Remnant quantum resources of collapsed macroscopic quantum superpositions

We consider the collapse of a macroscopic quantum superposition occurring due to the measurement which optimally distinguishes its branches. Given a macroscopic superposition of N spin-1/2 particles, we use such a Helstrom measurement to construct the local unitary operator which maximizes the usefulness of the superposition for Heisenberg-limited phase estimation (i.e., with quantum Cram\'{e}r Rao bound proportional to 1/N). In contrast, the collapsed state is not useful as a probe for phase estimation below the standard quantum limit. For the case N=2, we compute the entanglement entropy of the collapsed state and show that it is reduced below that of the initial superposition when the superposition is macroscopic. We consider the remnant quantum resources of collapsed hierarchical Schr\"{o}dinger cat states to show that collapsed macroscopic superpositions can still be useful for ultraprecise quantum metrology.