Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 33D (Principles of Quantum Mechanics)

In the spin-$z$ orthonormal basis, $|\uparrow ⟩=(1,0{)}^T$ and $|\downarrow ⟩=(0,1{)}^T$. The spin angular momentum operators are $s_i=(\hslash /2){\sigma }_i$. The Pauli matrices obey $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$, soThe second identity is the correct $s(s+1){\hslash }^2$ eigenvalue for $s=1/2$. Matrix multiplication givesThus $s_x$ and $s_y$ flip the spin-$z$ basis states, with different complex phases.

Put $a=\hslash /2$. Operators on distinct tensor product factors commute. Acting with ${\sigma }_x\otimes {\sigma }_y\otimes {\sigma }_y$ on $|\uparrow \uparrow \uparrow ⟩$ gives $-|\downarrow \downarrow \downarrow ⟩$, while acting on $|\downarrow \downarrow \downarrow ⟩$ gives $-|\uparrow \uparrow \uparrow ⟩$. Therefore their difference is an eigenvector with eigenvalue $+1$. The same computation applies to the permutations $yxy$ and $yyx$. The product $xxx$ flips the two basis tensors without a phase and so sends their difference to its negative.

Restoring the three factors of $a$, the GHZ state hasThese are certain measurement outcomes for the four product observables in this state.

A putative simultaneous assignment of classical values ${\tilde{s}}_x^{(i)},{\tilde{s}}_y^{(i)}\in \{\pm a\}$ reproducing the first three certain quantum outcomes must obey, almost surely,Multiply the three equalities. Each $y$ value occurs twice and its square is $a^2$, leaving $a^6{\tilde{s}}_x^{(1)}{\tilde{s}}_x^{(2)}{\tilde{s}}_x^{(3)}=a^9$. Thuscontrary to the quantum prediction $-a^3$.

This is the GHZ theorem against predetermined local measurement values. Experimentally prepare many copies of the three-spin state and choose independently among the four measurement settings on separated subsystems. The three mixed correlations force the classical prediction for the fourth, whereas quantum mechanics predicts the opposite sign. Since $s_x,s_y$ on the same subsystem do not commute, the four settings are measured on separate copies; the classical contradiction relies precisely on the common assignment of values across settings.