So I have recently learned how to time evolve systems subject to non-Hermitian Hamiltonians using left and right eigenvectors. Basically to construct my non-Hermitian Hamiltonian, I use all real eigenvalues as well as eigenvectors that are linearly independent but not necessarily orthogonal. However I am having trouble understanding how to check to see which symmetries my Hamiltonians obey. I believe that the commutation relations $[C,H]=0$, $[P,H]=0$, and $[T,H]=0$ are used to check for charge conjugation, parity, and time reversal symmetry, respectively, with $H$ being the Hamiltonian. I would think that these commutation relations can be calculated (using parity symmetry as an example) as $[P,H]|\Psi\rangle=PH|\Psi\rangle-HP|\Psi\rangle$ where $|\Psi\rangle$ is an arbitrary wave vector. As I understand it, for a 2 qubit system, the parity operator would be represented by the matrix:
$\begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\end{bmatrix}$.
However this doesn't seem to work for the equation testing for parity symmetry expressed above. If I just input the matrix describing the parity operator in the expression above, then it seems to say that various Hamiltonians do not obey parity symmetry when in fact they should. Am I using the wrong parity operator or is there some operation that I should be performing on the Hamiltonian.
I believe that I have the time reversal operator down. The time reversal operator is given by $T=UK$ where $K$ performs complex conjugation on the terms that come before it and $U$ is a unitary that reverses the time evolution. So for the check for time reversal symmetry we have $[T,H]|\Psi\rangle=U_2KHU_1|\Psi\rangle-HU_2KU_1|\Psi\rangle=U_2(HU_1|\Psi\rangle)^*-HU_2(U_1|\Psi\rangle)^*$. Here, the $*$ performs complex conjugation for everything inside the parentheses, $U_1$ evolves the system forward in time, and $U_2$ evolves the system backward in time. This actually seems to work for checking for time reversal symmetry for both the cases where the Hamiltonian should and should not break time reversal symmetry.
I have not really started working on checking for charge conjugation symmetry. I know that $P^2=C^2=1$ and I also observe that an infinite number of solutions to the square root of the identity matrix takes the form:
$\begin{bmatrix} a & b \\ c & -a\end{bmatrix}$
where $a^2+bc=1$. $b=c=1$ is what is used for the parity operator and a natural solution for the charge conjugation operator would be $a=1$. And so if this were the case, the charge conjugation operator for a single qubit would become:
$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
But I don't even know for sure if I have the parity operator written correctly and so I have no idea if this is actually the charge conjugation operator.
