Random Math notes

Random Math notes

• Trace of any operator $A$ is preserved under unitary change of basis $U$ because in the new basis, $A' = U^{\dagger}AU$ and $Tr\{AB\}=Tr\{BA\}$ implies $Tr\{A'\}=Tr\{A\}$.
• Note that $e^{A+B}=e^A . e^ B$ only if A and B commute. If they don't commute, the RHS will have terms of the type $C_{n,r} A^{n-r}B^r$ while the similar terms on LHS will be of the form $A^{n-r}B^r + A^{n-r-1}BAB^{r-1} + \cdots + B^{r}A^{n-r}$. If the terms don't commute, then these similar terms on LHS and RHS are not really equivalent. If they commute, then they are equal.