Quasiprobability distribution

A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Although 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, they all violate the σ-additivity axiom, because regions integrated under them do not represent probabilities of mutually exclusive states. To compensate, some quasiprobability distributions also counterintuitively have regions of negative probability density, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis, and elsewhere. In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ρ ^ {displaystyle {widehat { ho }}} ) of the system. The density operator is defined with respect to a complete orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove that the density can always be written in a diagonal form, provided that it is with respect to an overcomplete basis. When the density operator is represented in such an overcomplete basis, then it can be written in a manner more resembling of an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i.e. right eigenstates of the annihilation operator a ^ {displaystyle {widehat {a}}} serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property: They also have some additional interesting properties. For example, no two coherent states are orthogonal. In fact, if |α〉 and |β〉 are a pair of coherent states, then Note that these states are, however, correctly normalized with〈α | α〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete. Click to show an informal proof. In the coherent states basis, however, it is always possible 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: 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 P representation because of the following key statement: