@Sina is making a perfectly good point with this question.

The statement is this:

*The measurement problem is not really a problem, because the result of a measurement is not only dependent on the system being "measured", but also on the observer doing the "measuring"*.

To actually realize this dichotomy between an "internal" and "external" observer, the size of the observer's Hilbert space, given by its dimension $\dim(H_O)$, must be comparable to $\dim(H_S)$ - the dimension of the Hilbert space corresponding to the system under observation. Instead, what we generally encounter is $\dim(H_O) \gg \dim(H_S)$ as is the case for, say, an apparatus with a vacuum chamber and other paraphernalia which is being used to study an atomic scale sample.

In this case the apparatus is not described by the three states $\{|\text{ready}\rangle, |\text{up}\rangle, |\text{down}\rangle\}$, but by the large (infinite?) family of states $\{|\text{ready};\alpha\rangle, |\text{up};\alpha\rangle, |\text{down};\alpha\rangle\}$ where $\alpha$ parametrizes the "helper" degrees of freedom of the apparatus which are not directly involved in generating the final output, but are nevertheless present in any interaction. Examples of these d.o.f are the states of the electrons in the wiring which transmits data between the apparatus and the system.

So if we consider the apparatus to be "internal" then the Hilbert space of the total "system+apparatus" is:

$$ H_{S+O} = H_S \otimes H_O $$

which has as its basis vectors

$$ \{ \left|\uparrow\right\rangle|\text{ready};\alpha\rangle,
\left|\uparrow\right\rangle|\text{up};\alpha\rangle,
\left|\uparrow\right\rangle|\text{down}; \alpha\rangle;
\left|\downarrow\right\rangle|\text{ready};\alpha\rangle,
\left|\downarrow\right\rangle|\text{up};\alpha\rangle,
\left|\downarrow\right\rangle|\text{down};\alpha\rangle \} $$

It is the states of the form $\left|\uparrow\right\rangle|\text{down};\alpha\rangle$ and $\left|\downarrow\right\rangle|\text{up};\alpha\rangle$ which are "counterfactual" (and you neglected to mention in the question).

At this point I wave my hands and say when an external *super-observer* looks at the states of the system described by $H_{S+O}$, the conterfactual states of the type mentioned above will interfere destructively due to the presence of the numerous $\alpha$ d.o.f; leaving only the "consistent" states of the type $\left|\uparrow\right\rangle|\text{up};\alpha\rangle$ and $ \left|\downarrow\right\rangle|\text{down};\alpha\rangle$ as the ones with non-negligible amplitudes. So in such cases, their is no contradiction between what the super-observer sees and whatever output the apparatus yields.

It is when the observer and observed systems become comparable in size that we run into all kinds of problems. As far as I know no apparatus has yet been constructed which is described by the same number of d.o.f as the system it is supposed to measure. But we are rapidly approaching that limit with nanotechnology and then this measurement dichotomy will have to dealt with head on.

I hope this answer makes sense. However, such questions always lie in treacherous territory. So if I've made some tautological error which invalidates everything I have said, please point it out !

This post imported from StackExchange Physics at 2014-04-01 16:20 (UCT), posted by SE-user user346