Using pre-shared entangled states between the encoder and the decoder, we provide a previously unreported coding-theoretic framework for constructing entanglement-assisted stabilizer codes over qudits of dimension \(p^k\) from first principles, where p is prime and \(k \in {\mathbb {Z}}^+\). We introduce the concept of mathematically decomposing a qudit of dimension \(p^k\) into k subqudits, each of dimension p. Our contributions toward the entanglement-assisted stabilizer coding framework over qudits are multi-fold as follows: (a) We study the properties of the code and derive an analytical expression for the minimum number of pre-shared entangled subqudits required to construct the code. (b) We provide a code construction procedure that involves obtaining the explicit form of the stabilizers of the code. (c) We show that the proposed entanglement-assisted qudit stabilizer codes are analogous to classical additive codes over \({\mathbb {F}}_{p^k}\). (d) We provide the quantum coding bounds, such as the quantum Hamming bound, the quantum Singleton bound, and the quantum Gilbert–Varshamov bound for non-degenerate entanglement-assisted stabilizer codes over qudits. (e) We finally demonstrate that the error correction capability of the code can be increased with entanglement assistance. The proposed framework is useful for realizing coded quantum computing and communication systems over \(p^k\)-dimensional qudits.

使用编码器和解码器之间的预共享纠缠状态，我们提供了一个以前未报告的编码理论框架，用于根据第一原理在维度为\(p^k\) 的qudits 上构建纠缠辅助稳定器代码，其中p是素数，并且\ (k \in {\mathbb {Z}}^+\)。我们引入了数学上将维度为\(p^k\)的量子分解为k个子量子的概念，每个子量子的维度为p. 我们对基于 qudits 的纠缠辅助稳定器编码框架的贡献是多方面的，如下所示：（a）我们研究了代码的属性，并推导出了构建代码所需的最小预共享纠缠子量子数的解析表达式。(b) 我们提供了一个代码构建程序，它涉及获得代码稳定器的显式形式。(c) 我们表明提出的纠缠辅助 qudit 稳定器代码类似于\({\mathbb {F}}_{p^k}\) 上的经典加法代码. (d) 我们提供了量子编码边界，​​例如量子汉明边界、量子单例边界和量子吉尔伯特-瓦尔沙莫夫边界，用于非简并纠缠辅助稳定器代码在 qudits 上。(e) 我们最终证明了可以通过纠缠辅助来提高代码的纠错能力。所提出的框架可用于在\(p^k\)维 qudits 上实现编码量子计算和通信系统。