Near-optimal incoherent tomography of low-rank quantum channels

Abstract

We study tomography for quantum channels with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, to within diamond norm error $\varepsilon$, using adaptive experiments that retain no quantum memory between channel queries. - For quantum channels whose non-zero Choi eigenvalues are bounded below by $\Omega(d_1/r)$, we establish optimal query upper and lower bounds $\Theta(d_1d_2r^2/\epsilon^2)$. The upper bound is achieved by a nonadaptive algorithm that uses the estimator from [Surawy-Stepney et al., Quantum (2022)], together with a new diamond-norm analysis. The lower bound applies to arbitrary adaptive incoherent protocols and follows from a new local family of channels and a uniform one-query Fisher-information bound. - For general channels, we establish an upper bound $O(d_1d_2r^2\log(2d_1)/\epsilon^2)$, nearly matching the above lower bound $\Omega(d_1d_2r^2/\varepsilon^2)$. To achieve this, we generalize the above nonadaptive algorithm by adapting the input state over $O(\log(2d_1))$ rounds with the Matrix Multiplicative Weight Update algorithm.