Alternating minimization for computing doubly minimized Petz Renyi mutual information

2025/07/07 by Burri, Laura
#FOS: Computer and information sciences #FOS: Physical sciences #Information Theory (cs.IT) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2507.05205

Abstract

The doubly minimized Petz Renyi mutual information (PRMI) of order α is defined as the minimization of the Petz divergence of order α of a fixed bipartite quantum state ρ_AB relative to any product state σ_A⊗ τ_B. To date, no closed-form expression for this measure has been found, necessitating the development of numerical methods for its computation. In this work, we show that alternating minimization over σ_A and τ_B asymptotically converges to the doubly minimized PRMI for any α∈ ((1)/(2),1)∪ (1,2], by proving linear convergence of the objective function values with respect to the number of iterations for α∈ (1,2] and sublinear convergence for α∈ ((1)/(2),1). Previous studies have only addressed the specific case where ρ_AB is a classical-classical state, while our results hold for any quantum state ρ_AB.

Citations

• Min-reflected entropy = doubly minimized Petz Renyi mutual information of order 1/2
• Linear Convergence in Hilbert's Projective Metric for Computing Augustin Information and a Rényi Information Measure
• Doubly minimized Petz and sandwiched Renyi mutual information: Operational interpretation from binary quantum state discrimination
• Doubly minimized Petz and sandwiched Renyi mutual information: Properties
• Tight One-Shot Analysis for Convex Splitting with Applications in Quantum Information Theory
• A remainder term for H "older's inequality for matrices and quantum\n entropy inequalities
• Birkhoff's version of Hilbert's metric and its applications in analysis

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