  Quantum networks play an indispensable role in quantum information tasks such
as secure communications, enhanced quantum sensing, and distributed computing.
Among the most mature and promising platforms for quantum networking are
nitrogen-vacancy centers in diamond and other color centers in solids. One of
the challenges in using these systems for networking applications is to
controllably manipulate entanglement between the electron and the nuclear spin
register despite the always-on nature of the hyperfine interactions, which
makes this an inherently many-body quantum system. Here, we develop a general
formalism to quantify and control the generation of entanglement in an
arbitrarily large nuclear spin register coupled to a color center electronic
spin. We provide a reliable measure of nuclear spin selectivity, by exactly
incorporating into our treatment the dynamics with unwanted nuclei. We also
show how to realize direct multipartite gates through the use of dynamical
decoupling sequences, drastically reducing the total gate time compared to
protocols based on sequential entanglement with individual nuclear spins. We
quantify the performance of such gate operations in the presence of unwanted
residual entanglement links, capturing the dynamics of the entire nuclear spin
register. Finally, using experimental parameters of a well-characterized 27
nuclear spin register device, we show how to prepare with high fidelity
entangled states for quantum error correction.
