Related papers: Real and complex unbiased Hadamard matrices
In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases…
We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional…
We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…
Constructions of Hadamard matrices from smaller blocks is a well-known technique in the theory of real Hadamard matrices: tensoring Hadamard matrices and the classical arrays of Williamson, Ito are all procedures involving smaller order…
Certifying entanglement is an important step in the development of many quantum technologies, especially for higher-dimensional systems, where entanglement promises increased capabilities for quantum communication and computation. A key…
Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of N+1 mutually unbiased bases in Hilbert spaces of prime…
Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…
We construct new, previously unknown parametric families of complex conference matrices and of complex Hadamard matrices of square orders and related them to complex equiangular tight frames.
We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1-norm on O(N). Our study includes a…
This paper proves the existence of nonmeasurable dense sets with additional properties using combinatorial techniques.
A class of $(2n)^2\times(2n)^2$ multiparameter braid matrices are presented for all $n$ $(n\geq 1)$. Apart from the spectral parameter $\theta$, they depend on $2n^2$ free parameters $m_{ij}^{(\pm)}$, $i,j=1,...,n$. For real parameters the…
All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of…
A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\ast} = nI_{n}$. We write $\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map…
Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$,…
Assume that $H$ is a circulant Hadamard matrix of order $n\geq 4$. We consider an appropriate stochastic matrix $S$ of order $n$ depending on $H$. This allows us to prove that $n = 4$. Thus, there are only $10$ circulant Hadamard matrices.
We describe combinatorial properties of the defining row of a circulant Hadamard matrix by exploiting its orthogonality to subsequent rows, and show how to exclude several particular forms of these matrices.
In this paper we present new Hadamard matrices and related combinatorial structures. In particular, it is constructed 5202 inequivalent Hadamard matrices of order 36 as well as 180538 Hadamard symmetric designs with 35 points in addition to…
There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way.…