{"ID":2898681,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.02461","arxiv_id":"2507.02461","title":"Maximal entropy in the moment body","abstract":"A moment body is a linear projection of the spectraplex, the convex set of trace-one positive semidefinite matrices. Determining whether a given point lies within a given moment body is a problem with numerous applications in quantum state estimation or polynomial optimization. This moment body membership oracle can be addressed with semidefinite programming, for which several off-the-shelf interior-point solvers are available. In this paper, inspired by techniques from quantum information theory, we argue analytically and geometrically that a much more efficient approach consists of minimizing globally a smooth strictly convex log-partition function, dual to a maximum entropy problem. We analyze the curvature properties of this function and we describe a neat geometric pre-conditioning algorithm. A detailed complexity analysis reveals a cubic dependence on the matrix size, similar to a few eigenstructure computations. Basic numerical experiments illustrate that dense (i.e. non-sparse) projections of size 1000 of a dense semidefinite matrix of size 1000-by-1000 can be routinely handled in a few seconds on a standard laptop, thereby moving the main bottleneck in large-scale semidefinite programming almost entirely to efficient gradient storage and manipulation.","short_abstract":"A moment body is a linear projection of the spectraplex, the convex set of trace-one positive semidefinite matrices. Determining whether a given point lies within a given moment body is a problem with numerous applications in quantum state estimation or polynomial optimization. This moment body membership oracle can be...","url_abs":"https://arxiv.org/abs/2507.02461","url_pdf":"https://arxiv.org/pdf/2507.02461v1","authors":"[\"Didier Henrion\"]","published":"2025-07-03T09:18:40Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}