Quantum Coding with Finite Resources

TL;DR

This paper investigates the practical limits of quantum communication by characterizing the trade-offs between rate, coherence, and fidelity for finite resources, providing bounds and exact regions for specific quantum channels.

Contribution

It introduces a finite-resource framework for quantum coding, deriving bounds and exact regions for key channels, and highlights the importance of channel dispersion in quantum communication.

Findings

Characterized the achievable region of rate, coherence, and fidelity for qubit dephasing and erasure channels.

Developed efficient bounds applicable to all finite-dimensional quantum channels.

Established a lower bound on coherent channel uses to observe super-additivity of coherent information.

Abstract

The quantum capacity of a memoryless channel is often used as a single figure of merit to characterize its ability to transmit quantum information coherently. The capacity determines the maximal rate at which we can code reliably over asymptotically many uses of the channel. We argue that this asymptotic treatment is insufficient to the point of being irrelevant in the quantum setting where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. For all practical purposes we should instead focus on the trade-off between three parameters: the rate of the code, the number of coherent uses of the channel, and the fidelity of the transmission. The aim is then to specify the region determined by allowed combinations of these parameters. Towards this goal, we find approximate and exact characterizations of the region of allowed triplets for the qubit dephasing channel and for the erasure channel with classical post-processing assistance. In each case the region is parametrized by a second channel parameter, the quantum channel dispersion. In the process we also develop several general inner (achievable) and outer (converse) bounds on the coding region that are valid for all finite-dimensional quantum channels and can be computed efficiently. Applied to the depolarizing channel, this allows us to determine a lower bound on the number of coherent uses of the channel necessary to witness super-additivity of the coherent information.