DarMM said:
Counterfactual refers to reasoning about the results of experiments unperformed. It's not directly the same as not realizable. Superobservers might not be realizable but the reasons have little to do with counterfactuals.The Interpretation of Quantum Mechanics.
But isn't the upshot with "superobservers" usually that you claim they could "know" something due to measurements that are not changing the system, and then you think about a measurement (or rather a preparation) that's not realizable at all according to QM since it involves the simultaneous preparation in eigenstates of incompatible observables?
Isn't this the key mistake of the EPR argument, where you have a state of two particles, described by a two-particle wave function of the type
$$\Psi(\vec{r},\vec{P})=\psi(\vec{r}) \tilde{\phi}(\vec{P})$$
where ##\psi## is a sharply peaked relative-position wave function (##\vec{r}=\vec{x}_1-\vec{x}_2##) and ##\tilde{\phi}## a sharply peaked total-momentum wave function (##\vec{P}=\vec{p}_1+\vec{p}_2##). Note that these two observables are compatible and of course one can prepare such a state.
If you now measure ##\vec{x}_1## precisely, then also ##\vec{x}_2## is known precisely due to this preparation though, of course, nothing in measuring the position ##\vec{x}_1## disturbs the far distant particle at position ##\vec{x}_2=\vec{x}_1-\vec{r}##. Then, of course neither ##\vec{p}_1## nor ##\vec{p}_2## can be known very precisely.
Now EPR argue that you could have measured ##\vec{p}_1## very precisely without disturbing particle 2 in any way (which is of course true), and then you know also ##\vec{p}_2=\vec{P}-\vec{p}_1## very well. The mistake by EPR now simply is to conclude that does you'd know both ##\vec{x}_1## and ##\vec{p}_1## precisely though that violates the HUP.
The mistake simply is that this is counterfactual: If you measure ##\vec{p}_1## precisely you cannot also determine ##\vec{x}_1## precisely and thus also not know ##\vec{x}_2## precisely, while you in fact know ##\vec{p}_2## precisely. So choosing which measurement (position or momentum) you perform on particle 1 also determines what's precisely known about particle 2 (either position or momentum, respectively).
As Peres puts it: "unperformed measurements have no result".
It's a simple exercise to calculate the corresponding Fourier transformations for position or momentum measurements on either particle, e.g.,
$$\tilde{Psi}(\vec{x}_1,\vec{x}_2)=\psi(\vec{x}_1-\vec{x}_2) \int_{\mathbb{R}^3} \mathrm{d}^3 P \exp[\mathrm{i} \vec{P}(\vec{x}_1+\vec{x}_2)/2]/\sqrt{(2 \pi)^3} \tilde{\phi}(\vec{P})=\psi(\vec{x}_1-\vec{x}_2) \phi[(\vec{x}_1+\vec{x}_2)/2].$$
Indeed since ##\tilde{\phi}## is sharply peaked in ##\vec{P}##, ##\phi## is a wide distribution in ##\vec{R}=(\vec{x}_1+\vec{x}_2)/2##. Looking at all particle pairs, the position distribution of particle 1 is given by
$$P_1(\vec{x}_1)=\int_{\mathbb{R}^3} \mathrm{d}^3 x_2 |\tilde{\Psi}(\vec{x}_1,\vec{x}_2)|^2$$
which of course is a wide distribution.
Yet determining ##\vec{x}_2## to be in a small region around ##\vec{x}_{20}##, the corresponding probability distribution is
$$\tilde{P}_1(\vec{x}_1|\vec{x}_2 \simeq \vec{x}_{20}) \simeq \tilde{\Psi}(\vec{x}_1,\vec{x}_{20}),$$
which is sharply peaked in ##\vec{x}_1##, because ##\psi(\vec{x}_1-\vec{x}_{20})## is sharply peaked.
The analogous arguments can be made with the momenta, using the momentum representation of ##\Psi##.
What's well determined about particle 1 through measurements on particle 2 depends in this example (a) on the original preparation, and in this case the positions ##\vec{x}_1## and ##\vec{x}_2## are entangled in a specific sense though both alone are quite undetermined as well as the momenta ##\vec{p}_1## and ##\vec{p}_2## are entangled though also their individual values are quite indetermined, and on (b) what's measured (either ##\vec{x}_2## or ##\vec{p}_2##). Though not disturbing particle 1 in any way by the measurement on particle 2, you cannot know both ##\vec{x}_2## and ##\vec{p}_2## by a feasible measurement on particle 1.
Ironically what EPR call "realistic" is in fact utmost inrealistic according to QT, because they envoked a "counterfactual argument". Of course in their time, they could well claim that QT is incomplete (answering the question in the title of this infamous article with "yes" based on the counterfactual argument), because at this time neither the Bell inequality as a consequence of local deterministic HV theories were known nor the corresponding experiments have been performed. With all these achievements in the last 85 years, of course, such an excuse is mute since the corresponding zillions of "Bell tests" have confirmed the predictions of QT, and the stronger-than-classically-possible correlations described by entanglement, can be described nevertheless with microcausal relativisitc QFT, i.e., there's indeed no hint at "spooky actions at a distance".
The EPR paper becomes the more disfactory when one takes into account that Einstein himself didn't like it, because his own real quibble with QT was precisely the "inseparability" issue, i.e., the possibility of long-ranged correlations described by entanglement.
