A Concise Guide to Complex Hadamard Matrices

@article{Tadej2006ACG,
  title={A Concise Guide to Complex Hadamard Matrices},
  author={Wojciech Tadej and Karol Życzkowski},
  journal={Open Systems \& Information Dynamics},
  year={2006},
  volume={13},
  pages={133-177}
}
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. 

Parametrizing complex Hadamard matrices

Decoherence in quantum walks – a review

  • V. Kendon
  • Physics
    Mathematical Structures in Computer Science
  • 2007
TLDR
This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.

On discrete structures in finite Hilbert spaces

We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and

Finite-Function-Encoding Quantum States

We introduce finite-function-encoding (FFE) states which encode arbitrary d-valued logic functions, i.e., multivariate functions over the ring of integers modulo d, and investigate some of their

Classification of difference matrices and complex Hadamard matrices

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Pekka H. J. Lampio Name of the doctoral dissertation Classification of difference matrices and complex Hadamard matrices Publisher

Mutually Unbiased Product Bases

A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are

Tiling properties of spectra of measures

We investigate tiling properties of spectra of measures, i.e., sets $$\Lambda $$Λ in $$\mathbb {R}$$R such that $$\{e^{2\pi i \lambda x}{:}\, \lambda \in \Lambda \}$${e2πiλx:λ∈Λ} forms an orthogonal

Techniques de transmission et de réception MU-MIMO pour la génération suivante de standards de télécommunications cellulaires

Dans cette these, nous etudions les systemes multi-utilisateurs (MU), ou l'emetteur dispose M antennes et sert K utilisateurs mono-antenne, appeles MU-MISO. Base sur des informations de l'etat de

Construction, classification and parametrization of complex Hadamard matrices

The intended purpose of this work is to provide the reader with a comprehensive, state-of-the art presentation of the theory of complex Hadamard matrices, or at least report on the very recent
...

References

SHOWING 1-10 OF 63 REFERENCES

On the non-existence of generalized Hadamard matrices

Orthogonal Maximal Abelian *-Subalgebras of the N×n Matrices and Cyclic N-Roots

It is proved that for n = 5, there is up to isomorphism only one pair of orthogonal maximal abelian-subalgebras (MASA's) in the n n-matrices. The same result holds trivially for n = 2 and n = 3, but

Existence of continuous families of complex hadamard matrices of certain prime dimensions and related results

One proves the existence of continuous families of complex Hadamard matrices of certain prime dimensions, n = 7, 13, 19, 31, 79. This result implies the existence in the corresponding matrix algebras

A finiteness result for commuting squares of matrix algebras

We consider a condition for non-degenerate commuting squares of matrix algebras (finite dimensional von Neumann algebras) called the \emph{span condition}, which in the case of the $n$-dimensional

Relations Among Generalized Hadamard Matrices, Relative Difference Sets, and Maximal Length Linear Recurring Sequences

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1,

Geometrical description of quantal state determination

Under the assumption that every quantal measurement may give data about the post-measurement state of the inspected ensemble, the problem of the state determination is reconsidered. It is shown that

Error Correcting Codes Associated with Complex Hadamard Matrices

On the asymptotic existence of complex Hadamard matrices

Optimal state-determination by mutually unbiased measurements

...