The theory of decoherence aims to explain how macroscopic quantum objects become effectively classical. Understanding this process could help in the search for the quantum theory underlying gravity, and suggest new schemes for preserving the coherence of technological quantum devices. The process of decoherence is best understood in terms of information flow within a quantum system, and between the system and its environment. We develop a novel way of characterizing this information, and give a sufficient condition for its classicality. These results generalize previous models of decoherence, clarify the process by which a phase-space based on non-commutative quantum variables can emerge, and provide a possible explanation for the universality of the phenomenon of decoherence. In addition, the tools developed in this approach generalize the theory of quantum error correction to infinite-dimensional Hilbert spaces. We characterize the nature of the information preserved by a quantum channel by the observables which exist in its image (in the Heisenberg picture). The sharp observables preserved by a channel form an operator algebra which can be characterized in terms of the channel's elements. The effect of the channel on these observables can be reversed by another physical transformation. These results generalize the theory of quantum error correction to codes characterized by arbitrary von Neumann algebras, which can represent hybrid quantum-classical information, continuous variable systems, or certain quantum field theories. The preserved unsharp observables (positive operator-valued measures) allow for a finer characterization of the information preserved by a channel. We show that the only type of information which can be duplicated arbitrarily many times consists of coarse-grainings of a single POVM. Based on these results, we propose a model of decoherence which can account for the emergence of a realistic classical phase-space. This model supports the view that the quantum-classical correspondence is given by a quantum-to-classical channel, which is another way of representing a POVM.

The theory of decoherence aims to explain how macroscopic quantum objects become effectively classical. Understanding this process could help in the search for the quantum theory underlying gravity, and suggest new schemes for preserving the coherence of technological quantum devices. The process of decoherence is best understood in terms of information flow within a quantum system, and between the system and its environment. We develop a novel way of characterizing this information, and give a sufficient condition for its classicality. These results generalize previous models of decoherence, clarify the process by which a phase-space based on non-commutative quantum variables can emerge, and provide a possible explanation for the universality of the phenomenon of decoherence. In addition, the tools developed in this approach generalize the theory of quantum error correction to infinite-dimensional Hilbert spaces. We characterize the nature of the information preserved by a quantum channel by the observables which exist in its image (in the Heisenberg picture). The sharp observables preserved by a channel form an operator algebra which can be characterized in terms of the channel's elements. The effect of the channel on these observables can be reversed by another physical transformation. These results generalize the theory of quantum error correction to codes characterized by arbitrary von Neumann algebras, which can represent hybrid quantum-classical information, continuous variable systems, or certain quantum field theories. The preserved unsharp observables (positive operator-valued measures) allow for a finer characterization of the information preserved by a channel. We show that the only type of information which can be duplicated arbitrarily many times consists of coarse-grainings of a single POVM. Based on these results, we propose a model of decoherence which can account for the emergence of a realistic classical phase-space. This model supports the view that the quantum-classical correspondence is given by a quantum-to-classical channel, which is another way of representing a POVM.

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of "good codes" are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes - nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.