Fig. 6: The performance of ϑ(N, ϵ) assessed through Monte Carlo simulations.

We conduct 10,000 numerical experiments, where each experiment involves selecting a random initial state ρ and obtaining results. a–d Demonstrate the value of ϑ(N, ϵ) for N = 10, 20, 50 and 100, respectively. The dashed blue line in the figures represents the average descent ratio ϑ(N, ϵ) computed across all runs. Here, N denotes the number of randomly generated full-ranked quantum states used to construct τ (see Eq. (24)). The dashed red line represents the case where ϑ(N, ϵ) = 0, above which indicates a strict improvement compared to using the maximally mixed state \({\mathbb{I}}/{d}^{2}\) in constructing convex-split-lemma-assisted teleportation.