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Entropy Stability and Spectral Concentration under Convex Block Constraints

Hassan Nasreddine

TL;DR

This work establishes a general, dimension-free stability principle for entropy minimizers under convex block-diagonal constraints. By decomposing entropy into classical and internal components, the authors prove a sharp quadratic lower bound: a state with entropy within ε of the constrained minimum lies within O(√ε) in trace norm of the minimizers, and this exponent is optimal. The results hold for finite-dimensional Hilbert spaces with purely geometric assumptions on the block structure and constraint sets, without requiring dynamics or spectral gaps. An explicit constrained Gibbs-ensemble application demonstrates the practicality of the bound, and the work connects to majorization, Pinsker-type inequalities, and convex-geometry perspectives on entropy. Overall, the paper provides a robust, intrinsic stability mechanism for entropy minimization under structured linear spectral constraints, with broad implications for quantum information and statistical mechanics.

Abstract

We investigate entropy minimization problems for quantum states subject to convex block-diagonal constraints. Our principal result is a quantitative stability theorem: if a state has entropy within epsilon of the minimum possible value under a fixed block constraint, then it must lie within O(epsilon^{1/2}) in trace norm of the manifold of entropy minimizers. We show that this rate is optimal. The analysis is entirely finite-dimensional and relies on a precise decomposition of entropy into classical and internal components, together with sharp relative entropy inequalities. As an application, we study finite additive operators whose spectral decomposition induces natural block constraints. In this setting, the stability theorem yields quantitative non-concentration bounds for induced spectral measures. The framework is abstract and independent of arithmetic input. It provides a general stability principle for entropy minimizers under linear spectral constraints.

Entropy Stability and Spectral Concentration under Convex Block Constraints

TL;DR

This work establishes a general, dimension-free stability principle for entropy minimizers under convex block-diagonal constraints. By decomposing entropy into classical and internal components, the authors prove a sharp quadratic lower bound: a state with entropy within ε of the constrained minimum lies within O(√ε) in trace norm of the minimizers, and this exponent is optimal. The results hold for finite-dimensional Hilbert spaces with purely geometric assumptions on the block structure and constraint sets, without requiring dynamics or spectral gaps. An explicit constrained Gibbs-ensemble application demonstrates the practicality of the bound, and the work connects to majorization, Pinsker-type inequalities, and convex-geometry perspectives on entropy. Overall, the paper provides a robust, intrinsic stability mechanism for entropy minimization under structured linear spectral constraints, with broad implications for quantum information and statistical mechanics.

Abstract

We investigate entropy minimization problems for quantum states subject to convex block-diagonal constraints. Our principal result is a quantitative stability theorem: if a state has entropy within epsilon of the minimum possible value under a fixed block constraint, then it must lie within O(epsilon^{1/2}) in trace norm of the manifold of entropy minimizers. We show that this rate is optimal. The analysis is entirely finite-dimensional and relies on a precise decomposition of entropy into classical and internal components, together with sharp relative entropy inequalities. As an application, we study finite additive operators whose spectral decomposition induces natural block constraints. In this setting, the stability theorem yields quantitative non-concentration bounds for induced spectral measures. The framework is abstract and independent of arithmetic input. It provides a general stability principle for entropy minimizers under linear spectral constraints.

Paper Structure

This paper contains 66 sections, 29 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Block-diagonal constraints
  4. Main result (informal)
  5. Relation to existing work
  6. Structure of the paper
  7. Preliminaries and Notation
  8. Quantum states and norms
  9. Von Neumann entropy and relative entropy
  10. Entropy inequalities
  11. Purity and effective dimension
  12. Constrained entropy minimization
  13. Convex Block Constraints and Geometric Structure
  14. Orthogonal block decompositions
  15. Block-diagonal states
  16. ...and 51 more sections

Key Result

Lemma 2.1

Let be an orthogonal decomposition, and let $\rho\in\mathcal{S}(\mathcal{H})$ be block-diagonal with respect to this decomposition: where $p_i\ge0$, $\sum_i p_i=1$, and $\rho_i\in\mathcal{S}(\mathcal{H}_i)$. Then where $H(p):=-\sum_i p_i\log p_i$ is the Shannon entropy.

Theorems & Definitions (64)

  • Lemma 2.1: Entropy decomposition for block-diagonal states
  • proof
  • Lemma 2.2: Pinsker inequality
  • Remark 2.3
  • Lemma 2.4: Entropy--purity bound
  • proof
  • Remark 2.5
  • Definition 3.1: Block-diagonal algebra
  • Definition 3.2: Block-diagonal states
  • Definition 3.3: Block-convex constraint set
  • ...and 54 more