A complex Hadamard matrix is any complex
N
×
N
matrix
H
satisfying two conditions:
unimodularity (the modulus of each entry is unity):
|
H
j
k
|
=
1
f
o
r
j
,
k
=
1
,
2
,
…
,
N
orthogonality:
H
H
†
=
N
I
,
where
†
denotes the Hermitian transpose of H and
I
is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix
H
can be made into a unitary matrix by multiplying it by
1
N
; conversely, any unitary matrix whose entries all have modulus
1
N
becomes a complex Hadamard upon multiplication by
N
.
Complex Hadamard matrices arise in the study of operator algebras 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
[
F
N
]
j
k
:=
exp
[
(
2
π
i
(
j
−
1
)
(
k
−
1
)
/
N
]
f
o
r
j
,
k
=
1
,
2
,
…
,
N
belong to this class.
Two complex Hadamard matrices are called equivalent, written
H
1
≃
H
2
, if there exist diagonal unitary matrices
D
1
,
D
2
and permutation matrices
P
1
,
P
2
such that
H
1
=
D
1
P
1
H
2
P
2
D
2
.
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
N
=
2
,
3
and
5
all complex Hadamard matrices are equivalent to the Fourier matrix
F
N
. For
N
=
4
there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,
F
4
(
1
)
(
a
)
:=
[
1
1
1
1
1
i
e
i
a
−
1
−
i
e
i
a
1
−
1
1
−
1
1
−
i
e
i
a
−
1
i
e
i
a
]
w
i
t
h
a
∈
[
0
,
π
)
.
For
N
=
6
the following families of complex Hadamard matrices are known:
a single two-parameter family which includes
F
6
,
a single one-parameter family
D
6
(
t
)
,
a one-parameter orbit
B
6
(
θ
)
, including the circulant Hadamard matrix
C
6
,
a two-parameter orbit including the previous two examples
X
6
(
α
)
,
a one-parameter orbit
M
6
(
x
)
of symmetric matrices,
a two-parameter orbit including the previous example
K
6
(
x
,
y
)
,
a three-parameter orbit including all the previous examples
K
6
(
x
,
y
,
z
)
,
a further construction with four degrees of freedom,
G
6
, yielding other examples than
K
6
(
x
,
y
,
z
)
,
a single point - one of the Butson-type Hadamard matrices,
S
6
∈
H
(
3
,
6
)
.
It is not known, however, if this list is complete, but it is conjectured that
K
6
(
x
,
y
,
z
)
,
G
6
,
S
6
is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.