Bloch sphere representation of uncertainty

Question

If we consider the Bloch sphere in quantum mechanics, which is a two level representation of a quantum mechanical system, then any state can be represented as $$| \psi \rangle = \cos\left(\theta/2\right)|0 \rangle + e^{i \phi} \sin\left(\theta/2\right) | 1 \rangle.$$ We can show that for any state $| \psi \rangle$, the graphical representation of the expectation values $\langle \hat{S}_z \rangle$, $\langle \hat{S}_x \rangle$ and $\langle \hat{S}_y \rangle$ which are the projections onto the $z,x$ and $y$ axis of the Bloch sphere, respectively. Is there any clear graphical representation for the uncertainty in measurments of $\hat{S}_z, \hat{S}_z$ and $\hat{S}_y$?

Answer 1

Yes, there is something, and it holds even for mixed states, not only for pure states.

Take an arbitrary density matrix ${\hat \rho}$ corresponding to a point A = $(a_x, a_y, a_z)$ within the unit Bloch sphere, $|{\vec a}| \le 1$, such that $$ {\hat \rho} = \frac{1}{2} \Big( {\hat I} + {\vec a}\cdot {\hat {\vec \sigma}}\Big) $$ Since the spin averages along directions $x$, $y$, $z$ are the components $a_x$, $a_y$, $a_z$, $$ \langle {\hat \sigma}_i \rangle = a_i\;, \;\;\; i = x,\;y\;,z $$ the corresponding uncertainties read $$ \langle (\Delta {\hat \sigma}_i)^2 \rangle = Tr [({\hat\sigma}_i^2 - \langle {\hat \sigma}_i \rangle^2) {\hat \rho}] = 1 - a_i^2\;, \;\;\; i = x,\;y\;,z $$ Consider for example $\langle (\Delta {\hat \sigma}_x)^2 \rangle = 1 - a_x^2$. Slice the Bloch sphere with a plane perpendicular to the x-axis and passing through point A. Then the radius of the resulting circular cut is $\sqrt{1-a_x^2} = \sqrt{\langle (\Delta {\hat \sigma}_x)^2 \rangle}$.

For pure states, when $|{\vec a}| =1$ and $$ \langle (\Delta {\hat \sigma}_i)^2 \rangle = 1 - a_x^2 = a_y^2 + a_z^2 $$ the circular cut crosses through point A, and $\sqrt{a_y^2 + a_z^2}$ is just the distance from A to the x-axis. Similarly for the other axes. So in general,

For a pure state $\psi$ the spin uncertainty along any direction ${\vec n}$ is the distance from its representative point A = $(\langle {\hat \sigma}_x \rangle, \langle {\hat \sigma}_y\rangle, \langle {\hat \sigma}_z\rangle)$ on the Bloch sphere to that axis.

Obviously the only direction for which this distance and the corresponding uncertainty is null is the direction of ${\vec a}$ itself.