Quantum Entropy Derived from First Principles

Ugo Farinelli, A. Gamba

David Sigtermans

We show that the mathematical structure of Hilbert space—typically postulated in quantum theory—emerges as a consequence of two fundamental principles derived within the TEQ framework: (1) entropy as a generative structural principle; (2) a minimal principle selecting stable distinctions. These principles, and the resulting entropy-weighted action functional, are derived from first principles in prior work. Here, we analyze the entropy-weighted path integral and the geometry of entropy curvature, and prove that the space of entropy-stabilized modes satisfies the axioms of a Hilbert space: linearity, inner product structure, norm completeness, and probabilistic interpretation. In this view, the Born rule itself emerges from entropy-weighted path selection. Importantly, this result shows that the TEQ framework fully recovers the standard Hilbert space formalism—not by postulate, but as a structural consequence of its core principles—thus preserving compatibility with quantum theory while providing it with a deeper generative foundation. Our results strengthen the explanatory power of the TEQ framework and offer a structural derivation of core quantum features from a common thermodynamic-geometric basis.

Dėnes Petz

Kevin Vanslette

This thesis synthesizes probability and entropic inference with Quantum Mechanics and quantum measurement [1-6]. It is shown that the standard and quantum relative entropies are tools \emph{designed} for the purpose of updating probability distributions and density matrices, respectively [1]. The derivation of the standard and quantum relative entropy are completed in tandem following the same inferential principles and design criteria. This provides the first design derivation of the quantum relative entropy while also reducing the number of required design criteria to two.

Jianhao M. Yang

Abstract Recently it is shown that the non-relativistic quantum formulations can be derived from an extended least action principle Yang (2023). In this paper, we apply the principle to massive scalar fields, and derive the Schrödinger equation of the wave functional for the scalar fields. The principle extends the least action principle in classical field theory by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a field needs to exhibit in order to be observable. Second, there are constant random field fluctuations. A novel method is introduced to define the information metrics to measure additional observable information due to the field fluctuations, which is then converted to the additional action through the first assumption. Applying the variation principle to minimize the total actions allows us to elegantly derive the transition probability of field fluctuations, the uncertainty relation, and the Schrödinger equation of the wave functional. Furthermore, by defining the information metrics for field fluctuations using general definitions of relative entropy, we obtain a generalized Schrödinger equation of the wave functional that depends on the order of relative entropy. Our results demonstrate that the extended least action principle can be applied to derive both non-relativistic quantum mechanics and relativistic quantum scalar field theory. We expect it can be further used to obtain quantum theory for non-scalar fields.