  Transmitting data reliably over noisy communication channels is one of the
most important applications of information theory, and well understood when the
channel is accurately modelled by classical physics. However, when quantum
effects are involved, we do not know how to compute channel capacities. The
capacity to transmit quantum information is essential to quantum cryptography
and computing, but the formula involves maximising the coherent information
over arbitrarily many channel uses. This is because entanglement across channel
uses can increase the coherent information, even from zero to non-zero!
However, in all known examples, at least to detect whether the capacity is
non-zero, two channel uses already suffice. Maybe a finite number of channel
uses is always sufficient? Here, we show this is emphatically not the case: for
any n, there are channels for which the coherent information is zero for n
uses, but which nonetheless have capacity. This may be a first indication that
the quantum capacity is uncomputable.
