We derive an algebraic framework which identifies the minimal information required to
assess how well a quantum device implements a desired quantum operation. Our approach
is based on characterizing only the unitary part of an open system’s evolution. We
show that a reduced set of input states is sufficient to estimate the average fidelity
of a quantum gate, avoiding a sampling over the full Liouville space. Surprisingly,
the minimal set consists of only two input states, independent of the Hilbert space
dimension. The minimal set is, however, impractical for device characterization since
one of the states is a totally mixed thermal state and extracting bounds for the
average fidelity is impossible. We therefore present two further reduced sets of input
states that allow, respectively, for numerical and analytical bounds on the average
fidelity.