Optimal complex conjugation of unknown isometry channels
Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry $\overline{V}$ from $n$ uses of an unknown isometry channel $V: \mathbb{C}^d\to\mathbb{C}^D$. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity $n=\Theta(d[(D-d)/\epsilon+1])$ for achieving infidelity $\epsilon$. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity $O(\mathrm{poly}(D,1/\epsilon))$. This task is extended to the multi-copy case $V^{\otimes n}\mapsto \overline{V}^{\otimes k}$. For fixed $d<D$ and $k$, we show that the optimal fidelity for the multi-copy case is $1-kd(D-d)/n+o(n^{-1})$, and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-$r$ quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank $r$ is constant.