Protocol for bidirectional quantum teleportation using scrambling in the Dicke model

Implementing a universal SWAP for collective spins beyond highly polarized states remains challenging, even though SWAP is elementary for qubits. We present a practical protocol that performs a probabilistic, bidirectional teleportation between generic collective-spin states by leveraging scrambling dynamics generated by the Dicke Hamiltonian in a Penning trap. Two macroscopic spins couple sequentially to a shared bosonic mode that acts as an entangling bus: a first interaction scrambles and delocalizes the information, and a time-reversed interaction, followed by a projective readout of the bus, effects the exchange of states without pre-shared entanglement. The success probability scales inversely with the square of the collective-spin Hilbert-space dimension, and numerical simulations of the Dicke model show high fidelities at early times that are much shorter than the decoherence time relevant to Penning-trap platforms. This provides an AMO route to a SWAP gate for collective spins, using scrambling-generated entanglement as the resource, and establishes a broadly applicable scrambling-based primitive for many-body quantum information processing.