Quantum Theory as Symmetry Broken by Vitality

TL;DR

The paper addresses why finite-dimensional quantum theory emerges from a principle of vitality that rejects intrinsic hidden variables but permits a reference measurement. It develops a QBist, bilinear probabilistic framework in which quantum probabilities are computed via the urgleichung $q(j) = \sum_i \left[(d+1)p(i) - \frac{1}{d}\right] r(j|i)$, with the reference measurement encoded by the matrix $\Phi$ and the SIC-based geometry of qplexes. Through van Fraassen's reflection principle and the structure of Hilbert qplexes, the work narrows the algebraic options to the complex Hilbert space (via $N = d^2$ and $q=2$), arguing away real, quaternionic, and most octonionic alternatives and tying SIC existence to the viability of the framework. The paper then articulates a geometric and algebraic reconstruction program—defining Hilbert qplexes, the QBic constraint, and a spectral decomposition with at most $d$ maximally distant states—to argue that complex quantum theory is the maximally symmetric probabilistic theory compatible with quantum vitality, with implications for foundational research and SIC existence.

Abstract

I summarize a research program that aims to reconstruct quantum theory from a fundamental physical principle that, while a quantum system has no intrinsic hidden variables, it can be understood using a reference measurement. This program reduces the physical question of why the quantum formalism is empirically successful to the mathematical question of why complete sets of equiangular lines appear to exist in complex vector spaces when they do not exist in real ones. My primary goal is to clarify motivations, rather than to present a closed book of numbered theorems, and consequently the discussion is more in the manner of a colloquium than a PRL.

Figures (1)

• Figure 1: A choice between two experiments. In one scenario (solid line), a system is fed directly into a measuring appartus. In the other (dashed line), the system is sent through the reference measurement first. Probability theory does not itself enforce a relation between an agent's probabilities for these two scenarios. Different conditions, different probabilities! The classical intuition that the reference measurement just reads off the system's intrinsic degrees of freedom leads to using the Law of Total Probability to relate expectations between the two scenarios. Quantum theory, on the other hand, provides its own relation. Our goal is to identify exactly what physical principle implies the quantum relation. (After DeBrota:2018a.)