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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.12010 |
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| _version_ | 1866908601162924032 |
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| author | Keenan, Nathan Goold, John Nico-Katz, Alex |
| author_facet | Keenan, Nathan Goold, John Nico-Katz, Alex |
| contents | Quantum state tomography (QST), the process of reconstructing some unknown quantum state $\hatρ$ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error $Δ\hatρ= \hatρ-\hatρ^\prime$ in a QST reconstruction $\hatρ^\prime$ is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors $Δ\hatρ$ and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of $Δ\hatρ$ can be evaluated by substituting samples of an appropriate GUE for realizations of $Δ\hatρ$. This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance $\|Δ\hatρ\|_\mathrm{Tr}$ (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Random Matrix Theory of Pauli Tomography Keenan, Nathan Goold, John Nico-Katz, Alex Quantum Physics Statistics Theory Quantum state tomography (QST), the process of reconstructing some unknown quantum state $\hatρ$ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error $Δ\hatρ= \hatρ-\hatρ^\prime$ in a QST reconstruction $\hatρ^\prime$ is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors $Δ\hatρ$ and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of $Δ\hatρ$ can be evaluated by substituting samples of an appropriate GUE for realizations of $Δ\hatρ$. This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance $\|Δ\hatρ\|_\mathrm{Tr}$ (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future. |
| title | A Random Matrix Theory of Pauli Tomography |
| topic | Quantum Physics Statistics Theory |
| url | https://arxiv.org/abs/2506.12010 |