[NCTS TG1.1 Seminar] When Additivity Meets Lorenz Geometry: Bounds and Uniqueness for Quantum Divergences

• Event Date: 2026-09-17
• Chen, Jhan Yu 陳詹喻
• Quantum Information and Quantum Computing Quantum computing and interdisciplinary applications
• Speaker: Prof. Gilad Gour (Technion - Israel Institute of Technology)  /  Host: Prof. Hao-Chung Cheng
  Place: Rm. 307, 3F, Cosmology Hall, NTU

Title: [NCTS TG1.1 Seminar] When Additivity Meets Lorenz Geometry: Bounds and Uniqueness for Quantum Divergences
Speaker: Prof. Gilad Gour (Technion - Israel Institute of Technology)
Time: 2026/9/17 (Thurs.) 14:30
Place: Rm. 307, 3F, Cosmology Hall, NTU
 

Abstract

Quantum divergences quantify distinguishability, resource conversion, and one-shot information processing, but their structure depends strongly on which operational comparisons are imposed. In this talk, I will discuss two related ways in which additivity interacts with Lorenz-type geometry. The first concerns sharp universal bounds for smoothed quantum divergences: classically, smoothing is governed by a clipped probability vector, independently of the particular divergence, leading to optimal state-independent and dimension-independent bounds for smoothed Rényi divergences of arbitrary order and for the hypothesis-testing divergence. The second concerns quantum Lorenz divergences, defined as monotones of the binary testing preorder associated with a pair of quantum states. Under a natural Lorenz-continuity assumption on the classical restriction, this Lorenz geometry determines at most one quantum extension. Combining this extension rigidity with additivity yields a uniqueness theorem: among normalized additive quantum Lorenz divergences, the Umegaki relative entropy is singled out. Unlike the classical case, where additive monotones allow mixtures of Rényi divergences, quantum noncommutativity collapses this freedom to the order-one case.