# The Quantum Question Equality

The Quantum Question (QQ) Equality is an interesting restriction on the predictions of simple quantum models of order effects in binary choice. It was originally presented by Busemeyer and Wang (2013), and received spectacular experimental confirmation in a paper by Wang et al (2014).

For two binary events A and B, let [latex]p(A,B)[/latex] denote the probability that A and then B happens, [latex]p(An,B)[/latex] denote the probability that not A and then B happens etc. Then the QQ Equality reads,

[latex]p(A,B) + p(An,Bn)-p(B,A)-p(Bn,An)=0[/latex]

(This form is slightly different from that presented in Busemeyer and Wang, but I like this form because it shows the essential symmetry better.)

The fact that this relationship was predicted from quantum models, has been observed in experiments, and does not appear to follow from alternative models of order effects has been taken to provide powerful evidence for quantum models. However in the above papers the derivation of the QQ Equality was rather long and not very transparent. This is important because it is far from clear what feature of quantum theory leads to the QQ Equality, and therefore it’s hard to know what it teaches us about cognition.

In my recent tutorial paper I provided a much more streamlined derivation of the QQ Equality, but a standalone proof can be found here. One interesting thing that emerges is that it’s very unlikely that the QQ Equality can be generalised to more than two measurements, or measurements with more than two outcomes.

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