Past exam of the laser science and quantum information processing course of the University of Oxford / 2016 / 7 / Solution

Encode $|0⟩=|H⟩$ and $|1⟩=|V⟩$ in two orthogonal linear polarizations. Then $|\pm ⟩=(|H⟩\pm |V⟩)/\sqrt{2}$ are linear polarizations at opposite forty-five-degree angles, while $(|H⟩\pm i|V⟩)/\sqrt{2}$ are the two circular polarizations. Their right/left names depend on the viewing convention. A half-wave plate with its axis at $22.5^{\circ }$ to H implements H up to a common phase, since its Jones matrix has entries $\cos 2\theta ,\sin 2\theta$ and $-\cos 2\theta$.

In principle, a nonlinear interaction producing $e^{i\pi n_{1V}{n}_{2V}}$ supplies a deterministic Controlled-Z gate: only the joint VV component acquires a minus sign. A strongly coupled matter mediator can also provide the required conditional interaction. At optical single-photon energies the direct nonlinear interaction is weak; loss, unwanted noise and maintaining indistinguishable spatial and temporal modes make a high-fidelity deterministic gate difficult. Passive linear optics by itself does not supply this deterministic entangling interaction; measurement-based optical gates introduce additional resources and success conditions.

For the printed circuit, the first three H gates create $|+{⟩}^{\otimes 3}$. The two controlled-Z operations from the first qubit to the others giveThe final H gates on the last two qubits therefore produce the GHZ stateDefine $|s_X⟩=(|0⟩+s|1⟩)/\sqrt{2}$ and $|s_Y⟩=(|0⟩+is|1⟩)/\sqrt{2}$ for $s=\pm 1$. In any of the XYY, YXY and YYX bases, direct overlaps givewith the same coefficient formula when the X position is permuted. Hence the three explicit expansions can be written compactly asIn each sum the four sign triples are $(+,+,-),(+,-,+),(-,+,+),(-,-,-)$. They have equal probabilities $1/4$. Thus two Y outcomes determine the X outcome with certainty: their product times the X outcome is $-1$.

In the XXX basis,The four allowed triples are $(+,+,+),(+,-,-),(-,+,-),(-,-,+)$, so the XXX outcome product is $+1$.

The GHZ theorem follows without a statistical inequality. If local predetermined values $X_i,Y_i\in \{\pm 1\}$ reproduced the three mixed-basis correlations, multiplying them would givebecause every Y value is squared. Quantum mechanics instead predicts the XXX product $+1$. This is an all-versus-nothing contradiction with local realism using ideal perfect correlations. Standard two-qubit Bell-state tests compare statistical correlations through a Bell inequality; the GHZ construction makes the inconsistency visible already at the level of these deterministic parity constraints.