  Entangled coherent states for multiple bosonic modes, also referred to as
multimode cat states, not only are of fundamental interest, but also have
practical applications. The nonclassical correlation among these modes is well
characterized by the violation of the Mermin-Klyshko inequality. We here study
Mermin-Klyshko inequality violations for such multi-mode entangled states with
rotated quantum-number parity operators. Our results show that the
Mermin-Klyshko signal obtained with these operators can approach the maximal
value even when the average quantum number in each mode is only 1, and the
inequality violation exponentially increases with the number of entangled
modes. The correlations among the rotated parities of the entangled bosonic
modes are in distinct contrast with those among the displaced parities, with
which a nearly maximal Mermin-Klyshko inequality violation requires the size of
the cat state to be increased by about 15 times.
