Minimal Projective Measurement Settings for Estimating Multiple Observables

We study the minimum number of orthogonal projective measurement settings needed to estimate multiple observables without bias for arbitrary input states. We derive structural bounds that go beyond dimension counting and combine them with explicit constructions to certify minimum setting counts for selected target spaces. A classical variational algorithm searches for shared measurement bases and optimizes their readout coefficients and sample allocation. For the collective first and second moments of two qubits, we certify that four projective settings are necessary and sufficient. For this task, a numerically optimized four-setting design achieves a Haar-averaged variance within approximately $0.0341\%$ of a lower bound valid for any number of projective settings in our fixed-allocation, unbiased linear estimation model. Numerical examples show that shared measurements can improve precision relative to the comparison protocols. For finite-depth circuits, we account for reconstruction bias and identify finite-sample regimes in which this advantage persists.