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

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

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.

A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on complex parameters. The method we use is via doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into complex Hadamard matrices, and in this way we find new parametrizations of Hadamard matrices for dimensions n = 8, 10, and 12.

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.

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.

Hadamard equiangular tight frames

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.

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 $$v \le 46$$ .

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.

In this letter, ternary semi-bent functions and a complex Hadamard matrix are used to construct spreading sequences for code-division multiple-access systems. An efficient assignment of these sequences to a lattice of regular hexagonal cells, which are mutually orthogonal within each cell and the adjacent cells, is achieved and most notably the number of sequences per cell is maximum possible. The large sequence reuse distances, being √27, and low cross-correlation between non-orthogonal sequences (in non-adjacent cells) make these sequences appealing for practical applications.

On the realizability of the critical points of a realizable list

<p>As a powerful technique for holding relations in things, combinatorics has experienced rapidly development in the past century, particularly,enumeration of configurations, combinatorial design and graph theory. How-<br /> ever, the main objective for mathematics is to bring about a quantitative analysis for other sciences, which implies a natural question on combinatorics. Thus, how combinatorics can contributes to other mathematical sciences, not just in discrete mathematics, but metric mathematics and physics? After a long time speculation, I brought the CC conjecture for advancing mathematics by combinatorics, i.e., any mathematical science can be reconstructed from or made by combinatorialization in my postdoctoral report for Chinese Academy of Sciences in 2005, and then reported it at a few conferences of China. Clearly,<br /> CC conjecture is in fact a combinatorial notion and holds by a philosophical law, i.e., all things are inherently related, not isolated. The main purpose of this report is to survey the roles of CC conjecture in developing mathematical sciences with notions, such as those of its contribution to algebra, topology,Euclidean geometry and differential geometry, non-solvable differential equations or classical mathematical systems with contradictions to mathematics,quantum fields after it appeared 10 years ago. All of these show the importance of combinatorics to mathematical sciences in the past and future.</p>

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.