Schmidt-number benchmark for genuine quantum memories and gates

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.