Lower Bounds on Pauli Manipulation Detection Codes

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 $R \leq 1 - \frac{2}{n}\log_q\left(\frac{1}{\epsilon}\right) + o(1)$, where $\epsilon$ is the error parameter.