  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.
