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.
Let us assume that there are two projections (the general case works similarly). So assume we have $p_1, p_2$ with $p_1+p_2\leq id$. Let $H_1$ be the space on which $p_1$ projection and $H_1^\perp$ be the orthogonal complement, similarly for $H_2, H_2^\perp$. We want to show that $H_2\subset H_1^\perp$. Let us assume not. So choose $v\in H_2\setminus H_1^\perp$ with $\|v\|_2=1$ where $\|\cdot\|_2$ denotes the norm in the Hilbert space. Now we can create an orthonormal basis for the hilbert space call it $\{v_1, v_2, ...\}$, where $v=v_1$. Then $p_2(v)=v$ and $p_1(v)$=$a_1v_1+a_2v_2+\cdots$. Without loss of generality we can assume that $a>0$, if not we can multiply $v$ by a scalar with absolute value one so that this is true. Now since the $v_i$ are orthogonal we get
$\|p_1(v)+p_2(v)\|^2_2=\|(1+a)v_1\|_2^2+\sum_{i=2}^\infty\|a_2v_2\|_2^2>1$. This contradicts that $p_1+p_2\leq id$.