Growing Schrödinger's cat states by local unitary time evolution of product states

We envisage many-body systems described by quantum spin-chain Hamiltonians featuring a trivial separable eigenstate. For generic Hamiltonians, such a state represents a quantum scar. We show that, typically, a macroscopically entangled state naturally grows after a single projective measurement of just one spin in the quantum scar. Moreover, we identify a condition under which what is growing is a "Schr & ouml;dinger's cat state." Our analysis does not reveal any particular requirement for the entangled state to develop, provided that the quantum scar does not minimize/maximize / maximize a local conservation law. We study two explicit examples: systems described by generic Hamiltonians where spins interact in pairs, and models that exhibit a U(1) hidden symmetry. The latter can be reinterpreted as a two-leg ladder in which the interactions along the legs are controlled by the local state on the other leg through transistorlike building blocks.