On the Capacity of Vector Linear Computation over a Noiseless Quantum Multiple Access Channel with Entangled Transmitters

TL;DR

This paper investigates the capacity of linear function computation over a noiseless quantum multiple access channel with entangled transmitters, providing a complete solution for three transmitters and optimal coding schemes.

Contribution

It fully characterizes the optimal communication cost for three transmitters in quantum MACs with entanglement, advancing understanding of quantum network coding.

Findings

Optimal coding schemes based on the N-sum box protocol are identified.

The problem is fully solved for three transmitters; higher numbers remain open.

Entanglement significantly enhances the capacity for linear computation over quantum channels.

Abstract

Network function computation is an active topic in network coding, with much recent progress for linear (over a finite field) computations over broadcast (LCBC) and multiple access (LCMAC) channels. Over a quantum multiple access channel (QMAC) with quantum-entanglement shared among transmitters, the linear computation problem (LC-QMAC) is non-trivial even when the channel is noiseless, because of the challenge of optimally exploiting transmit-side entanglement through distributed coding. Given an arbitrary linear function of data streams defined in a finite field $\mathbb{F}_d$, the LC-QMAC problem seeks the optimal communication cost (minimum number of qudits that need to be sent by the transmitters to the receiver, per computation instance) over a noise-free QMAC, when the independent input data streams originate at the corresponding transmitters, who share quantum entanglement in advance. As our main result, we fully solve this problem for $K=3$ transmitters ($K\geq 4$ settings remain open). Coding schemes based on the $N$-sum box protocol (along with time-sharing and batch-processing) are shown to be information theoretically optimal in all cases.