Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables

Understand

An important class of fermionic observables, relevant in tasks such as fermionic partial tomography and estimating energy levels of chemical Hamiltonians, are the binary measurements obtained from the product of anti-commuting Majorana operators.

• In this work, we investigate efficient estimation strategies of these observables based on a joint measurement which, after classical post-processing, yields all sufficiently unsharp (noisy) Majorana observables of even-degree.
• By exploiting the symmetry properties of the Majorana observables, as described by the braid group, we show that the incompatibility robustness, i.e., the minimal classical noise necessary for joint measurability, relates to the spectral properties of the Sachdev-Ye-Kitaev (SYK) model.
• In particular, we show that for an $n$ mode fermionic system, the incompatibility robustness of all degree-$2k$ Majorana observables satisfies $\Theta(n^{-k/2})$ for $k\leq 5$.