On small perturbations of coherent information

TL;DR

This paper develops perturbative techniques to analyze how small changes in input states affect the coherent information of quantum channels, providing criteria for suboptimality, superadditivity, and capacity gaps, with applications to various quantum channels.

Contribution

It introduces novel perturbative methods and criteria for assessing coherent information behavior in quantum channels, aiding capacity analysis.

Findings

Criteria for suboptimal coherent information

Conditions for superadditivity of one-shot quantum capacity

Detection of capacity gaps in specific quantum channels

Abstract

Quantum capacity quantifies the amount of quantum information that can be transmitted by a quantum channel with an arbitrary small probability of error. Mathematically, the quantum capacity is given by an asymptotic formula involving the one-shot quantum capacity of the associated channel, which, due to purely quantum effects such as superadditivity of one-shot quantum capacity, is rarely computable. The one-shot quantum capacity is mathematically characterized in terms of optimizing an entropic quantity referred to as coherent information over all possible input states of a channel, the computation of which also tends to be intractable due to the difficulty of optimizing the coherent information. In this work, we develop perturbative methods for analyzing the behavior of coherent information of a quantum channel with respect to small perturbations of the input state. By doing so, we are able to derive three general criteria for determining whether an input state yields suboptimal coherent information. We then show how our criteria yield sufficient conditions for superadditivity of one-shot quantum capacity, and also for detecting a positive gap between one-shot private capacity and one-shot quantum capacity. The utility of our criteria is illustrated through examples, which yield new results regarding the one-shot quantum capacity of qubit depolarizing channels, Pauli channels and dephrasure channels.