  Passing a photon number state through a balanced beam splitter will produce
an entangled state in which the phases of the two output beams are highly
correlated. This entangled state can be viewed as a generalized form of a
Schrodinger cat state where there is an equal probability amplitude for all
possible phases. We show that Bell's inequality can be violated using this
entangled state and two distant measuring devices that consist of a
single-photon interferometer with a Kerr medium in one path, a set of
single-photon detectors, and postselection based on a homodyne measurement.
These entangled states are sensitive to photon loss and a violation of Bell's
inequality requires either that the losses are inherently small or that their
effects have been minimized using linear optics techniques [M. Micuda et al.,
Phys. Rev. Lett. 109, 180503 (2012)]. Somewhat surprisingly, the use of the
fair sampling assumption is not required for a violation of Bell's inequality
despite the use of postselection if the measurements are made in the correct
order.
