gentzen said:
What do you mean by position?
pines-demon said:
Take some state like ##|\psi\rangle=\int \mathrm d x\, |x\rangle|-x\rangle## (if you are worried that this is not normalized correctly, limit it to a given space), in momentum it looks something like ##|\psi\rangle=\int \mathrm d p\, |p\rangle|p\rangle##
OK, what has this to do with my question, or with anything you wrote in your answer?
gentzen said:
The position where the particle was created? Or the position where the particle hits a spherical screen of given radius r around the expected position of particle creation?
pines-demon said:
The position of detection, keep it 1D for now.
Even in 1D, it makes an important difference whether position means to one at the moment of creation, or at some other time.
But I see your general issue with my question "What ...? ...? Or ...?": You interpret it as three separate questions, not as one question with some alternatives what you could have meant. I suggest you answer my questions again, and if staying in 1D makes this easier for you, no problem:
What do you mean by position? ...
What do you mean by a mixture of position and momentum? ...
In that case, do you believe ... would "in principle" allow to make measurements showing Bell's inequality violations?
Because I still have an urge to suggest concrete answers, here I go again, this time without question marks:
- Position could mean position at the moment of creation, it could mean the center of gravity of the entire system, it could mean position at a given time since the moment of creation, and it could also mean something like position at a "given distance" from the expected position of particle creation.
- Mixture of position and momentum could mean measurement of a complex superposition of position and momentum, i.e. ##(\alpha|x\rangle+\beta|p\rangle)(\alpha\langle x|+\beta\langle p|)## with complex numbers ##\alpha, \beta## with ##|\alpha|^2+|\beta|^2=1##. It could mean some actual physical measurement procedure, like those I suggested in my initial post.
- For a concrete experimental arrangement, like the one described, the interesting question is whether somebody is willing to at least "guess" that it could work "in principle". Or any concrete scenario could always end in a retreat to abstract math.
pines-demon said:
No. It is only ##x## and ##p=p_x## the ones that do not commute.
I hope I already clarified above that my question should not have been interpreted as yes/no question, but as a request for general clarification. And to be perfectly clear: your answer above did not deliver that clarification for me. What it did deliver is "your mental model" of "how I think, and why that is wrong", but it did not deliver what you want to measure, neither in the abstract math, nor in concrete experimental arrangements.
pines-demon said:
I mean in a simple "Mermin like" Bell test you need three angles to measure spin. Here I mean to measure linear combinations of ##x## and ##p##. Note that you can render all this binary à la Bell by binning results (left vs right located), (positive vs negative ##p##).
So I get that you thought a bit more about what you are proposing, and noticed that position and momentum give continuous results, but "measurements showing Bell's inequality violations" work with binary results. So you suggest binning. My guess it that binning will probably destroy the delicate quantum entanglement. For a sufficiently well specified (thought) experiment, it should be possible to check whether this is the case or not.
pines-demon said:
We can discuss how hard is to measure linear combinations of ##x## and ##p## but that is not my point to solve. You are discussing this as if it is somehow proven that the original EPR cannot be Bell tested, why?
Let me check my words, whether they suggested anything like "it is somehow proven":
gentzen said:
I have not analyzed the EPR setup for how to preserve or destroy entanglement, but it is unclear to me which degrees of freedom you could measure that would keep their entanglement.