Codes for entanglement-assisted classical communication

TL;DR

The paper introduces an explicit entanglement-assisted classical communication (EACC) scheme that uses $n$ channel uses and $c$ entangled pairs to correct up to $d-1$ erasures, by reducing to a classical mixed-alphabet code and implementing a mixed alphabet Reed-Solomon construction. Entanglement enables $c$ uses of super-dense coding, leading to a reduction to a classical code over the mixed alphabets and a code family denoted $\mathcal{C}=[n,k,d;c]_q$ with a tunable minimum distance $d$. The authors derive a block-error bound $k \le 2(n_2-(d-1))+n_1$ (when $n_2\ge d-1$) and a quantum bound $k \le (n-d+1)(1+c/n)$, identifying parameter regimes where the scheme is optimal and outperforming classical limits. They show that entanglement can increase both distance and rate, enabling finite-length codes with $R>1$ in certain cases and providing a practical pathway to implementable EACC without quantum memory, using current super-dense coding demonstrations.

Abstract

Entanglement-assisted classical communication (EACC) aims to enhance communication systems using entanglement as an additional resource. However, there is a scarcity of explicit protocols designed for finite transmission scenarios, which presents a challenge for real-world implementation. In response we introduce a new EACC scheme capable of correcting a fixed number of erasures/errors. It can be adjusted to the available amount of entanglement and sends classical information over a quantum channel. We establish a general framework to accomplish such a task by reducing it to a classical problem. Comparing with specific bounds we identify optimal parameter ranges. The scheme requires only the implementation of super-dense coding which has been demonstrated successfully in experiments. Furthermore, our results shows that an adaptable entanglement use confers a communication advantage. Overall, our work sheds light on how entanglement can elevate various finite-length communication protocols, opening new avenues for exploration in the field.

Figures (2)

• Figure 1: Schematic representation of the encoding and decoding process. The mixed alphabet classical code maps $k$ input symbols to $n$ outputs from different alphabets. Then, $n-c$ of the symbols are transmitted using direct coding and $c$ symbols are transmitted using super-dense coding. The encoder $\mathcal{E}_{D}$ maps one classical symbol from $\mathbb{F}_q$ onto a qudit which is transmitted across a noisy quantum channel $\mathcal{N}$ and measured in the corresponding basis. This process is repeated $n-c$ times as illustrated by the direct coding block. The encoder $\mathcal{E}_{S}$ encodes a pair of classical symbols (identifying $\mathbb{F}_{q^2}$ and $(\mathbb{F}_q)^2$) onto one half of a maximally entangled state $|\Phi\rangle$. This encoded state is then transmitted across a noisy quantum channel $\mathcal{N}$. The other half of the maximally entangled state $|\Phi\rangle$ is shared with the receiver noiselessly. Upon reception, the receiver performs a measurement in the Bell basis on the received pair of states. This process is repeated $c$ times as illustrated by the super-dense coding block. Finally, the decoder corrects errors and outputs a classical message.
• Figure 2: The code rate $R=k/n$ of our mixed-alphabet Reed-Solomon code is plotted as a function of the normalised minimum distance $d/n$ in the asymptotic limit.