Completeness of quantum mechanics and the EPR paradox

Question

I am reading through the EPR paper and follow most of it. The authors argue that either QM must be incomplete (let's call this statement A), or incompatible observables can not have simultaneous physical reality (statement B). The authors define what they mean by complete and physically real. They go on to show that rejecting A forces us to also reject B. Since one of these statements must be true, it follows that must accept A, i.e. QM must be incomplete.

I am having trouble seeing exactly where the authors invoke the !A to demonstrate !B. They construct a joint quantum state and show that by measuring the position or momentum of one particle the position or momentum (respectively) of the other can be known perfectly. It is not clear to me that the assumed completeness of QM is exploited anywhere in this argument.

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update: In Arthur Fine's analysis of the EPR argument he writes,

Indeed what EPR proceed to do is odd. Instead of assuming completeness and on that basis deriving that incompatible quantities can have real values simultaneously, they simply set out to derive the latter assertion without any completeness assumption at all. This “derivation” turns out to be the heart of the paper and its most controversial part.

This articulates my confusion about/objection to the EPR argument well.

Answer 1

The idea of proposition B is that when observables $P$ and $Q$ do not commute, EPR consider that the observable effectively measured has an "element of reality" as counterpart at the time of measurement, while the other observable lacks this "element of reality" at the time of measurement (else $P$ and $Q$ would commute).

They then consider a compound system $1+2$ where the fact that subsystems $1$ and $2$ did interact in the past implies that, at a later time when they are not interacting anymore, $P1$ cannot be independent from $P2$ and similarly $Q1$ cannot be independent from $Q2$ (a situation we now call "entanglement").

Because of that entanglement, by choosing to measure $P1$ or $Q1$ we are in the position to predict respectively $P2$ or $Q2$ at will.

Now according to EPR criterion of reality, which says that an "element of reality" has to be associated to any 100% certain prediction of the theory (this is explicited in part 1 of the paper), it then seems that both $P2$ and $Q2$ must be simultaneous real, and this is how EPR come to the conclusion that "two physical quantities, with noncommuting operators, can have simultaneous reality" (which is proposition !B).

Taking QM as complete (proposition !A) is at the core of the above contradiction because it is this consideration that forbids $P2$ and $Q2$ to be both associated to elements of reality simultaneously (due to $P$ and $Q$ not commuting, as we saw first when discussing proposition B).