On the Incompatibility of Quantum State Geometry and Fuzzy Metric Spaces: Three No-Go Theorems
Nicola Fabiano
TL;DR
The paper investigates whether fuzzy metric spaces can replicate the geometric structure of quantum state space. Using Gaussian states and the Hilbert-space distance $d_Q$, it proves three no-go results: (1) fuzzy systems cannot model destructive interference due to phase insensitivity, (2) there is no faithful, distance-preserving embedding of quantum state geometry into any fuzzy metric space, and (3) fuzzy logic cannot distinguish symmetric versus antisymmetric combinations of concepts. Collectively, these results argue that fuzzy frameworks are structurally incompatible with the inner-product and phase-sensitive geometry of quantum mechanics, while quantum state geometry provides a complete language for intrinsic uncertainty. The work suggests that quantum-inspired representations offer a more faithful foundation for modeling uncertain or context-dependent knowledge, with potential extensions to alternative logics and decoherence-based explanations.
Abstract
We prove three structural impossibility results demonstrating that fuzzy metric spaces cannot capture essential features of quantum state geometry. First, we show they cannot model destructive interference between concepts due to phase insensitivity. Second, we prove there is no distance-preserving embedding from quantum state space into any fuzzy metric space. Third, we establish that fuzzy logic cannot distinguish symmetric from antisymmetric concept combinations -- a fundamental limitation for modeling structured knowledge. These theorems collectively show that fuzzy frameworks are structurally incapable of representing intrinsic uncertainty, where quantum mechanics provides a superior, geometrically coherent alternative.