So we have ($G,\{\rho_i\}$) and descend in symmetry along $\rho_k$ to get say in this case only one ($K1,\{\sigma_j\}$). Now I want to know if there is generic map from the set \{\rho_i\}/\{\rho_k\} to \{\sigma_j\}
@MatheinBoulomenos: A simple example would be that one part of this map is always the trivial representation of $G$ to the trivial representation in $K_1$.
@MatheinBoulomenos: I doubt this. But could that be in case it exists a peculiarity of point groups?
Suppose you and someone on social media both like to follow the local teams: basketball, baseball, and football. (Note that I"m pretending you're American for the purpose of this :P)
Each night, each of you choose to watch one of the games and report on social media whether the home team one or not. If you both watch the same game, then you'll both of course report the same outcome (which I'll call + or -)
@MatheinBoulomenos: So the point is that sometimes a higher dimensional irrep splits into several new irreps in the stabilizer. And I am now searching for a functional relation between functions of the two.
So if you were to keep track of your various social media accounts, you'd find that P(first person saw the home team win, second person saw the home team win | first person watched hockey, second person watched hockey) = 1
@MatheinBoulomenos: say $\lambda_{\rho_k}(\rho_j) = (\sigma_a\oplus\sigma_b)$. Then I have conjecture connecting $\rho_k \otimes \rho_k$ and $\sigma_a\otimes\sigma_b$
I got into a spat with my learning partner today, he wanted to write the problem statement using the standard basis but I claimed you didn't gain any insight or anything useful for the context of the problem by fixing a basis at all
feel free to think of it like that: each of you get to watch one of three coin flips.
if you watch the same one, you get the same outcome
and so forth
anyways. since the partner's choice of which game they watch shouldn't matter as to whether the coin is fair, that requirement further imposes that P(+|H) = P(++|HB) + P(+-|HB) = 1/2
For instance, take three coin flip variables X1,X2,X3 (outcomes {+1,-1} with equal probability) and take them to be independent. then they've all got zero mean, unit variance, and zero covariance e.g. <X1.X2> = 0
and then construct the variables Y1 = X1 - (X1+X2+X3)/3, etc.
you'll find that the new variables have zero mean, variance 1/3, and covariance -2/3
So... I am guessing, the source of the correlation is the reporting here?
> what's not obvious are again the cross-cases. suppose they all work out like this: HB, BH, HF, FH, BF, FB: ++ (12.5%) +- (37.5%) -+ (37.5%) ++ (12.5%)
I recall one explanation on Bell inequality as some pressing button to light up lamp game explains how because of the richer structure of the operators that appeared in the expectation values (compared to just scalars in the classical case), it provide the wiggle room needed to achieve lager correlations thus give rise to entanglement
Actually, I wonder if your above sports example can be used to describe Bell inequality
To get Bell's inequality, note that the outcomes of the games are intended to be some random vector $(X_1,X_2,X_3)=(\pm 1,\pm 1,\pm 1)$
in order for the outcomes to not show any bias, you need +++ and --- to show up equally often
and similarly for the pairs {+--, -++}, {-+-,+-+} , {--+,++-}
if you now consider the products, you see that the three pairwise products are +1 for +++ and ---
whereas for the rest you have one +1 and two -1
consequently when you add the products together you get +3 in the first two cases and -1 in the rest. as such, the expected value of X1.X2+X2.X3+X1.X3 has to be between 3 and -1
whereas in the previous case the expected value of the products worked out to be -1/2 for each of them
well, note that p12 = <X1.X2> in this case defines the correlation coefficient of X1 and X2. (the mean values of X1,X2 are zero and their stddev's are one, so the usual definition simplifies a bit)
so the point is to look at c12+c23+c13
however, the motivation is a bit harder to explain. I guess you can partly relate it to what I was doing earlier re: (X1+X2+X3)^2
i.e. the expected values of X1^2, X2^3, X3^2 are all 1, so the expected value of their squared sum is determined by the sum of those product-moments
<H^2>+<F^2>+<B^2> +2<HB>+2<BF>+2<FH> hmmm, it feels like the expression X1.X2+X2.X3+X1.X3 is summing up all the correlations that are present in the system
Say I have an $n \times n$ covariance matrix for a sample set of $n$ random variables. Is there any meaning if the sum of the rows of this matrix? Is it a meaningful measurement of the contribution of each random variable to the variance of sum of the random variables?
the main point is that, if you only work with sets of games whose outcomes are +1/-1, then the pairwise correlations are constrained more than they'd be otherwise
(As pointed out by helpful users, one of the bolded claims is interpretation dependent. Therefore, unless otherwise specified, Copenhagen interpretation is used throughout)
Recall that for classical correlations, the observables are already determined even before a measurement, and thus knowing ...
We have the ring $R=\mathbb{Z}_5[x]$ and the element $b=(x-4)(x-3)^2$ of $R$. Let $A$ the set of $a$ in $R$ such that $at=b$ has a solution $t\in R$. How can we determine the number of elements of $A$ ? Could you give me a hint?
hmm, so I am guessing, the extra correlation beyond entanglement in the PR box is that equation AB=(a+b) mod 2 that forces both inputs and outputs to correlate with each other even though the inputs can be randomly chosen between 0 or 1
Say I have an $n \times n$ covariance matrix for a sample set of $n$ random variables. Is there any meaning if the sum of the rows of this matrix? Is it a meaningful measurement of the contribution of each random variable to the variance of sum of the random variables?
