Detecting positive quantum capacities of quantum channels

TL;DR

This paper introduces a simple perturbative method to determine if quantum channels can reliably transmit quantum information by analyzing the dimensions of their input, output, and environment spaces, simplifying capacity detection.

Contribution

The authors develop an elementary, dimension-based test to detect positive quantum capacities, applicable to a wide range of quantum channels, simplifying the capacity analysis process.

Findings

The method successfully detects positive capacities in various channels.

Single-copy coherent information is positive in all tested examples.

The approach links channel capacity to input, output, and environment dimensions.

Abstract

Determining whether a noisy quantum channel can be used to reliably transmit quantum information at a non-zero rate is a challenging problem in quantum information theory. This is because it requires computation of the channel's coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel's ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels' input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.