Distillation and simulation in quantum information

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We use the techniques of convex optimization, especially semidefinite programming, to study two kinds of fundamental tasks, i.e., distillation and simulation in quantum information theory. We investigate these tasks in a unified framework of resource theory and focus on their computation and characterization with finite resources. Particularly we study the tradeoff among relevant parameters such as the number of resource copies, resource transformation rate, error tolerance and success probability. In the first part, we study the task of distillation for two different resources, maximally entangled state and maximally coherent state, representing nonlocal and local “quantumness” respectively. For entanglement distillation, we derive an efficiently computable second-order estimation of the distillation rate for general quantum states, which are tight for quantum states of practical interest. Our study overcomes the limitation of conventional research either focusing on the asymptotic rate or ignoring the computability. For the coherence distillation, we perform finite analysis for both deterministic and probabilistic scenarios. Our results unveil several new features of coherence from a resource theoretic viewpoint and contribute to an increased understanding of the fundamental properties of different sets of free operations. In the second part, we investigate the resource cost of simulating a quantum channel via quantum coherence or another quantum channel. We introduce the channel’s analogs of max-relative entropy, logarithmic robustness and max-information of quantum states, providing their operational interpretation with the channel simulation cost via different resources. Particularly, we establish the asymptotic equipartition property of the channel’s max-information, that is, it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. As applications, this asymptotic equipartition property implies the quantum reverse Shannon theorem in the presence of non-signalling correlations. From the perspective of resource theory, the worth of a resource can usually be characterized by the minimum distance to a set of useless resources under a proper distance measure. We give such characterization for all the tasks studied in this thesis, and find that the distance measure for the distillation and simulation process naturally corresponds to the quantum hypothesis testing relative entropy and the max-relative entropy, respectively.
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