Consider an optical experiment with photons or light pulses.
Is there an optical element that acts in the polarisation degree of freedom like the unitary $$ U = \frac 1 {\sqrt 2} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}\quad \text ? $$ I choosed the basis such that $$ |H\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\;,\quad |V\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} $$
If yes, then a photon that passes this device twice would expirience a flip in the polarisation: $$ U^2 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\;, $$ like $U|H\rangle = |V\rangle$ and $U|V\rangle = -|H\rangle$.
Does it exist?