Quantum information theory

This dissertation investigates a theory of the manipulation of quantum states and their protection against noise. This theory shows similarities to classical information theory, and crucial differences arising from fundamental differences between quantum and classical states. The classical theory plays an important role in computer science and communications engineering; quantum information theory will have a similar role in the emerging fields of quantum computation, communication, and cryptography. Like classical information theory, it may also prove relevant to basic physics, from quantum chaos to statistical mechanics and the foundations of quantum mechanics.; The first two chapters introduce fundamental ideas and technical tools of classical and quantum information theory. Since quantum information theory has to do with how much we know about the quantum state of a system, Chapter Three investigates the structure of the space of these states. A geometry whose line element represents distinguishability of nearby states is used; its geodesics and associated optimal measurements are derived, and a way of generalizing the notion of superposition to mixed states is presented. Chapter Four explores the tradeoff between disturbance and information gain in quantum measurement. Chapter Five treats compression of quantum states, extending the quantum noiseless coding theorem to allow general decodings. Chapter Six discusses quantum error correcting codes and the reversal of quantum operations, while Chapter Seven derives an upper bound on capacity for transmission of a state through a quantum channel while preserving its entanglement with a reference system. Chapter Eight shows equivalence between this task and that of transmitting all states in a given subspace with high fidelity. Chapter Nine bounds the capacity of noisy quantum channels for sending ensembles of quantum states with high average fidelity. Chapter Ten shows that the encodings required for quantum transmission may be restricted in a plausible way. Chapter Eleven introduces quantum rate-distortion theory, and derives a lower bound on the rate-distortion function. Chapter Twelve reexamines the idea of a quantum information theory in light of the results of the dissertation, concludes that quantum information theory is shaping up as a worthy analogue of classical information theory, and highlights the significant similarities and the fascinating differences between the two.