General Scheme for Perfect Quantum Network Coding with Free Classical Communication

TL;DR

This paper demonstrates that perfect quantum network coding with free classical communication is achievable whenever classical network coding is possible, providing explicit quantum operations and bounds on classical communication costs.

Contribution

It establishes a constructive method for perfect quantum network coding based on classical coding schemes and provides bounds on classical communication requirements.

Findings

Quantum network coding is possible with free classical communication.

Explicit quantum coding operations are provided for each network node.

An upper bound on classical communication cost is derived.

Abstract

This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general -- indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with $k$ source-target pairs if there exists a classical linear (or even vector linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of $k$, the maximal fan-in of any network node, and the size of the network.