Bounded information as a foundation for quantum theory

Abstract

The purpose of this paper is to formalize the concept that best synthesizes our intuitive understanding of quantum mechanics -- that the information carried by a system is limited -- and, from this principle, to construct the foundations of quantum theory. In our discussion, we also introduce a second important hypothesis: if a measurement closely approximates an ideal one in terms of experimental precision, the information it provides about a physical system is independent of the measurement method and, specifically, of the system's physical quantities being measured. This principle can be expressed in terms of metric properties of a manifold whose points represent the state of the system. These and other reasonable hypotheses provide the foundation for a framework of quantum reconstruction. The theory presented in this paper is based on a description of physical systems in terms of their statistical properties, specifically statistical parameters, and focuses on the study of estimators for these parameters. To achieve the goal of quantum reconstruction, a divide-and-conquer approach is employed, wherein the space of two discrete conjugate Hamiltonian variables is partitioned into a binary tree of nested sets. This approach naturally leads to the reconstruction of the linear and probabilistic structure of quantum mechanics.