By name: majorization-entropy-inequalities description: "Majorization lattice framework for proving entropy inequalities in classical and quantum information theory. Covers supermodularity and subadditivity of all sum-concave entropies (Shannon, Rényi, Tsallis) via structural majorization relations. Use when analyzing entropy inequalities, information-theoretic bounds, quantum state entropy comparisons, or proving subadditivity/supermodularity results." license: Complete terms in LICENSE.txt metadata: arxiv_id: "2605.30331" published: "2026-05-31" category: "cs.IT, quant-ph" tags: ["information-theory", "quantum-information", "entropy", "majorization", "lattice-theory"]
Majorization Entropy Inequalities
Core Concepts
The majorization lattice provides a unified framework for proving entropy inequalities across all sum-concave entropy measures simultaneously.
Key Insight
Two structural majorization relations on the majorization lattice serve as precursors to:
- Supermodularity: f(x ∨ y) + f(x ∧ y) ≥ f(x) + f(y)
- Subadditivity: f(x ⊕ y) ≤ f(x) + f(y)
Since Shannon, Rényi, and Tsallis entropies are all sum-concave, proving inequalities at the majorization lattice level automatically implies them for ALL these entropy measures.
Mathematical Framework
Majorization order: For vectors x, y ∈ Rⁿ, x ≺ y iff partial sums of sorted components satisfy the majorization condition.
Lattice operations:
- Join (∨): least upper bound in majorization lattice
- Meet (∧): greatest lower bound
Entropy inheritance: If F is sum-concave and respects majorization order, supermodularity and subadditivity follow from lattice structure.
Usage Patterns
Pattern 1: Proving Entropy Inequalities
- Identify the probability distributions or quantum state eigenvalues
- Construct join (∨) and meet (∧) in the majorization lattice
- Verify structural majorization conditions
- Conclude inequality holds for all sum-concave entropies
Pattern 2: Quantum Information Applications
For quantum states ρ, σ:
- Use eigenvalue distributions λ(ρ), λ(σ) as classical analogs
- Apply same majorization lattice structure to spectra
- Entropy inequalities transfer to von Neumann entropy S(ρ) = -Tr(ρ log ρ)
Pattern 3: Unified Multi-Entropy Analysis
Instead of separate proofs for Shannon/Rényi/Tsallis:
- Prove at majorization lattice level
- All sum-concave entropies inherit the result
- Specialize if specific bounds are needed
When to Use
- Proving entropy inequalities (Shannon, Rényi, Tsallis, von Neumann)
- Analyzing quantum state entropy bounds
- Information-theoretic security proofs
- Comparing probability distributions under majorization
- Studying subadditivity or supermodularity properties
- Quantum cryptography conditional entropy analysis
Error Handling
- Non-sum-concave functions: Majorization approach may not apply; decompose or use alternative techniques
- Incomparable distributions: Consider ε-majorization or Lorenz curve methods
Related Skills
- quantum-information-protocol-analyzer
- quantum-probability-statistics
- quantum-fisher-information-duality