Why are quantum correlations limited?

The question in perspective

Two separated observers choose settings \(x,y\in\{0,1\}\) and record values \(a,b\in\{-1,+1\}\). The question is which joint probabilities \(p(a,b|x,y)\) nature permits. Quantum theory violates some classical constraints without allowing controllable faster-than-light messages. Why does its strength stop where it does?

Three different bounds

A local hidden-variable model, with setting-independent variable \(\lambda\), has the form

\[p(a,b|x,y)=\int d\lambda\,q(\lambda)p(a|x,\lambda)p(b|y,\lambda).\]

For deterministic local responses \(A_0,A_1,B_0,B_1\in\{-1,+1\}\), the combination \(A_0(B_0+B_1)+A_1(B_0-B_1)\) is always \(+2\) or \(-2\): one bracket vanishes and the other is \(\pm2\). Averaging therefore gives the CHSH inequality

\[\begin{aligned}|S|&\leq2,\\S&=E_{00}+E_{01}+E_{10}-E_{11},\\E_{xy}&=\sum_{a,b}ab\,p(a,b|x,y).\end{aligned}\]

Quantum observables obey \(A_x^2=B_y^2=I\). Operators on the two laboratories act on separate tensor factors. For their CHSH operator \(\mathcal B\), direct multiplication gives

\[\mathcal B^2=4I-[A_0,A_1]\otimes[B_0,B_1].\]

Here \([A,B]=AB-BA\). Each commutator has operator norm at most two, so \(\|\mathcal B\|\leq2\sqrt2\). A maximally entangled pair and suitable measurement directions attain this value. This bound follows from the quantum operator structure.

No-signalling requires \(\sum_b p(a,b|x,y)\) to be independent of \(y\), and similarly for the other observer. Let binary outcomes \(u,v\) satisfy \(u\oplus v=xy\), assigning probability one half to each allowed pair. The marginals are uniform, yet \(S=4\) after setting \(a=(-1)^u,b=(-1)^v\). Thus no-signalling alone does not recover the quantum bound.

What is established

The Bell nonlocality review explains the inequalities, quantum violations, and experimental assumptions. A violation rejects the stated local model; it does not independently identify which philosophical account of quantum states is correct.

Information causality supplies an additional operational restriction on accessible information and rules out important stronger-than-quantum correlations. Recovering a familiar bound in one scenario is different from characterizing the full quantum correlation set.

What remains unresolved

The deeper target is a physical explanation of the allowed set across settings, outcomes, and multiple parties. A proposed principle must specify which experiments it constrains and whether it is necessary, sufficient, or both. Numerical agreement for a few inequalities cannot establish a complete reconstruction.

A concrete next investigation

Reproduce: mix the binary box above with uniform noise, \(p_v=v p_{\rm box}+(1-v)/4\). Verify uniform marginals and derive \(S=4v\). The CHSH local bound is exceeded for \(v>1/2\); the quantum CHSH bound is exceeded for \(v>1/\sqrt2\).

Check: explicitly build qubit measurements attaining \(2\sqrt2\), rather than assuming every no-signalling table is realizable.

Extend: compare a chosen information principle with quantum realizability in a specified larger scenario. A new separation requires either an explicit admissible counterexample or a proof with stated dimension and modelling assumptions.

Selected references

1. Bell nonlocality

   Nicolas Brunner et al.. 2013. Review

   Introduces Bell correlations and their role in the foundations and applications of quantum information.

2. Information Causality as a Physical Principle

   Marcin Pawłowski et al.. 2009. Research paper

   An operational restriction on accessible information beyond no-signalling.

Years refer to first arXiv submission. Publication details are available in the linked records.