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The Hadamard is a two by two matrix:

$\begin{equation} \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \end{equation} $

The Kronecker product of the Hadamard with itself is $H \otimes H$ or $H^{\otimes 2}$

Let X be a $2 \times 4$ complex valued matrix, and let $\overline{X}$ be its conjugate transpose.

Is it possible to write $H^{\otimes 2}$ as $ \overline{X} H X$ ?

More generally, is there a way to find a $ 4 \times 8$ matrix $Y$ such that $ H^{\otimes 3} = \overline{Y} \space\overline{X} H X \space Y$

and so on for higher order Kronecker products of the Hadamard?

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$H\otimes H$ has rank $4$, but $\operatorname{rank}(PHQ)\le\operatorname{rank}(H)=2$.

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