Complex Hadamard Matrices Of Order Research Articles

Multi-Unitary Complex Hadamard Matrices

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.

Two-unitary complex Hadamard matrices of order 36

Abstract A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

Complex Hadamard graphs and Butson matrices

This article introduces complex Hadamard graphs and studies their properties. Using the complete subgraphs of these complex Hadamard graphs, complex Hadamard matrices of order n are generated, where n is a multiple of four. An algorithm to construct strongly regular graphs with n 2 vertices using complex Hadamard matrices of order n is also discussed in this article. MAGMA codes to construct the adjacency matrices of these strongly regular graphs are also developed.

Quaternary complex Hadamard matrices of order 18

AbstractA technique for classifying Butson‐type Hadamard matrices over the complex 4th roots of unity is developed. The technique resembles the approach used by Kharaghani and Tayfeh‐Rezaie in their classification of the (real) Hadamard matrices, where the search is split into parts according to matrix types. A classification of the Butson‐type Hadamard matrices over the complex 4th roots of unity is carried out; the number of such objects up to Hadamard equivalence is 1 955 625.

BCCB complex Hadamard matrices of order 9, and MUBs

A new type of complex Hadamard matrices of order 9 are constructed. The studied matrices are symmetric, block circulant with circulant blocks (BCCB) and form an until now unknown non-reducible and non-affine two-parameter orbit. Several suborbits are identified, including a one-parameter intersection with the Fourier orbit F9(4). The defect of this new type of Hadamard matrices is observed to vary, from a generic value 2 to the anomalous values 4 and 10 for some sub-orbits, and to 12 and 16 for some single matrices. The latter matrices are shown to be related to complete sets of MUBs in dimension 9.

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.

Small circulant complex Hadamard matrices of Butson type

We study the circulant complex Hadamard matrices of order n whose entries are lth roots of unity. For n=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pq with p,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,l.

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.

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.

Unbiased complex Hadamard matrices and bases

We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is attained for n?=?1, 5, 9. Furthermore, we prove that certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. As a consequence we generate a new pair of unbiased real Hadamard matrices of order 36.

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.

Towards a Classification of 6 × 6 Complex Hadamard Matrices

Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţă's matrix D6. This answers a recent question of Bengtsson et al. [3].

Orthogonal maximal abelian [formula omitted]-subalgebras of the [formula omitted] matrices

We construct new pairs of orthogonal maximal abelian *-subalgebras of M6(C), by classifying all self-adjoint complex Hadamard matrices of order 6. In particular, we exhibit a non-affine one-parameter family of non-equivalent Hadamard matrices of order 6.

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.

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.

Complex Hadamard matrices related to Bent sequences

Hadamard matrices are often used for some applications, such as error-correcting codes and spread sequences. This article gives the construction of complex Hadamard matrices of order p/sup n/, where p is prime and n is even. The complex Hadamard matrices include bi-phase Hadamard matrices whose elements take {-1, +1}, and four-phase Hadamard matrices whose elements take {/spl plusmn/1, /spl plusmn/j} with j=/spl radic/(-1).