Complex Hadamard Matrices from Sylvester Inverse Orthogonal Matrices

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c ≠ 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.KeywordsHadamard matrixTradeRankMathematics Subject Classification (2010):05B2015B34

Hadamard equiangular tight frames

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.)

This paper discusses the factorization of a bent function type complex Hadamard matrix of order N=p/sup n/ with a prime p. The bent function type complex Hadamard matrix is the generalization of the bent function type binary Hadamard matrix. It is shown that the bent function type complex Hadamard matrix is closely related to the complex Hadamard matrix generalized the Sylvester type binary Hadamard matrix, and can be expressed by using 2n-2 matrices of order N with pN non-zero elements at most. The application of the bent function type complex Hadamard matrix to synchronous spread spectrum communication is also discussed.

The A DFT (Discrete Fourier Transform) has seen studied and applied to signal processing and communication theory. The relation between the Fourier matrix and the Hadamard transform was developed in [Ahmed & Rao, 1975; Whelchel & Guinn, 1968] for signal representation and classification and the Fast Fourier-Hadamand Transform(FFHT) was proposed. This idea was further investigated in [Lee & Lee, 1998] as an extension of the conventional Hadamard matrix. Lee et al [Lee & Lee, 1998] has proposed the Reverse Jacket Transform(RJT) based on the decomposition of the Hadamard matrix into the Hadamard matrix(unitary matrix) itself and a sparse matrix. Interestingly, the Reverse Jacket(RJ) matrix has a strong geometric structure that reveals a circulant expansion and contraction properties from a basic 2x2 sparse matrix. The discrete Fourier transform (DFT) is an orthogonal matrix with highly practical value for representing signals and images [Ahmed & Rao, 1975; Lee, 1992; Lee, 2000]. Recently, the Jacket matrices which generalize the weighted Hadamard matrix were introduced in [Lee, 2000], [Lee & Kim, 1984, Lee, 1989, Lee & Yi, 2001; Fan & Yang, 1998]. The Jacket matrix1 is an abbreviated name of a reverse Jacket geometric structure. It includes the conventional Hadamard matrix [Lee, 1992; Lee, 2000; Lee et al., 2001; Hou et. al., 2003], but has the weights,ω , that are j or 2 , where k is an integer, and 1 j = − , located in the central part of Hadamard matrix. The weighted elements' positions of the forward matrix can be replaced by the non-weighted elements of its inverse matrix and the signs of them do not change between the forward and inverse matrices, and they are only as element inverse and transpose. This reveals an interesting complementary matrix relation. Definition 1: If a matrix m J ⎡ ⎤ ⎣ ⎦ of size m m × has nonzero elements

Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex projective space to Cd(d+1)/2. Applying this result with d = 3, we give constructions of complex Hadamard matrices, and more generally, type-II matrices, in the Bose–Mesner algebra of a certain 3-class symmetric association scheme. In particular, we recover the complex Hadamard matrices of order 15 found by Ada Chan. We compute the Haagerup sets to show inequivalence of resulting type-II matrices, and determine the Nomura algebras to show that the resulting matrices are not decomposable into generalized tensor products.

In this paper we classify complex Hadamard matrices contained in the Bose-Mesner algebra of nonsymmetric 3-class association schemes. As a consequence of our classification, we have two infinite families and some small examples of complex Hadamard matrices contained in the Bose-Mesner algebra of a self-dual fission of a complete multipartite graph.

McNulty and Weigert (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908–25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any 6 × 6 complex Hadamard matrix whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of 2 × 2 complex Hadamard submatrices of any H 2-reducible matrix is not 10 , … , 16 , 18 . We also put forward some possible directions for further development.

Pairs of complementary binary or quaternary sequences of length v such as Golay pairs, complex Golay pairs and periodic Golay pairs may be used to construct Hadamard matrices and complex Hadamard matrices of order 2v. We generalize these and define unitary Golay pairs and phased unitary Golay pairs of length v with entries in the kth roots of unity for any $$k \ge 2$$ . This leads to a construction of Butson Hadamard matrices of order 2v over the kth roots of unity for even k. Ito conjectured that a central relative (4v, 2, 4v, 2v)-difference set exists in a dicyclic group of order 8v for all $$v \ge 1$$ , and this is known to imply the Hadamard conjecture. With our construction we prove that Ito’s conjecture also implies the stronger complex Hadamard conjecture. As a consequence, with this method we construct a complex Hadamard matrix of order 2v for any v for which Ito’s conjecture is verified, in particular, any $$v \le 46$$ .

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.

On the asymptotic existence of partial complex Hadamard matrices and related combinatorial objects

Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.

Polynomial construction of complex Hadamard matrices with cyclic core

In this paper, a novel spectral technique based on the transform constructed from a Complex Hadamard matrix is introduced. The technique finds potential applications in various synthesis and optimization processes of logic circuits. Classification of switching function is shown here as one of the possible applications. The technique uses the Complex Hadamard matrix as the spectral transform, manipulates the complex spectral coefficients of the functions and classifies them into the NPN-equivalent and linearly separable functions. The method has proven to be a more efficient classification scheme than the existing classification method using the standard spectral technique based on the Walsh Transform. For efficient manipulation of switching functions, some new decision diagrams to represent Complex Hadamard matrices and their spectra of integervalued functions are proposed. With the intrinsic properties of the new transform in handling complex numbers, the presented decision diagrams lead to a more efficient and compact representation of both switching and complex functions.

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.