  We present quantum fidelity benchmarks for continuous-variable (CV) quantum
devices to outperform quantum channels which can transmit at most
$k$-dimensional coherences for positive integers $k$. We determine an upper
bound of an average fidelity over Gaussian distributed coherent states for
quantum channels whose Schmidt class is $k$. This settles fundamental fidelity
steps where the known classical limit and quantum limit correspond to the two
endpoints of $k=1$ and $k= \infty $, respectively. It turns out that the
average fidelity is useful to verify to what extent an experimental CV gate can
transmit a high dimensional coherence. The result is also extended to be
applicable to general quantum operations. While the fidelity is directly
associated with heterodyne measurements in quantum optics, we can also obtain
similar criteria based on mean square quadrature deviations via homodyne
measurements.
