  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.
