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

The Einstein–Podolsky–Rosen criterion of reality says that if, without disturbing a system, one can predict a physical quantity with certainty, then an element of physical reality corresponds to that quantity.

For two settings on each wing and outcomes in $\{-1,1\}$, the CHSH inequality for every local hidden-variable theory is

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Choose all four axes in one plane, at anglesTheir relevant unsigned separations areThe specified correlation function therefore gives $E(a,b)=E(a,b^{\prime })=E(a^{\prime },b)=1$ and $E(a^{\prime },b^{\prime })=-1$. Hencethe algebraic maximum and thus a maximal violation of the CHSH inequality.

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Suppose local elements of reality predetermined outputs $x(a,\lambda ),x(a^{\prime },\lambda ),y(b,\lambda ),y(b^{\prime },\lambda )\in \{-1,1\}$. For every hidden state $\lambda$,equals $2$ or $-2$, because exactly one bracket vanishes and the other equals $\pm 2$. Averaging over $\lambda$ proves the CHSH inequality. Part (b) instead gives $4$, so no such locally predetermined response functions can reproduce the device. This is the standard Bell theorem obstruction.

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For each pair of inputs, define the joint probabilitiesThey are nonnegative because $|f|\le 1$, sum to one, and have the required correlation:Both marginals are uniform,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.

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