Clarification on the Premises of the Einstein-Podolsky-Rosen Argument In their famous EPR paper, Einstein, Podolsky and Rosen argue that quantum mechanics does not provide a complete description of physical reality. To do this, they make two key assumptions:

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*in a complete theory, every element of physical reality must have a corresponding element in the theory


*if we can predict with certainty (i.e. with probability 1) the value of a physical quantity of some system without disturbing that system, then there must exist an element of physical reality corresponding to that physical quantity
The authors stress that the second condition is taken to be a sufficient condition for the reality of a physical quantity but that it need not be a necessary one. It is claimed that this is in agreement with classical and quantum mechanical ideas of reality. However, supposing that a complete theory exists, then for the condition to not be necessary for the reality of a physical quantity, it must, according to the first assumption, be possible for an element of physical reality to have a corresponding physical quantity in the complete theory but for it not to be possible to predict with certainty the value of this quantity from the theory without disturbing the system.
It appears to me then that according to the usual quantum mechanical idea of reality, the condition would in fact be necessary for the reality of a physical quantity. Indeed, later in the paper, the authors do appear to take the condition to be necessary for the physical reality of a quantity in quantum mechanics when they use the fact that no quantum state can be a simultaneous eigenstate of the position and momentum operators to claim that the position of a particle with known momentum has no physical reality.
Is it true that the usual quantum mechanical idea of reality takes the given condition as sufficient but not necessary for the physical reality of some quantity? If so, how does the argument that the position of a particle with definite momentum has no physical reality follow?
 A: 
"supposing that a complete theory exists, then for the condition to not be necessary for the reality of a physical quantity, it must, according to the first assumption, be possible for an element of physical reality to have a corresponding physical quantity in the complete theory but for it not to be possible to predict with certainty the value of this quantity from the theory without disturbing the system."

Yes. One way this could be is when the complete theory is like classical theory with random variables; there is a property of the system that we regard as element of reality, so it has representation in the theory, but nothing else in that theory determines its value. For example, a random variable like fluctuating electric field acting on the charged particles in the system. We know it is there, it is element of reality, but nothing else in the theory determines its value, it is an independent quantity.

It appears to me then that according the classical and quantum mechanical ideas of reality, the condition would in fact be necessary for the reality of a physical quantity.

Some people would regard the condition as necessary for quantity being real (the Copenhagen interpretation, a kind of scientific positivism based on rejection of classical physics thinking), others don't (EPR, statistical interpretation, an extension of classical physics thinking). Unfortunately, there is no agreed upon concept of what is / is not an element of reality. The Copenhagen school usually admits only those quantities as really existing that can be predicted, not created via measurement, with 100% success, EPR simply assume there are such things regardless of our predictive abilities and they have some properties (independence of people thoughts, intentions, abilities, distant events like measurements etc.)

"later in the paper, the authors do appear to take the condition to be necessary for the physical reality of a quantity in quantum mechanics when they use the fact that no quantum state can be a simultaneous eigenstate of the position and momentum operators to claim that the position of a particle with known momentum has no physical reality."

If you mean the EPR text after formula (6), I don't think authors mean what you describe. The text there becomes somewhat incoherent and harder to follow, but I think what they are saying is that usually in quantum theory expositions, the mathematical fact that no eigenfunction of $p$ can be also eigenfunction of $x$, is believed to imply that when state is that describe by eigenfunction of $p$, position $x$ cannot be an element of reality.
EPR do not believe this is necessarily correct, because they have a different conception of what is real than the usual quantum theory expositions. They suspect non-commuting quantities can be simultaneously real, which would be possible if quantum theory was not complete. Which is a possibility the usual expositions did not discuss. They propose this stance in the EPR paper and defend it by trying to show there are elements of reality (because predictable with 100% success) not captured by the quantum theory.

Is it true that the usual quantum mechanical idea of reality takes the given condition as sufficient but not necessary for the physical reality of some quantity?

The usual used to be the Copenhagen stance described above, so it used to be "yes it is necessary" or "don't talk about real, talk about predictions and measurements". As decades passed, with help of Bohm and Bell and others, people got more open to EPR thinking and we got also "it is sufficient, but there are other possibilities, maybe there are real things for which we just don't have the best theory". Nowadays there are other viewpoints on what is real as well, such as the pretty absurd Everett theory, where the real thing is the psi function of the Universe with large number of arguments and our observations and measurements are just a random subset of properties of this extremely complicated mathematical object.
