Local Solution of the EPR-Paradox
The EPR Paradox will be cleared up in two rounds. The first one is dedicated exclusively to the refutation of the conviction that it is impossible to reproduce the quantum mechanical predictions for measurements on entangled systems by a strictly local theory. To this effect it is sufficient to present such a theory – the physical implications resulting from it can be ignored for the moment.
To understand what the paradox is about, only a few facts are needed:
1. Generally, the quantum mechanical description of an object determines for some attributes not a definite value but only the probability distribution of possible measurement values.
2. This applies also to the case of two spatially separated objects which interacted in the past or which originate from the decay of an object.
3. Between the outcomes of certain measurements on these two objects there will then be a connection that is called "entanglement". E.g. in the case of two identical particles A and B which come from the decay of an object at rest and depart into opposite directions, the measurement values of the two momentums are interconnected in the same way as in classical physics, which means that in any case pA = – pB . Another example: If a spin 0 system decays into two photons, then the measured polarization directions of the photons are rectangular to each other.
That's all there is to it! What is paradoxical about it? This is quickly explained, too:
Let us assume as yet no measurement has been performed. Thus only the probability distribution of the measurement values is known. But if now the momentum of particle A is measured, then, because of (3), at the same moment also the momentum of B is known, and the same applies to the case of the photon polarizations.
Now one can argue with Einstein, Podolsky and Rosen in the following way:
B is at an arbitrarily great distance from A. Therefore, the measurement on A cannot have influenced B. Thus we can state: if B has a definite momentum after the measurement on A, then it must have had this momentum also already before the measurement on A – otherwise the measurement on A would have caused a change of the state of B. However, since the quantum mechanical description does not contain this momentum, it must be considered incomplete. (In this case, the momentum would be a so-called hidden parameter.)
That sounds like a reasonable argument! Indeed the alternative would be to assume a nonlocal connection between the two measurements, that is a connection which requires either a faster-than-light transmission or which exists without any mediating process at all and must simply be accepted as such.
But now follows the paradox: Exactly this plausible EPR-assumption – that the result of the measurement on B is already determined before the measurement on A, because it corresponds to an objectively existing attribute of a single system – is a necessary and sufficient condition for the derivation of Bell's Inequality, from which then follows that a local description of the world, which conforms to the experimentally verified predictions of quantum theory, is impossible. Thus, in the end, exactly the argument by which EPR intended to proof the incompleteness of quantum theory serves to reduce their own intention to absurdity, to describe the world in an objective and local way.
Therefore we seem to be compelled to resign ourselves to the nonlocality of the world. At least this is the current state of affairs.