Abstract It is quite common to use the generalized probabilistic theories (GPTs) as generic models to reconstruct quantum theory from a few basic principles and to gain a better understanding of the probabilistic or information theoretic foundations of quantum physics and quantum computing. A variety of symmetry postulates was introduced and studied in this framework, including the transitivity of the automorphism group (1) on the pure states, (2) on the pairs of orthogonal pure states [these pairs are called 2-frames] and (3) on any frames of the same size. The second postulate is Müller and Ududec’s bit symmetry , which they motivate by quantum computational needs. Here we explore these three postulates in the transition probability framework, which is more specific than the GPTs since the existence of the transition probabilities for the quantum logical atoms is presupposed either directly or indirectly via a certain geometric property of the state space. This property for compact convex sets was introduced by the author in a recent paper. We show that bit symmetry implicates the symmetry of the transition probabilities between the atoms. Using a result by Barnum and Hilgert, we can then conclude that the third rather strong symmetry postulate rules out all models but the classical cases and the simple Euclidean Jordan algebras.Abstract It is quite common to use the generalized probabilistic theories (GPTs) as generic models to reconstruct quantum theory from a few basic principles and to gain a better understanding of the probabilistic or information theoretic foundations of quantum physics and quantum computing. A variety of symmetry postulates was introduced and studied in this framework, including the transitivity of the automorphism group (1) on the pure states, (2) on the pairs of orthogonal pure states [these pairs are called 2-frames] and (3) on any frames of the same size. The second postulate is Müller and Ududec’s bit symmetry , which they motivate by quantum computational needs. Here we explore these three postulates in the transition probability framework, which is more specific than the GPTs since the existence of the transition probabilities for the quantum logical atoms is presupposed either directly or indirectly via a certain geometric property of the state space. This property for compact convex sets was introduced by the author in a recent paper. We show that bit symmetry implicates the symmetry of the transition probabilities between the atoms. Using a result by Barnum and Hilgert, we can then conclude that the third rather strong symmetry postulate rules out all models but the classical cases and the simple Euclidean Jordan algebras.

arXiv:2411.18589Quantum Information and ComputationDec 1, 20250 citations

Generalized “transition probability”

Vittorio Cantoni

Communications in Mathematical PhysicsJun 1, 197570 citations

Twisted Abelian Fracton Models: Subdimensional Particles Constrained by Point Group Symmetries — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Twisted Abelian fracton models in 3D generalize lattice gauge theories with subdimensional particle mobility, constrained by point group symmetries. | MATH: Fracton models as generalized Abelian lattice gauge theories; point group constraints restrict particle mobility to lower-dimensional subspaces (e.g., lineons, planons). Key constants/ratios not explicitly given, but lattice structure implies discrete symmetry groups (e.g., cubic, tetrahedral). | CONNECTION: Crystallographic point groups (e.g., Oh, Td) and root systems (e.g., A3, B3) naturally constrain subdimensional excitations; lattice geometry dictates mobility patterns—fractons immobile, lineons move along lines, planons in planes. Ratios like 0.618, 1.618 may appear in quasicrystalline analogs but not in this Abelian setting. | DEPTH: 7 — Novel topological phases with constrained mobility, linking gauge theory, point group symmetry, and lattice geometry; deep implications for quantum matter and emergent spacetime. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Twisted Abelian fracton models in 3D generalize lattice gauge theories with subdimensional particle mobility, constrained by point group symmetries. | MATH: Fracton models as generalized Abelian lattice gauge theories; point group constraints restrict particle mobility to lower-dimensional subspaces (e.g., lineons, planons). Key constants/ratios not explicitly given, but lattice structure implies discrete symmetry groups (e.g., cubic, tetrahedral). | CONNECTION: Crystallographic point groups (e.g., Oh, Td) and root systems (e.g., A3, B3) naturally constrain subdimensional excitations; lattice geometry dictates mobility patterns—fractons immobile, lineons move along lines, planons in planes. Ratios like 0.618, 1.618 may appear in quasicrystalline analogs but not in this Abelian setting. | DEPTH: 7 — Novel topological phases with constrained mobility, linking gauge theory, point group symmetry, and lattice geometry; deep implications for quantum matter and emergent spacetime. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 23, 20260 citations

TRANSITION AMPLITUDES AND PROBABILITIES

H. Muirhead

Elsevier eBooksJan 1, 19710 citations

Transition probability

Institute of Physics eBooksNov 26, 20040 citations

Topological and conventional phases of a three-dimensional electronic glass

Prateek Mukati, Adhip Agarwala, Subhro Bhattacharjee

We investigate a symmetry-protected ${Z}_{2}$ topological electron glass, a glassy equivalent of the ${Z}_{2}$ topological band insulator in crystalline systems, and uncover associated quantum phase transitions in this three-dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the ${Z}_{2}$ glass is characterized by an anomalous electromagnetic response, i.e., it can host dyons with $\frac{1}{2}$ electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the ${Z}_{2}$ glass into a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and ``spectral inversion'' which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases, where strong structural disorder interplays with symmetry-protected topological order.We investigate a symmetry-protected ${Z}_{2}$ topological electron glass, a glassy equivalent of the ${Z}_{2}$ topological band insulator in crystalline systems, and uncover associated quantum phase transitions in this three-dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the ${Z}_{2}$ glass is characterized by an anomalous electromagnetic response, i.e., it can host dyons with $\frac{1}{2}$ electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the ${Z}_{2}$ glass into a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and ``spectral inversion'' which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases, where strong structural disorder interplays with symmetry-protected topological order.

arXiv:1907.10098Physical review. B./Physical review. BJan 27, 202025 citations

We investigate a symmetry-protected

{Z}_{2}

topological electron glass, a glassy equivalent of the

{Z}_{2}

topological band insulator in crystalline systems, and uncover associated quantum phase transitions in this three-dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the

{Z}_{2}

glass is characterized by an anomalous electromagnetic response, i.e., it can host dyons with

\frac{1}{2}

electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the

{Z}_{2}

glass into a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and ``spectral inversion'' which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases, where strong structural disorder interplays with symmetry-protected topological order.