A quasiprobability distribution is a mathematical object similar to a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
but which relaxes some of
Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, ''the ability to yield
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
s 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. Indeed, quasiprobability distributions also have regions of
negative probability density, counterintuitively, contradicting the
first axiom. Quasiprobability distributions arise naturally in the study of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
when treated in
phase space formulation
The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
, commonly used in
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
,
time-frequency analysis, and elsewhere.
Introduction
In the most general form, the dynamics of a
quantum-mechanical system are determined by a
master equation in
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
: an equation of motion for the
density operator
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
(usually written
) of the system. The density operator is defined with respect to a ''complete''
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For exam ...
. 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 operator can always be written in a ''
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
'' 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
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
, i.e. right
eigenstates of the
annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property,
:
They also have some further 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 state
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an imp ...
s, 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:
:*
(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 optical equivalence theorem in quantum optics asserts an equivalence between the expectation value of an operator in Hilbert space and the expectation value of its associated function in the phase space formulation with respect to a quasiprobabi ...
).
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
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study qua ...
, which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the
particle number operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.
The number operator acts on Fock space. Let
:, \Psi\rangle_\nu=, \phi_1,\phi_ ...
, is naturally expressed in
normal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...
. In that case, the corresponding representation of the phase space distribution is the
Glauber–Sudarshan P representation The Sudarshan-Glauber P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which obse ...
. 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
The Husimi Q representation, introduced by Kôdi Husimi in 1940, is a quasiprobability distribution commonly used in quantum mechanics to represent the phase space distribution of a quantum state such as light in the phase space formulation. It ...
, 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 function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
s,
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 transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s of the above characteristic functions. That is,
:
Here
and
may be identified as
coherent state
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
amplitudes in the case of the Glauber P and Q distributions, but simply
c-numbers for the Wigner function. Since differentiation in normal space becomes multiplication in Fourier space, moments can be calculated from these functions in the following way:
*
*
*
Here
denotes symmetric ordering.
These representations are all interrelated through
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
by
Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
s,
Weierstrass transform
In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''.
Specifically, it is the function defined ...
s,
*
*
or, using the property that convolution is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
,
*
It follows that
*
an often divergent integral, indicating ''P'' is often a distribution. ''Q'' is always broader than ''P'' for the same density matrix.
For example, for a thermal state,
:
one has
:
Time evolution and operator correspondences
Since each of the above transformations from to the distribution functions is
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear re ...
, the equation of motion for each distribution can be obtained by performing the same transformations to
. Furthermore, as any
master equation which can be expressed in
Lindblad form is completely described by the action of combinations of
annihilation and creation operators on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions.
[C. W. Gardiner, ''Quantum Noise'', Springer-Verlag (1991).]
For instance, consider the annihilation operator
acting on . For the characteristic function of the P distribution we have
:
Taking the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
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 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.
Examples
Coherent state
By construction, ''P'' for a coherent state
is simply a delta function:
:
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
is
:
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 ''L
n'' is the nth
Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:
xy'' + (1 - x)y' + ny = 0
which is a second-order linear differential equation. This equation has nonsingular solutions only ...
, ''W'' is
:
which can go negative but is bounded.
''Q'', by contrast, 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
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
,
:
where ''κ'' = 0, 1/2, 1 for the ''P'', ''W'', and ''Q'' representations, respectively.
If the system is initially in the coherent state
, then this equation has the solution
:
References
Particle distributions
Quantum optics
Exotic probabilities