Symmetric $N \to M$ telecloning and remote quantum state inference
Teleporting unknown quantum states between distant nodes of a network is a significant feature of quantum communication. Few studies, however, have been conducted on $N \to M$ telecloning, in which $N$ copies of an unknown quantum state are optimally teleported to $M \geq N$ receivers. Previous work requires global POVMs on all copies and auxiliaries; here, we show that symmetric $N \to M$ telecloning can be probabilistically performed using only sequential (or parallel) Bell state measurements (BSMs), allowing the copies to be spatially separated. When successful, the fidelity of each receiver's reduced state saturates the bound imposed by the no-cloning theorem, with the probability of success being independent of $M$. In the case of an `unsuccessful' BSM, we show that the teleportation fidelity is likely to remain high, with the measurement outcome also providing information about the closest Pauli eigenbasis to the unknown state being telecloned. In this sense, each receiver can remotely infer the unknown quantum state with a level of confidence that scales with the number of copies, even if the fidelity of their own reduced state is sub-optimal. We additionally analyze the required resources to ensure the classical-communication fidelity bound is surpassed, both in terms of two-qubit inseparability and varying classes of multipartite entanglement. Surprisingly, we uncover that, at a given iteration of the protocol, pairwise entanglement is not always necessary to increase the teleportation fidelity and is never required to beat the classical bound for $N \geq 2$.