Complex Hadamard matrix

A complex Hadamard matrix is any complex ${\displaystyle N\times N}${\displaystyle N\times N} matrix ${\displaystyle H}${\displaystyle H} satisfying two conditions:

• unimodularity (the modulus of each entry is unity): ${\displaystyle |H_{jk}|=1{\text{ for }}j,k=1,2,\dots ,N}${\displaystyle |H_{jk}|=1{\text{ for }}j,k=1,2,\dots ,N}
• orthogonality: ${\displaystyle HH^{\dagger }=NI}${\displaystyle HH^{\dagger }=NI},

where ${\displaystyle \dagger }${\displaystyle \dagger } denotes the Hermitian transpose of ${\displaystyle H}${\displaystyle H} and ${\displaystyle I}${\displaystyle I} is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix ${\displaystyle H}${\displaystyle H} can be made into a unitary matrix by multiplying it by ${\displaystyle {\frac {1}{\sqrt {N}}}}${\displaystyle {\frac {1}{\sqrt {N}}}}; conversely, any unitary matrix whose entries all have modulus ${\displaystyle {\frac {1}{\sqrt {N}}}}${\displaystyle {\frac {1}{\sqrt {N}}}} becomes a complex Hadamard upon multiplication by ${\displaystyle {\sqrt {N}}.}${\displaystyle {\sqrt {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 number ${\displaystyle N}${\displaystyle N} (compare with the real case, in which Hadamard matrices do not exist for every ${\displaystyle N}${\displaystyle N} and existence is not known for every permissible ${\displaystyle N}${\displaystyle N}). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),

${\displaystyle [F_{N}]_{jk}:=\exp[2\pi i(j-1)(k-1)/N]{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N}${\displaystyle [F_{N}]_{jk}:=\exp[2\pi i(j-1)(k-1)/N]{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N}

belong to this class.

Two complex Hadamard matrices are called equivalent, written ${\displaystyle H_{1}\simeq H_{2}}${\displaystyle H_{1}\simeq H_{2}}, if there exist diagonal unitary matrices ${\displaystyle D_{1},D_{2}}${\displaystyle D_{1},D_{2}} and permutation matrices ${\displaystyle P_{1},P_{2}}${\displaystyle P_{1},P_{2}} such that

${\displaystyle H_{1}=D_{1}P_{1}H_{2}P_{2}D_{2}.}${\displaystyle 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 ${\displaystyle N=2,3}${\displaystyle N=2,3} and ${\displaystyle 5}${\displaystyle 5} all complex Hadamard matrices are equivalent to the Fourier matrix ${\displaystyle F_{N}}${\displaystyle F_{N}}. For ${\displaystyle N=4}${\displaystyle N=4} there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,

${\displaystyle F_{4}^{(1)}(a):={\begin{bmatrix}1&1&1&1\1円&ie^{ia}&-1&-ie^{ia}\1円&-1&1&-1\1円&-ie^{ia}&-1&ie^{ia}\end{bmatrix}}{\quad {\rm {with\quad }}}a\in [0,\pi ).}${\displaystyle F_{4}^{(1)}(a):={\begin{bmatrix}1&1&1&1\1円&ie^{ia}&-1&-ie^{ia}\1円&-1&1&-1\1円&-ie^{ia}&-1&ie^{ia}\end{bmatrix}}{\quad {\rm {with\quad }}}a\in [0,\pi ).}

For ${\displaystyle N=6}${\displaystyle N=6} the following families of complex Hadamard matrices are known:

• a single two-parameter family which includes ${\displaystyle F_{6}}${\displaystyle F_{6}},
• a single one-parameter family ${\displaystyle D_{6}(t)}${\displaystyle D_{6}(t)},
• a one-parameter orbit ${\displaystyle B_{6}(\theta )}${\displaystyle B_{6}(\theta )}, including the circulant Hadamard matrix ${\displaystyle C_{6}}${\displaystyle C_{6}},
• a two-parameter orbit including the previous two examples ${\displaystyle X_{6}(\alpha )}${\displaystyle X_{6}(\alpha )},
• a one-parameter orbit ${\displaystyle M_{6}(x)}${\displaystyle M_{6}(x)} of symmetric matrices,
• a two-parameter orbit including the previous example ${\displaystyle K_{6}(x,y)}${\displaystyle K_{6}(x,y)},
• a three-parameter orbit including all the previous examples ${\displaystyle K_{6}(x,y,z)}${\displaystyle K_{6}(x,y,z)},
• a further construction with four degrees of freedom, ${\displaystyle G_{6}}${\displaystyle G_{6}}, yielding other examples than ${\displaystyle K_{6}(x,y,z)}${\displaystyle K_{6}(x,y,z)},
• a single point - one of the Butson-type Hadamard matrices, ${\displaystyle S_{6}\in H(3,6)}${\displaystyle S_{6}\in H(3,6)}.

It is not known, however, if this list is complete, but it is conjectured that ${\displaystyle K_{6}(x,y,z),G_{6},S_{6}}${\displaystyle 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.

• For an explicit list of known ${\displaystyle N=6}${\displaystyle N=6} complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices

• Matrices (mathematics)