Conjecture on optimal input and measurement angles for Pauli channel tomography

Let $\underline{\lambda}$ denote the channel contraction parameters and let $N$ be the number of channel uses. Let $\tilde h(\underline{\lambda},\underline{0},\underline{\tau},\underline{\vartheta},N)$ be the loss function for estimating the angle parameters, with input and measurement angle parameters $\underline{\tau}$ and $\underline{\vartheta}$. For a minimizer $(\underline{\tau}_{opt},\underline{\vartheta}_{opt})$, the conjecture asserts the following. Optimal-angle conjecture. For any fixed parameters $\underline{\lambda}$ and $N$, if $\tilde h(\underline{\lambda},\underline{0},\underline{\tau},\underline{\vartheta},N)$ is minimal at $(\underline{\tau}_{opt},\underline{\vartheta}_{opt})$, then

$$\underline{\tau}_{opt}=\underline{\vartheta}_{opt}.$$

Moreover, the estimation strategies with

$$\underline{\tau}_1=\underline{\vartheta}_1=\left(\frac{\pi}{4},\frac{\pi}{4},0\right),\qquad \underline{\tau}_2=\underline{\vartheta}_2=\left(\frac{\pi}{4},0,\frac{\pi}{4}\right)$$

are nearly optimal. The claim is based on numerical optimization because the general six-variable minimization problem cannot be solved analytically in the source. The precise meaning of “nearly optimal” and a general proof are not supplied.