Order-Six CHMs Containing Exactly Three Distinct Elements

Yanzu Huang; Mengfan Liang; Lin Chen

doi:10.2478/qic-2026-0001

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Order-Six CHMs Containing Exactly Three Distinct Elements

Quantum Information & Computation

Volume 26 (2026): Issue 1 (March 2026)

By: Yanzu Huang, Mengfan Liang and Lin Chen

Open Access

|Jun 2026

Abstract

Complex Hadamard matrices (CHMs) are intimately related to the number of distinct matrix elements. We investigate CHMs containing exactly three distinct elements, which is also the least number of distinct elements. In this paper, we show that such CHMs can only be complex equivalent to two kinds of matrices, one is H₂-reducible and the other is the Tao matrix. Using our result one can further narrow the range of MUB trio (a set of four MUBs in ℂ⁶ consists of an MUB trio and the identity) since we find that neither of the two CHMs belong to any MUB trio. Our results may lead to the more complete classification of 6 × 6 CHMs whose elements in the first row are all 1.

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Articles in this issue

DOI: https://doi.org/10.2478/qic-2026-0001 | Journal eISSN: 3106-0544 | Journal ISSN: 1533-7146

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Language: English

Page range: 1 - 19

Submitted on: Sep 19, 2025

Accepted on: Oct 22, 2025

Published on: Jun 4, 2026

Published by: Cerebration Science Publishing Co., Limited

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Complex Hadamard matrices,

MUB problem

Related subjects:

Physics,

Computer sciences in the humanities,

Computer sciences,

Computer sciences in natural sciences

© 2026 Yanzu Huang, Mengfan Liang, Lin Chen, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 26 (2026): Issue 1 (March 2026)