5 ms·

She mentions correlation. What could she mean by correlation? Please, I'd prefer the most detailed and mathematically precise and advanced view.

drdeca 4y ago
I believe this is referring to the entanglement between parts of the system
bowsamic 4y ago
Correlation between two quantum observables indicates how, on average, measurements of them are related. Positive correlation means that generally you measure both variables to have the same sign, more strongly correlated being closer to a straight line, i.e. for perfect correlation a measurement of one variable totally determines the other. Negative correlation means that if you measure one to be positive you will probably measure the other one to be negative and vice versa. For two quantum operators a and b, for simplicity on the same Hilbert space H for which one state is |psi>, which is also chosen to be the current state of the system, the correlation is given by, <(a - <a>) (b - <b>)> / (sigma_a sigma_b) where <o> = <psi|o|psi> for any operator o, and sigma_o = sqrt(<o^2> - <o>^2) is the standard deviation of the quantum operator
- graycat 4y ago
  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!
coderenegade 4y ago
You have two sensors, P and Q, used to measure some property of two entangled particles. When P and Q have the same setting, they always disagree, because the entangled particle that Q measures has a state opposite to the particle measured by P. Similarly, when Q has a setting opposite to that of P, P and Q always agree. For any other setting, P and Q will sometimes agree, and sometimes disagree. The extent to which the agree is what quantum physicists refer to as the correlation. If they always agree, the correlation is 1; If they always disagree, the correlation is -1; when they sometimes agree you'll get a value between -1 and 1. It's really just the percentage agreement between the two detectors.

drdeca 4y ago

I believe this is referring to the entanglement between parts of the system

bowsamic 4y ago

Correlation between two quantum observables indicates how, on average, measurements of them are related. Positive correlation means that generally you measure both variables to have the same sign, more strongly correlated being closer to a straight line, i.e. for perfect correlation a measurement of one variable totally determines the other. Negative correlation means that if you measure one to be positive you will probably measure the other one to be negative and vice versa. For two quantum operators a and b, for simplicity on the same Hilbert space H for which one state is |psi>, which is also chosen to be the current state of the system, the correlation is given by, <(a - <a>) (b - <b>)> / (sigma_a sigma_b) where <o> = <psi|o|psi> for any operator o, and sigma_o = sqrt(<o^2> - <o>^2) is the standard deviation of the quantum operator

graycat 4y ago
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!

coderenegade 4y ago

You have two sensors, P and Q, used to measure some property of two entangled particles. When P and Q have the same setting, they always disagree, because the entangled particle that Q measures has a state opposite to the particle measured by P. Similarly, when Q has a setting opposite to that of P, P and Q always agree. For any other setting, P and Q will sometimes agree, and sometimes disagree. The extent to which the agree is what quantum physicists refer to as the correlation. If they always agree, the correlation is 1; If they always disagree, the correlation is -1; when they sometimes agree you'll get a value between -1 and 1. It's really just the percentage agreement between the two detectors.