A generalization of the Kullback–Leibler divergence and its properties

2009/02/28 by Takuya Yamano · 17 citations
Environmental Science · Physics and Astronomy · #Absolute continuity #Advanced Thermodynamics and Statistical Mechanics #Connection (principal bundle) #Divergence (linguistics) #Entropy (arrow of time) #Generalization #Information theory #Kullback–Leibler divergence #Probability distribution #Statistical Mechanics and Entropy #Sustainability and Ecological Systems Analysis #cond-mat.stat-mech

paper · pdf · doi:10.1063/1.3116115

published in Journal of Mathematical Physics 50(4) (American Institute of Physics) · revised, 10 pages

openalex publication_date 2009/04/01 · arxiv created 2009/04/30 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

A generalized Kullback–Leibler relative entropy is introduced starting with the symmetric Jackson derivative of the generalized overlap between two probability distributions. The generalization retains much of the structure possessed by the original formulation. We present the fundamental properties including positivity, metricity, concavity, bounds, and stability. In addition, a connection to shift information and behavior under Liouville dynamics are discussed.

Citations

• On Information and Sufficiency
• A single quantum cannot be cloned
• Communication by EPR devices
• Classical no-cloning theorem under Liouville dynamics by non-Csiszár f-divergence
• The role of relative entropy in quantum information theory