Topics in Estimation of Quantum Channels

TL;DR

This thesis explores bounds on quantum channel estimation, introduces a related Riemannian metric, and proposes improved methods for estimating parameters of unitary channels with minimal resources.

Contribution

It answers key questions about the Sarovar-Milburn Fisher information bound, introduces a new quantum state metric, and presents a resource-efficient phase estimation method.

Findings

Bounded Fisher information for quantum channels is characterized.

Combining multiple non-identical unitary channels improves estimation accuracy.

A simple, resource-efficient phase estimation method is developed.

Abstract

A quantum channel is a mapping which sends density matrices to density matrices. The estimation of quantum channels is of great importance to the field of quantum information. In this thesis two topics related to estimation of quantum channels are investigated. The first of these is the upper bound of Sarovar and Milburn (2006) on the Fisher information obtainable by measuring the output of a channel. Two questions raised by Sarovar and Milburn about their bound are answered. A Riemannian metric on the space of quantum states is introduced, related to the construction of the Sarovar and Milburn bound. Its properties are characterized. The second topic investigated is the estimation of unitary channels. The situation is considered in which an experimenter has several non-identical unitary channels that have the same parameter. It is shown that it is possible to improve estimation using the channels together, analogous to the case of identical unitary channels. Also, a new method of phase estimation is given based on a method sketched by Kitaev (1996). Unlike other phase estimation procedures which perform similarly, this procedure requires only very basic experimental resources.