majorization-supermodularity-information - SKILL.md Agent Skill

name: majorization-supermodularity-information description: "Majorization lattice supermodularity and subadditivity framework — two structural majorization relations (precursors) underlying supermodularity and subadditivity of all sum-concave functions including Tsallis, Rényi, and Shannon entropies on the majorization lattice. Applies to quantum information theory, entropy inequalities, lattice theory. Activation: majorization, supermodularity, subadditivity, majorization lattice, Tsallis entropy, Rényi entropy, sum-concave, information theory lattice, 信息论格, 优超格" arxiv_id: "2605.30331" arxiv_date: "2026-05-28"

Majorization Lattice Supermodularity & Subadditivity Framework

Source

arXiv: 2605.30331 — "Majorization precursors to supermodularity and subadditivity on the majorization lattice"
Authors: Alexander Stévins, Michael G. Jabbour, Serge Deside, Nicolas J. Cerf
Category: Information Theory (cs.IT)
Date: 2026-05-28

Core Concept

This paper establishes two structural majorization relations called "precursors" that underlie the properties of supermodularity and subadditivity on the lattice induced by majorization. These precursors immediately imply that all sums of concave functions ("sum-concave functions") are supermodular and subadditive on the majorization lattice.

Key Results

1. Majorization Precursors

Two structural relations on the majorization lattice that serve as foundations for:

Supermodularity: f(x ∨ y) + f(x ∧ y) ≥ f(x) + f(y)
Subadditivity: f(x ∨ y) ≤ f(x) + f(y) - f(x ∧ y)

2. Entropy Families Covered

Tsallis entropies (for all α): proven supermodular and strictly subadditive
Rényi entropies (for all α): proven supermodular and strictly subadditive
Shannon entropy: recovered as special case, proven strictly supermodular and strictly subadditive

3. Strengthened Inequalities

(i) All entropic functionals are strictly subadditive on the majorization lattice
(ii) Tsallis entropies (and Shannon entropy) are strictly supermodular on the majorization lattice

Mathematical Framework

Majorization Lattice

The majorization preorder induces a lattice structure on probability distributions
For distributions x, y: x ∧ y = meet (greatest lower bound), x ∨ y = join (least upper bound)
Sum-concave functions: functions that are sums of concave functions applied to individual components

Precursor Relations

The two structural majorization relations serve as "precursors" — stronger statements that immediately imply supermodularity and subadditivity for the entire class of sum-concave functions.

Applications

Quantum Information Theory

Nicolas J. Cerf is a leading quantum information theorist
Majorization is fundamental in quantum state transformation, entanglement theory
These results strengthen the mathematical foundation of quantum entropy inequalities
Applicable to quantum resource theories where majorization determines state convertibility

Classical Information Theory

Strengthens known entropy inequalities (Shannon, Tsallis, Rényi)
Provides lattice-theoretic understanding of entropy properties
Unifies treatment of different entropy families under a single framework

Machine Learning & Optimization

Sum-concave functions appear in regularization, information-theoretic objectives
Supermodularity enables efficient optimization (greedy algorithms with guarantees)
Subadditivity bounds for information-theoretic generalization bounds

Reusable Patterns

Pattern 1: Lattice-Theoretic Entropy Analysis

Problem: Prove entropy inequality on probability distributions
Approach: 
  1. Identify the relevant lattice structure (majorization lattice)
  2. Find structural "precursor" relations on the lattice
  3. Show the entropy function is sum-concave
  4. Apply precursor → supermodularity/subadditivity follows immediately

Pattern 2: Unification via Sum-Concavity

Problem: Multiple entropy families need separate proofs
Approach:
  1. Show all target functions are sum-concave (sum of concave functions)
  2. Prove precursor relation holds on the lattice
  3. All sum-concave functions inherit the property simultaneously
  4. Strengthens to strict inequality where possible

Pattern 3: Strengthening Known Results

Problem: Existing inequality is non-strict
Approach:
  1. Start with known non-strict inequality
  2. Analyze equality conditions on the lattice
  3. Show equality holds only in trivial/degenerate cases
  4. Conclude strict inequality for all non-trivial inputs

Connections to Existing Skills

quantum-entropy-inequalities: This provides the lattice-theoretic foundation for many quantum entropy inequalities
majorization-quantum-state-transform: Majorization is the core ordering relation in quantum state transformation theory
information-theoretic-generalization: Subadditivity of entropy connects to information-theoretic generalization bounds

Activation Keywords

majorization, supermodularity, subadditivity, majorization lattice, Tsallis entropy, Rényi entropy, Shannon entropy, sum-concave function, information theory, lattice theory, quantum information theory, entropy inequalities, 优超, 次可加性, 超模性, 格理论