Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, the ability to yield expectation values with respect to the weights of the distribution.
However, they can violate the σ-additivity axiom: integrating over them does not necessarily yield probabilities of mutually exclusive states.
Quasiprobability distributions also have regions of negative probability density, counterintuitively, contradicting the first axiom.
The density operator is defined with respect to a complete orthonormal basis.
However, it is possible to prove[2] that the density operator can always be written in a diagonal form, provided that it is with respect to an overcomplete basis.
Integration over the complex plane can be written in terms of polar coordinates with
Where exchanging sum and integral is allowed, we arrive at a simple integral expression of the gamma function: Clearly, one can span the Hilbert space by writing a state as On the other hand, despite correct normalization of the states, the factor of π > 1 proves that this basis is overcomplete.
In the coherent states basis, however, it is always possible[2] to express the density operator in the diagonal form where f is a representation of the phase space distribution.
This function f is considered a quasiprobability density because it has the following properties: 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,[4] which is related to symmetric operator ordering.
[5] The quasiprobabilistic nature of these phase space distributions is best understood in the P representation because of the following key statement:[6] If the quantum system has a classical analog, e.g. a coherent state or thermal radiation, then P is non-negative everywhere like an ordinary probability distribution.
If, however, the quantum system has no classical analog, e.g. an incoherent Fock state or entangled system, then P is negative somewhere or more singular than a delta function.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.
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
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.
: 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.
may be identified as coherent state amplitudes in the case of the Glauber P and Q distributions, but simply c-numbers for the Wigner function.
For the characteristic function of the P distribution we have Taking the Fourier transform with respect to
to find the action corresponding action on the Glauber P function, we find By following this procedure for each of the above distributions, the following operator correspondences can be identified: Here κ = 0, 1/2 or 1 for P, Wigner, and Q distributions, respectively.
is Since for n>0 this is more singular than a delta function, a Fock state has no classical analog.
Q, by contrast, always remains positive and bounded, Consider the damped quantum harmonic oscillator with the following master equation, This results in the Fokker–Planck equation, where κ = 0, 1/2, 1 for the P, W, and Q representations, respectively.