General Scheme for Perfect Quantum Network Coding with Free Classical Communication

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.

Efficient Quantum Transmission in Multiple-Source Networks

“…The proposed quantum transmission attains the optimal transmission capacity under the assumption of restricted maximum-flow; in this respect, it is superior to other proposed approaches9101112131415. Unlike quantum k -pair tasks with enlarged linear encodings1314 or nonlinear encodings15, our encoding is linear and is not completed in an enlarged space. And classical channels are assumed to exist only for the common quantum channels and not all quantum channels15.…”

“…And classical channels are assumed to exist only for the common quantum channels and not all quantum channels15. Moreover, the new scheme extends the solvability of the quantum k -pair problem, and is more general than previous schemes131415 which depend on the solvability of the classical k -pair problem. One typical example is presented in the Figure 1(c); this example is unsolvable525354 in terms of the classical 2-pair problem using network coding.…”

“…Unlike the entanglement quantization of a classical channel910111213141516, it may be quantized with a continuous physical channel2627282930313233343536, and the transmission information may be quantized with quantized electromagnetic fields of identical frequencies. To distinguish the decoded quantum states for different nodes, special phase-shift operations may be designed to index different incoming quantum states, using phase shifters [coupling the optical fields to driven Duffing oscillators] or gauge transformations33.…”

Analysis of quantum network coding for realistic repeater networks

“…It is expected that network coding also allows the same resolution in a quantum network. In recent years, a number of researchers have studied quantum network coding [21][22][23][24][25][26][27] . However, all of these studies presuppose the use of pure states and perfect local gates.…”

Implementing controlled-unitary operations over the butterfly network

“…In [1], it is shown that perfect quantum 2-pair communication over the butterfly network is possible if we add two maximally entangled qubits between the two inputs and allow each channel to use either one qubit or two bits of communication. If classical communications are not restricted and only quantum communications are restricted, it is shown that any quantum k-pair problem has a solution if the corresponding classical k-pair problem has a solution [4] for the network. …”