicly available real-world datasets on more than 80000 image pairs.
  arXiv  Detail & Related papers  (2022-03-15T14:16:39Z)
• Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation [71.27989298860481]
  We address the non- optimisation problem of finding a matrix on the Stiefel manifold that maximises a quadratic objective function. We propose a simple yet effective sparsity-promoting algorithm for finding the dominant eigenspace matrix.
  arXiv  Detail & Related papers  (2021-09-30T19:17:35Z)
• Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion [101.83262280224729]
  We develop a relative error bound for nuclear norm regularized matrix completion. We derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix.
  arXiv  Detail & Related papers  (2015-04-26T13:12:16Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.