What does simultaneous reality and non-commuting operators mean?

[elbeasto]

"if Quantum Mechanics (QM) is complete (and there are no "hidden variables"), then there cannot be simultaneous reality to non-commuting operators" - Taken from http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

I am trying to understand this sentence but I do not fully comprehend 'non-commuting operators'. Wikipedia uses an example of

"physical variables are represented by linear operators such as x (meaning multiply by x), and d/dx. These two operators do not commute as may be seen by considering the effect of their products x (d/dx) and (d/dx) x on a one-dimensional wave function ψ(x):"

is it the 'ψ(x)' that makes this non-commuting? Also, based on reading this, am I to assume that this means that the answer I arrive at is based on solely on the variable I measure first? I thought the idea was no matter what I measure, the other will reflect my observation. For example, I measure 'up' spin on variable A, therefore variable B must be 'down'. Regardless of the variable I measure first, the end result is the same: A is up and B is down.

Also, so I can try to nail down a clear definition, when EPR says 'simultaneous reality' are they referring to identical simultaneous instance of a variable?

 

[DrClaude]

is it the 'ψ(x)' that makes this non-commuting?

No, the non-commutativity is a property of the operators themselves.

Also, based on reading this, am I to assume that this means that the answer I arrive at is based on solely on the variable I measure first? I thought the idea was no matter what I measure, the other will reflect my observation. For example, I measure 'up' spin on variable A, therefore variable B must be 'down'. Regardless of the variable I measure first, the end result is the same: A is up and B is down.

This is confusing measurements on the same particle with different (eventually non-commutating) operators and independent measurements on two entangled particles.

Also, so I can try to nail down a clear definition, when EPR says 'simultaneous reality' are they referring to identical simultaneous instance of a variable?

Consider spin along different directions. If one knows the spin along z, one knows nothing about the spin along x. So there is no simultaneous reality for spin along z and spin along x, they are not defined at the same time.