Conjectured optimal scaling for mixed Gaussian state tomography

Conjecture that learning an n‑mode mixed bosonic Gaussian state to ε trace distance with success probability at least 2/3 requires and is achievable with tilde{Theta}(n^2/ε^2) copies.

Background

Building on lower bounds and new upper bounds for special subclasses, the authors hypothesize that the general mixed‑state case shares the same sample‑complexity scaling as pure Gaussian states.

If true, this would close the gap between known lower bounds and current best algorithms for mixed states.

References

We conjecture that the correct scaling is \widetilde{\Theta}(n2/\varepsilon2), thereby saturating the general lower bound established in Theorem~\ref{th:main_lo_any} and coinciding with the sample complexity for learning pure Gaussian states established in Theorem~\ref{th:main_up_pure}.

— 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")