because it has the following properties: (normalization) If is an operator that can be expressed as a power series of the creation and annihilation operators in an ordering Ω, then its expectation value is (optical equivalence theorem). The function f is not unique. There exists a family of different representations, each connected to a different ordering Ω. The most popular in the general physics literature and historically first of these is the Wigner quasiprobability distribution, which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation. The quasiprobabilistic nature of these phase space distributions is best understood in the representation because of the following key statement: This sweeping statement is inoperative in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog. In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation, which is useful when operators are in anti-normal order. More recently, the positive representation and a wider class of generalized representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function. Characteristic functions Analogous to probability theory, quantum quasiprobability distributions can be written in terms of characteristic functions, from which all operator expectation values can be derived. The characteristic functions for the Wigner, Glauber P and Q distributions of an N mode system are as follows: Here and are vectors containing the annihilation and creation operators for each mode of the system. These characteristic functions can be used to directly evaluate expectation values of operator moments. The ordering of the annihilation and creation operators in these moments is specific to the particular characteristic function. For instance, normally ordered (annihilation operators preceding creation operators) moments can be evaluated in the following way from : In the same way, expectation values of anti-normally ordered and symmetrically ordered combinations of annihilation and creation operators can be evaluated from the characteristic functions for the Q and Wigner distributions, respectively. The quasiprobability functions themselves are defined as Fourier transforms of the above characteristic functions. That is, Here