Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 57 / 2 / a

The EPR criterion of reality says that a physical quantity has an element of reality if its value can be predicted with certainty without physically disturbing the system. The proposed criterion distinguishes what the system possesses from what one happens to measure.

A local hidden-variable theory supplements the preparation by a variable $\lambda$, drawn from a distribution $\mu (\lambda )$ independent of the later measurement settings. Its locality assumption is conditional factorization:Alice's local response does not depend on Bob's setting $y$, and Bob's does not depend on Alice's $x$. Correlations can arise from the shared past variable $\lambda$. In a deterministic local hidden-variable model, the response probabilities are point masses at functions $a=A_x(\lambda )$ and $b=B_y(\lambda )$. In particular, all possible local settings have definite values for a fixed $\lambda$, even when only one setting is used in a trial.

The setting-independence assumption is needed for comparing these predetermined values across different experiments. Conditional factorization is stronger than quantum no-signalling: no-signalling constrains observed marginal probabilities, whereas local hidden-variable theory constrains their decomposition at fixed $\lambda$. This distinction is what Bell theorem tests.