A Concise Guide to Complex Hadamard Matrices

Experimental realization of any discrete unitary operator

Reck, M., Zeilinger, A., Bernstein, H. J., & Bertani, P. (1994) · Physical Review Letters · doi:10.1103/physrevlett.73.58 · Cited by 2,155

Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation

Jarlskog, C. (1985) · Physical Review Letters · doi:10.1103/physrevlett.55.1039 · Cited by 1,657

Optimal state-determination by mutually unbiased measurements

Wootters, W. K., & Fields, B. D. (1989) · Annals of Physics · doi:10.1016/0003-4916(89)90322-9 · Cited by 1,458

Geometrical description of quantal state determination

Ivonovic, I. D. (1981) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/14/12/019 · Cited by 732

Fuglede’s conjecture is false in 5 and higher dimensions

Tao, T. (2004) · Mathematical Research Letters · doi:10.4310/mrl.2004.v11.n2.a8 · Cited by 392

LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers

Sylvester, J. J. (1867) · The London Edinburgh and Dublin Philosophical Magazine and Journal of Science · doi:10.1080/14786446708639914 · Cited by 341

Constructions of Mutually Unbiased Bases

Klappenecker, A., & Rötteler, M. (2004) · Lecture Notes in Computer Science · doi:10.1007/978-3-540-24633-6_10 · Cited by 288

On Orthogonal Arrays

Seiden, E., & Zemach, R. (1966) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177699280 · Cited by 285

All Teleportation and Dense Coding SchemesPreprint

Werner, R. F. (2000) · Cited by 271

All teleportation and dense coding schemes

Werner, R. F. (2001) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/34/35/332 · Cited by 270

On constructing Hermitian unitary matrices with prescribed moduli

Smoktunowicz, A., & Tadej, W. (2008) · Journal of Generalized Lie Theory and Applications · doi:10.4303/jglta/s080328 · Cited by 2

A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

Egan, R. (2024) · Discrete Mathematics · doi:10.1016/j.disc.2024.114201 · Cited by 3

Complex Hadamard transform of digital signals: properties and applications

Mironov, R., & Kountchev, R. K. (2012) · International Journal of Reasoning-based Intelligent Systems · doi:10.1504/ijris.2012.046487 · Cited by 2

A characterization of complex Hadamard matrices appearing in families of MUB triplets

Matszangosz, Á. K., & Szöllősi, F. (2024) · Designs Codes and Cryptography · doi:10.1007/s10623-024-01503-w · Cited by 1

Maximal determinants of matrices over the roots of unity

Ponasso, G. N. (2025) · Linear Algebra and its Applications · doi:10.1016/j.laa.2025.05.024 · Cited by 3

Discrete Variable (DV) QKD

Djordjević, I. B. (2025) · Springer eBooks · doi:10.1007/978-3-031-88372-9_8 · Cited by 1

Information-Theoretic Interpretation of Quantum Formalism

Feldmann, M. (2023) · Foundations of Physics · doi:10.1007/s10701-023-00690-1 · Cited by 2

TWISTED COMMUTING SQUARES

Nicoară, R., & WHITE, J. (2013) · International Journal of Mathematics · doi:10.1142/s0129167x13500365 · Cited by 1

Schmidt-Correlated States, Weak Schmidt Decomposition and Generalized Bell Bases Related to Hadamard Matrices

Hua, B., Fei, S.-M., Jost, J., & Li‐Jost, X. (2014) · Reports on Mathematical Physics · doi:10.1016/s0034-4877(14)60059-8 · Cited by 1

Построение $R$-матриц Темперли - Либа и обобщенные матрицы Адамара

Аван, Ж., Avan, J., Фонсека, Т., Fonseca, T., Frappat, L., Kulish, P. P., Рагуси, Э., Ragoucy, E., Ролле, Ж., & Rollet, G. (2014) · Теоретическая и математическая физика · doi:10.4213/tmf8564 · Cited by 1

Encrypted quantum state tomography with phase estimation for quantum Internet

Gülbahar, B. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-04034-w · Cited by 1

The non-$$H_2$$-reducible matrices and some special complex Hadamard matrices

Liang, M., Chen, L., Long, F., & Qiu, X. (2024) · Quantum Information Processing · doi:10.1007/s11128-024-04482-y · Cited by 1

In-Depth Analysis of Ka-Band Integrated Commercial Beamformer Non-Idealities in Phased Array Calibration and Pattern Prediction Methods for 5G Scenarios

Maeso, J. C., Sun, X., Barrado, A. M., Varela, F. R., Olivares, P. S., González, J. M. F., & Castañer, M. S. (2025) · IEEE Access · doi:10.1109/access.2025.3555140 · Cited by 1

Pseudo Jacket Matrix and Its MIMO SVD Channel

컴퓨터공학과, 대., Yang, J., Kim, J.-S., & Lee, M.-H. (2015) · 한국인터넷방송통신학회 논문지 · doi:10.7236/jiibc.2015.15.5.39 · Cited by 1

Isolated Partial Hadamard Matrices and Related Topics

Banica, T., Özteke, D., & Pittau, L. (2018) · Open Systems & Information Dynamics · doi:10.1142/s1230161218500099 · Cited by 1

On the existence of complex Hadamard submatrices of the Fourier matrices

Bond, B. M., Glickfield, R. A., & Herr, J. E. (2018) · Demonstratio Mathematica · doi:10.1515/dema-2019-0001 · Cited by 1

The Birkhoff theorem for unitary matrices of prime-power dimension

Vos, A. D., & Baerdemacker, S. D. (2019) · Linear Algebra and its Applications · doi:10.1016/j.laa.2019.05.007 · Cited by 1

Complementarity and Stochastic Independence

Accardi, L., & Lu, Y. G. (2019) · Springer optimization and its applications · doi:10.1007/978-3-030-12661-2_1 · Cited by 1

A Birkhoff Connection Between Quantum Circuits and Linear Classical Reversible Circuits

Vos, A. D., & Baerdemacker, S. D. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-21500-2_2 · Cited by 1

Informing the structure of complex Hadamard matrix spaces using a flow

Motta, F. C., & Shipman, P. D. (2019) · Discrete and Continuous Dynamical Systems - S · doi:10.3934/dcdss.2019147 · Cited by 1

Second order deformations of group commuting squares and Hadamard matrices

Nicoară, R., & White, J. (2020) · Proceedings of the American Mathematical Society · doi:10.1090/proc/15025 · Cited by 1

Subfactors and Applications

Bisch, D., Gannon, T. J., Jones, V. F. R., & Kawahigashi, Y. (2020) · Oberwolfach Reports · doi:10.4171/owr/2019/49 · Cited by 1

Defect and Equivalence of Unitary Matrices. The Fourier Case. Part II

Tadej, W. (2019) · Open Systems & Information Dynamics · doi:10.1142/s1230161219500033 · Cited by 1

Local translations associated to spectral sets

Dutkay, D. E., & Haussermann, J. (2014) · Contemporary mathematics - American Mathematical Society · doi:10.1090/conm/626/12504 · Cited by 1

On Hadamard matrices at roots of unityPreprint

Banica, T., & Schlenker, J. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0707.4175 · Cited by 1

The 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer

Simone Severini, B. (2010) · Journal of Physics Conference Series · doi:10.1088/1742-6596/254/1/012012 · Cited by 4

Sinkhorn-based synthesis of an arbitrary quantum circuitPreprint

Vos, A. D., & Baerdemacker, S. D. (2015) · arXiv (Cornell University) · Cited by 1

Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces

Rajchel-Mieldzioć, G., Gąsiorowski, A., & Życzkowski, K. (2018) · Mathematics in Computer Science · doi:10.1007/s11786-018-0384-y

Mutually Unbiased Product Bases

McNulty, D. (2013) · White Rose eTheses Online (University of Leeds, The University of Sheffield, University of York)

Complex Hadamard matrices and equiangular tight frames

Szöllősi, F. (2011) · Linear Algebra and its Applications · doi:10.1016/j.laa.2011.05.034

MUTUALLY UNBIASED BASES 1Preprint

Bengtsson, I. (2006) · arXiv (Cornell University)

Formula for unbiased bases

Kibler, M. (2010) · Czech digital mathematics library

Constructions for complementary code sets

Coxson, G. E. (2014) · IEEE Radar Conference (RadarConf) · doi:10.1109/radar.2014.6875689

Complex Hadamard matrices from Sylvester inverse orthogonal matrices

Dit, P. (2009)

Continuous Spectra For Substitution-Based Sequences

Chan, L. K. L. (2018) · Open Research Online (The Open University) · doi:10.21954/ou.ro.0000d19d

Maximum coherence under noisy channels

Ding, Z., Yang, H., Liu, C.-C., Sun, W., He, J., & Ye, L. (2018) · Laser Physics Letters · doi:10.1088/1612-202x/aad98e

On the Construction of Several Types of BCCB Complex Hadamard Matrices of Order n2

徐, 登. (2018) · Pure Mathematics · doi:10.12677/pm.2018.86088

Deformed Fourier matrices in bistochastic formPreprint

Banica, T. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1901.04941

FBG-Based Multidimensional QKD

Djordjević, I. B. (2018) · International Conference on Transparent Optical Networks · doi:10.1109/icton.2018.8473590

Defect of a Kronecker product of unitary matricesPreprint

Tadej, W. (2010) · arXiv (Cornell University) · doi:10.48550/arxiv.1009.4037

Modelling questions for transitive quantum groupsPreprint

Banica, T., & Chirvăsitu, A. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1711.03206

Identifying complex Hadamard submatrices of the Fourier matrices via primitive sets

Herr, J. E., & Wiegand, T. M. (2021) · Linear Algebra and its Applications · doi:10.1016/j.laa.2021.01.017

Multiparameter estimation for qubit states with collective measurements.Preprint

Len, Y. L. (2021) · arXiv (Cornell University)

K-sparse Pure State Tomography with Phase EstimationPreprint

Gülbahar, B. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2111.04359

Acquisition Complete Session

(2017) · Expanded abstracts with biographies/SEG technical program expanded abstracts · doi:10.1190/segam2017-acq

Dual bounds for the positive definite functions approach to mutually unbiased bases

Bandeira, A. S., Doppelbauer, N., & Kunisky, D. (2022) · Sampling Theory Signal Processing and Data Analysis · doi:10.1007/s43670-022-00033-7

Hadamard bent sequences-survey and new resultsPreprint

Shi, M., Xiao, X., & Solé, P. (2022) · Research Square · doi:10.21203/rs.3.rs-2182737/v1

Discrimination of geometrically uniform states and the no-signaling principle

Lü, X. (2023) · Physica Scripta · doi:10.1088/1402-4896/accee3

Improving the Reconstruction Efficiency of Dice- Stagewise Weak Orthogonal Matching Pursuit

He, S., Xiao, L., Gao, S., & Zhu, Y. (2023) · International Conference on Computer Communication and Artificial Intelligence (CCAI) · doi:10.1109/ccai57533.2023.10201278

Report on scipost_202308_00029v1

Mesty\'an, M., Pozsgay, B., & Wanless, I. M. (2023) · SciPost Physics · doi:10.21468/scipost.report.8139

Report on scipost_202308_00029v1

Mesty\'an, M., Pozsgay, B., & Wanless, I. M. (2023) · SciPost Physics · doi:10.21468/scipost.report.8082

New families of quaternionic Hadamard matrices

Acevedo, S. B., Dietrich, H., & Lionis, C. (2024) · Designs Codes and Cryptography · doi:10.1007/s10623-024-01401-1

A Parametric Class of Mutually Unbiased Bases Using Resolvable Block Designs

Kumar, A., Kumar, R., & Maitra, S. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-80308-6_16

Quantum Computing with the Subgroup YU(n) of the Unitary Group U(n)

Vos, A. D. (2025) · Open Systems & Information Dynamics · doi:10.1142/s1230161225500027

Centraliser algebras of monomial representations and applications in combinatorics

Acevedo, S. B., Catháin, P. Ó., Dietrich, H., & Egan, R. (2025) · Algebraic Combinatorics · doi:10.5802/alco.426 · Cited by 1

Vector space constructions of Kochen-Specker sets using Hadamard matrices

Lisoněk, P., & Salt, L. (2025) · Physical Review A · doi:10.1103/bdrr-q25l

On Obtaining New MUBs by Finding Points on Complete Intersection Varieties over $$\mathbb {R}$$

Banerjee, A., Das, K. K., Kumar, A., Kumar, R., & Maitra, S. (2025) · International Journal of Theoretical Physics · doi:10.1007/s10773-025-06094-3

Quasiorthogonality of commutative algebras, complex Hadamard matrices, and mutually unbiased measurements

Kim, S., Kribs, D. W., Lozano, E., Pereira, R., & Plosker, S. (2025) · Linear Algebra and its Applications · doi:10.1016/j.laa.2025.09.010

Mutually equibiased bases

Akhtarshenas, S. J., Karimi, S., & Salehi, M. (2025) · Physical Review A · doi:10.1103/lg7s-yrzt

Equivalence of mutually unbiased bases via orbits: general theory and a d = 4 case study

Te’eni, A., & Cohen, E. (2026) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ae4301

Certifying bipartite entangled states with few local measurements: from separable stabilizers to applications

Ahiable, J., & Winter, A. (2026) · Quantum Science and Technology · doi:10.1088/2058-9565/ae4e54

Eigenvalues of Two-Parameter Complex Hadamard Matrices of Order Six

Huang, Y., & Chen, L. (2026) · Quantum Information and Computation · doi:10.2478/qic-2026-0010

Equivalence of complex Hadamard matrices

Östergård, P. R. J., & Valtonen, T. (2026) · Journal of Algebraic Combinatorics · doi:10.1007/s10801-026-01568-x

Constructions of Complex Hadamard Matrices via Tiling Abelian Groups

Matolcsi, M., Réffy, J., & Szöllősi, F. (2007) · Open Systems & Information Dynamics · doi:10.1007/s11080-007-9050-6 · Cited by 26

Parametrizing complex Hadamard matrices

Szöllősi, F. (2007) · European Journal of Combinatorics · doi:10.1016/j.ejc.2007.06.009 · Cited by 37

Towards a Classification of 6 × 6 Complex Hadamard Matrices

Matolcsi, M., & Szöllősi, F. (2008) · Open Systems & Information Dynamics · doi:10.1142/s1230161208000092 · Cited by 29

Constructions of complex Hadamard matrices via tiling abelian groups, Open SystPreprint

Matolcsi, M., Réffy, J., & Szöllősi, F. (2012)

Construction, classification and parametrization of complex Hadamard matrices

Szöllősi, F. (2012) · Open Research (Auckland University of Technology) · doi:10.60644/3263