Geometry of quantum states

Jincheng Zhang

Quantum Geometry – Geometric Quantum State Collapse Core Claim: Quantify the geometric properties of quantum states – specifically, the 'collapse' of quantum states – by developing a mathematical model where the state's geometry directly influences its collapse. Core Mechanism: Model the collapse as a phase transition – where the state's geometry changes drastically, triggering a change in the quantum state. This will be based on a new understanding of entanglement and the relationship between geometry and quantum mechanics. Why It Is New: Current quantum mechanics models often treat quantum states as purely probabilistic. This proposes a direct, geometric connection.

Samuel L. Braunstein, Carlton M. Caves

Rowan Brad Quni-Gudzinas

This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.

Julian Schwinger

José Ignacio Rosado

The geometry of the Quantum State Space, described by Bloch vectors, is a very intricate one. A deeper understanding of this geometry could lead to the solution of some difficult problems in Quantum Foundations and Quantum Information such as the existence of SIC-POVMs and the cardinality of the maximal set of MUBs. In this paper we show that the geometry of quantum states can be described by the probability distributions that quantum states induce over the outcomes of symmetric POVMs not necessarily of arbitrary rank or informationally complete. We also describe the properties of these symmetric POVMs.

Samuel L. Braunstein, Carlton M. Caves