We present a lower bound for Pauli Manipulation Detection (PMD) codes, a class of quantum codes that detect every Pauli error with high probability. Our lower bound reveals the first trade-off between the error parameter and the coding rate. Specifically, we show that every $q$-ary PMD code of length $n$ and coding rate $R$ must satisfy $R \leq 1 - \frac{2}{n}\log_q\left(\frac{1}ε\right) + o(1)$, where $ε$ is the error parameter.

翻译：我们给出了Pauli操控检测(PMD)码的下界，这是一类能够以高概率检测所有Pauli错误的量子码。我们的下界揭示了误差参数与编码率之间的首个权衡关系。具体而言，我们证明了任意长度为$n$、编码率为$R$的$q$进制PMD码必须满足$R \leq 1 - \frac{2}{n}\log_q\left(\frac{1}ε\right) + o(1)$，其中$ε$为误差参数。