Fig. 4: The comparison of \({n}_{\min }({{{{\mathcal{S}}}}}_{{\mathbb{I}}/{d}^{2}},\epsilon )\) and \({n}_{\min }(N,\epsilon )\).

Here we choose the initial state as the one considered in Eq. (45) and take N full-ranked states randomly, to construct τ (see Eq. (24)) and compare the number of copies that are needed to achieve higher fidelity in (a). Investigation of descent ratio ϑ(N, ϵ) (see Eq. (49)) with respect to different N is shown in (b).