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.
Contents
Introduction
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
The coherent states, i.e. right eigenstates of the annihilation operator
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:
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 unavailable 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 P representation and a wider class of generalized P 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
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
Here
These representations are all interrelated through convolution by Gaussian functions, Weierstrass transforms,
or, using the property that convolution is associative,
Time evolution and operator correspondences
Since each of the above transformations from ρ to the distribution functions is linear, the equation of motion for each distribution can be obtained by performing the same transformations to
For instance, consider the annihilation operator
Taking the Fourier transform with respect to
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. In this way, master equations can be expressed as an equations of motion of quasiprobability functions.
Coherent state
By construction, P for a coherent state
The Wigner and Q representations follows immediately from the Gaussian convolution formulas above:
The Husimi representation can also be found using the formula above for the inner product of two coherent states:
Fock state
The P representation of a Fock state
Since for n>0 this is more singular than a delta function, a Fock state has no classical analog. The non-classicality is less transparent as one proceeds with the Gaussian convolutions. If Ln is the nth Laguerre polynomial, W is
which can go negative but is bounded. Q always remains positive and bounded:
Damped quantum harmonic oscillator
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. If the system is initially in the coherent state