{"ID":2857847,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.07699","arxiv_id":"2510.07699","title":"Optimal lower bounds for quantum state tomography","abstract":"We show that $n = Ω(rd/\\varepsilon^2)$ copies are necessary to learn a rank $r$ mixed state $ρ\\in \\mathbb{C}^{d \\times d}$ up to error $\\varepsilon$ in trace distance. This matches the upper bound of $n = O(rd/\\varepsilon^2)$ from prior work, and therefore settles the sample complexity of mixed state tomography. We prove this lower bound by studying a special case of full state tomography that we refer to as projector tomography, in which $ρ$ is promised to be of the form $ρ= P/r$, where $P \\in \\mathbb{C}^{d \\times d}$ is a rank $r$ projector. A key technical ingredient in our proof, which may be of independent interest, is a reduction which converts any algorithm for projector tomography which learns to error $\\varepsilon$ in trace distance to an algorithm which learns to error $O(\\varepsilon)$ in the more stringent Bures distance.","short_abstract":"We show that $n = Ω(rd/\\varepsilon^2)$ copies are necessary to learn a rank $r$ mixed state $ρ\\in \\mathbb{C}^{d \\times d}$ up to error $\\varepsilon$ in trace distance. This matches the upper bound of $n = O(rd/\\varepsilon^2)$ from prior work, and therefore settles the sample complexity of mixed state tomography. We pro...","url_abs":"https://arxiv.org/abs/2510.07699","url_pdf":"https://arxiv.org/pdf/2510.07699v1","authors":"[\"Thilo Scharnhorst\",\"Jack Spilecki\",\"John Wright\"]","published":"2025-10-09T02:36:48Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.CC\",\"cs.DS\"]","methods":"[]","has_code":false}