Optimal sample complexity for learning mixed Gaussian states

Determine the optimal sample complexity for learning an n‑mode mixed bosonic Gaussian state to ε trace distance with success probability at least 2/3, and establish whether this can match the lower bound of order n^2/ε^2 achieved for pure Gaussian states.

Background

The paper proves lower bounds of order n^2/ε^2 for arbitrary measurements and n^3/ε^2 for Gaussian-only measurements, and gives nearly matching upper bounds for two important subclasses: pure and passive Gaussian states. However, for general mixed Gaussian states, no protocol achieving the n^2/ε^2 scaling is known.

The authors identify closing this gap—achieving n^2/ε^2 for mixed states—as the central outstanding question in their program toward sample‑optimal Gaussian state tomography.