quantum mechanics - Heisenberg choice of the observables and different outcomes

Somebody could help me to clarify how its possible that different choices of the observable to measure can lead to different outcomes of the observed system state?

Answer

While it is trivially true that measuring different things gives us different kinds of information, there is a more interesting sense in which the state of the system is affected by what we choose to measure. In the real world, the assumption that a measurable property exists whether or not we measure it is inconsistent with the experimental facts.

Here's a relatively simple example. Suppose we have four observables, $A,B,C,D$, each of which has two possible outcomes. (For example, these could be single-photon polarization observables.) For mathematical convenience, label the two possible outcomes $+1$ and $-1$, for each of the four observables. Suppose for a moment that the act of measurement merely reveals properties that would exist anyway even if they were not measured. If this were true, then any given state of the system would have definite values $a,b,c,d$ of the observables $A,B,C,D$. Each of the four values $a,b,c,d$ could be either $+1$ or $-1$, so there would be $2^4=16$ different possible combinations of these values. Any given state would have one of these 16 possible combinations.

Now consider the two quantities $a+c$ and $c-a$. The fact that $a$ and $c$ both have magnitude $1$ implies that one of these two quantities must be zero, and then the other one must be either $+2$ or $-2$. This, in turn, implies that the quantity $$ (a+c)b+(c-a)d $$ is either $+2$ or $-2$. This is true for every one of the 16 possible combinations of values for $a,b,c,d$, so if we prepare many states, then the average value of this quantity must be somewhere between $+2$ and $-2$. In particular, the average cannot be any larger than $+2$. This gives the CHSH inequality $$ \langle{AB}\rangle +\langle{CB}\rangle +\langle{CD}\rangle -\langle{AD}\rangle\leq 2, $$ where $\langle{AB}\rangle$ denotes the average of the product of the values of $a$ and $b$ over all of the prepared states.

In the real world, the CHSH inequality can be violated, and quantum theory correctly predicts the observed violations. The quantity $\langle{AB}\rangle +\langle{CB}\rangle +\langle{CD}\rangle -\langle{AD}\rangle$ can be as large as $2\sqrt{2}$. Here are a few papers describing experiments that verify this:

The fact that the CHSH inequality is violated in the real world implies that the premise from which it was derived cannot be correct. The CHSH inequality was derived above by assuming that the act of measurement merely reveals properties that would exist anyway even if they were not measured. The inequality is violated in the real world, so this assumption must be wrong in the real world. Measurement plays a more active role.