Schmidt-number benchmarks for continuous-variable quantum devices

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.