Mermin inequalities for perfect correlations in many-qutrit systems
Jay Lawrence · Physical Review A · 2017
The existence of Greenberger-Horne-Zeilinger (GHZ) contradictions in many-qutrit systems was a long-standing theoretical question until its (affirmative) resolution in 2013. To enable experimental tests, we derive Mermin inequalities from concurrent observable sets identified in those proofs. These employ a weighted sum of observables, called $\mathcal{M}$, in which every term has the chosen GHZ state as an eigenstate with eigenvalue unity. The quantum prediction for $\mathcal{M}$ is then just the number of concurrent observables, and this grows asymptotically as ${2}^{N}/3$ as the number of qutrits $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The maximum classical value falls short for every $N\ensuremath{\ge}3$, so that the quantum to classical ratio (starting at 1.5 when $N=3$) diverges exponentially ($\ensuremath{\sim}1.{064}^{N}$) as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, where the system is in a Schr\"odinger-cat-like superposition of three macroscopically distinct states.