Capacity Bounds via Operator Space Methods

TL;DR

This paper introduces operator space methods to derive bounds on quantum capacity, providing new insights into channel capacities and identifying conditions where bounds are tight or optimal.

Contribution

It reformulates quantum capacity approximations using operator space norms and interpolation, offering new bounds and conditions for specific classes of quantum channels.

Findings

Upper and lower bounds on quantum capacity within a factor of 2.

Identification of channels satisfying a 'comparison property' on entropy and capacity.

Optimality of the hashing bound for certain channel classes.

Abstract

Quantum capacity, as the ultimate transmission rate of quantum communication, is characterized by regularized coherent information. In this work, we reformulate approximations of the quantum capacity by operator space norms and give both upper and lower bounds on quantum capacity, and potential quantum capacity using interpolation techniques. We identify a situation in which nice classes of channels satisfy a "comparison property" on entropy, coherent information and capacities. The paradigms for our estimates are so-called conditional expectations. These generally non-degradable channels admit a strongly additive expression for $Q^{(1)}$. We also identify conditions on channels showing that the "hashing bound" is optimal for the cb-entropy. These two estimates combined give upper and lower bounds on quantum capacity on our "nice" classes of channels, which differ only up to a factor 2, independent of the dimension. The estimates are discussed for certain classes of channels, including group channels, Pauli channels and other high-dimensional channels.