Classical-quantum network coding: a story about tensor

TL;DR

This paper investigates the capabilities of quantum versus classical network coding, introducing a tensor-based formalism to characterize distribution tasks and demonstrating that some tasks are uniquely achievable with quantum protocols.

Contribution

It develops a tensor factorization formalism to analyze quantum and classical network coding, revealing fundamental differences and the existence of tasks achievable only with quantum methods.

Findings

Quantum network coding can succeed where classical cannot.

A tensor formalism unifies the analysis of classical and quantum distribution tasks.

Some distribution tasks are exclusively achievable in the quantum setting.

Abstract

We study here the conditions to perform the distribution of a pure state on a quantum network using quantum operations which can succeed with a non-zero probability, the Stochastic Local Operation and Classical Communication (SLOCC) operations. In their pioneering 2010 work, Kobayashi et al. showed how to convert any classical network coding protocol into a quantum network coding protocol. However, they left open whether the existence of a quantum network coding protocol implied the existence of a classical one. Motivated by this question, we characterize the set of distribution tasks achievable with non zero probability for both classical and quantum networks. We develop a formalism which encompasses both types of distribution protocols by reducing the solving of a distribution task to the factorization of a tensor with complex coefficients or real positive ones. Using this formalism, we examine the equivalences and differences between both types of distribution protocols exhibiting several elementary and fundamental relations between them as well as concrete examples of both convergence and divergence. We answer by the negative to the issue previously left open: some tasks are achievable in the quantum setting, but not in the classical one. We believe this formalism to be a useful tool for studying the extent of quantum network ability to perform multipartite distribution tasks.