Classic and Quantum Network Coding

Abstract

Ahlswede, Cai, Li, and Yeung (IEEE Trans. Inform. Theory, 2000) showed that the fundamental law for network flow, the max-flow min-cut theorem, no longer applies for “digital information flow.” The simple, nice example they gave is called the Butterfly network illustrated in Fig. 1. The capacity of each directed link is all one and there are two source-sink pairs s ₁ to t ₁ and s ₂ to t ₂. Notice that both paths have to use the single link from s ₀ to t ₀ and hence the total amount of (conventional commodity) flow in both paths is bounded by one, say, 1/2 for each. In the case of digital information flow, however, the protocol shown in Fig. 2 allows us to transmit two bits, x and y, simultaneously. Thus, we can effectively achieve larger channel capacity than can be achieved by simple routing. This is known as network coding since this seminal paper and has been quite popular as a mutual interest of theoretical computer science and information theory.

Supported in part by Scientific Research Grant, Ministry of Japan, 1609211 and 16300003.