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

TL;DR

This paper introduces two new definitions of 'almost i.i.d.' quantum states using quantum Wasserstein distance and k-body marginals, analyzing their properties and hierarchical relations.

Contribution

It proposes alternative, rigorous definitions of almost i.i.d. states and establishes their hierarchical relationships with explicit examples.

Findings

Hierarchical relation among the definitions is proven.

Explicit examples demonstrate strict separation between notions.

The new definitions extend the understanding of almost i.i.d. states in quantum information.

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.