Investigating controlled teleportation capability of quantum states with respect to $k$-separability

Quantum teleportation is an essential application of quantum entanglement. The examination of teleportation fidelity in two-party standard teleportation schemes reveals a critical threshold distinguishing separable and entangled states. For separable states, their teleportation fidelities cannot exceed the threshold, emphasizing the significance of entanglement. We extend this analysis to multi-party scenarios known as controlled teleportation. Our study provides thresholds that $N$-qudit $k$-separable states cannot exceed in a controlled teleportation scheme, where $N \ge 3$ and $2 \le k \le N$. This not only establishes a standard for utilizing a given quantum state as a resource in controlled teleportation but also enhances our understanding of the influence of the entanglement structure on controlled teleportation performance. In addition, we show that genuine multipartite entanglement is not a prerequisite for achieving a high controlled teleportation capability.