Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 325 / 2

The outputs satisfyand each allowed pair occurs with probability $1/2$. This is a Popescu–Rohrlich box. For either party, summing over the remote output gives a uniform local bit:independently of $y$, and similarly for Bob independently of $x$. The device is therefore a no-signalling box and does not necessarily permit superluminal signalling, despite correlations stronger than quantum theory allows.

For input pairs other than $11$, either local output is the equiprobable ensemble $\{|+⟩,|-⟩\}$; for input $11$, it is $\{|0⟩,|1⟩\}$. Both ensembles have the same density operator,Every local quantum measurement therefore has the same statistics for every remote input. The joint outputs have unusual correlations, but observing them requires the parties to compare results through an ordinary causal channel. Hence this device also does not necessarily allow superluminal signalling.

For the stated pure product inputs, each local classical output is uniformly $c({\psi }_X)$ or $c({\psi }_X^{\perp })$, independently of the remote bit and state. Those cases alone are therefore non-signalling.

The behavior on entangled inputs is not fixed by the specification, because a qubit entangled with another system has an improper mixed state rather than its own pure state vector. A naive extension that reports a remotely steered pure-state decomposition would permit signalling: one party could choose a measurement basis on half of an entangled pair, and the other party's infinite-precision descriptions would distinguish the resulting ensembles even though they have the same reduced density matrix.

That extension is not forced. For example, the boxes may base outputs only on the local quantum state under objective collapse, or use a fixed ensemble determined solely by the local reduced density matrix; they may also reject inputs that are not pure local states. If one or both input qubits are entangled, such a local rule can output a description of the same local mixed state, a fixed basis ensemble for it, or a designated invalid-input result, all independently of spacelike-separated choices. The device therefore does not *necessarily* allow superluminal signalling, although nonlocal pure-state-readout extensions would do so.