Entropy Concentration and Universal Typicality for Weakly Almost i.i.d. Quantum Sources

TL;DR

This paper establishes concentration principles for weakly almost i.i.d. quantum sources, enabling a unified approach to quantum information tasks beyond traditional i.i.d. assumptions.

Contribution

It introduces noncommutative concentration principles for such sources, facilitating new proofs and bounds in quantum information theory.

Findings

Proves a noncommutative weak law of large numbers for empirical observables.

Shows asymptotic entropy concentration on subspaces governed by von Neumann entropy.

Provides applications to universal quantum compression and hypothesis testing.

Abstract

Weakly almost i.i.d. quantum sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We establish two concentration principles for such sources: a noncommutative weak law of large numbers for empirical observables, and a universal entropy-concentration principle showing asymptotic concentration on subspaces of exponential dimension governed by the von Neumann entropy of the reference state. These concentration principles provide a unified and conceptually transparent approach to several information-theoretic applications beyond the i.i.d. setting, including direct proofs of universal compression within classes of weakly almost i.i.d. sources sharing a common reference state, asymmetric quantum hypothesis-testing bounds, concentration results for macroscopic observables in quantum many-body systems including generalized Gibbs ensembles and for repeated local measurement statistics, as well as bounds on smooth- and spectral entropy quantities.