Random quantum codes from Gaussian ensembles and an uncertainty relation

TL;DR

This paper demonstrates that random Gaussian quantum codes can reliably transmit entanglement through noisy channels, using an information-uncertainty relation and large deviations techniques to establish achievable rates.

Contribution

It introduces a novel proof that random Gaussian codes achieve the coherent information rate for entanglement transmission, leveraging an information-uncertainty relation.

Findings

Random Gaussian codes achieve the coherent information rate.

Classical data in conjugate bases can be decoded with low error.

Quantum mutual information remains high for entanglement-encoded data.

Abstract

Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender's input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement, and concluding, which ensures the existence of a decoding by the receiver.