Bipartite entanglement in a nuclear spin register mediated by a quasi-free electron spin

Quantum networks will rely on photons entangled to robust, local quantum registers for computation and error correction. We demonstrate control of and entanglement in a fully connected three-qubit $^{13}\mathrm{C}$ nuclear spin register in diamond. The register is coupled to a quasi-free electron spin-1/2 of a silicon-vacancy center (SiV). High strain decouples the SiVs electron spin from spin-orbit interaction reducing the susceptibility to phonons at liquid helium temperature. As a result, the electron spin lifetime of hundreds of milli seconds enables sensing of nuclear-nuclear couplings down to few hertz. To detect and control the register we leverage continuous decoupling using shaped, low-power microwave and direct radio frequency driving. Furthermore, we implement a nuclear spin conditional phase-gate on the electron spin to mediate bipartite entanglement. This approach presents an alternative to dynamically decoupled nuclear spin entanglement, not limited by the electron spin's 1/2 nature, opening up new avenues to an optically-accessible, solid-state quantum register.