Using Weak Measurements to Synthesize Projective Measurement of Nonconserved Observables of Weakly Coupled Nuclear Spins

Projective quantum measurement is fundamental for quantum technologies. Here we propose a method for construction of projective measurement of nonconserved quantity of nuclear spin via a weakly cou-pled auxiliary electron spin. The key idea is to apply suitable quantum control on the electron to construct a weak quantum nondemolition (QND) measurement on the nuclear spin and then cascade a sequence of such measurements into a projective one. We identify a set of tunable parameters to select the QND observables and control the strength of the weak QND measurement. We also find that the QND measure-ment can be stabilized against realistic experimental control errors. As a demonstration of our method, we design projective measurement on a 13C nuclear spin weakly coupled to a nitrogen-vacancy center electron spin in diamond.