The Detection Loophole in Bell Test Experiments

Caroline Thompson's Chaotic Ball*

Continuing the discussion on Bell inequalities, supporters of local realism challenged the experimental verification by means of several experimental loopholes. One of these loopholes, the detection loophole, is very interesting because it can precisely reproduce quantum mechanics predictions in an EPR-B experiment.

The key idea is that spin measurements can have three outcomes: +1, -1, and no detection. In 1970, Philip Pearle found such an example and computed a minimum no-detection limit of 14% required to reproduce the minus cosine correlation, but the math there is cumbersome and it was not explored further. (What this shows is that detector efficiency needs to exceed 86% to close the detection loophole). I will not discuss Pearl’s model, but I will show instead an intuitive (but inexact) model found by Caroline Thompson in 1996 (http://xxx.lanl.gov/abs/quant-ph/9611037 ).

Let us start with the minus cosine correlation between spin measurement in EPR-B. In The EPR-B experiment, a source of two electrons initially in spin zero state emits two electrons in opposite directions and their spin is measured on two directions A and B making an angle α between them. In any experimental model (quantum mechanics, classical mechanics, and local hidden variable models) there are three fixed correlation values: -1 for α = 0, 0 for α = 90 degrees, +1 for α = 180 degrees. Quantum mechanics formalism and experiments show that the measurement outcome correlation is –cos α. But what is so special about this? When α = 0 by conservation of spin, if we measure the left electron on a direction and obtain an outcome, we naturally expect that if we measure the right electron on the same direction we obtain the opposite outcome.

However, the catch is in the slope of the correlation: since the differential with respect to α of –cos α is +sin α, the tangent to the correlation curve at α = 0 is zero. Let’s think of this for a minute of what it means: if we have a slight deviation in the two measurement angles, the correlation stays the same. In quantum mechanics this is true, because a Bell state is a superposition of two wavefunctions. So what? What this has to do with anything? In the classical case, the electron has a definite direction of spin independent of measurement and the correlation curve has a constant slope of 1 (the three fixed points are connected by a straight line). In quantum mechanics, the correlation curve slope at α = 0 is zero because there is a compensation effect in measurement outcomes due to superposition.

Now back to the detection loophole: can we imagine a simple classical system where not all measurements generate an experimental outcome, and still the correlation curve at α = 0 is zero? Late CarolineThompson came with such a simple system, and is as follows:

Consider a uniformed colored ball which spins randomly around its center. Pick two opposite points and write N and S on the ball (for North and South Pole). Let the ball spin chaotically and look at the ball from two different directions A and B and at certain time intervals. The two experimentalists write down what they see: N, S, or nothing.  If  the observers are close to the ball, due to the reduction in the field of vision, there are some bands on the ball which nobody can see and they generate the “nothing outcome”.

To compute the correlation, the experimentalists have to discard the cases where nothing was detected by one or both observers and the surprise is that the correlation exhibits a flat correlation curve at 0 and 180 degrees.

As I stated earlier the model is not exact, but it raises the question of the detector efficiency in experimental tests and questions the validity of the observed correlation as an argument against local realism because the observed correlations can be an artifact of incomplete detection. In general to create realistic models of the EPR-B experiments using the detection loophole (and exact models do exist) one needs to have unfair sampling depending on the angle between the measurement direction and the intrinsic spin direction. Since the undetected outcomes are by their very nature hidden from the experimentalist, who is to say that Nature obeys fair sampling? After all we want to describe Nature as is, and not to force our preconceptions of fair sampling on the experiments.

Do not expect however to prove quantum mechanics wrong by a few clever hidden variable models exhibiting the EPR-B correlation using the detection loophole. No experiment to date contradicted quantum mechanics predictions. Also in a few years it is expected that loophole free experiments confirming Bell theorem will become feasible.  

* I thank Richard Gill of making me aware of Caroline Thompson's work