Quantum interval-valued probability: Contextuality and the Born rule

We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and the Born rule. While foundational results such as the Kochen-Specker and Gleason theorems are valid in the context of infinite precision, they fail to hold in general in a world with limited resources. Here we employ an interval-valued framework to establish bounds on the validity of those theorems in realistic experimental environments. In this way, not only can we quantify the idea of finite-precision measurement within our theory, but we can also suggest a possible resolution of the Meyer-Mermin debate on the impact of finite-precision measurement on the Kochen-Specker theorem.

Figure 1

Region to the left of the vertical line at $\delta =\frac{1}{3}$ is where we assume small measurement degradation; in that region our extension of the KS theorem definitely demonstrates contextuality ($\mathsf{C}$). In the region to the right, the degradation of the data is large and our extension of the KS theorem no longer refutes other explanations for the experimental data.