Information-Theoretic Models for Physical Observables.

Area of Science:

• Information theory
• Quantum mechanics
• Information geometry

Background:

• John Archibald Wheeler's concept of the physical as information-theoretic.
• The need for a mathematical framework to connect information theory and quantum mechanics.

Purpose of the Study:

• To introduce a novel information-theoretic framework using information geometry.
• To explore the representation and distribution of information using quantum harmonic oscillators.
• To investigate the connection between estimator variance, quantum energy levels, and the Cramér-Rao lower bound.

Main Methods:

• Utilizing information geometry with the Fisher information metric.
• Applying the time-independent Schrödinger's equation to stationary states.
• Modeling information sources as quantum harmonic oscillators.
• Employing Bayes' theorem for posterior probability distribution calculations.

Main Results:

• Information is represented and distributed across quantum harmonic oscillators.
• Estimator variance is quantized and equals oscillator energy levels, with minimum variance at minimum energy.
• Quantum harmonic oscillators achieve the Cramér-Rao lower bound at the lowest energy level.
• The collective mode's probability density function at the lowest energy matches the posterior probability distribution.

Conclusions:

• The physical world can be decomposed into informational elements represented by quantum harmonic oscillators.
• The square modulus of the collective mode at the lowest energy signifies the most probable reality.
• This supports the view that the universe is fundamentally composed of informational, not physical, parts.