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(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 information of one subsystem will immediately let you know the corresponding information in another subsystem.

For correlations in entanglement, suppose I take the singlet bell state $\frac{1}{\sqrt{2}}(\lvert 01\rangle-\lvert 10\rangle)$. Then upon preparing the bell state, the correlations of the 2 subsystems are already established thus the whole system is described by one wavefunction. In addition, unlike classical correlations, since the outcome of the measurement is probabilistic, knowing information about one subsystem tells you nothing about the other subsystem, and one will not see a correlation exists until the two parties compare their measurement results. In such case we will find $\langle \hat{\sigma}_x\rangle=\langle \hat{\sigma}_y\rangle=\langle \hat{\sigma}_z\rangle=0$ indicating the 50:50 result of measure either eigenvalue of the spins along each component, but $\langle \hat{\sigma}_x^A\hat{\sigma}_x^B\rangle=\langle \hat{\sigma}_y^A\hat{\sigma}_y^B\rangle=\langle \hat{\sigma}_z^A\hat{\sigma}_z^B\rangle=-1$ indicating perfect anticorrelation.

Comparing these results with the CHSH correlation function

$$C_{CHSH}(\hat{a},\hat{b};\hat{a}',\hat{b}')=\langle \hat{a}\hat{b}\rangle-\langle \hat{a}'\hat{b}\rangle+\langle \hat{a}\hat{b}'\rangle+\langle \hat{a}'\hat{b}'\rangle$$

and Tsirelson's proof, we knew that any classical correlations cannot exceed 2, while bell states can push correlation up to $2\sqrt{2}$, the Tsirelson bound. From the examples above, it is thus clear the extra $\sqrt{2}$ corresponds to the entanglement correlations.

Now for PR boxes, which is defined to receive two inputs A,B and give two outputs a,b with a joint probability of 0.5 if AB=(a+b) mod 2 and 0 otherwise, that is

$$P(a,b\vert A,B)=\left\{\begin{matrix}\frac{1}{2},a+b \mod 2 = AB \\ 0, \textrm{otherwise}\end{matrix}\right.$$

. Once can then plug the joint porbabilities into the CHSH and get 4 as expected. However, what is not clear to me is what is physically the step beyond entanglement correlations. To summarise, we knew the following:

Classical correlations: Observables of the two subsystems are correlated, and knowing information from one subsystem tells you the corresponding information of the other. The correlation is on the level of some predetermined observables.

Entanglement correlations: Observables of the two subsystems are correlated, but knowing information from one subsystem does not tell you much (or anything for bell states) about the other subsystem. The correlation is on the level of the wavefunction, hence the spectrum of observables that can be obtained from the system, along with the joint probabilities (For operationalists, the correlation is on the level of observables and the probabilities of getting them given some initial instruction in the preparation, and the observables are encoded by the outputs of the measuring device)

PR correlations: The observables are correlated to two, not just one inputs, with probabilistic outcomes. Similar to entanglement, information learnt from one subsystem tell you nothing about information in the other subsystem, and (?). The correlation is on the level of (?).

What is this (?) phenomenon that is absent in both classical and entanglement correlations, but unique to PR correlations?

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