Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 58 / 4 / Solution

The EPR criterion of reality proposes that a quantity has an element of physical reality if its value can be predicted with certainty without disturbing the system. To use remote predictions this way, one additionally assumes locality: the choice and execution of a measurement in quantum mechanics on separated particles cannot disturb the local physical properties of the remaining particle. The contradiction below is with this combination of the criterion and locality, rather than with a definition of reality alone.

Use the computational basis $|0⟩=|\uparrow ⟩$, $|1⟩=|\downarrow ⟩$ and the conventional Pauli matricesFor each site $j$, let $O_j$ be the tensor product of $X$ at that site and $Y$ at all other sites. For $N=4M+3$, the number of $Y$ factors is $N-1=4M+2$, soThe minus sign in the given GHZ state then impliesHence every one-$X$, remaining-$Y$ product has measurement outcome $+1$ with certainty, while the all-$X$ product has outcome $-1$ with certainty. These are experimentally separate local Pauli measurements on identically prepared copies, not joint measurements of $X$ and $Y$ on the same particle.

For a chosen site, the other $N-1$ sites can determine its $X$ outcome with certainty by measuring $Y$ and using the corresponding $O_j$ relation. They can determine its $Y$ outcome with certainty by selecting an $O_{\ell }$ with $\ell \ne j$, measuring $X$ at $\ell$ and $Y$ at all other remote sites. Since $N\ge 3$, both choices are available. Locality and the EPR criterion of reality therefore assign each site definite values $x_j,y_j\in \{+1,-1\}$, independent of which remote choice is made, even though quantum theory does not jointly measure the local $X$ and $Y$.

The certain one-$X$ correlations require those assigned values to obeyMultiplying these scalar equations givessince $N-1$ is even and $y_k^2=1$. Thus the all-$X$ product would necessarily be $+1$. But its quantum value is $-1$, givingThe equations cannot hold even for one assignment of values, so allowing a probability distribution over assignments cannot remove the contradiction. This is the GHZ contradiction for N congruent to 3 modulo 4 for every $M\ge 0$. Multiplying classical assigned values is crucial: one must not commute the incompatible local Pauli operators as though they were ordinary numbers. Their noncommutativity is exactly what permits the quantum correlations to evade the classical product argument.