# The log log jam in Gaussian state tomography

arXiv: https://arxiv.org/abs/2607.12983 · HTML: https://mlsanity.com/paper/2607.12983/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2026.

## Built on by (most cited first, 2 of 2)

- [When are bosonic Gaussian states classical to learn?](https://mlsanity.com/paper/2609.26705/) (2026)
- [Finding Gaussian Structure in Bosonic States](https://mlsanity.com/paper/2610.06810/) (2026)

## Abstract (arXiv)

Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with log log E, where E is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a log log E dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn n-mode pure Gaussian states with O(n2 / ε2) samples, independent of E. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with O(1/ε2) samples, again independent of E. Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.