A complex Hadamard matrix is any
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
satisfying two conditions:
*unimodularity (the modulus of each entry is unity):
*
orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
:
,
where
denotes the
Hermitian transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex co ...
of
and
is the identity matrix. The concept is a generalization of the
Hadamard matrix
In mathematics, a 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. In geometric terms, this means that each pair of rows ...
. Note that any complex Hadamard matrix
can be made into a
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
by multiplying it by
; conversely, any unitary matrix whose entries all have modulus
becomes a complex Hadamard upon multiplication by
.
Complex Hadamard matrices arise in the study of
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of ...
s and the theory of
quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
. Real Hadamard matrices and
Butson-type Hadamard matrices In mathematics, a complex Hadamard matrix ''H'' of size ''N'' with all its columns (rows) mutually orthogonal, belongs to the Butson-type ''H''(''q'', ''N'') if all its elements are powers of ''q''-th root of unity,
:: (H_)^q=1 j,k=1,2,\dot ...
form particular cases of complex Hadamard matrices.
Complex Hadamard matrices exist for any natural
(compare the real case, in which existence is not known for every
). For instance the Fourier matrices (the complex conjugate of the
DFT matrices without the normalizing factor),
:
belong to this class.
Equivalency
Two complex Hadamard matrices are called equivalent, written
, if there exist diagonal
unitary matrices
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose ...
and
permutation matrices
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...
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:
* 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.
References
*U. Haagerup, Orthogonal maximal abelian *-subalgebras of the n×n matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), 1996 (Cambridge, MA: International Press) pp 296–322.
*P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys. A: Math. Gen. 37, 5355-5374 (2004).
*F. Szollosi, A two-parametric family of complex Hadamard matrices of order 6 induced by hypocycloids, preprint
arXiv:0811.3930v2[math.OA]
*W. Tadej and Karol Życzkowski, K. Życzkowski, A concise guide to complex Hadamard matrices Open Systems & Infor. Dyn. 13 133-177 (2006)
External links
*For an explicit list of known
complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 se
Catalogue of Complex Hadamard Matrices
{{Matrix classes
Matrices