Consider quantum channels with input dimension $d_1$, output dimension $d_2$ and Kraus rank at most $r$. Any such channel must satisfy the constraint $rd_2\geq d_1$, and the parameter regime $rd_2=d_1$ is called the boundary regime. In this paper, we show an optimal query lower bound $Ω(rd_1d_2/\varepsilon^2)$ for quantum channel tomography to within diamond norm error $\varepsilon$ in the away-from-boundary regime $rd_2\geq 2d_1$, matching the existing upper bound $O(rd_1d_2/\varepsilon^2)$. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions $d_1=d_2=d$ with $r\geq 2$, in sharp contrast to the unitary case $r=1$ where Heisenberg scaling $Θ(d^2/\varepsilon)$ is achievable.

翻译：考虑输入维度为$d_1$、输出维度为$d_2$且Kraus秩至多为$r$的量子信道。任何此类信道必须满足约束条件$rd_2\geq d_1$，而参数区域$rd_2=d_1$被称为边界区域。本文证明，在远离边界区域$rd_2\geq 2d_1$中，为达到钻石范数误差$\varepsilon$的量子信道层析，其查询复杂度存在最优下界$Ω(rd_1d_2/\varepsilon^2)$，与已有上界$O(rd_1d_2/\varepsilon^2)$相匹配。特别地，该下界完全确定了输入输出维度相等$d_1=d_2=d$且$r\geq 2$这一常见情况下的查询复杂度，与幺正情形$r=1$可实现海森堡标度$Θ(d^2/\varepsilon)$的结果形成鲜明对比。