{"ID":6626561,"CreatedAt":"2026-07-15T02:56:36.47817413Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.12983","arxiv_id":"2607.12983","title":"The log log jam in Gaussian state tomography","abstract":"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 $O(n^2 / ε^2)$ 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 $O(1/ε^2)$ 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.","short_abstract":"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 finit...","url_abs":"https://arxiv.org/abs/2607.12983","url_pdf":"https://arxiv.org/pdf/2607.12983v1","authors":"[\"Sitan Chen\",\"Weiyuan Gong\",\"Qi Ye\",\"Zhihan Zhang\"]","published":"2026-07-14T17:25:43Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.DS\",\"math-ph\"]","methods":"[]","has_code":false}