IntU.jl Documentation

Welcome to the documentation for IntU.jl, a generic symbolic integration engine for random matrices and quantum states.

Getting Started

INTU.jl is designed to make "back-of-the-envelope" calculations in Random Matrix Theory and Quantum Information exact and automated.

Installation

using Pkg
Pkg.add(url="https://github.com/iitis/IntU.jl")

The @integrate Macro

For more intuitive symbolic integration, IntU.jl provides the @integrate macro. It automatically identifies random matrices based on the measure and manages variable declarations.

@integrate expr measure

[!TIP] Symbol Scope and Redefinition: The macro manages a persistent symbolic state. If a symbol is used in one context (e.g., as a random matrix for dU) and then in another (e.g., as a constant for dO), the macro automatically re-binds it to the correct type. This "Safety Rebind" prevents silent mathematical errors during sequential execution.

Common Interface

The primary way to interact with IntU.jl is through the integrate function. It provides a unified interface for all supported groups and ensembles.

integrate(expr, measure)

• expr: A symbolic expression (e.g., product of SymbolicMatrix elements) or an array of such expressions.
• measure: A measure object defining the group/ensemble and its dimension (e.g., dU(d), dO(d), dGUE(d)).

Manual

• Unitary Integration: Core functionality for $U(d)$.
• Special Unitary Group: Integration over SU(d).
• Diagonal Unitaries: Integration over the Torus group.
• Orthogonal & Symplectic: Integration over real groups.
• Gaussian Ensembles: GUE, GOE, and GSE.
• Circular Ensembles: COE, CUE, and CSE.
• Permutation Groups: Symmetric and centered permutations.
• Unitary Designs: Exact moments of $t$-designs.
• Pure States: Integration over random vectors.
• Symbolic Trace Logic: Index-free matrix integration.
• Asymptotic Expansions: Large-$d$ limit analysis.
• Integral Library: Pre-computed standard results.
• QI Helpers: Tools for Purity, Fidelity, etc.
• ITensors Integration: Symmetric integration of tensor networks.

API Reference

See the API Reference for detailed function signatures.