We introduce an open quantum system model to describe the statistical fluctuations of a spin ensemble in NMR. The model considers an ensemble measurement where the detection coil does not distinguish spins, and accounts for the state update rule. The analysis brings clarity and accuracy in describing the notion of spin noise and derives a correct statistical distribution and correlation function for the spin noise signal.
We propose a proof-of-principle experiment to encode one logical qubit in the noise protected subspace of three identical spins in a methyl group. We use a symmetry analysis to derive the dipole moment allowed transitions, which enable us to access the noiseless subsystem. We further analyze the symmetry of the hetronuclear dipolar relaxation, which is one of the responsible mechanisms for observing a noise protected state. Our analytical calculations predict features of of the NMR peaks that are in agreement with the experimental observations.