A systematic analysis of equivalence in multistage networks

Many approaches to switching in optoelectronic and optical networks decompose the switching function across multiple stages or hops. This paper addresses the problem of determining whether two multistage or multihop networks are functionally equivalent. Various ad-hoc methods have been used in the past to establish such equivalences. A systematic method for determining equivalence is presented based on properties of the link permutations used to interconnect stages of the network. This method is useful in laying out multistage networks, in determining optimal channel assignments for multihop networks, and in establishing the routing required in such networks. A purely graphical variant of the method, requiring no mathematics or calculations, is also described

General Scheme for Perfect Quantum Network Coding with Free Classical Communication

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.Comment: 12 pages, 2 figures, generalizes some of the results in arXiv:0902.1299 to the k-pair problem and codes over rings. Appeared in the Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP'09), LNCS 5555, pp. 622-633, 200