1. I say there is 1 degree of freedom in the spin of an entangled singlet system as we have been discussing. Not 2, because for there to be 2, the particles would need to be separable. And they aren't if entangled.1. Yes. Mathematically, the operators ##Z_A## and ##X_A## don't commute. Nor do the operators ##Z_B## and ##X_B##. But none of those by themselves can be the operators we are discussing, because none of them operate on the two-qubit Hilbert space.
Meaning, "spin of Alice's qubit" is only one degree of freedom? Yes, that's right.
2. Not if those operators are the ones I wrote down; those operators manifestly commute. Nor do I see why that is a problem, since "spin of Alice's qubit [in the ]" and "spin of Bob's qubit" are different degrees of freedom, and the operations "measure z-spin of Alice's qubit" and "measure x-spin of Bob's qubit" therefore operate on different degrees of freedom. The fact that those degrees of freedom are entangled in the particular state of the two-qubit system we are talking about does not change any of this.
If you think the operations of "measure z-spin on Alice's qubit" [##Z_A##] and "measure x-spin of Bob's qubit" [##X_B##]are not described by the operators I wrote down, then what operators do you think should describe them?
3. We are not going to resolve this discussion unless we express what we are saying with precise math.
2. This is as specific as I can make it - and I am applying the same idea of the HUP as is formulated in EPR-B. The purpose of this representation is because we are analyzing the following statement by Weinberg, which summarizes my position in this thread as clearly and concisely as possible: ...according to present ideas a measurement in one subsystem does change the state vector for a distant isolated subsystem..." A sharp measurement on ##Z_A## here leads to ##Z_B## changing to match expectation, and a subsequent sharp measurement on ##Z_B## supports that.
Any particle called A:
$$ \sigma A_p \space \sigma A_x \geq \frac \hbar 2 $$
A pair of entangled particles (your example of A and B):
$$ \sigma A_p \space \sigma B_x \geq \frac \hbar 2 $$
Therefore, returning to the discussion on entanglement swapping/quantum teleportation (photons 1 to 4): When a measurement on photons 2 & 3 projects 1 & 4 into an entangled state, the 1 & 4 state vector changes quantum non-locally into one which could not possibly have existed prior to the swap. There is no revealing of a pre-existing entangled state for 1 & 4, because the swap was required to make that happen.
3. It seems awkward to me that I quote EPR, Weinberg et al verbatim to support the language I use. And yet you disagree, and show me representations of the math that appear to be diametrically the opposite of what I said. I am combing over what you are saying trying to pick out the source differences in our positions, but I admit it is as much a struggle for me as it is frustrating for you. Thanks for sticking with it.