The log log jam in Gaussian state tomography

In continuous-variable quantum systems, Gaussian states are defined by their first and second moments, yet previous state-of-the-art tomography protocols exhibited a sample complexity that scaled with the system's energy (E). The authors investigate whether this energy dependence is a fundamental physical limitation or merely an artifact of existing measurement protocols, and whether non-Gaussian measurements can provide a more efficient alternative.

To address this, the authors employ a Fisher information-based framework to analyze the limits of Gaussian measurements. They prove that any protocol restricted to Gaussian measurements—even those utilizing adaptive strategies or entanglement across copies—must incur a log log E dependence. They then explore the trade-off between the number of adaptive rounds and energy dependence. Finally, they move beyond the Gaussian regime by constructing protocols using non-Gaussian measurements, specifically leveraging the Siegel disk parametrization for pure Gaussian states and the canonical phase POVM for single-mode states.

1. Gaussian Limitation: The authors prove that the log log E dependence is an inherent bottleneck for any protocol relying solely on Gaussian measurements. They establish a formal trade-off showing how limited adaptivity influences this energy dependence.
2. Non-Gaussian Advantage: By employing highly entangled, non-Gaussian measurements, the authors achieve an energy-independent sample complexity of O(n^2/ε^2) for n-mode pure Gaussian states, resolving a significant open question in the field.
3. Simplified Protocols: For single-mode pure Gaussian states, they demonstrate that a simple, non-adaptive protocol using the canonical phase POVM achieves energy-independent learning, providing a practical path forward for specific experimental setups.