Abstract: This paper considers the commutative ralations of the generalized number operator N_h and the quantum Bernoulli noise {ə_σ,ə^*_σ:σ∈Γ} indexed by Γ, such as Lie bracket, the expressions of the composition of N_h and ə_σ(ə^*_σ), the commutative relation of N_h and ə_σə^*_σ(ə^*_σə_σ). The family of bounded linear operators {ə_σ,ə^*_σ:σ∈Γ} on L2(M) satisfies the canonical anticommutative relation, nilpotence and the composition are commutative if the intersection of the index is empty. Especially, {ə_σ,ə^*_σ:σ∈Γ} satisfy “absorbing commutative relation”. In the following, the paper considers the commutative relations of N_h and {ə_σ,ə^*_σ:σ∈Γ}. For any nonnegative function h on N, the Lie bracket of N_h and the σ-creation ə^*_σ(σ-annihilation ə_σ)are just #_h(σ)ə^*_σ(#_h(σ)ə_σ). Especially, if the support of h is not N, then N_h is commutative with some special kind of ə^*_σ(ə_σ). If the support of h is a finite subset of N, the composition of N_h and a special kind of ə^*_σ(ə_σ) are just the creation type(annihilation type)operators. Moreover, the paper obtains that N_h is commutative with {ə_σə^*_σ,ə^*_σə_σ:σ∈Γ}.

Key words: generalized number operator N_h, quantum Bernoulli noise indexed by Γ, commutative ralation