Quantifying Contextuality of Empirical Models in Terms of Trace-Distance

In order to quantify contextuality of empirical models, the quantity of contextuality (QoC) of empirical models is introduced in terms of the trace-distance. Let Q C(e) denote the QoC of an empirical model e. The following conclusions are proved. (i) An empirical model e is non-contextual if and only if Q C(e)=0, and then it is contextual if and only if Q C(e)>0; (ii) the QoC function QC is convex, contractive and continuous. Finally, the QoC of some famous models is computed, including PM-isotropic boxes P M ^α , M-isotropic boxes M ^α , C H _n -isotropic boxes \(CH_{n}^{\alpha }\) as well as K box, where α∈0,1]. Moreover, P M ^α is non-contextual if and only if \(\alpha \in \frac {1}{6},\frac {5}{6}]\); M ^α is non-contextual if and only if \(\alpha \in 0,\frac {4}{5}]\); when n is even, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in \frac {1}{n},\frac {n-1}{n}]\), and when n is odd, \(CH_{n}^{\alpha }\) is non-contextual if and only if \(\alpha \in 0,\frac {n-1}{n}]\). The most important thing is that it is very easy to compare the QoC of any two isotropic boxes discussed in the above.