New approaches to almost i.i.d. information theory

2026/05/31 by Filippo Girardi, Giacomo De Palma, Ludovico Lami
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.MP

paper · pdf

25 pages, 1 figure. Sections 3.4 and 3.5 are new

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

Independent and identically distributed (i.i.d.) states are ubiquitous in quantum information theory. However, in a practical setting, the i.i.d. assumption is too stringent, and possibly not realistic. A physically more compelling class of 'almost i.i.d.' sources was recently proposed by [Mazzola/Sutter/Renner, arXiv:2603.15792]. In this paper, we introduce two alternative definitions of almost i.i.d. states, based on the normalised quantum Wasserstein distance and on the idea of looking at the average k-body marginal. We explore some basic properties of these notions and prove a strict hierarchical relation among them, with Mazzola et al.'s notion being the strictest, the one based on k-body marginals the loosest, and the one based on the quantum Wasserstein distance in between. Strict separation is established by means of explicit examples.

Citations

• A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature
• A doubly composite Chernoff-Stein lemma and its applications
• A solution of the generalised quantum Stein's lemma
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach
• Quantum Differential Privacy: An Information Theory Perspective
• Exploring pure quantum states with maximally mixed reductions
• A Generalization of Quantum Stein's Lemma
• Optimal Transport
• Symmetry implies independence
• A sharp continuity estimate for the von Neumann entropy
• One-and-a-Half Quantum de Finetti Theorems
• A de Finetti representation for finite symmetric quantum states
• The private classical capacity and quantum capacity of a quantum channel
• Unknown Quantum States: The Quantum de Finetti Representation
• Coding Theorem and Strong Converse for Quantum Channels
• The Capacity of Quantum Channel with General Signal States
• Capacity of the noisy quantum channel
• Concentrating Partial Entanglement by Local Operations
• Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
• A continuity property of the entropy density for spin lattice systems
• Logical foundations and measurement of subjective probability