Second order asymptotics of visible mixed quantum source coding via universal codes

TL;DR

This paper establishes the second order asymptotic rates for fixed-length visible quantum source coding of mixed sources with memory, extending classical results and introducing universal quantum source codes.

Contribution

It derives the second order asymptotics for quantum sources with memory and develops universal codes based on Hayashi's classical construction.

Findings

First second order asymptotics for quantum sources with memory

Universal quantum source codes achieving these rates

Extension of classical second order results to quantum setting

Abstract

The simplest example of a quantum information source with memory is a mixed source which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of second order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han [IEEE Trans. on Inf. Th. 59.1 (2013), pp. 1-16]. To prove the achievability part of our main result, we introduce universal quantum source codes achieving second order asymptotic rates. These are obtained by an extension of Hayashi's construction [IEEE Trans. on Inf. Th. 54.10 (2008), pp. 4619-4637] of their classical counterparts.