# A Concise Guide to Complex Hadamard Matrices

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}
}
• Published 19 December 2005
• Mathematics
• Open Systems & Information Dynamics
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.
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## References

SHOWING 1-10 OF 63 REFERENCES

### 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

• Mathematics
• 1997
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