Simultaneous precise measurement of the non-commuting observables spin angle and spin amplitude is achieved by directing the error due to quantum measurement back-action into an unmeasured spin component.
Measurement of spin precession is central to extreme sensing in physics1,2, geophysics3, chemistry4, nanotechnology5and neuroscience6, and underlies magnetic resonance spectroscopy7. Because there is no spin-angle operator, any measurement of spin precession is necessarily indirect, for example, it may be inferred from spin projectors at different times. Such projectors do not commute, and so quantum measurement back-action—the random change in a quantum state due to measurement—necessarily enters the spin measurement record, introducing errors and limiting sensitivity. Here we show that this disturbance in the spin projector can be reduced belowN1/2—the classical limit forNspins—by directing the quantum measurement back-action almost entirely into an unmeasured spin component. This generates a planar squeezed state8that, because spins obey non-Heisenberg uncertainty relations9,10, enables simultaneous precise knowledge of spin angle and spin amplitude. We use high-dynamic-range optical quantum non-demolition measurements11,12,13applied to a precessing magnetic spin ensemble to demonstrate spin tracking with steady-state angular sensitivity 2.9 decibels below the standard quantum limit, simultaneously with amplitude sensitivity 7.0 decibels below the Poissonian variance14. The standard quantum limit and Poissonian variance indicate the best possible sensitivity with independent particles. Our method surpasses these limits in non-commuting observables, enabling orders-of-magnitude improvements in sensitivity for state-of-the-art sensing15,16,17,18and spectroscopy19,20.