Quantum optics theory needs an advanced method to tackle density operator’ various physical quantities, such as expec-tation value, variance, cumulant, etc. To be specific, since photon creation and annihilation operators do not commute, we need to deal with the problems of how to convert normally ordered operators into anti-normally ordered operators, and how to convert anti-normally ordered operators into normally ordered operators. In short, the operator re-ordering problem is often encountered in quantum optics theory. In this paper we employ the generating function of two-variable Hermite polynomials to derive two basic operator identities. The first basic operator identity is aⁿa^+m=(-i)^m+n：H_m,n(ia⁺, ia): , which converts anti-normally ordered operators into normally ordered operators. As an application of the basic operator identity we compute and get a^m|n>=√ (n!/(n-m)!)|n-m>, meanwhile, we give the commutation relation of [a^m,a⁺ⁿ]. The second basic operator identity is a^+maⁿ=H_n,m(a+,a), which converts normally ordered operators into anti-normally ordered operators. When m= n, in virtue of laguerre's polynomials we get the equality H_n,n(x,y)=(-1)ⁿn!L_n (xy). We derive a formula for the transformation between normal product and the anti-normal product in the end. The two basic operator identities are easily remembered and useful in quantum optics. The application of two-variable Hermite polynomials, such as for studying quantum entangled state representation, is greatly developed by Fan Hong-yi in recent years. One can also apply the new basic operator identities to develop binomial and negative- binomial theory which involves twovariable Hermite polynomials.