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The '''Husimi Q representation''', introduced by {{ill|ja|Kôdi Husimi|伏見康治}} in 1940,<ref>Kôdi Husimi (1940). "Some Formal Properties of the Density Matrix", ''Proc. Phys. Math. Soc. Jpn.'' '''22''': 264-314 .</ref> is a [[quasiprobability distribution]] commonly used in [[quantum mechanics]]<ref name=Dirac>{{cite book
|author=[[Paul Dirac|PAM Dirac]]
|title=The principles of quantum mechanics
|year=1982
|edition=Fourth Edition
|page=18 ff
|isbn=0-19-852011-5
|publisher=Oxford University Press
|location=Oxford UK
|url=http://books.google.com/books?id=XehUpGiM6FIC&printsec=frontcover&dq=intitle:quantum+intitle:mechanics+inauthor:dirac&lr=&as_brr=0&sig=mRVsWMu1RsjbysOw2sG2CK_mNpc#PPA20,M1}}
</ref> to represent the [[phase space]] distribution of a [[quantum state]] such as [[light]] in the [[phase space formulation]].<ref>Ulf Leonhardt (1997). ''Measuring the Quantum State of Light'', Cambridge Studies in Modern Optics. ISBN 0521497302 , ISBN 978-0521497305.</ref> It is used in the field of [[quantum optics]]<ref>H. J. Carmichael (2002). ''Statistical Methods in Quantum Optics I: Master Equations and Fokker-Planck Equations'', Springer-Verlag. ISBN 978-3-540-54882-9</ref> and particularly for [[tomography|tomographic]] purposes. It is also applied in the study of [[quantum]] effects in [[Type-I superconductor|superconductors]].<ref>{{cite journal |
journal = Nuclear Physics B|
volume=344|
year=1990|
pages=627–645|
title=On the remarkable structure of the superconducting intermediate state|
doi=10.1016/0550-3213(90)90672-Z|
author=[[David J E Callaway|DJE Callaway]] |bibcode = 1990NuPhB.344..627C }}</ref>

[[File:Husimi distribution squeezed state.jpg|thumb|Husimi distribution of the squeeezed coherent state]]
[[File:Hussimi distribution function.gif|thumb|Husimi distribution function of three coherent states merged together]]

==Definition and properties==

The Husimi Q distribution (called Q-function in the context of [[quantum optics]]) is one of the simplest distributions of quasiprobability in [[phase space]]. It is constructed in such a way that observables written in [[normal order|''anti''-normal order]] follow the [[optical equivalence theorem]]. This means that it is essentially the [[density matrix]] put into [[normal order]]. This makes it relatively easy to calculate compared to other quasiprobability distributions through the formula

: <math> Q(\alpha)=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle, </math>

which is effectively a [[trace (linear algebra)|trace]] of the density matrix over the basis of [[coherent states]] <math>\{|\alpha\rangle\}</math>. It produces a pictorial representation of the state ''ρ'' to illustrate several of its mathematical properties.<ref>[[Cosmas Zachos|Cosmas K. Zachos]], [[David Fairlie|David B. Fairlie]], and [[Thomas Curtright|Thomas L. Curtright]] (2005). ''Quantum Mechanics in Phase Space'', (World Scientific, Singapore) ISBN 978-981-238-384-6 [http://www.worldscibooks.com/physics/5287.html].</ref> Its relative ease of calculation is related to its smoothness compared to other quasiprobability distributions. In fact, it can be understood as a smoothing of the [[Wigner quasiprobability distribution]] by a [[Gaussian filter]]:
:<math>Q(\alpha)= \frac{2}{\pi} \int W(\beta) e^{-2|\alpha-\beta|^2} \, d^2\beta.</math>
Such Gauss transforms being essentially invertible in the Fourier domain via the [[convolution theorem]], ''Q'' provides an equivalent description of quantum mechanics in phase space to that furnished by the Wigner distribution. Alternatively, one can compute the Husimi Q distribution by taking the [[Segal–Bargmann space#The Segal–Bargmann transform|Segal–Bargmann transform]] of the wave function and then computing the associated probability density.

''Q'' is normalized to unity,

: <math> \int Q(\alpha)\,d\alpha^2 = 1 </math>

and is ''non-negative definite''<ref>{{cite doi|10.1016/0378-4371(76)90145-X|noedit}}</ref> and ''bounded'':

: <math> 0 \leq Q(\alpha) \leq \frac{1}{\pi}. </math>

Despite the fact that ''Q'' is non-negative definite and bounded like a standard [[joint probability distribution]], this similarity is misleading because different coherent states are not orthogonal. Thus ''Q'' does ''not represent the probability of mutually exclusive states'', as needed in the [[probability axioms#Third axiom|third axiom of probability theory]].

==See also==
*[[Nonclassical light]]
*[[Glauber–Sudarshan P-representation]]
*[[Wehrl entropy]]

==References==
<references/>

{{DEFAULTSORT:Husimi Q Representation}}
[[Category:Quantum optics]]
[[Category:Particle statistics]]

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en>Cydebot: Robot - Moving category Trees (structure) to :Category:Trees (data structures) per CFD at Wikipedia:Categories for discussion/Log/2012 January 12.