Channel capacities via $p$-summing norms

TL;DR

This paper connects the metric theory of tensor products with channel capacities in Shannon and quantum information theory, showing that quantum channel capacities can be characterized using noncommutative p-summing norms.

Contribution

It introduces a novel approach to quantum channel capacities using noncommutative p-summing maps, bridging tensor product theory and quantum information.

Findings

Classical capacity expressed as derivative of p-summing norm

Extension of tensor product theory to quantum channels

Characterization of quantum capacities with restricted entanglement

Abstract

In this paper we show how \emph{the metric theory of tensor products} developed by Grothendieck perfectly fits in the study of channel capacities, a central topic in \emph{Shannon's information theory}. Furthermore, in the last years Shannon's theory has been generalized to the quantum setting to let the \emph{quantum information theory} step in. In this paper we consider the classical capacity of quantum channels with restricted assisted entanglement. In particular these capacities include the classical capacity and the unlimited entanglement-assisted classical capacity of a quantum channel. To deal with the quantum case we will use the noncommutative version of $p$-summing maps. More precisely, we prove that the (product state) classical capacity of a quantum channel with restricted assisted entanglement can be expressed as the derivative of a completely $p$-summing norm.