Uncertainty principle & simultaneous measurement

[stevendaryl]

Basically yes.

It's an operator O = ∑ yi |bi><bi|, yi real. It means the resolution of the identity |bi><bi| describes some observation such that the probability of outcome i depends only on the |bi><bi|. Given any resolution of the identity we can in principle find an observation that describes it. The yi are the arbitrary real numbers associated with each outcome and the operator from that association is O = ∑ yi |bi><bi|. Via the spectral theorem the two are in 1-1 correspondence meaning each implies the other.

In fact, via Gleason the above implies the Born rule.

For foundational issues its easier to consider finite spaces and extend it via the RHS formalism. That way subtle issues of what exactly is a Hermitian operator is avoided.

Thanks
Bill

Don't superselection rules imply that not all Hermitian operators are observables? For example, if $O$ is an operator whose eigenstates are superpositions of states with different total charges, then those eigenstates are not physically realizable, and $O$ is not observable.