PDF change of variables

The goal of this module is to numerically evaluate the probability density at a point on the Bloch sphere. The probability density is known analytically on \(\mathbb{R}^2\) for pairs of real variables \((q_1,q_2)\) as:

\[\begin{split}\begin{align} G(q_1,q_2)&=\frac{\epsilon}{4\pi}\exp\left\{-\frac{\epsilon(q_1^2+q_2^2+1)} {2}\right\}(\cosh(\epsilon(q_1+q_2))+\cosh(\epsilon(q_1-q_2))) \end{align}\end{split}\]

We also have an analytic expression for mapping points in \(\mathbb{R}^2\) to points on the Bloch sphere:

\[\begin{split}\begin{align} \cos\theta&=\frac{2}{\cosh(\epsilon(q_1+q_2))+\cosh(\epsilon(q_1-q_2))} \\ \phi&=\operatorname{atan2}\Big(2\sinh(\epsilon q_2),\, \sinh(\epsilon(q_1+q_2))+\sinh(\epsilon(q_1-q_2))\Big) \end{align}\end{split}\]

where \((\theta,\phi)\) are spherical coördinates about the y-axis:

\[\begin{split}\begin{align} z&=\sin\theta\cos\phi \\ x&=\sin\theta\sin\phi \\ y&=\cos\theta \end{align}\end{split}\]

Let \(\vec{\omega}:=(\cos\theta,\phi)\) and \(\vec{q}:=(q_1,q_2)\), and define \(\Omega:\vec{q}\mapsto\vec{\omega}\). The differentials in these two sets of variables transform as:

\[\begin{split}\begin{align} d(\cos\theta)d\phi&=\vert\operatorname{det}(\mathrm{D}\Omega)\vert dq_1dq_2 \end{align}\end{split}\]

where \(\mathrm{D}\Omega\) is the Jacobian:

\[\begin{split}\begin{align} \begin{bmatrix} \frac{d(\cos\theta)}{dq_1} & \frac{d(\cos\theta)}{dq_2} \\ \frac{d\phi}{dq_1} & \frac{d\phi}{dq_2} \end{bmatrix} \end{align}\end{split}\]

The probability density function we want is \(\tilde{G}\) such that:

\[\begin{split}\begin{align} \tilde{G}(\cos\theta,\phi)d(\cos\theta)d\phi&= G(\Omega^{-1}(\cos\theta,\phi))dq_1dq_2 \end{align}\end{split}\]

Using the change of variables formula we can write:

\[\begin{split}\begin{align} \tilde{G}(\cos\theta,\phi)&=\frac{G(\Omega^{-1}(\cos\theta,\phi))} {\vert\operatorname{det}(\mathrm{D}\Omega)\vert} \end{align}\end{split}\]

Jacobian

We calculate the matrix elements of the Jacobian to be:

\[\begin{split}\begin{align} \frac{d(\cos\theta)}{dq_1}&=-2\epsilon\frac{\sinh(\epsilon q_+)+ \sinh(\epsilon q_-)}{\big(\cosh(\epsilon q_+)+\cosh(\epsilon q_-)\big)^2} \\ \frac{d(\cos\theta)}{dq_2}&=-2\epsilon\frac{\sinh(\epsilon q_+)- \sinh(\epsilon q_-)}{\big(\cosh(\epsilon q_+)+\cosh(\epsilon q_-)\big)^2} \\ \frac{d\phi}{dq_1}&=-2\epsilon\frac{\sinh(\epsilon q_2)\big( \cosh(\epsilon q_+)+\cosh(\epsilon q_-)\big)}{\big(\sinh(\epsilon q_+)+ \sinh(\epsilon q_-)\big)^2+4\sinh^2(\epsilon q_2)} \\ \frac{d\phi}{dq_2}&=-2\epsilon\frac{\sinh(\epsilon q_2)\big( \cosh(\epsilon q_+)-\cosh(\epsilon q_-)\big)-\cosh(\epsilon q_2)\big( \sinh(\epsilon q_+)+\sinh(\epsilon q_-)\big)}{\big(\sinh(\epsilon q_+)+ \sinh(\epsilon q_-)\big)^2+4\sinh^2(\epsilon q_2)} \end{align}\end{split}\]

using the shorthand \(q_\pm:=q_1\pm q_2\).

Limiting case

For some of our plots, we want to calculate \(\tilde{G}(0,\phi)\). We can find \(\Omega^{-1}(0,\phi)\), since:

\[\begin{split}\begin{align} 0&=\theta&\Leftrightarrow \\ 0&=\big(\cosh(\epsilon(q_1+q_2))-1\big)+\big(\cosh(\epsilon(q_1-q_2))-1\big) &\Leftrightarrow \\ q_1&=q_2=0 \end{align}\end{split}\]

We easily calculate \(G(0,0)=\epsilon e^{-\epsilon/2}/2\pi\). However, \(\vert\operatorname{det}(D\Omega)\vert\) evaluates to the indeterminate form \(0/0\) at \(q_1=q_2=0\). The full expression is given below:

\[\vert\operatorname{det}(D\Omega)\vert=4\epsilon^2\left\vert \frac{2s_2(s_-c_+-s_+c_-)-(s_++s_-)^2c_2}{(c_++c_-)^2((s_++s_-)^2+4s_2^2)} \right\vert\]

where the notation uses further shorthands \(s_a:=\sinh(\epsilon q_a)\) and \(c_a:=\cosh(\epsilon q_a)\). We can rewrite in a different (perhaps more suggestive) form:

\[\vert\operatorname{det}(D\Omega)\vert=4\epsilon^2\frac{c_2}{(c_++c_-)^2} \left\vert 1-\frac{4s_2^2+2\frac{s_-c_+-s_+c_-}{c_2}s_2} {4s_2^2+(s_++s_-)^2}\right\vert\]

From this and numerical calculations, I believe \(\lim_{(q_1,q_2)\to(0,0)}=\epsilon^2\), but I haven’t managed to prove it yet.