  The so-called quantum Cheshire cat is a phenomenon in which an object,
identified with a "cat", is dissociated from a property of the object,
identified with the "grin" of the cat. We propose a thought experiment, similar
to this phenomenon, with an interferometric setup, where a property (a
component of polarization) of an object (photon) can be separated from the
object itself and can simultaneously be amplified when it is already decoupled
from its object. We further show that this setup can be used to dissociate two
complementary properties, e.g., two orthogonal components of polarization of a
photon and identified with the grin and the snarl of a cat, from each other and
one of them can be amplified while being detached from the other. Moreover, we
extend the work to a noisy scenario, effected by a spin-orbit-coupling -like
additional interaction term in the Hamiltonian for the measurement process,
with the object in this scenario being identified with a so-called confused
Cheshire cat. We devise a gedanken experiment in which such a "confusion" can
be successfully dissociated from the system, and we find that the dissociation
helps in the amplification of signals.
