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=                   Quasiprobability distribution                    =
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                            Introduction
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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 '�'-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 \widehat{\rho}) 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 \widehat{a} serve as the overcomplete basis in the
construction described above.  By definition, the coherent states have
the following property:

:\begin{align}\widehat{a}|\alpha\rangle&=\alpha|\alpha\rangle \\
\langle\alpha|\widehat{a}^{\dagger}&=\langle\alpha|\alpha^*.
\end{align}

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
:\langle\beta\mid\alpha\rangle=e^{-{1\over2}(|\beta|^2+|\alpha|^2-2\beta^*\alpha
)}\neq\delta(\alpha-\beta).
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.
!Proof of the overcompleteness of the coherent states
Integration over the complex plane can be written in terms of polar
coordinates with d^2\alpha=r \, dr \, d\theta.  Where exchanging sum
and integral is allowed, we arrive at a simple integral expression of
the gamma function:     :\begin{align}\int |\alpha\rangle\langle\alpha| \,
d^2\alpha       &= \int \sum_{n=0}^\infty\sum_{k=0}^\infty
e^{-{|\alpha|^2}} \cdot \frac{\alpha^n (\alpha^*)^k}{\sqrt{n!k!}}
|n\rangle \langle k| \, d^2\alpha \\    &= \int_0^\infty
\int_0^{2\pi} \sum_{n=0}^{\infty}\sum_{k=0}^\infty e^{-{r^2}} \cdot
\frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} |n\rangle \langle k| \,
d\theta \,dr \\ &= \sum_{n=0}^\infty \int_0^\infty
\sum_{k=0}^\infty \int_0^{2\pi} e^{-{r^2}} \cdot
\frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} |n\rangle \langle k| \,
d\theta \,dr \\ &= 2\pi \sum_{n=0}^\infty \int_0^\infty
\sum_{k=0}^\infty e^{-{r^2}} \cdot
\frac{r^{n+k+1}\delta(n-k)}{\sqrt{n!k!}} |n\rangle \langle k| \, dr \\
&= 2\pi \sum_{n=0}^\infty \int e^{-{r^2}} \cdot
\frac{r^{2n+1}}{n!} |n\rangle \langle n| \, dr \\       &= \pi
\sum_{n=0}^\infty \int e^{-u} \cdot \frac{u^n}{n!} |n\rangle \langle
n| \, du \\     &= \pi \sum_{n=0}^\infty |n\rangle \langle n| \\
&= \pi \widehat{I}.\end{align}      Clearly we can span the Hilbert
space by writing a state as     :|\psi\rangle = \frac{1}{\pi} \int
|\alpha\rangle\langle\alpha|\psi\rangle \, d^2\alpha.   On the other
hand, despite correct normalization of the states, the factor of
�>1 proves that this basis is overcomplete.

In the coherent states basis, however, it is always possible to
express the density operator in the diagonal form
:\widehat{\rho} = \int f(\alpha,\alpha^*) |\alpha\rangle \langle
\alpha| \, d^2\alpha
where 'f' is a representation of the phase space distribution.  This
function 'f' is considered a quasiprobability density because it has
the following properties:

:*\int f(\alpha,\alpha^*) \, d^2\alpha =
\operatorname{tr}(\widehat{\rho}) = 1  (normalization)
:*If g_\Omega (\widehat{a},\widehat{a}^\dagger) 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
:::\langle g_{\Omega} (\widehat{a},\widehat{a}^\dagger) \rangle = \int
f(\alpha,\alpha^*) g_\Omega(\alpha,\alpha^*) \, d\alpha \, d\alpha^*
(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 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
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
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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:

* \chi_W(\mathbf{z},\mathbf{z}^*)= \operatorname{tr}(\rho
e^{i\mathbf{z}\cdot\widehat{\mathbf{a}}+i\mathbf{z}^*\cdot\widehat{\mathbf{a}}^{
\dagger}})
* \chi_P(\mathbf{z},\mathbf{z}^*)= \operatorname{tr}(\rho
e^{i\mathbf{z}^*\cdot\widehat{\mathbf{a}}^{\dagger}}e^{i\mathbf{z}\cdot\widehat{
\mathbf{a}}})
* \chi_Q(\mathbf{z},\mathbf{z}^*)=\operatorname{tr}(\rho
e^{i\mathbf{z}\cdot\widehat{\mathbf{a}}}e^{i\mathbf{z}^*\cdot\widehat{\mathbf{a}
}^{\dagger}})

Here \widehat{\mathbf{a}} and \widehat{\mathbf{a}}^{\dagger} 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 \chi_P\,:

: \langle\widehat{a}_j^{\dagger m}\widehat{a}_k^n\rangle =
\frac{\partial^{m+n}}{\partial(iz_j^*)^m\partial(iz_k)^n}\chi_P(\mathbf{z},\math
bf{z}^*)\Big|_{\mathbf{z}=\mathbf{z}^*=0}

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,

: \{W\mid P\mid
Q\}(\mathbf{\alpha},\mathbf{\alpha}^*)=\frac{1}{\pi^{2N}}\int
\chi_{\{W\mid P\mid
Q\}}(\mathbf{z},\mathbf{z}^*)e^{-i\mathbf{z}^*\cdot\mathbf{\alpha}^*}e^{-i\mathb
f{z}
\cdot \mathbf{\alpha}} \, d^{2N}\mathbf{z}.

Here \alpha_j\, and \alpha^*_k 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. Since differentiation in
normal space becomes multiplication in Fourier space, moments can be
calculated from these functions in the following way:

* \langle\widehat{\mathbf{a}}_j^{\dagger
m}\widehat{\mathbf{a}}_k^n\rangle=\int
P(\mathbf{\alpha},\mathbf{\alpha}^*)\alpha_j^n\alpha_k^{*m} \,
d^{2N}\mathbf{\alpha}
* \langle\widehat{\mathbf{a}}_j^m\widehat{\mathbf{a}}_k^{\dagger
n}\rangle=\int
Q(\mathbf{\alpha},\mathbf{\alpha}^*)\alpha_j^m\alpha_k^{*n} \,
d^{2N}\mathbf{\alpha}
* \langle(\widehat{\mathbf{a}}_j^{\dagger
m}\widehat{\mathbf{a}}_k^n)_S\rangle=\int
W(\mathbf{\alpha},\mathbf{\alpha}^*)\alpha_j^m\alpha_k^{*n} \,
d^{2N}\mathbf{\alpha}
Here (\cdots)_S denotes symmetric ordering.

These representations are all interrelated through convolution by
Gaussian functions, Weierstrass transforms,
*W(\alpha,\alpha^*)= \frac{2}{\pi} \int P(\beta,\beta^*)
e^{-2|\alpha-\beta|^2} \, d^2\beta
*Q(\alpha,\alpha^*)= \frac{2}{\pi} \int W(\beta,\beta^*)
e^{-2|\alpha-\beta|^2} \, d^2\beta
or, using the property that convolution is associative,
*Q(\alpha,\alpha^*)= \frac{1}{\pi} \int P(\beta,\beta^*)
e^{-|\alpha-\beta|^2} \, d^2\beta ~.


            Time evolution and operator correspondences
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Since each of the above transformations from  to the distribution
functions is linear, the equation of motion for each distribution can
be obtained by performing the same transformations to \dot{\rho}.
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.


For instance, consider the annihilation operator \widehat{a}_j\,
acting on  . For the characteristic function of the P distribution we
have

: \operatorname{tr}(\widehat{a}_j\rho
e^{i\mathbf{z}\cdot\widehat{\mathbf{a}}}e^{i\mathbf{z}^*\cdot\widehat{\mathbf{a}
}^{\dagger}})
= \frac{\partial}{\partial(iz_j)}\chi_P(\mathbf{z},\mathbf{z}^*).

Taking the Fourier transform with respect to \mathbf{z}\, to find the
action corresponding action on the Glauber P function, we find

\widehat{a}_j\rho \rightarrow \alpha_j
P(\mathbf{\alpha},\mathbf{\alpha}^*).

By following this procedure for each of the above distributions, the
following
'operator correspondences' can be identified:

* \widehat{a}_j\rho \rightarrow \left(\alpha_j +
\kappa\frac{\partial}{\partial\alpha_j^*}\right)\{W\mid P\mid
Q\}(\mathbf{\alpha},\mathbf{\alpha}^*)
* \rho\widehat{a}^\dagger_j \rightarrow \left(\alpha_j^* +
\kappa\frac{\partial}{\partial\alpha_j}\right)\{W\mid P\mid
Q\}(\mathbf{\alpha},\mathbf{\alpha}^*)
* \widehat{a}^\dagger_j\rho \rightarrow \left(\alpha_j^* -
(1-\kappa)\frac{\partial}{\partial\alpha_j}\right)\{W\mid P\mid
Q\}(\mathbf{\alpha},\mathbf{\alpha}^*)
* \rho\widehat{a}_j \rightarrow \left(\alpha_j -
(1-\kappa)\frac{\partial}{\partial\alpha_j^*}\right)\{W\mid P\mid
Q\}(\mathbf{\alpha},\mathbf{\alpha}^*)

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.


Coherent state
================
By construction, 'P' for a coherent state |\alpha_0\rangle is simply a
delta function:
:P(\alpha,\alpha^*)=\delta^2(\alpha-\alpha_0).
The Wigner and 'Q' representations follows immediately from the
Gaussian convolution formulas above:

:W(\alpha,\alpha^*)=\frac{2}{\pi} \int \delta^2(\beta-\alpha_0)
e^{-2|\alpha-\beta|^2} \,
d^2\beta=\frac{2}{\pi}e^{-2|\alpha-\alpha_0|^2}
:Q(\alpha,\alpha^*)=\frac{1}{\pi} \int \delta^2(\beta-\alpha_0)
e^{-|\alpha-\beta|^2} \,
d^2\beta=\frac{1}{\pi}e^{-|\alpha-\alpha_0|^2}.

The Husimi representation can also be found using the formula above
for the inner product of two coherent states:

:Q(\alpha,\alpha^*)=\frac{1}{\pi}\langle
\alpha|\widehat{\rho}|\alpha\rangle =\frac{1}{\pi}|\langle
\alpha_0|\alpha\rangle|^2 = \frac{1}{\pi}e^{-|\alpha-\alpha_0|^2}


Fock state
============
The 'P' representation of a Fock state |n\rangle is
:P(\alpha,\alpha^*)=\frac{e^{|\alpha|^2}}{n!}
\frac{\partial^{2n}}{\partial\alpha^{*n}\,\partial\alpha^n}
\delta^2(\alpha).

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

:W(\alpha,\alpha^*) = (-1)^n\frac{2}{\pi} e^{-2|\alpha|^2}
L_n\left(4|\alpha|^2\right)  ~,

which can go negative but is bounded.  'Q' always remains positive and
bounded:

:Q(\alpha,\alpha^*)=\frac{1}{\pi}\langle
\alpha|\widehat{\rho}|\alpha\rangle =\frac{1}{\pi}|\langle
n|\alpha\rangle|^2 =\frac{1}{\pi n!}|\langle
0|\widehat{a}^n|\alpha\rangle|^2 = \frac{|\alpha|^{2n}}{\pi n!}
|\langle 0|\alpha\rangle|^2


Damped quantum harmonic oscillator
====================================
Consider the damped quantum harmonic oscillator with the following
master equation:

: \frac{d\widehat{\rho}}{dt} = i\omega_0
[\widehat{\rho},\widehat{a}^\dagger\widehat{a}] + \frac{\gamma}{2}
(2\widehat{a}\widehat{\rho}\widehat{a}^\dagger -
\widehat{a}^\dagger\widehat{a} \widehat{\rho} -
\rho\widehat{a}^\dagger \widehat{a}) + \gamma \langle n \rangle
(\widehat{a} \widehat{\rho} \widehat{a}^\dagger +
\widehat{a}^\dagger\widehat{\rho}\widehat{a} -
\widehat{a}^\dagger\widehat{a}\widehat{\rho}-\widehat{\rho}
\widehat{a} \widehat{a}^\dagger).

This results in the Fokker-Planck equation
:\frac{\partial}{\partial t} \{W\mid P\mid Q\}(\alpha,\alpha^*,t) =
\left[(\gamma+i\omega_0)\frac{\partial}{\partial \alpha}\alpha +
(\gamma-i\omega_0)\frac{\partial}{\partial \alpha^*}\alpha^* +
\frac{\gamma}{2}(\langle n \rangle +
\kappa)\frac{\partial^2}{\partial\alpha\,\partial\alpha^*}\right]\{W\mid
P\mid Q\}(\alpha,\alpha^*,t)
where 'κ' = 0, 1/2, 1 for the 'P', 'W', and 'Q' representations,
respectively.  If the system is initially in the coherent state
|\alpha_0\rangle, then this has the solution
:\{W\mid P\mid Q\}(\alpha,\alpha^*,t) = \frac{1}{\pi \left[\kappa +
\langle n \rangle\left(1-e^{-2\gamma t}\right)\right]}
\exp{\left(-\frac{\left|\alpha-\alpha_0 e^{-(\gamma +i\omega_0)
t}\right|^2}{\kappa + \langle n \rangle\left(1-e^{-2\gamma
t}\right)}\right)}


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