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Thanks. Okay, good to see that quantum mechanics defines its use of correlation as <(a - <a>) (b - <b>)> / (sigma_a sigma_b) Good. Glad to see the actual de

Thanks. Okay, good to see that quantum mechanics defines its use of correlation as <(a - <a>) (b - <b>)> / (sigma_a sigma_b) Good. Glad to see the actual definition. That is, with usual quantum mechanics notation, this is the usual Pearson correlation coefficient and, thus, appropriate to call correlation. Fine. Good to see that this is just what physics is talking about. My concern has been, does physics say "correlation" when really they mean some case of what we might call the usually much stronger functionally dependent? That is, for complex valued random variables X and Y, they are functionally dependent provided there exist functions f and g so that Y = f(X) and X = g(Y). With this functionally dependent situation, the correlation could be anything from -1 to 1 but usually we won't care because having either of X or Y implies we also have the other exactly. So, as I was reading quantum mechanics, I was wondering if there could be, and open to, two random variables functionally dependent but without correlation 1 or -1 so that, really, talking about mere correlation would be too weak and, thus, not really appropriate. Well, in the case of correlation 1 or -1, there exists non-zero, complex a so that Y = aX in which case instead of mere correlation it would be better to use the stronger functionally dependent. As I understand the situation, usually quantum mechanics uses correlation for examples of entanglement where it appears that, yes, the stronger functionally dependent would hold, and, thus, be more appropriate. That is, the usual uses of correlation outside of quantum mechanics, say, in statistics of medical research, economics, finance, social science, mean "two random variables not independent and also not functionally dependent" -- having the two random variables be independent (which implies correlation 0) or functionally dependent would be unusual.

bowsamic 4y ago
> As I understand the situation, usually quantum mechanics uses correlation for examples of entanglement where it appears that, yes, the stronger functionally dependent would hold, and, thus, be more appropriate. Not really, since things can be weakly entangled, and thus measuring one thing doesn't totally determine the other. The kind of entanglement where you have perfect correlation is usually only achievable in certain strange situations, like EPR entanglement. In that case it can be seen as perfect correlation though, which I don't see as being any different from the functional dependence you speak of. In most cases in quantum systems, we have some correlation but not perfect correlation (e.g. squeezed states of light), and since we have that whole spectrum of behaviour, with maximum entanglement (e.g. some kind of EPR) being the extreme, we choose to use correlation to quantify the entire spectrum of possibilities. Also from a more philosophical point of view, we expect to treat different degrees of freedom separately. If you have two particles but measuring the momentum of one gives you perfect knowledge of the other (i.e. EPR), it still doesn't really make sense to say p' = -p, or whatever, since conceptually p' and p are momenta belonging to two different subsystems. The most neutral way of stating it is that the momenta are perfectly correlated.
- graycat 4y ago
  Thanks!

bowsamic 4y ago

> As I understand the situation, usually quantum mechanics uses correlation for examples of entanglement where it appears that, yes, the stronger functionally dependent would hold, and, thus, be more appropriate. Not really, since things can be weakly entangled, and thus measuring one thing doesn't totally determine the other. The kind of entanglement where you have perfect correlation is usually only achievable in certain strange situations, like EPR entanglement. In that case it can be seen as perfect correlation though, which I don't see as being any different from the functional dependence you speak of. In most cases in quantum systems, we have some correlation but not perfect correlation (e.g. squeezed states of light), and since we have that whole spectrum of behaviour, with maximum entanglement (e.g. some kind of EPR) being the extreme, we choose to use correlation to quantify the entire spectrum of possibilities. Also from a more philosophical point of view, we expect to treat different degrees of freedom separately. If you have two particles but measuring the momentum of one gives you perfect knowledge of the other (i.e. EPR), it still doesn't really make sense to say p' = -p, or whatever, since conceptually p' and p are momenta belonging to two different subsystems. The most neutral way of stating it is that the momenta are perfectly correlated.

graycat 4y ago
Thanks!