Quantum Shannon theory made robust: a tale of three protocols for almost i.i.d. sources

TL;DR

This paper investigates the robustness of fundamental quantum information protocols under near-i.i.d. conditions, introducing new concepts like club distance to analyze their performance beyond ideal assumptions.

Contribution

It introduces protocols that maintain optimal rates with almost i.i.d. resources and develops the notion of club distance and almost i.i.d. processes.

Findings

Protocols remain effective under arbitrary almost i.i.d. resources.

Introduces the concept of club distance as a variant of diamond distance.

Analyzes hypothesis testing, data compression, and channel coding in near-i.i.d. settings.

Abstract

The asymptotic rates of information-theoretic protocols - including error exponents, compression rates, and channel capacities - are traditionally defined under the idealised assumption that the underlying resource (state or channel) is independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can lead to a drastic breakdown of these protocols. The asymptotic rates of information theoretic protocols - error exponents, compression rates, capacities - were originally evaluated taking for granted that the underlying source (state or channel) is i.i.d. Differently from what we might expect at first glance, it is not hard to exhibit instances of protocols that may drastically fail when the i.i.d. assumption holds only approximately rather than exactly. If the precise nature of the perturbation from the i.i.d. regime is known (e.g. a pointwise defect), we could design a bespoke protocol that compensates for the defect (for example, by discarding the corrupted subsystem). However, in any realistic setting, neither can the i.i.d. behaviour of the system be precisely guaranteed, nor can the deviations from the ideal regime be determined exactly. In this paper we answer the following question: are there protocols that can still achieve the optimal asymptotic rates when the i.i.d. resource is replaced by any arbitrary almost i.i.d. resource along it? What is the nature of the unknown perturbation under which protocols like these are possible? We focus, in particular, on hypothesis testing, data compression, and channel coding. As a by-product of our analysis, we introduce the notion of club distance, as a variant of the well-known diamond distance, and of an almost i.i.d. process, which may be of independent interest.