Complex Hadamard matrices
Encyclopedia
A complex Hadamard matrix is any complex
matrix
satisfying two conditions:
where
denotes the Hermitian transpose of H and 
is the identity matrix. The concept is a generalization of the Hadamard matrix
.
Complex Hadamard matrices arise in the study of operator algebra
s and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices.
Complex Hadamard matrices exist for any natural N (compare the real case, in which existence is not known for every N). For instance the Fourier matrices
belong to this class.
, if there exist diagonal unitary matrices 
and permutation matrices
such that
Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.
For
and 
all complex Hadamard matrices are equivalent to the Fourier matrix 
. For 
there exists
a continuous, one-parameter family of inequivalent complex Hadamard matrices,
For
the following families of complex Hadamard matrices
are known:
It is not known, however, if this list is complete, but it is conjectured that
is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
satisfying two conditions:- unimodularity (the modulus of each entry is unity):
- orthogonalityOrthogonal matrixIn linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
:
,
where
denotes the Hermitian transpose of H and is the identity matrix. The concept is a generalization of the Hadamard matrixHadamard matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...
.
Complex Hadamard matrices arise in the study of operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices.
Complex Hadamard matrices exist for any natural N (compare the real case, in which existence is not known for every N). For instance the Fourier matrices
belong to this class.
Equivalency
Two complex Hadamard matrices are called equivalent, written, if there exist diagonal unitary matrices and permutation matricesPermutation matrix
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
such that
Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.
For
and all complex Hadamard matrices are equivalent to the Fourier matrix . For there existsa continuous, one-parameter family of inequivalent complex Hadamard matrices,
For
the following families of complex Hadamard matricesare known:
- a single two-parameter family which includes
, - a single one-parameter family
, - a one-parameter orbit
, including the circulant Hadamard matrix 
, - a two-parameter orbit including the previous two examples
, - a one-parameter orbit
of symmetric matrices, - a two-parameter orbit including the previous example
, - a three-parameter orbit including all the previous examples
, - a further construction with four degrees of freedom,
, yielding other examples than 
, - a single point - one of the Butson-type Hadamard matrices,
.
It is not known, however, if this list is complete, but it is conjectured that
is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.External links
- For an explicit list of known
complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices