Classical-Quantum Transition as a Disorder-Order Process

We associate here the relationship between de-coherence to the statistical notion of disequilibrium with regards to the dynamics of a system that reflects the interaction between matter and a given field. The process is described via information geometry. Some of its tools are shown here to appropriately explain the process’ mechanism. In particular we gain some insight into what is the role of the uncertainty principle (UP) in the pertinent proceedings.

Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

Information geometry, trade-off relations, and generalized Glansdorff-Prigogine criterion for stability

We discuss a relationship between information geometry and the Glansdorff-Prigogine criterion for stability. For the linear master equation, we found a relation between the line element and the excess entropy production rate. This relation leads to a new perspective of stability in a nonequilibrium steady-state. We also generalize the Glansdorff-Prigogine criterion for stability based on information geometry. Our information-geometric criterion for stability works well for the nonlinear master equation, where the Glansdorff-Prigogine criterion for stability does not work well. We derive a trade-off relation among the fluctuation of the observable, the mean change of the observable, and the intrinsic speed. We also derive a novel thermodynamic trade-off relation between the excess entropy production rate and the intrinsic speed. These trade-off relations provide a physical interpretation of our information-geometric criterion for stability. We illustrate our information-geometric criterion for stability by an autocatalytic reaction model, where dynamics are driven by a nonlinear master equation.

Space from entanglement: An information-geometric perspective

In this paper, we study the construction of classical geometry from the quantum entanglement structure by using information geometry. In the information geometry of classical spacetime, the Fisher information metric is related to a blurred metric of a classical physical space. We first show that a local information metric can be obtained from the entanglement contour in a local subregion. This local information metric measures the fine structure of entanglement spectra inside the subregion, which suggests a quantum origin of the information-geometric blurred space. We study both the continuous and the classical limits of the quantum-originated blurred space by using the techniques from the statistical sampling algorithms, the sampling theory of spacetime and the projective limit. A scheme for going from a blurred space with quantum features to a classical geometry is also explored.

Information Geometry, Fluctuations, Non-Equilibrium Thermodynamics, and Geodesics in Complex Systems

Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance by quantifying how information unfolds over time as a probability density function (PDF) evolves in time. Here, we discuss some recent developments in information geometric theory focusing on time-dependent dynamic aspects of non-equilibrium processes (e.g., time-varying mean value, time-varying variance, or temperature, etc.) and their thermodynamic and physical/biological implications. We compare different distances between two given PDFs and highlight the importance of a path-dependent distance for a time-dependent PDF. We then discuss the role of the information rate Γ=dLdt and relative entropy in non-equilibrium thermodynamic relations (entropy production rate, heat flux, dissipated work, non-equilibrium free energy, etc.), and various inequalities among them. Here, L is the information length representing the total number of statistically distinguishable states a PDF evolves through over time. We explore the implications of a geodesic solution in information geometry for self-organization and control.