Advanced Topics
===============

*This section is under development.*

Entanglement with Pair Operators
---------------------------------

For studying entanglement, use pair operator bases that can create correlations.

High-Precision Derivatives (Duhamel)
-------------------------------------

The Duhamel integration method provides high-precision quantum derivatives.

Third-Order Cumulants
---------------------

The third cumulant :math:`\nabla G = \nabla \nabla \nabla \psi` describes how the Fisher metric varies.

Optimisation on Manifolds
--------------------------

Constrained optimization techniques for quantum state spaces.

See Also
--------

* :mod:`qig.duhamel` - High-precision derivatives
* :mod:`qig.pair_operators` - Entangling operators