Efficient quantification of experimental evidence against local realism

Tests of local realism and their applications aim for very high confidence in their results even in the presence of potentially adversarial effects. For this purpose, one can measure a quantity that reflects the amount of violation of local realism and determine a bound on the probability, according to local realism, of obtaining a violation at least that observed. In general, it is difficult to obtain sufficiently robust and small bounds. Here we describe an efficient protocol for computing such bounds from any set of Bell inequalities for any number of parties, measurement settings, or outcomes. The protocol can be applied to tests of other properties (such as entanglement or dimensionality) that are witnessed by linear inequalities.

Figure 1

Confidence-gain rates in the test of the CGLMP inequality $\langle I_d\left(X\right)\rangle \le 2$. Here, we use the quantum state and measurement settings of Ref. [19], Eqs. (15) and (9), respectively.

Figure 2

Confidence-gain rates in the test of LR with an unbalanced Bell state $\left|\psi \right(\theta )\rangle$. The measurement settings are chosen to maximize the violation of the CHSH inequality (1) given the state $\left|\psi \right(\theta )\rangle$. The gain rates achieved by the simplified PBR protocol using the CHSH inequality are shown as circles ($\circ$), while the gain rates by the same protocol using the CHSH inequality together with no-signaling conditions are shown as crosses ($+$).

Figure 3

An example of running log-$p$ values as functions of the number of trials $n$ in a test of the CGLMP inequality. The dashed and solid lines are the asymptotic lines for log-$p$ values based on gain rates achieved by the (full or simplified) PBR protocol and the martingale-based protocol, respectively. Repetitions of this Monte Carlo simulation show similar behavior.