Theory

Mathematical background for quantum information geometry and the inaccessible game.

• Quantum Exponential Families
  • Mathematical Framework
  • Properties
  • Why Duhamel Integrals Appear
  • What Is Special About Our Geometry?
  • See Also
• Fisher Information and BKM Metric
  • The Bogoliubov-Kubo-Mori Metric
  • Quantum Covariance
  • See Also
• The Inaccessible Game
  • Information-Geometric Framework
  • The Central Question
  • Geometric Interpretation
  • See Also
• Origin Regularisation
  • The North Pole Analogy
  • Valid Regularisation Matrices
  • Efficiency Implications
  • Physics vs Efficiency Trade-off
  • Block-Diagonal Fisher Information
  • Different Origins: bell_indices
  • Further Reading
• GENERIC Structure
  • General Equation for Non-Equilibrium Reversible-Irreversible Coupling
  • Decomposition
  • For Constrained Quantum Dynamics
  • When Duhamel Integrals Are Needed
  • See Also
• Symbolic Computation for Qutrits
  • Overview
  • Natural Parameter Interpretation
  • The su(9) Pair Basis
  • LME Block Decomposition (Key Breakthrough)
  • Available Methods
  • Validation
  • Current Status
  • See Also

Quantum Exponential Families

Coming soon: Mathematical theory of quantum exponential families

Topics: - Definition and properties - Log-partition function \(\psi(\theta)\) - Expectation parameters and duality - Quantum Fisher information

Fisher Information and the BKM Metric

Coming soon: The Bogoliubov-Kubo-Mori metric

Topics: - Definition via perturbation theory - Kubo-Mori inner product - Relationship to quantum Fisher information - Geometric properties

The Inaccessible Game Framework

Coming soon: Information-geometric perspective on constrained dynamics

Topics: - Marginal entropy constraints - The inaccessible region - Information gain vs accessibility - Relationship to thermodynamics

Origin Regularisation

The LME origin (product of Bell states) lies at the pure-state boundary where natural parameters θ → -∞. The regularisation matrix σ encodes the “direction of approach” to this boundary. Key topics:

• The north pole analogy (coordinate singularity at pure states)

• Valid σ requirements (Hermitian, PSD, unit trace)

• Efficiency implications (isotropic vs product vs general)

• Block-diagonal Fisher information for product states

See Origin Regularisation for full details.

GENERIC Structure

Coming soon: General Equation for Non-Equilibrium Reversible-Irreversible Coupling

Topics: - Decomposition of dynamics: \(\dot{\theta} = F(\theta) = F_{\text{rev}} + F_{\text{irr}}\) - Jacobian structure: \(M = S + A\) - Relationship to Hamiltonian and gradient flows - Entropy production