  Radical-ion pairs and their reactions have triggered the study of quantum
effects in biological systems. This is because they exhibit a number of effects
best understood within quantum information science, and at the same time are
central in understanding the avian magnetic compass and the spin transport
dynamics in photosynthetic reaction centers. Here we address radical-pair
reactions from the perspective of quantum metrology. Since the coherent spin
motion of radical-pairs is effected by an external magnetic field, these
spin-dependent reactions essentially realize a biochemical magnetometer. Using
the quantum Fisher information, we find the fundamental quantum limits to the
magnetic sensitivity of radical-pair magnetometers. We then explore how well
the usual measurement scheme considered in radical-pair reactions, the
measurement of reaction yields, approaches the fundamental limits. In doing so,
we find the optimal hyperfine interaction Hamiltonian that leads to the best
magnetic sensitivity as obtained from reaction yields. This is still an order
of magnitude smaller than the absolute quantum limit. Finally, we demonstrate
that with a realistic quantum reaction control reminding of Ramsey
interferometry, here presented as a quantum circuit involving the spin-exchange
interaction and a recently proposed molecular switch, we can approach the
fundamental quantum limit within a factor of 2. This work opens the application
of well-advanced quantum metrology methods to biological systems.
