Completeness forces orthogonality of measurement projections
= Completeness forces orthogonality of measurement projections
Let self-adjoint <orthogonal projections> $P_i$ sum to the identity. For $v$ in the range of $P_j$,
$$
\|v\|^2=\sum_i\langle v,P_iv\rangle=\sum_i\|P_iv\|^2.
$$
The $j$th term already equals $\|v\|^2$, so $P_iv=0$ for every $i\neq j$. Hence $P_iP_j=P_jP_i=0$ and the <projective measurement> operators commute.
// Target: quantum-theory.bigb