For each pair of inputs, define the joint probabilities
$$
\boxed{P(x,y|a,b)=\frac14[1+xyf(\theta(a,b))]},
\qquad x,y\in\{-1,1\}.
$$
They are nonnegative because $|f|\leq1$, sum to one, and have the required correlation:
$$
\sum_{x,y}xyP(x,y|a,b)=f(\theta(a,b)).
$$
Both marginals are uniform,
$$
\sum_yP(x,y|a,b)=\frac12,
\qquad
\sum_xP(x,y|a,b)=\frac12,
$$
independently of the remote input. The device is thus a <no-signalling box>: its superquantum correlation does not by itself transmit a message, so it is compatible with <relativistic causality>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 325 / 2 / d / Solution

A no-signalling box is a conditional probability distribution $P(x,y|a,b)$ whose marginal distribution for either party is independent of the other party's input. It can therefore exhibit correlations stronger than quantum correlations without transmitting information superluminally.


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