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

Table of contents
- Solution d

Solution

For each pair of inputs, define the joint probabilities

P (x, y ∣ a, b) = \frac{1}{4} [1 + x y f (θ (a, b))], x, y \in {- 1, 1} .

They are nonnegative because

∣ f ∣ \leq 1

, sum to one, and have the required correlation:

\sum_{x, y} x y P (x, y ∣ a, b) = f (θ (a, b)) .

Both marginals are uniform,

\sum_{y} P (x, y ∣ a, b) = \frac{1}{2}, \sum_{x} P (x, y ∣ a, b) = \frac{1}{2},

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.

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