§ Normal operators: Decomposition into Hermitian operators
Given a normal operator , we can always decompose it
where , , and .
This means that we can define 'complex measurements' using a normal operator,
because a normal operator has full complex spectrum. Since we can always
decompose such an operator into two hermitian operators
that commute, we can diagonalize simultaneously and thereby measure
So extending to "complex measurements" gives us no more power than staying
at "real measurements"
§ Decomposing a normal operator
Assume we have a normal operator . Write the operator in its eigenbasis .
This will allow us to write .
with each . Now write this as:
are simultaneously diagonalizable in the eigenbasis
and hence .