View source for Quasiprobability distribution

{{Use American English|date = February 2019}}
{{short description|Concept in statistics}}

A '''quasiprobability distribution''' is a mathematical object similar to a [[probability distribution]] but which relaxes some of [[probability axioms|Kolmogorov's axioms of probability theory]]. Quasiprobability distributions arise naturally in the study of [[quantum mechanics]] when treated in [[phase space formulation]], commonly used in [[quantum optics]], [[time-frequency analysis]],<ref>L. Cohen (1995), ''Time-frequency analysis: theory and applications'', Prentice-Hall,  Upper Saddle River, NJ,    {{isbn|0-13-594532-1}} </ref>  and elsewhere.

Quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, ''the ability to yield [[expectation value]]s with respect to the weights of the distribution''.  However, they can violate the [[probability axioms#Third axiom|''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 [[probability axioms#First axiom|first axiom]].

== Introduction ==
{{main|Coherent states}}
{{main|Optical phase space}}
In the most general form, the dynamics of a [[quantum mechanics|quantum-mechanical]] system are determined by a [[master equation]]  in [[Hilbert space]]:  an equation of motion for the [[density operator]] (usually written <math>\widehat{\rho}</math>) 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<ref name="Sudarshan">{{cite journal | last=Sudarshan | first=E. C. G. | title=Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=10 | issue=7 | date=1963-04-01 | issn=0031-9007 | doi=10.1103/physrevlett.10.277 | pages=277–279| bibcode=1963PhRvL..10..277S }}</ref> that the density operator can always be written in a ''[[diagonal matrix|diagonal]]'' form, provided that it is with respect to an ''[[overcompleteness|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 [[eigenstate]]s of the [[annihilation operator]] <math>\widehat{a}</math> serve as the overcomplete basis in the construction described above.  By definition, the coherent states have the following property,
:<math>\begin{align}\widehat{a}|\alpha\rangle&=\alpha|\alpha\rangle \\
\langle\alpha|\widehat{a}^{\dagger}&=\langle\alpha|\alpha^*. \end{align}</math>

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
:<math>\langle\beta\mid\alpha\rangle=e^{-{1\over2}(|\beta|^2+|\alpha|^2-2\beta^*\alpha)}\neq\delta(\alpha-\beta).</math>
Note that these states are, however, correctly [[unit vector|normalized]] with〈''α''&nbsp;|&nbsp;''α''〉 =&nbsp;1.  Owing to the completeness of the basis of [[Fock state]]s, the choice of the basis of coherent states must be overcomplete.<ref>{{cite journal | last=Klauder | first=John R | title=The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers | journal=Annals of Physics | publisher=Elsevier BV | volume=11 | issue=2 | year=1960 | issn=0003-4916 | doi=10.1016/0003-4916(60)90131-7 | pages=123–168| bibcode=1960AnPhy..11..123K }}</ref>  Click to show an informal proof.
{| class="toccolours collapsible collapsed" width="100%" style="text-align:left"
!Proof of the overcompleteness of the coherent states
|-
|
Integration over the complex plane can be written in terms of polar coordinates with <math>d^2\alpha=r \, dr \, d\theta</math>.  Where [[order of integration (calculus)|exchanging sum and integral]] is allowed, we arrive at a simple integral expression of the [[gamma function]]:
:<math>\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}</math>

Clearly,  one can span the Hilbert space by writing a state as
:<math>|\psi\rangle = \frac{1}{\pi} \int |\alpha\rangle\langle\alpha|\psi\rangle \, d^2\alpha.</math>

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<ref name="Sudarshan" /> to express the density operator in the diagonal form
:<math>\widehat{\rho} = \int f(\alpha,\alpha^*) |\alpha\rangle \langle \alpha| \, d^2\alpha</math>
where ''f'' is a representation of the phase space distribution.  This function ''f'' is considered a quasiprobability density because it has the following properties:
:*<math>\int f(\alpha,\alpha^*) \, d^2\alpha = \operatorname{tr}(\widehat{\rho}) = 1 </math> (normalization)
:*If <math>g_\Omega (\widehat{a},\widehat{a}^\dagger)</math> 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
:::<math>\langle g_{\Omega} (\widehat{a},\widehat{a}^\dagger) \rangle = \int f(\alpha,\alpha^*) g_\Omega(\alpha,\alpha^*) \, d\alpha \, d\alpha^*</math> ([[optical equivalence theorem]]).

There exists a family of different representations, each connected to a different ordering&nbsp;Ω.  The most popular in the general physics literature and historically first of these is the [[Wigner quasiprobability distribution]],<ref>{{cite journal | last=Wigner | first=E. | title=On the Quantum Correction For Thermodynamic Equilibrium | journal=Physical Review | publisher=American Physical Society (APS) | volume=40 | issue=5 | date=1932-06-01 | issn=0031-899X | doi=10.1103/physrev.40.749 | pages=749–759| bibcode=1932PhRv...40..749W }}</ref> 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]].<ref>{{cite journal | last=Glauber | first=Roy J. | title=Coherent and Incoherent States of the Radiation Field | journal=Physical Review | publisher=American Physical Society (APS) | volume=131 | issue=6 | date=1963-09-15 | issn=0031-899X | doi=10.1103/physrev.131.2766 | pages=2766–2788| bibcode=1963PhRv..131.2766G }}</ref>  The quasiprobabilistic nature of these phase space distributions is best understood in the {{mvar|P}} representation because of the following key statement:<ref>{{Citation
  | last1 = Mandel
  | first1 = L.
  | author-link = Leonard Mandel
  | last2 = Wolf
  | first2 = E.
  | author2-link = Emil Wolf
  | title = Optical Coherence and Quantum Optics
  | place = Cambridge UK
  | publisher = Cambridge University Press
  | year = 1995
  | isbn = 0-521-41711-2
  }}</ref>

{{Quotation|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 [[quantum entanglement|entangled system]], then ''P'' is negative somewhere or more singular than a [[Dirac delta function|delta function]].}}

This sweeping statement is inoperative in other representations.  For example, the Wigner function of the [[EPR paradox|EPR]] state is positive definite but has no classical analog.<ref>{{cite journal | last=Cohen | first=O. | title=Nonlocality of the original Einstein-Podolsky-Rosen state | journal=Physical Review A | publisher=American Physical Society (APS) | volume=56 | issue=5 | date=1997-11-01 | issn=1050-2947 | doi=10.1103/physreva.56.3484 | pages=3484–3492| bibcode=1997PhRvA..56.3484C }}</ref><ref>{{cite journal | last1=Banaszek | first1=Konrad | last2=Wódkiewicz | first2=Krzysztof | title=Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation | journal=Physical Review A | volume=58 | issue=6 | date=1998-12-01 | issn=1050-2947 | doi=10.1103/physreva.58.4345 | pages=4345–4347| arxiv=quant-ph/9806069 | bibcode=1998PhRvA..58.4345B | s2cid=119341663 }}</ref>

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]],<ref>{{cite conference| last=Husimi | first=Kôdi | title=Some Formal Properties of the Density Matrix | conference=Proceedings of the Physico-Mathematical Society of Japan|publisher=The Mathematical Society of Japan | volume=22 | issue=4 | issn=0370-1239 | doi=10.11429/ppmsj1919.22.4_264 | pages=264–314|doi-access=free}}</ref> which is useful when operators are in ''anti''-normal order.  More recently, the positive  {{mvar|P}}   representation and a wider class of generalized {{mvar|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 function (probability theory)|characteristic function]]s,
from which all operator expectation values can be derived. The characteristic
functions for the Wigner, [[Glauber–Sudarshan P-representation|Glauber P]] and Q distributions of an ''N'' mode system
are as follows:

* <math>\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}})</math>
* <math>\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}}})</math>
* <math>\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}})</math>

Here <math>\widehat{\mathbf{a}}</math> and <math>\widehat{\mathbf{a}}^{\dagger}</math> 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, [[normal order|normally ordered]] (creation operators preceding annihilation operators) moments can be evaluated in the following way from <math>\chi_P\,</math>:

: <math>\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},\mathbf{z}^*)\Big|_{\mathbf{z}=\mathbf{z}^*=0}</math>

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]]s of the above characteristic functions. That is,

: <math>\{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\mathbf{z} \cdot \mathbf{\alpha}} \, d^{2N}\mathbf{z}.</math>

Here <math>\alpha_j\,</math> and <math>\alpha^*_k</math> may be identified as [[coherent state]] amplitudes in the case of the Glauber P and Q distributions, but simply [[c-number]]s 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:
* <math>\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}</math>
* <math>\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}</math>
* <math>\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}</math>
Here <math>(\cdots)_S</math> denotes symmetric ordering.  

These representations are all interrelated through [[convolution]] by [[Gaussian function]]s, [[Weierstrass transform]]s,
*<math>W(\alpha,\alpha^*)= \frac{2}{\pi} \int P(\beta,\beta^*) e^{-2|\alpha-\beta|^2} \, d^2\beta</math>
*<math>Q(\alpha,\alpha^*)= \frac{2}{\pi} \int W(\beta,\beta^*) e^{-2|\alpha-\beta|^2} \, d^2\beta</math>
or, using the property that convolution is [[associative]],
*<math>Q(\alpha,\alpha^*)= \frac{1}{\pi} \int P(\beta,\beta^*) e^{-|\alpha-\beta|^2} \, d^2\beta ~.</math>

It follows that 
*<math>P(\alpha,\alpha^*)= \frac{1}{\pi^2} \int Q(\beta,\beta^*) e^{|\lambda|^2+\lambda^* ( \alpha-\beta) -\lambda  ( \alpha-\beta) ^*} \, d^2\beta ~d^2\lambda,</math>
an often divergent integral, indicating ''P'' is often a distribution. ''Q'' is always broader than ''P'' for the same density matrix. <ref>Wolfgang Schleich, ''Quantum Optics in Phase Space'',   (Wiley-VCH,  2001) {{isbn|978-3527294350}}</ref>

For example, for a thermal state, 
:<math>\hat \rho= \frac{1}{\bar n +1}\sum_{n=0}^\infty \left (\frac{\bar n}{1+\bar n }\right)^n |n\rangle \langle n|~~, </math>
one has 
:<math>P(\alpha)= \frac{1}{\pi \bar n } e^{-\frac{|\alpha|^2}{\bar n}},  \qquad
 Q(\alpha)= \frac{1}{\pi (1+ \bar n) } e^{-\frac{|\alpha|^2}{1+\bar n}}~~~.</math>

==Time evolution and operator correspondences==
Since each of the above transformations from {{mvar|ρ}} to the distribution functions is [[linear]], the equation of motion for each distribution can be obtained by performing the same transformations to <math>\dot{\rho}</math>. Furthermore, as any [[master equation]] which can be expressed in [[lindblad equation|Lindblad form]] is completely described by the action of combinations of [[creation and annihilation operators|annihilation and creation operators]] on the density operator, it is useful to consider the effect such operations have on each of the quasiprobability functions.<ref>H. J. Carmichael, ''Statistical Methods in Quantum Optics I: Master Equations and Fokker–Planck Equations'', Springer-Verlag (2002).</ref>
<ref>C. W. Gardiner, ''Quantum Noise'', Springer-Verlag (1991).</ref>

For instance, consider the annihilation operator <math>\widehat{a}_j\,</math> acting on  {{mvar|ρ}}. For the characteristic function of the P distribution we have
: <math>\operatorname{tr}(\widehat{a}_j\rho e^{i\mathbf{z}^*\cdot\widehat{\mathbf{a}}^{\dagger}} e^{i\mathbf{z}\cdot\widehat{\mathbf{a}}}) = \frac{\partial}{\partial(iz_j)}\chi_P(\mathbf{z},\mathbf{z}^*).</math>

Taking the [[Fourier transform]] with respect to <math>\mathbf{z}\,</math> to find the
action corresponding action on the Glauber P function, we find
:<math>\widehat{a}_j\rho \rightarrow \alpha_j P(\mathbf{\alpha},\mathbf{\alpha}^*).</math>

By following this procedure for each of the above distributions, the following
''operator correspondences'' can be identified:
* <math>\widehat{a}_j\rho \rightarrow \left(\alpha_j + \kappa\frac{\partial}{\partial\alpha_j^*}\right)\{W\mid P\mid Q\}(\mathbf{\alpha},\mathbf{\alpha}^*)</math>
* <math>\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}^*)</math>
* <math>\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}^*)</math>
* <math>\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}^*)</math>

Here {{math|κ {{=}} 0, 1/2}} or 1 for P, Wigner, and Q distributions, respectively.  In this way, [[master equation]]s can be expressed as an equations of
motion of  quasiprobability functions.

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

The Husimi representation can also be found using the formula above for the inner product of two coherent states,
:<math>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}</math>

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

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<sub>n</sub>'' is the nth [[Laguerre polynomial]], ''W'' is
:<math>W(\alpha,\alpha^*) = (-1)^n\frac{2}{\pi} e^{-2|\alpha|^2} L_n\left(4|\alpha|^2\right)  ~,</math>
which can go negative but is bounded.  

''Q'',  by contrast,  always remains positive and bounded,
:<math>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 ~.</math>

===Damped quantum harmonic oscillator===

Consider the damped quantum harmonic oscillator with the following master equation,
: <math>\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).</math>

This results in the [[Fokker–Planck equation]],
:<math>\frac{\partial}{\partial t} \{W\mid P\mid Q\}(\alpha,\alpha^*,t) = \left[(\frac{\gamma}{2}+i\omega_0)\frac{\partial}{\partial \alpha}\alpha + (\frac{\gamma}{2}-i\omega_0)\frac{\partial}{\partial \alpha^*}\alpha^* + \gamma(\langle n \rangle + \kappa)\frac{\partial^2}{\partial\alpha\,\partial\alpha^*}\right]\{W\mid P\mid Q\}(\alpha,\alpha^*,t), </math>
where ''κ''&nbsp;=&nbsp;0, 1/2, 1 for the ''P'', ''W'', and ''Q'' representations, respectively.  

If the system is initially in the coherent state <math>|\alpha_0\rangle</math>, then this equation has the solution
:<math>\{W\mid P\mid Q\}(\alpha,\alpha^*,t) = \frac{1}{\pi \left[\kappa + \langle n \rangle\left(1-e^{-\gamma t}\right)\right]} \exp{\left(-\frac{\left|\alpha-\alpha_0 e^{-(\frac{\gamma}{2} +i\omega_0) t}\right|^2}{\kappa + \langle n \rangle\left(1-e^{-\gamma t}\right)}\right)}~~.</math>

==See also==
* [[Krein space]]

==References==
<references/>

[[Category:Particle distributions]]
[[Category:Quantum optics]]
[[Category:Exotic probabilities]]