| name | loewner-order-algorithm |
| description | Iterative algorithm to compute minimal upper bounds in the Loewner order on Hermitian matrices. Use for quantum information, convex optimization, operator theory, and numerical linear algebra tasks involving matrix inequalities. |
| metadata | {"arxiv_id":"2606.18173","published":"2026-06-16","authors":"Adam Humeniuk, Gabriel Jarry-Bolduc, Patrick Pascua","tags":["quantum-information","matrix-analysis","optimization","linear-algebra","loewner-order"]} |
Loewner Order Minimal Upper Bound Algorithm
Description
An iterative method to exactly compute a minimal upper bound (MUB) for any finite collection of n×n Hermitian matrices under the Loewner order (positive semidefiniteness). The algorithm terminates in at most n iterations and applies to quantum information, control theory, numerical linear algebra, and operator theory.
Activation Keywords
- loewner order, loewner upper bound
- hermitian matrix bound, matrix partial order
- quantum information matrix, matrix inequality
- positive semidefinite bound, minimal upper bound
- 算子理论, 半正定矩阵
- 量子信息矩阵不等式
Core Mathematical Framework
Loewner Order
For Hermitian matrices A, B: A ≤ B iff B - A is positive semidefinite (PSD).
Key property: Unlike scalar ordering, the Loewner order is a partial order — two Hermitian matrices may have multiple incomparable minimal upper bounds.
Minimal Upper Bound (MUB)
A matrix M is a MUB of {A₁, ..., Aₖ} if:
- M ≥ Aᵢ for all i (upper bound)
- No M' < M satisfies M' ≥ Aᵢ for all i (minimal)
Algorithm (at most n iterations)
Input: Hermitian matrices {A₁, ..., Aₖ} ∈ ℂⁿˣⁿ
Initialize: M₀ = max(λ_max(Aᵢ)) · I (spectral upper bound)
For t = 0, 1, ..., n-1:
1. Compute Dₜ = Mₜ - A_j for the "tightest" constraint j
2. Find the smallest eigenvalue λ_min(Dₜ) and eigenvector v
3. If λ_min(Dₜ) ≥ 0: Mₜ is already PSD for all constraints → done
4. Update: Mₜ₊₁ = Mₜ - α·vv† where α minimizes the update
Output: Minimal upper bound M
Key Theoretical Result
- Algorithm terminates in at most n iterations
- The MUB is characterized algebraically by Stott's conditions
- Self-contained proof provided in the paper
Usage Patterns
Pattern 1: Quantum State Comparison
Compare quantum states via density matrix ordering — find minimal states that dominate a set of given density matrices.
Pattern 2: Convex Optimization
Use MUB computation as a subroutine in semidefinite programming where constraints are defined by Loewner ordering.
Pattern 3: Control Theory
Compute minimal Lyapunov function bounds for stability analysis of linear systems.
Error Handling
Numerical Instability
If eigenvalue computations become unstable near termination:
- Use high-precision arithmetic for final iterations
- Apply small regularization ε·I to ensure strict PSD
Non-uniqueness
MUBs are not unique in general — the algorithm finds ONE minimal bound. For comprehensive analysis, iterate with different initializations.
References
- arXiv: 2606.18173 — "An algorithm to exactly compute minimal upper bounds in the Loewner order"
