QIP2021 | No-go theorems and limitations for quantum resource purification (Liu&Regula)

Authors: Kun Fang and Zi-Wen Liu // Bartosz Regula and Ryuji Takagi
Affiliations: University of Waterloo | Perimeter Institute for Theoretical Physics // Nanyang Technological University | Nanyang Technological University

Abstract
The manipulation of quantum resources such as entanglement and coherence lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to purify the quantum resources, since they are inevitably contaminated by noises and thus often lost their power or become unreliable for direct usage. In these two works, we establish a theory of the universal limitations on the accuracy and efficiency of resource purification tasks which apply to any well-behaved resource theory, for both state (static) and channel (dynamical) resources. The general results bring new insights and imply various forms of fundamental limits to a broad range of problems of great theoretical and practical importance, including magic state distillation and fault tolerant quantum computing, quantum error correction, quantum Shannon theory, and quantum circuit synthesis.
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We establish universal limitations on the manipulation of quantum channel resources under the most general transformation protocols. Focusing in particular on the class of distillation tasks -- which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels -- we develop fundamental restrictions on the error necessarily incurred in such transformations. Our results are applicable to the study of general quantum resources under any physical manipulation scheme, which includes general adaptive protocols with or without a definite causal order. We introduce comprehensive lower bounds for the overhead of any distillation protocol in terms of required channel uses, imposing strong limitations on the practical efficiency and cost of channel resource manipulation. In the asymptotic setting, our results yield broadly applicable strong converse bounds for the rates of distillation. As a special case, our methods apply to the manipulation of quantum states, in which case they significantly improve on and extend previous approaches. We demonstrate our results through explicit applications to quantum communication, where we recover in particular a number of strong converse bounds for the quantum capacity of channels assisted by different classes of operations, as well as to fault-tolerant quantum computation, where we obtain improved bounds for the overhead cost of magic state distillation and gate synthesis.

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