Homodyne measurement with a Schr\"odinger cat state as a local oscillator

Homodyne measurements are a widely used quantum measurement. Using a coherent state of large amplitude as the local oscillator, it can be shown that the quantum homodyne measurement limits to a field quadrature measurement. In this work, we give an example of a general idea: injecting non-classical states as a local oscillator can led to non-classical measurements. Specifically we consider injecting a superposition of coherent states, a Schr\"odinger cat state, as a local oscillator. We derive the Kraus operators and the positive operator-valued measure (POVM) in this situation.