I'm having a little trouble understanding what positive-operator valued measure (POVM) are- in particular why/how they are non-negative. For instance, if they just represent measurements, what about something like measuring spin- it can take on a negative value, but I imagine it is also a POVM. What am I missing?
Answer
Let me expand a bit on the intuition part and write down an example. This is all essentially already covered by yuggib's answer.
Your confusion about positive operator valued measures, as also pointed out, is that they are not to be confused with measurement outcomes. The problem with measurement outcomes is that they are rather arbitrary. Often, they rely on a scale (e.g. position relies on a frame of reference), hence you can fix that scale differently and change the measurement outcomes. They are, in a way, something to be determined within an experiment.
Thus, as a theoretician, you are not really interested in the measurement outcomes most of the time, but you are just interested in knowing when you obtain different outcomes and most importantly in knowing their probabilities. And this is, where POVMs come in.
Let's suppose we are on a Hilbert space. A positive operator-valued measure is a map that takes some Borel set (mostly the reals as outcomes of the experiment) and maps it to positive operators. They must be positive (semi)definite, because given a state, there is a measure (i.e. a map from the Borel set to $[0,1]$) associated to the operator via the scalar product.
For instance, let's take your spin experiment: We want to measure the spin of an electron. The outcomes are $-1/2,1/2$. We can define a positive operator valued measure in the following way:
Let $\mathcal{H}$ be our Hilbert space with our state being $\rho\in\mathcal{B}(\mathcal{H})$ (the corresponding density matrix in the bounded operators - here, we could just take $\mathbb{C}^2$ for the spin, but maybe we want to have the whole density matrix with a lot of other information in it). Let $\mathcal{B}$ be the Borel sigma algebra of the set $\{-1/2,1/2\}$, i.e. the set of subsets of this set. Then the positive operator valued measure is a map
$$ \mathcal{P}:\mathcal{B}\to \mathcal{B}(\mathcal{H}) $$
and it is defined via:
$$ \mathcal{P}(\{-1/2\})=P,\quad \mathcal{P}(\{1/2\})=1-P\quad \mathcal{P}(\emptyset)=0 \quad \mathcal{P}(\{-1/2,1/2\})=1$$
where $P$ is some positive operator associated to the spin measurement (maybe the Pauli $Z$ when we consider $\mathbb{C}^2$) The measure associated to this for the given state $\rho$ is
$$ \mu_{\rho}(U):=\operatorname{tr}(\mathcal{P}(U)\rho) \qquad \forall U\in \mathcal{B}$$
This is supposed to be a probability measure. If you for example take the subset $\{-1/2\}$ then $\mu_{\rho}(\{-1/2\})$ gives you the probability that if you measure $\rho$, you'll get the outcome $-1/2$. This explains our definitions above: The empty set should be mapped to zero, the whole space should be mapped to 1 and everything should be nonnegative and between zero and one. Therefore, the operators must be positive and sum up to one!
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