Quantum Interval-Valued Probability: Contextuality and the Born Rule

TL;DR

This paper introduces a quantum interval-valued probability framework to analyze how experimental imperfections affect foundational quantum principles like contextuality and the Born rule, providing bounds and potential resolutions for finite-precision scenarios.

Contribution

It develops a novel interval-valued probability approach to quantify the impact of finite measurement precision on quantum theorems such as Kochen-Specker and Gleason.

Findings

Establishes bounds on the validity of foundational theorems under finite precision.

Quantifies finite-precision measurement effects within the new framework.

Proposes a resolution to the Meyer-Mermin debate on measurement precision and contextuality.

Abstract

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.