Fig. 8: Comparisons across different catalytic teleportation protocols.

a, c Illustrate the minimum dimension of embezzling catalysts needed to achieve a given performance of teleportation with random initial states ρ (see Fig. 1), with fidelities of 0.75 and 0.8, as presented in Supplementary Table 1. Here, τ^E indicates the catalyst dimension based on embezzling states (see Eq. (50)), while \({\tau }^{CS}({n}_{\min }(100,\epsilon ))\) and \({\tau }^{CS}({n}_{\min }({{{{\mathcal{S}}}}}_{{\mathbb{I}}/{d}^{2}},\epsilon ))\) denote the catalyst dimensions constructed using the convex-split lemma (see Lem. 1). The former employs a selection of 100 randomly chosen full-ranked states for constructing τ (see Eq. (24)), whereas the latter utilizes maximally mixed states \({\mathbb{I}}/{d}^{2}\). To enhance readability, we adjusted the proportions according to the varying average fidelity values across these figures. b, d Demonstrate the consumption of embezzling catalysts during the teleportation process in terms of purified distance. Specifically, the blue line denotes the upper bound of the consumption for the embezzling catalyst τ^CS (see Eq. (38)), while the pink line represents the exact consumption of the catalyst τ^E in catalytic teleportation (see Eq. (70)). These comparisons indicate that the superior performance of the embezzling-state-assisted protocol, with the same dimension as the convex-split-lemma-assisted protocol, comes at the cost of a greater change from its original form before catalytic teleportation.