## 6.7 Loopholes in Bell tests

When we introduced the idea of hidden variable theories in Section 6.1, we made some assumptions to simplify the exposition, but these have a big impact on the practical reality of violating Bell tests. Any test that does not satisfy one or more of these assumptions is said to have a loophole. For verifying fundamental physics, we are not so worried about these loopholes — it feels very unlikely that the putative classicality of the experiment is hiding in whatever loophole might be available in a given system. But for cryptographic purposes, an adversary will use and loophole at their disposal to try to trick you!

There are three types of key assumptions that we will talk about here, and for each one we provide some exercises to work through in order to explore them further:

• detector efficiency (Exercise 6.8.7)
• locality (Exercise 6.8.8)
• free will (Exercise 6.8.9).

Let’s start with the first, which gives rise to the detector loophole. When we make a measurement with a real-life device, in practice it doesn’t always work — maybe it just fails to notice a photon flying past. Each detector has a parameter \eta known as its efficiency: \eta is the probability that the measurement succeeds. For testing fundamental physics, it seems reasonable to assume that the successful measurements are a fair sample of what’s really going on. But if there’s an adversary, they might substitute our detectors for completely perfect one, and then deliberately choose to fake a failure whenever their eavesdropping attempts fail.

The next crucial assumption in the CHSH test is that Alice and Bob are separated by a “large enough” distance in space and time. More precisely, if they are physical at distance L from each other, then their random choices of measurement setting, followed by their corresponding carrying out of the measurement, and receipt of the answers, should all be accomplished within a time approximately137 L/c of each other, where c is the speed of light. If Alice and Bob are not far enough away from each other, then they are said to be within each other’s locality, and so this is known as the locality loophole.

The final important assumption that we will mention here involves the availability of true randomness, and emphasises the importance of randomness expansion. It asserts that Alice and Bob must be able to choose their measurement settings randomly. This freedom to make their own choices is glibly referred to as them having “free will”, and so this is known as the free-will loophole. Resolving the locality loophole puts extremely tight constraints on how quickly choices must be made,138 to the extent that Alice and Bob cannot make those choices manually — they need to use random number generators. But then they need to able to trust that these generators are indeed random, not merely pseudorandom, otherwise somebody else could know the origin of the “random” numbers and use that information to their advantage.

1. We say “approximately” here because we are avoiding being specific about how we actually define distance.↩︎

2. If we say that Alice and Bob are L=30\,\mathrm{km} apart from each other, then we’re talking of timescales on the order of 10^{-4}\,\mathrm{s}, which is not very long!↩︎