defect

defect as the dimension of an eigenspace of a unitary operator

The undephased defect $D(H)$ is the dimension of the complex eigenspace

$$\mathbb{V}_1^{{\small \frac{1}{\sqrt{N}}}H},$$

associated with eigenvalue $1$, of the operator

$$\mathcal{J}_{{\small \frac{1}{\sqrt{N}}}H} = \mathcal{C}^{-1}_{{\small \frac{1}{\sqrt{N}}}H}\mathcal{D}_{{\small \frac{1}{\sqrt{N}}}H}$$

on the space of all $N\times N$ complex matrices, where

$H$ is an $N\times N$ complex Hadamard matrix, hence ${\small \frac{1}{\sqrt{N}}}H$ is unitary
for an $N\times N$ unitary $U$ and any $N\times N$ complex matrix $F$ $$\mathcal{C}_U(F) = \left(F\circ U\right)U^{\ast}, \qquad \mathcal{D}_U(F) = U\left(\bar{F}\circ U\right)^{\ast},$$ where $\circ$ is the entrywise (Hadamard) product and $\ast$ is the Hermitian transposition
$\mathcal{C}_U$ and $\mathcal{D}_U$ are isomorphisms on $\mathbb{C}^{N\times N}$, and $\mathcal{J}_U = \mathcal{C}^{-1}_U\mathcal{ D}_U$ is unitary with respect to the weighted inner product $$\langle F, G\rangle_U=\sum_{i,j=1,...,N} |U_{i,j}|^2F_{i,j}\bar{G}_{i,j} \ .$$

The real $D(H)$ dimensional space of imaginary eigenvectors

$$\left(\mathbb{V}_1^{{\small \frac{1}{\sqrt{N}}}H}\right)^{Im} = \Big\{X\in\mathbb{V}_1^{{\small \frac{1}{\sqrt{N}}}H} : \bar{X} = -X\Big\}$$

parametrizes the feasible space $\mathbb{D}_H$:

$$\mathbb{D}_H = \left(\mathbb{V}_1^{{\small \frac{1}{\sqrt{N}}}H}\right)^{Im}\circ H.$$

$\mathbb{V}_1^{{\small \frac{1}{\sqrt{N}}}H}$ (even $\mathbb{V}_1^{U}$ for any unitary $U$) always contains the $2N-1$ dimensional complex space $$\Big\{[a_i+b_j]_{i,j=1,...,N}:a_x,b_x\in\mathbb{C}\Big\},$$ therefore $D(H)\geqslant 2N-1$ and $d(H)\geqslant 0$. For more details see sec. 2 and 5 in [143] and sec. 3 in [140].