The information-carrying capacity of certain quantum channels

TL;DR

This thesis analyzes the classical information capacity of specific quantum channels, deriving formulas for product-state capacities, demonstrating minimal ensembles for optimal encoding, and establishing capacity equivalences for channels with memory.

Contribution

It provides a detailed calculation of the product-state capacity for the qubit amplitude-damping channel and proves classical capacity equals product-state capacity for certain quantum channels with memory.

Findings

Product-state capacity of amplitude-damping channel determined by a transcendental equation

Minimal ensemble of non-orthogonal pure states achieves capacity

Classical capacity equals product-state capacity for channels with memory

Abstract

In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the "single-letter" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. Next we consider the classical capacity of two quantum channels with memory, namely a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof by Winter for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.