Increasing the Statistical Significance of Entanglement Detection in Experiments

Entanglement is often verified by a violation of an inequality like a Bell inequality or an entanglement witness. Considerable effort has been devoted to the optimization of such inequalities in order to obtain a high violation. We demonstrate theoretically and experimentally that such an optimization does not necessarily lead to a better entanglement test, if the statistical error is taken into account. Theoretically, we show for different error models that reducing the violation of an inequality can improve the significance. Experimentally, we observe this phenomenon in a four-photon experiment, testing the Mermin and Ardehali inequality for different levels of noise. Furthermore, we provide a way to develop entanglement tests with high statistical significance.

Figure 1

Significance $\mathcal{S}$ for the Mermin (red, solid) and the Ardehali inequality (blue, dashed) for bit-flip noise. On the horizontal axes, we show the bit-flip probability and the corresponding fidelity with respect to a perfect GHZ state. We assumed that the experimenter prepares 8000 instances of a GHZ state and chooses either to measure the eight terms of the Mermin inequality (each term with 1000 realizations of the state) or the 16 terms of the Ardehali inequality with 500 states per correlation term. See text for further details.

Figure 2

Scheme of the experimental setup. (a) The setup to generate the required four-photon GHZ state. Femtosecond laser pulses ($\approx 200\mathrm{fs}$, 76 MHz, 788 nm) are converted to ultraviolet pulses through a frequency doubler ${\mathrm{LiB}}_3{\mathrm{O}}_5$ (LBO) crystal (not shown). The pulses go through two main $\beta$-barium borate (BBO) crystals (2 mm), generating two pairs of photons. The observed twofold coincidence count rates are about $1.6\times {10}^4/\mathrm{s}$ with a visibility of 96% (94%) in the $H/V$ ($+/-$) basis. (b) Setup for engineering the bit-flip noise. (c) The measurement setup.