Partition of unity with mixed quantum states

The completeness of quantum state space, is usually expressed as \sum_{m=0}^{\infty}|m><m|=1, where {|m>} is selected set of quantum states (basis). Density matrix |m><m| describes a pure quantum state. In this paper, by virtue of the summation method within the normally ordered product of operators we propose and show that the completeness relation can also be represented or partitioned in terms of some mixed states, such as binomial states and negative binomial states. Thus the view on the structure of Fock space is widen and the connotation of Fock space is enriched. See the matter in this sight, experimentalists may have interests to prepare the binomial- and negative binomial states.