Violation of the Leggett-Garg Inequality in Neutrino Oscillations

The Leggett-Garg inequality, an analogue of Bell’s inequality involving correlations of measurements on a system at different times, stands as one of the hallmark tests of quantum mechanics against classical predictions. The phenomenon of neutrino oscillations should adhere to quantum-mechanical predictions and provide an observable violation of the Leggett-Garg inequality. We demonstrate how oscillation phenomena can be used to test for violations of the classical bound by performing measurements on an ensemble of neutrinos at distinct energies, as opposed to a single neutrino at distinct times. A study of the MINOS experiment’s data shows a greater than $6\sigma$ violation over a distance of 735 km, representing the longest distance over which either the Leggett-Garg inequality or Bell’s inequality has been tested.

Figure 1

The survival probability of ${\nu }_{\mu }$ as measured by the MINOS experiment. The solid (blue) curve indicates the prediction for oscillations assuming global values of $\mathrm{\Delta }{m}_{\mathrm{atm}}^2$, ${\sin }^{2}2{\theta }_{\mathrm{atm}}$ [33], while the dashed (red) curve indicates the prediction fitting directly to the measured MINOS values of $P_{\mu \mu }$. The red band indicates a $1\sigma$ confidence interval around the fitted prediction. The data are taken from Ref. [32].

Figure 2

(Left) The number of $K_3$ values that violate the LGI bound. The red curve indicates the expected classical distribution, while the indigo curve indicates the quantum expectation. The arrow indicates the observed number of violations. (Right) The distribution of $K_4$ versus the sum of the phases $\sum _a{\psi }_a$ as reconstructed from $P_{\mu \mu }$ at various energies. The data (black points) show a clear clustering above the LGI bound. Also shown are the expected distributions for the classical (red dots) and quantum (blue dots) theoretical predictions. Note that $K_4$ can attain multiple values for a given relative phase, because there are many triplets of phase points that add up to a given relative phase. The shown points have high statistical correlations.