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.
Highlights
The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel
The quantum capacity is at most a proxy for the answer to this question, and we show with concrete examples that it is often not a very good one
We develop a more precise approximate characterization of the performance of optimal coding schemes that takes into account finite size effects. We find that these effects are succinctly described by a second channel parameter, which we name quantum channel dispersion
Summary
The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. It is natural to ask how well quantum coding schemes perform when we restrict the size of the quantum devices used for encoding the channel inputs and decoding its outputs This is equivalent to considering communication with only a fixed number of channel uses. We find that in the order of a 1,000 qubits are required to get within 90% of the quantum capacity of a typical qubit dephasing channel To overcome this issue, we develop a more precise approximate characterization of the performance of optimal coding schemes that takes into account finite size effects. We develop a more precise approximate characterization of the performance of optimal coding schemes that takes into account finite size effects We find that these effects are succinctly described by a second channel parameter (besides its capacity), which we name quantum channel dispersion. Our work generalizes recent progress in the study of cc over quantum channels[12,13]
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