A Linearly Convergent Algorithm for Computing the Petz-Augustin Information

2025-02-10

AI Summary

This work addresses the lack of efficient, non-asymptotic convergent algorithms for computing the Petz–Augustin information over classical-quantum channels for α ∈ (1/2, 1) ∪ (1, ∞). We propose the first iterative algorithm with an explicit linear convergence guarantee. Our method leverages the density operator structure and integrates matrix analysis with quantum information theory to drastically reduce computational complexity: initialization costs O(nd³), and each iteration costs O(nd² + d³). The convergence rate is explicitly governed by |1 − 1/α|, yielding exponential error decay at rate O(|1 − 1/α|ᵀ), thereby overcoming prior algorithms’ inability to characterize convergence rates. Experimental results demonstrate that our algorithm significantly enhances the computability and practicality of the Petz–Augustin information for high-dimensional quantum channels.

Technology Category

Machine Learning: Information TheorySearch and Optimization: Non-convex OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We propose an iterative algorithm for computing the Petz-Augustin information of order $alphain(1/2,1)cup(1,infty)$. The optimization error is guaranteed to converge at a rate of $Oleft(vert 1-1/alpha vert^T ight)$, where $T$ is the number of iterations. Let $n$ denote the cardinality of the input alphabet of the classical-quantum channel, and $d$ the dimension of the quantum states. The algorithm has an initialization time complexity of $Oleft(n d^{3} ight)$ and a per-iteration time complexity of $Oleft(n d^{2}+d^3 ight)$. To the best of our knowledge, this is the first algorithm for computing the Petz-Augustin information with a non-asymptotic convergence guarantee.

Problem

• Computing Petz-Augustin information efficiently
• Ensuring linear convergence rate
• Handling classical-quantum channel complexity

Innovation

• Iterative algorithm for Petz-Augustin information
• Linear convergence rate guaranteed
• Efficient initialization and iteration complexity