Energy dependence of tomography algorithms

Ascertain whether the doubly logarithmic dependence on the energy constraint in current Gaussian‑state tomography algorithms is fundamental or can be removed.

Background

The best current protocols for mixed and pure Gaussian states exhibit only a mild, doubly logarithmic dependence on the energy bound, achieved via adaptive Gaussian measurements.

It is not known whether this residual dependence is required by information‑theoretic limitations or is an artifact of existing techniques.

References

A related open question concerns the role of energy constraints. The best-known tomography protocols for both mixed Gaussian states and pure Gaussian states (see Theorem~\ref{th:main_up_pure}) exhibit a mild double-logarithmic dependence on the energy. It remains unclear whether this mild dependence reflects a fundamental limitation or is merely an artifact of existing techniques.

— Towards sample-optimal learning of bosonic Gaussian quantum states  (2603.18136 - Chen et al., 18 Mar 2026) in Open problems, Section 5.2 ("Open problems")