# Plotting quadrature uncertainties in phase space

In most books like in the picture given below, the uncertainties regarding quantum states like coherent and squeezed states are represented in phase space plot by some area enclosed within a circle or as an ellipse depending on the squeezed quadrature.

While the representation is intuitive, I do not understand the precise framework that allows to plot uncertainties. For example, what does the position or momentum axis represent? how can I plot this in a precise way or is it only for a representational purpose? I had a brief look on notions like Wigner quasi probability distribution in phase space but I fail to see how it is related and it looks quite advanced to my level at the moment. It will be very much appreciated if someone could either explain what it is or perhaps direct me to any references that cover this.

## 1 Answer

Plots like these are known as Wigner quasi-probability functions. They are not proper probability distributions, as for some states they can be negative. However, for a set of states known as Gaussian states, they completely positive. The diagrams you have linked to are 2D representations of these Wigner functions. The 2D 'blobs' in your diagrams are actually 3D 'hills'.

The Wigner function $$W(x,p)$$ when integrated over one quadrature, gives the probability of the result that would be obtained if we measured the other quadrature (up to a factor of a half). It does not make sense to ask what the probability distribution of both quadratures simultaneously, since they cannot both be measured at the same time, due to the uncertainty principle. For example if we evaluate $$W(x, p)$$ at $$x = x_0$$ and integrate over the $$p$$ quadrature, for a state $$\rho$$ we get:

$$\frac{1}{2} \int^{+\infty}_{-\infty} dp W(x_0, p) = \langle x_0 |\rho | x_0 \rangle$$

Which is the probability of a measurement on the $$x$$ quadrature giving the result $$x_0$$. (Where I have used $$x$$ and $$p$$ as the quadratures, your diagram has them labelled $$x_1$$ and $$x_2$$ but it makes no difference).

Hope this gives you an intuition behind what's going on. The actual derivation of Wigner functions is not too hard, but a bit fiddly, so I won't write here! It is derived in chapter 4 of 'Quantum Optics' by Walls and Milburn or chapter 4 of 'Quantum Continuous Variables' by Serafini, if you're interested in looking it up!

• The answer indeed gives me some intuition but could you please give me some clarification regarding the interpretation of the integration? In the example that you gave, isn't integrating over p quadrature for a given state equivalent to probability of measurement of a value for p around $x_0$ ? – rahul rj Feb 15 '19 at 9:05
• I'm not quite sure what you mean by 'measurement of a value for $p$ around $x_0$'. We can't measure $x$ and $p$ for the same state due to the uncertainty principle. If we measure the $x$ quadrature, we forgo the ability to know what the value of $p$ would be if we had measured that instead. In the Wigner representation, this is captured by integrating over all values of $p$. The right hand side of the equation $\langle x_0 |\rho | x_0 \rangle$ is the probability that we get result $x_0$ (where $x_0$ is a specific value of $x$) if we measure the $x$ quadrature. – asph Feb 15 '19 at 10:37
• Actually, what I've just said isn't quite true. Technically, the right hand side is not a probability, but a probability distribution $p(x) = \langle x |\rho | x \rangle$ over $x$, since $x$ is a continuous variable, so can take a continuum of values. It doesn't make sense to talk about the 'probability of getting outcome $x_0$'. I should have said the 'probability density' or the 'probability of getting a result between $x_0$ and $x_0 + dx$. – asph Feb 15 '19 at 10:42
• Alright that makes much sense. Could you please edit your answer with what you mentioned just now so that it will be more clear? – rahul rj Feb 15 '19 at 11:09