defect
defect as the dimension of the intersection of tangent spaces
The undephased defect
$D(H)$ is the dimension of the (real) feasible space
$\mathbb{D}_H = T_H\left(\sqrt{N}\mathcal{U}\right)\cap T_H\mathcal{M}$, where
- $H$ is an $N\times N$ complex Hadamard matrix
- $\mathcal{U}$ is the $N^2$ dimensional (real) manifold consisting of all the $N\times N$
unitary matrices, and $\sqrt{N}\mathcal{U}$ is obtained from $\mathcal{U}$ by
multiplying all its elements by $\sqrt{N}$
- $T_H\left(\sqrt{N}\mathcal{ U}\right) = \{EH : E \ \ \text{antihermitian}\}$ is
the $N^2$ dimensional (real) space tangent to $\sqrt{N}\mathcal{U}$ at $H$, as $H\in\sqrt{N}\mathcal{U}$
by $T_X\mathcal{Y}$, for $X\in\mathcal{Y}$, $\mathcal{Y}\subset\mathbb{R}^k$,
we understand the set of the
derivatives $\gamma'(0)$ of all the differentiable curves $\gamma:(a,b)\to\mathcal{Y}$ such that
$0\in(a,b)$ and $\gamma(0)=X$
- $\mathcal{M}$ is the $N^2$ dimensional (real) manifold composed of all the $N\times N$
matrices with all their entries unimodular
- $T_H\mathcal{M} = \big\{iR\circ H : R \ \ \text{real matrix}\big\}$ is the $N^2$ dimensional (real)
space tangent to $\mathcal{M}$ at $H$, as $H\in\mathcal{M}$, where $\circ$
is the entrywise (Hadamard) product
$\mathbb{D}_H$ always contains the $2N-1$ dimensional space
$$\Big\{i[a_j+b_k]_{j, k=1,...,N}\circ H : a_x, b_x\in\mathbb{R}\Big\}$$
tangent to $\big\{D_rHD_c:D_r, D_c \ \ \text{unitary diagonal}\big\}$ at $H$,
therefore $D(H)\geqslant 2N-1$ and $d(H) \geqslant 0$.
See sec. 2 in [140].