The general setting is the study of positive operator measures in quantum mechanics,
instead of the projector operator measures.
Going from the PVM to the POVM is just saying that our bunch of operators are not pairwise orthogonal.
SO the question is:
In a matrix algebra or the bounded operators on a Hilbert space, how to prove that if a bunch of projectors sums to the identity [or their sum is less than or equal to the identity], they must be mutually orthogonal ?
Can we prove this when we only know that
- a projector is an operator $p$ such that
$
p = p^* = p^2
$ - a projector is always positive, i.e.,
$$
\forall\,\,{ \lvert \psi \rangle} \qquad 0 \leq \langle\psi \lvert P \lvert \psi \rangle.
$$
I cannot find a proof that if $p, q$ are (positive) projectors then $pq$ is again positive.
We know that in general, if two operators commute and are positive, then their product is again positive.
But how the constraint by the identity forces them to be orthogonal in our setting ?
My book on quantum mechanics says,
$
\sum_j p_j \leq \text{Id} \iff \forall{ i \neq j} \quad p_i \leq \text{Id} – p_j
$
without proof unfortunately.
Best Answer
One can prove the claim without using that the $p$ are operators, but just that they are projections in a C$^*$-algebra. Also, no equality is needed, only that they are below the identity.
Suppose $$\sum_{k=1}^Np_k\leq Id.$$ Fix indices $i$ and $j$ with $i\ne j$. We have $$\sum_{k\ne i}p_k\leq Id-p_i.$$Then $$ 0\leq p_ip_jp_i\leq p_i\left(\sum_{k\ne i} p_k\right)p_i\leq p_i(Id-p_i)p_i=0. $$ So $p_ip_jp_i=0$. But then $$ 0=p_ip_jp_i=p_ip_j^2p_i=(p_jp_i)^*p_jp_i, $$ and then $p_jp_i=0$.