Unbounded number of channel uses are required to see quantum capacity

Abstract

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.