Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

We prove that learning an unknown quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ to diamond distance $\epsilon$ requires $\mathrm{\Omega }\text{\hspace{0.17em}⁣}\left(\frac{d_A{d}_{B}r}{\epsilon \log (d_{B}r\mathrm{/}\epsilon )}\right)$ channel queries when $d_A=r{d}_B$, and $\mathrm{\Omega }\text{\hspace{0.17em}⁣}\left(\frac{d_A{d}_{B}r}{{\epsilon }^2\log (d_{B}r\mathrm{/}\epsilon )}\right)$ channel queries when $d_A\le r{d}_B\mathrm{/}2$. These lower bounds improve upon the best previous $\mathrm{\Omega }(d_A{d}_{B}r)$ bound by introducing explicit, near-optimal $\epsilon$-dependence. Moreover, when $d_A\le r{d}_B\mathrm{/}2$, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.