Optimizing lossy state preparation for quantum sensing using Hamiltonian engineering

One of the most prominent platforms for demonstrating quantum sensing below the standard quantum limit is the spinor Bose-Einstein condensate. While a quantum advantage using several tens of thousands of atoms has been demonstrated in this platform, it faces an important challenge: atom loss. Atom loss is a Markovian error process modelled by Lindblad jump operators, and a no-go theorem, which we also show here, states that the loss of atoms in all spin components reduces the quantum advantage to a constant factor. Here, we show that this no-go theorem can be circumvented if we constrain atom losses to a single spin component. Moreover, we show that in this case, the maximum quantum Fisher information with $N$ atoms scales as $N^{3/2}$, establishing that a scalable quantum advantage can be achieved despite atom loss. Although Lindblad jump operators are generally non-Hermitian and non-invertible, we use their Moore-Penrose inverse to develop a framework for constructing several states with this scaling of Fisher information in the presence of losses. We use Hamiltonian engineering with realistic Hamiltonians to develop experimental protocols for preparing these states. Finally, we discuss possible experimental techniques to constrain the losses to a single spin mode.