  We propose to apply the notion of the Schmidt number in order to show that a
quantum memory or gate process is capable of maintaining a genuine multi-level
quantum coherence. We present a simple criterion in terms of an average gate
fidelity with respect to the input states of two mutually unbiased bases and
demonstrate the coherence of the gate operation in several existing
experiments. The Schmidt-number benchmark naturally includes the quantum
benchmark as a specific case.

Quantum Physics