Systematic construction of unit-range joint estimators

Construct, for an arbitrary family of unit-range nonlinear state functionals, jointly unbiased estimators whose estimator range remains one.

Background

The paper constructs a commuting family of multi-replica observables that jointly estimates all squared Pauli moments with estimators bounded in the unit interval. The authors emphasize that commutativity is sufficient but not necessary for unit-range joint estimation, motivating the quantum Bernstein norm as a general incompatibility measure for nonlinear functionals.

The unresolved problem is to develop a general method that, given an arbitrary family of nonlinear functionals individually bounded by one, produces a single measurement and unbiased post-processing rules with the same unit estimator range.

References

This result, however, still leaves its systematic construction open: Given an arbitrary family of unit-range nonlinear functionals, how can we systematically construct jointly unbiased estimators with unit range?

— Pauli-resolved virtual distillation  (2609.24132 - Chen et al., 21 Sep 2026) in Section “Squared Pauli moments”