Quantum Blind Data Compression and Structure of Quantum Operations Approximately Preserving Quantum States

Abstract

In this thesis, we investigate a variation of quantum information processing tasks, blind data compression, and we analyze an approximation of a structure of a set of quantum states closely related to the task, which is called the Koashi-Imoto (KI) decomposition.

While various quantum information processing tasks have been extensively investigated in the framework of quantum Shannon theory, a problem called blind quantum data compression is considered as one of the most general forms of quantum data compression. It is known that its optimal compression rate within an asymptotically vanishing error is given by using the KI decomposition. However, it is also argued that allowing even an extremely small approximation causes a significant change in the compression rate. The sensitivity of the compression rate to approximations originates from the sensitivity of the KI decomposition. In this thesis, taking advantage of the sensitivity, we construct a novel protocol for blind quantum data compression that may perform remarkably well under the existence of finite approximations. Furthermore, to acquire insights into the instability of the KI decomposition and to analyze an approximation of the KI decomposition with finite approximations allowed, we investigate a structure of quantum channels that may lead to further understanding of an approximate structure of quantum states that is essential for more sophisticated error analysis of blind compression.

Our results shed light on an instability of the rate of blind quantum data compression against approximations. Our compression protocol makes the data transmission with approximations much more efficient. Furthermore, our results on the approximation of the KI decomposition provides us with insights into an approximate KI decomposition of quantum states that is essential to conduct more rigorous and general analysis of blind data compression, as well as contributes to foundation of quantum mechanics from the perspective of what restrictions are imposed to quantum operations when they cause a small disturbance. We believe that our work paves the way to further investigation of blind quantum data compression with finite approximations, and our results make substantial progress towards the general analysis of approximate KI decomposition, which is essential not only for the study of blind quantum data compression but also for investigation of other quantum phenomena characterized by the KI decomposition.