Random and not-so-random codes for quantum channels

Summary form only given. Recently, after being open for almost a decade, a complete proof of the quantum channel coding theorem was given by Devetak, using a peculiar code construction described as `random CSS code'. Unlike in it's classical analogue, Shannon's channel coding, the code depends not only on a test source but also on the channel. Recently, M Horodecki, S Lloyd, P Shor and me found not only one but several random code families which are described only in terms of a test source. It turns out that the error analysis for these codes is extremely simple and conceptually interesting. An overview of these code constructions and their applicability will be given in the talk. However, it has been understood for a while that random codes only achieve the quantum capacity if test sources of arbitrary block length are considered; hence, no single-letter formula for the quantum capacity is known. I will discuss what is known regarding this peculiarity, connected to the nondegeneracy of random quantum codes, and present a single-letter upper bound on the quantum capacity (derived in joint work with G Smith)