title: Quantum coding with finite resources.
abstract: The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. In practical settings, we should instead focus on the optimal trade-off between three parameters: the rate of the code, the size of the quantum devices at the encoder and decoder, and the fidelity of the transmission. We find approximate and exact characterizations of this trade-off for various channels of interest, including dephasing, depolarizing and erasure channels. In each case, the trade-off is parameterized by the capacity and a second channel parameter, the quantum channel dispersion. In the process, we develop several bounds that are valid for general quantum channels and can be computed for small instances.
authors.affiliation: School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia.
authors.forename: Marco
authors.identifier: 0000-0001-5410-3329
authors.initials: M
authors.lastname: Tomamichel
authors.affiliation: Institute for Quantum Information and Matter, Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125, USA.
authors.forename: Mario
authors.initials: M
authors.lastname: Berta
authors.affiliation: Department of Physics, Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland.
authors.forename: Joseph M
authors.initials: JM
authors.lastname: Renes