Theory

Mathematical background for quantum information geometry and the inaccessible game.

• Matrix 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
• Hamiltonian Extraction from Antisymmetric GENERIC Flow
  • Gibbs-Locked Frames
  • Mathematical Background
  • Extraction Algorithm
  • Implementation
  • Validation and Limitations
  • See Also
  • References
• 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

Matrix Exponential Families

Coming soon: Mathematical theory of Matrix 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

Decomposition of dynamics into reversible and irreversible parts: \(M = S + A\). Covers Lie closure cancellation, when Duhamel integrals are required, and the practical consequences for qig computations.

See GENERIC Structure for full details.

Hamiltonian Extraction

How the antisymmetric sector \(A\) of the GENERIC Jacobian is mapped to an explicit effective Hamiltonian \(H_\text{eff}(\theta) = \sum_c \eta_c F_c\) via the Lie structure constants \(f_{abc}\). Explains the extraction algorithm, the Kubo-Mori kernel subtlety, and validated properties.

See Hamiltonian Extraction from Antisymmetric GENERIC Flow for full details.