  We formulate the density matrices of a quantum state obtained by first adding
multi-photons to and then subtracting multi-photons from any arbitrary state as
well as performing the same process in the reverse order. Considering the field
to be initially in a thermal (or in an even coherent) state, we evaluate the
photon number distribution, Wigner function and Mandel's $Q$ parameter of the
resulting field. We show graphically that in which order multi-photons are
added and subtracted has a noticeable effect on the temporal behavior of these
statistical properties.
