scholarly journals Informing the structure of complex Hadamard matrix spaces using a flow

2019 ◽  
Vol 12 (8) ◽  
pp. 2349-2364
Author(s):  
Francis C. Motta ◽  
◽  
Patrick D. Shipman ◽  
Keyword(s):  
Hadamard Matrix ◽  
Complex Hadamard Matrix ◽  
Matrix Spaces

scholarly journals Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

2018 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Takuya Ikuta ◽  
Akihiro Munemasa
Keyword(s):  
Association Scheme ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Association Schemes ◽  
Roots Of Unity ◽  
Galois Rings ◽  
Type Complex ◽  
Complex Hadamard Matrix

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

scholarly journals Complex Hadamard Matrix-Aided Generalized Space Shift Keying Modulation

IEEE Access ◽  
2017 ◽  
Vol 5 ◽  
pp. 21139-21147 ◽  
Author(s):  
Han Hai ◽  
Xue-Qin Jiang ◽  
Wei Duan ◽  
Jun Li ◽  
Moon Ho Lee ◽  
...  
Keyword(s):  
Hadamard Matrix ◽  
Shift Keying ◽  
Complex Hadamard Matrix

On C-Matrices of Arbitrary Powers

1971 ◽  
Vol 23 (3) ◽  
pp. 531-535 ◽  
Author(s):  
Richard J. Turyn
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Main Diagonal ◽  
Quadratic Character ◽  
Symmetric Complex ◽  
Complex Hadamard Matrix ◽  
Square Matrix

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m > 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

scholarly journals Construction of the Outer Automorphism of $${\mathcal {S}}_{6}$$ S 6 via a Complex Hadamard Matrix

2018 ◽  
Vol 12 (4) ◽  
pp. 453-458
Author(s):  
Neil I. Gillespie ◽  
Padraig Ó Catháin ◽  
Cheryl E. Praeger
Keyword(s):  
Hadamard Matrix ◽  
Outer Automorphism ◽  
Complex Hadamard Matrix

Yates Quaternary Complex Hadamard Matrix

2017 ◽  
Vol 50 (5) ◽  
pp. 298-306
Author(s):  
Naseer Ahmad Malik ◽  
◽  
Hrishikesh Mahato
Keyword(s):  
Hadamard Matrix ◽  
Quaternary Complex ◽  
Complex Hadamard Matrix

scholarly journals Constructions of Complex Hadamard Matrices via Tiling Abelian Groups

2007 ◽  
Vol 14 (03) ◽  
pp. 247-263 ◽  
Author(s):  
Máté Matolcsi ◽  
Júlia Réffy ◽  
Ferenc Szöllősi
Keyword(s):  
Information Theory ◽  
Necessary Conditions ◽  
Hadamard Matrix ◽  
Abelian Groups ◽  
Quantum Tomography ◽  
Hadamard Matrices ◽  
Quantum Information Theory ◽  
Current Interest ◽  
Parametric Families ◽  
Complex Hadamard Matrix

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent, very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabó [8], we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue [10] of complex Hadamard matrices of small order.

scholarly journals Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three

2020 ◽  
Vol 476 (2235) ◽  
pp. 20190754
Author(s):  
Mengyao Hu ◽  
Lin Chen ◽  
Yize Sun
Keyword(s):  
Open Problem ◽  
Unitary Transformation ◽  
Quantum Physics ◽  
Hadamard Matrix ◽  
Mutually Unbiased Bases ◽  
Unitary Operation ◽  
Local Unitary Transformation ◽  
Zero Entry ◽  
Schmidt Rank ◽  
Complex Hadamard Matrix

Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2  ⊗  V and I 2  ⊗  W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.

scholarly journals An infinite family of Williamson matrices

1977 ◽  
Vol 24 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Edward Spence
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Infinite Family ◽  
Williamson Matrices

AbstractIn this paper the following result is proved. Suppose there exists a C-matrix of order n + 1. Then if n≡1 (mod 4) there exists a Hadamard matrix of order 2nr(n + 1), while if n≡3 (mod 4) there exists a Hadamard matrix of order nr(n + 1) for all r ≧0. If n≡1 (mod 4) is a prime power, the method is adapted to prove the existence of a Hadamard matrix of the Williamson type, of order 2nr(n + 1), for all r ≧0.

scholarly journals A skew-Hadamard matrix of order 92

1971 ◽  
Vol 5 (2) ◽  
pp. 203-204 ◽  
Author(s):  
Jennifer Wallis
Keyword(s):  
Hadamard Matrix

There is a skew-Hadamard matrix of order 92.Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below. The orders < 200 which are now undecided are 100, 116, 148, 156, 172, 188, 196; see [2], [3]. The existence of any Hadamard matrix of order 92 was unknown until 1962 [1].

scholarly journals Linear/additive preservers of ranks 2 and 4 on alternate matrix spaces over fields

2004 ◽  
Vol 392 ◽  
pp. 25-38 ◽  
Author(s):  
Xian Zhang
Keyword(s):  
Additive Preservers ◽  
Matrix Spaces
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