Understanding the relationship between Phase Space Distributions (Wigner vs Glauber-Sudarshan P vs Husimi Q)

I am moving into a new field and after thorough literature research need help appreciating what is out there.

In the continuos variable formulation of optical state space. (Quantum mechanical/Optical) states are represented by quasi-probability distributions in this picture. There are at least three famous distributions in use to represent these states: The Wigner vs Glauber-Sudarshan P vs Husimi Q.

The Wigner distribution is defined as $$W(x,p)=:\frac{1}{\pi\hbar}\int \int \psi(x+y)^{*} \psi(x-y) e^{\frac{2ipy}{\hbar}} dx dp$$ The Glauber-Sudarshan $P$ distribution is $$P(\alpha)=\frac{e^{|\alpha|^{2}}}{\pi}\int \langle-\beta|\rho|\beta \rangle e^{|\beta|^{2}-\beta \alpha^{*}+\beta^{*}\alpha} d\alpha d\beta$$ The Husimi $Q$ distribution is $$Q(\alpha) = \langle \alpha | \rho | \alpha \rangle$$ One thing i have noticed already the difference between $\alpha$ and $x,p$ seems crucial. From my understanding $x,p$ are mathematically treated just as two parameters. They are identified with position x and momentum p in physics.

In contrast $\alpha$ is not really a scalar, but a parameter defining a so called coherent state. It seems that somehow the other distributions are expressed in terms of theses coherent states. (see wikipedia for more on coherent states: http://en.wikipedia.org/wiki/Coherent_states )

I admit not even properly understanding these definitions entirely. As far as i understand, they should be equivalent, even though this is not obvious to me.

First Question: How are these quite different looking distributions the same?

Furthermore once this is understood, i expect them to have advantages and disadvantages else they would not be there. I fail to see how they are related and why one representation is favourable over the others.

Second Question: Could someone help me understand why a certain representation is chosen over the other?

• You are piling up 3 different problems. First, you are dealing with W in a coordinate picture instead of the coherent state picture; once you get W in it, your own link tells you the Q is a Weirstrass transform of W. – Cosmas Zachos Jul 1 '16 at 0:37
• Alternatively, you can forfeit coherent space and express all three in phase space, and hit the books: Measuring the Quantum State of Light, Ulf Leonhardt, chapter 3.2; or Quantum Optics in Phase Space, Wolfgang P. Schleich; or ours p.58. They all indicate how W,P, and Q are equivalent ordering prescriptions of the operators that got into their definition out of the density matrix. There is lots of algebra, and the answer would be worth book chapters, not this venue, unless you were technically experienced. – Cosmas Zachos Jul 1 '16 at 0:46
• @Cosmas Zachos: Thank you for the literature recommendation! I was under the impression, that shifting from x,p representation to a basis representation of coherent states, is somehow the main idea behind the P distribution, but maybe that is an error...... – ckrk Jul 1 '16 at 12:09

Let me actually collect most of my comments in an answer attempting to be more coherent than they, or your labile question. In fact, you are piling up three different questions, logically distinct, but with strong and natural connections, so it might be worth splitting them apart, before bringing them back together in the final coda.

• First, there is plain Hilbert space, $|x\rangle$, or $|p\rangle$, whose parameter space is plain phase space, with parameters $x$ and $p$, eigenvalues of the respective operators. Then there is the Fock space of creation and annihilation operators, where now $\alpha$ is a complex parameter, eigenvalue of the annihilation operator on coherent states, with its complex conjugate $\alpha^*$, often said to comprise optical phase space. They are equivalent, but the overcomplete coherent states have gaussian overlaps with each other, and conversions require inordinate care. Basically, in natural units $\hbar=1$, the overlaps with position eigenstates are gaussian Schroedinger wavepackets,
$$\langle x | \alpha\rangle= \frac{1}{\pi^{1/4}} ~ e^{\sqrt{2}\alpha x - x^2/2- \alpha \Re (\alpha)}~.$$

• Secondly, selected transforms of the density matrix ρ to phase space via the Wigner transform, or the Glauber-Sudarshan transform, or the Husimi transform, yield the equivalent quasiprobability distribution functions W,P, or Q. The three equivalent transforms differ by the operator orderings involved in the characteristic functions of the distributions: Weyl ordering, normal ordering, or antinormal ordering, respectively. Q is an invertible Weierstrass transform, a convolution with a Gaussian, of W, as you can find in standard books: Measuring the Quantum State of Light, Ulf Leonhardt, chapter 3.2; or Quantum Optics in Phase Space, Wolfgang Schleich; or ours p.58. These books opt for phase-space variables, x and p, where everything is easy, and the equivalence conversions among them are straightforward. (The general one-parameter conversion systematics among them is called "Cohen's classification theory".)

But if you must work in optical phase space, which I don't really love, to compare apples with apples, you need the W function represented, schematically, not in the phase space form you have, but as the Royer expectation value of the parity operator, that is, as something like $$W(\alpha)= \frac{1}{\pi^2}\int d^2 z ~ \operatorname{tr}(\rho e^{iz(\widehat{a} -\alpha) +iz^*(\widehat{a}^{\dagger}-\alpha^*)} ).$$ Then, as you see here , by dint of the orderings in their characteristic functions, these representations are all interrelated also through Weierstrass transforms, now in optical phase space, $$W(\alpha,\alpha^*)= \frac{2}{\pi} \int P(\beta,\beta^*) e^{-2|\alpha-\beta|^2} \, d^2\beta$$ $$Q(\alpha,\alpha^*)= \frac{2}{\pi} \int W(\beta,\beta^*) e^{-2|\alpha-\beta|^2} \, d^2\beta ,$$ or, using the associativity of convolutions, $$Q(\alpha,\alpha^*)= \frac{1}{\pi} \int P(\beta,\beta^*) e^{-|\alpha-\beta|^2} \, d^2\beta ~.$$

• Thirdly, the relative merits of each variant are roughly these. W is best for plain phase space: there, it is the only one that does not need a star product in the expectation value integrals—the analog of the cartesian coordinate system. However, if no multiple strings of star products are required (only!) there is a convenient kludge, the optical equivalence theorem, to empower P to compute expectation values for normal-ordered operators, in coherent states’ space, starlessly; but things go terribly south if strings of star products are involved. So the user must carefully normal-order everything before calculating. Q does the same for anti-normal ordered operators (if you ran into them and failed to easily normal order them!). So, in a way, your misconception is well-founded: it pays to use W in plain phase space and possibly P in optical phase space, even though you don't need to.

One parting comment: Often, and quite wrongly, people contrast the properties of such distributions, making something of the fact Q is positive semidefinite, implying, wrongly, again, that it can “therefore” come closer to serving as a bona-fide probability distribution. This is a dangerous illusion, as, firstly, expectation values in plain phase space with Q but without star products are plain wrong (they are uncontrollable semiclassical approximations). With star products, they are correct--—simple dysfunctional changes of variables of the Wigner picture, as all of the above, as we saw, are glorified changes of variables and pictures.

Most importantly, last but not least, all of the above cannot be probability distributions, even if positive semidefinite, by failing Kolmogorov’s third axiom on the disjoint contingency of different points in the parameter (phase-)space: the uncertainty principle tells you two points x,p and x’,p’ are confused by the uncertainty principle if they are close enough in $\hbar$ units (here, 1). In fact, the uncertainty principle ensures that the solidly negative puddles of W be small in area units of $\hbar$, as detailed in our book, cited above. Now and forever, these, all three, are merely quasi-probability distributions.

• thank you for this excellent answer! I just stumbled upon this question, but it has great value to me since I only just started studying quantum optics and this answer almost reads like a guide to what the structure of this distribution topic is. +1 – Wolpertinger Jul 5 '16 at 20:52
• thanks you also Cosmas, i also think the collecting your comments into an answer is an excellent idea and apprecieate it! +1 – ckrk Jul 6 '16 at 8:55