Coherent information of a quantum channel or its complement is generically positive

TL;DR

This paper demonstrates that for most quantum channels with larger output spaces than their environment, a single copy suffices to determine if they can transmit quantum information, simplifying the assessment of their quantum capacity.

Contribution

It shows that the coherent information of a single randomly chosen quantum channel is almost surely positive under certain conditions, reducing the need for multiple copies to assess quantum capacity.

Findings

Single-copy coherent information is positive for most channels with larger output than environment.

Channels with zero coherent information are measure zero in the specified subset.

Results also apply to the channel's complement when the environment exceeds the output space.

Abstract

The task of determining whether a given quantum channel has positive capacity to transmit quantum information is a fundamental open problem in quantum information theory. In general, the coherent information needs to be computed for an unbounded number of copies of a channel in order to detect a positive value of its quantum capacity. However, in this Letter, we show that the coherent information of a single copy of a randomly selected channel is positive almost surely if the channel's output space is larger than its environment. Hence, in this case, a single copy of the channel typically suffices to determine positivity of its quantum capacity. Put differently, channels with zero coherent information have measure zero in the subset of channels for which the output space is larger than the environment. On the other hand, if the environment is larger than the channel's output space, identical results hold for the channel's complement.