Complex Hadamard matrix

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

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

where ${\displaystyle \dagger }$ denotes the Hermitian transpose of ${\displaystyle H}$ and ${\displaystyle I}$ is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix ${\displaystyle H}$ can be made into a unitary matrix by multiplying it by ${\displaystyle {\frac {1}{\sqrt {N}}}}$; conversely, any unitary matrix whose entries all have modulus ${\displaystyle {\frac {1}{\sqrt {N}}}}$ becomes a complex Hadamard upon multiplication by ${\displaystyle {\sqrt {N}}.}$

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 ${\displaystyle N}$ (compare with the real case, in which Hadamard matrices do not exist for every ${\displaystyle N}$ and existence is not known for every permissible ${\displaystyle N}$). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),

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

belong to this class.