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Using random Gaussian vectors and an information-uncertainty relation, wegive a proof that the coherent information is an achievable rate forentanglement transmission through a noisy quantum channel. The codes are randomsubspaces selected according to the Haar measure, but distorted as a functionof the sender's input density operator. Using large deviations techniques, weshow that classical data transmitted in either of two Fourier-conjugate basesfor the coding subspace can be decoded with low probability of error. Arecently discovered information-uncertainty relation then implies that thequantum mutual information for entanglement encoded into the subspace andtransmitted through the channel will be high. The monogamy of quantumcorrelations finally implies that the environment of the channel cannot besignificantly coupled to the entanglement, and concluding, which ensures theexistence of a decoding by the receiver.

Random quantum codes from Gaussian ensembles and an uncertainty relation