Exact Purification Rates for Alternating Qubit Measurements and Their Blind Spot at Right Angles

<jats:p>The usual way to quantify measurement incompatibility is noise robustness: how much noise must be mixed in before some joint measurement becomes possible. That is an extremal, worst-case quantity, computed by optimization, and it says little about what a real experiment would actually record if it just kept running. We ask a plainer question. A qubit is measured, over and over, in two projective bases separated by a Bloch angle \(\theta\). Once the resulting chain settles into steady state, how far does the state typically move between one measurement and the next? For sharp measurements this rate, \(r{(\theta)}\), turns out to have the closed form \(\frac{1}{2}{\sin{\theta\sqrt{1 + {\sin\theta}}}}\). It agrees with the standard robustness threshold \({\nu{(\theta)}} = {1 - {1/\sqrt{1 + {\sin\theta}}}}\) to first order near \(\theta = 0\), then departs from it everywhere else. For unsharp (POVM) measurements in the weak-measurement limit \(\lambda\rightarrow 0\), full-rank Kraus operators preserve purity exactly, and this collapses the problem onto a one-dimensional diffusion on the Bloch circle. The resulting rate is \({c{(\theta)}} = {{\sin{({\theta/2})}}/{\lbrack{\sqrt{2}E{(m)}}\rbrack}}\) with \(m = {1 - {\tan^{2}{({\theta/2})}}}\) and \(E\) the complete elliptic integral of the second kind, giving the clean value \({c{({\pi/2})}} = {1/\pi}\) at the orthogonal point. A blind spot shows up along the way: at \(\theta = {\pi/2}\), \(r{(\theta)}\) stops responding to noise entirely, for any Pauli-type or amplitude-damping channel, simply because \({\sin 45^{\circ}} = {\cos 45^{\circ}}\). That is unfortunate news for anyone hoping to use \(r{(\theta)}\) as a calibration check at the angle experimentalists tend to prefer. Purification under repeated or simultaneous incompatible measurement has by now a sizable literature, but in every case we examined the angle \(\theta\) itself is held fixed and something else, measurement strength or the number of axes, is varied instead. Treating \(\theta\) as the free parameter and solving for it in closed form is the gap this paper tries to close. Past the qubit, the same construction becomes a random walk on \({\mathbb{C}}{\mathbb{P}}^{d - 1}\) rather than on a circle, and its closed form we leave open.</jats:p>