Families Of Complex Hadamard Matrices Research Articles

Complex Hadamard matrices with noncommutative entries

We axiomatize and study the matrices of type $H\in M_N(A)$, having unitary entries, $H_{ij}\in U(A)$, and whose rows and columns are subject to orthogonality type conditions. Here $A$ can be any $C^*$-algebra, for instance $A=\mathbb C$, where we obtain the usual complex Hadamard matrices, or $A=C(X)$, where we obtain the continuous families of complex Hadamard matrices. Our formalism allows the construction of a quantum permutation group $G\subset S_N^+$, whose structure and computation is discussed here.

Analytic deformations of group commuting squares and complex Hadamard matrices

Let G be a finite group and denote by CG the commuting square associated to G. The defect of the group G, given by the formula d(G)=∑g∈G|G|order(g), was introduced in [9] as an upper bound for the number of linearly independent directions in which CG can be continuously deformed in the class of commuting squares. In this paper we show that this bound is actually attained, by constructing d(G) analytic families of commuting squares containing CG.In the case G=Zn, the defect d(Zn) can be interpreted as the dimension of the enveloping tangent space of the real algebraic manifold of n×n complex Hadamard matrices, at the Fourier matrix Fn (in the sense of [14,1]). The dimension of the enveloping tangent space gives a natural upper bound on the number of continuous deformations of Fn by complex Hadamard matrices, of linearly independent directions of convergence. Our result shows that this bound is reached, which is rather surprising. In particular our construction yields new analytic families of complex Hadamard matrices stemming from Fn.In the last section of the paper we use a compactness argument to prove non-equivalence (i.e. non-isomorphism as commuting squares) for dephased versions of the families of Hadamard matrices constructed throughout the paper.

New Families of Complex Hadamard Matrices

The main result of this paper is a construction for complex Hadamard matrices: for [Formula: see text] any prime power and [Formula: see text] the size of a real Hadamard matrix, this construction yields a family of complex Hadamard matrices of order [Formula: see text] with [Formula: see text] parameters, including Butson-type matrices of even type [Formula: see text] a divisor of [Formula: see text]. Only a few lowdimensional examples and the real Hadamard matrices obtained by this construction are already known. Also a small extension of Diţa’s construction (cf. Lemma 1) is given.

Nested Construction of Families of Complex Hadamard Matrices

We present a new parametrization of families of complex Hadamard matrices stemming from the Fourier matrices in every prime power dimension. We connect continuous Abelian groups with families of complex Hadamard matrices and conjecture that the constructed families are maximal. Also, we derive new relations for complex Hadamard matrices in every prime power dimension and prove that some real Hadamard matrices can be written as a product of an arbitrarily large number of real Hadamard matrices.

A new method to construct families of complex Hadamard matrices in even dimensions

We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi's method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

We study the problem of constructing mutually unbiased bases in dimension 6. This approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions. Our technique suggests the existence of previously unknown symmetries in Karlsson's non-affine family K(2)6 which we confirm analytically. Also, we obtain strong evidence that not more than three mutually unbiased bases can be constructed from pairs which contain members of some non-affine families of complex Hadamard matrices.

Families of complex Hadamard matrices

What is the dimension of a smooth family of complex Hadamard matrices including the Fourier matrix? We address this problem with a power series expansion. Studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6, or a power of a prime. In general the answer depends critically on the prime number decomposition of the dimension. Our results suggest that a general theory is possible. We discuss the case of dimension 12 in detail, and argue that the solution consists of two 13-dimensional families intersecting in a previously known 9-dimensional family. A precise conjecture for all dimensions equal to a prime times another prime squared is formulated.

The quaternary complex Hadamard matrices of orders 10, 12, and 14

A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10×10 and 12×12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14×14 matrix that there cannot be a general method for introducing parameters into these types of matrices.

Complex Hadamard matrices of order 6: a four-parameter family

In this paper, we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the new, generic family G6(4) together with Karlsson's degenerate family K6(3) and Tao's spectral matrix S(0)6 form an exhaustive list of complex Hadamard matrices of order 6. Such a complete characterization might finally lead to the solution of the problem of mutually unbiased bases in ℂ6.

Complex Hadamard matrices and equiangular tight frames

In this paper, we give a new construction of parametric families of complex Hadamard matrices of square orders, and connect them to equiangular tight frames. The results presented here generalize some of the recent ideas of Bodmann et al. [3] and extend the list of known equiangular tight frames. In particular, a (36,21)-frame coming from a nontrivial cube root signature matrix is obtained for the first time.

Three-parameter complex Hadamard matrices of order 6

A three-parameter family of complex Hadamard matrices of order 6 is presented. It significantly extends the set of closed form complex Hadamard matrices of this order, and in particular contains all previously described one- and two-parameter families as subfamilies.

A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids

We construct a 2 2 -parameter family of complex Hadamard matrices of order 6 6 by a natural block construction. We combine this family with an earlier result of Zauner to derive a 2 2 -parameter family of triplets of mutually unbiased bases (MUBs) in C 6 \mathbb C^6 . This invalidates some numerical evidence given by Brierley and Weigert and sheds new light on the problem of determining the maximal number of MUBs in C 6 \mathbb C^6 .

Two-parameter complex Hadamard matrices for N=6

A new, two-parameter, nonaffine family of complex Hadamard matrices of order 6 is reported. It interpolates between the two Fourier families and contains as one-parameter subfamilies the Diţǎ family, a symmetric family, and an almost (up to equivalence) self-adjoint family.

Constructions of Complex Hadamard Matrices via Tiling Abelian Groups

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.

Parametrizing complex Hadamard matrices

The purpose of this paper is to introduce new parametric families of complex Hadamard matrices in two different ways. First, we prove that every real Hadamard matrix of order N ≥ 4 admits an affine orbit. This settles a recent open problem of Tadej and Życzkowski [W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006) 133–177], who asked whether a real Hadamard matrix can be isolated among complex ones. In particular, we apply our construction to the only (up to equivalence) real Hadamard matrix of order 12 and show that the arising affine family is different from all previously known examples listed in [W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006) 133–177]. Second, we recall a well-known construction related to real conference matrices, and show how to introduce an affine parameter in the arising complex Hadamard matrices. This leads to new parametric families of orders 10 and 14. An interesting feature of both of our constructions is that the arising families cannot be obtained via Diţă’s general method [P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A 37 (20) (2004) 5355–5374]. Our results extend the recent catalogue of complex Hadamard matrices [W. Tadej, K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006) 133–177], and may lead to direct applications in quantum-information theory.

Mutually unbiased bases and Hadamard matrices of order six

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

A Concise Guide to Complex Hadamard Matrices

Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2,…,16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12, 14 and 16, which we could not find in the existing literature.