Operator–Scalar Curvature Gap for the Depolarising Lindbladian on Mn(C) and NESS Foundations

Mathematica Imagnis: Mathematical and Physical Foundations of Quantum Curvature and Non-Equilibrium Steady States presents a unified geometric framework for the analysis of quantum Markov semigroups across both mathematical and physical domains. The work is organized into two complementary components. The mathematical part develops a theory of quantum Ricci curvature based on the symmetric s=12s=\tfrac{1}{2}s=21 GNS inner product, establishing curvature–dimension inequalities and their connection to the Modified Logarithmic Sobolev Inequality (MLSI). The physical part applies this geometric perspective to boundary-driven quantum systems, with particular emphasis on the XX spin chain, demonstrating the thermodynamic persistence of scalar curvature and providing hypocoercive convergence rates to non-equilibrium steady states (NESS). By integrating methods from quantum optimal transport, functional inequalities, and open quantum system theory, this volume bridges microscopic dissipative dynamics with macroscopic transport phenomena. The repository includes both standalone manuscripts and a unified monolithic edition, offering a comprehensive resource for researchers in mathematical physics and quantum information science.