Experimentally Faking the Violation of Bell’s Inequalities

Entanglement witnesses such as Bell inequalities are frequently used to prove the nonclassicality of a light source and its suitability for further tasks. By demonstrating Bell inequality violations using classical light in common experimental arrangements, we highlight why strict locality and efficiency conditions are not optional, particularly in security-related scenarios.

Figure 1

Testing a Bell inequality in an intercept-resend scenario. (a) Experimental setup. Pairs of polarization-entangled photons are generated via spontaneous parametric down-conversion and sent for polarization analysis (PA) in two conjugate linearly polarized bases on both sides using polarizing beam splitters (PBS), half-wave plates ($\lambda /2$), and nonpolarizing beam splitters (BS); details are described in Ref. 24. An intercept-resend system (Eve) is inserted in the fiber line that carries one of the photons. Eve consists of a PA and faked-state generator (FSG), and generates measurement results that are copied into the receiver Bob. Key elements of the FSG are laser diodes (LD), polarization controllers (PC), and attenuators (Att); see details in Ref. 15. Detection events are recorded with time-stamp units (TU) for later correlations. (b) From the correlations between the measurement results, Alice and Bob can test a CHSH Bell inequality. With the photodetectors used for this experiment, they obtain the same result (i.e., a violation of a Bell inequality) with and without the presence of Eve, and would conclude that they witnessed entanglement in a pair of photons.

Figure 2

Bell test with the source replaced by a twin FSG. (a) Optical outputs of the FSG are connected to two polarization analyzers at Alice and Bob. (b) Implementation of the twin FSG: A preprogrammed list of correlation events is sent to two single FSGs at random times defined by a photocounting detector illuminated with a constant intensity. (c) Polarization analyzer with a passive choice of measurement basis, and (d) with an active choice, where a half-wave plate ($\lambda /2$) is rotated by a motor according to the value of an independent random bit source.

Figure 3

Difference between observed correlation function $S$ and expected $S_p$ in a CHSH Bell test, as a function of the programmed value $S_p=S(q)=4q$. The error bars were assigned through propagated Poissonian counting statistics.

Figure 4

Experimentally obtained coincidences between pairs of detection events (0 or 1) at receivers Alice and Bob for their different measurement bases ($HV$ or $\pm$) as a function of different faked-state pairs of bright pulses sent to both sides. Each faked-state pair is made up by a combination of two pulses for horizontal ($H$), vertical ($V$), and $\pm 45°$ linear polarization ($+$ and $-$). Since bases for Alice and Bob are rotated in a Bell test by 22.5°, we denote the faked-state polarizations and measurement bases on Bob’s side with a tilde. Each cell’s area is filled proportionally to the number of coincidence events; empty cell corresponds to 0 coincidence events, while a completely filled one corresponds to 40 229 coincidences. The level of unwanted clicks is always below 1.4% ($\le 566$ coincidences).