Optimal lower bounds for quantum state tomography
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914082349645824 |
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| author | Scharnhorst, Thilo Spilecki, Jack Wright, John |
| author_facet | Scharnhorst, Thilo Spilecki, Jack Wright, John |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal lower bounds for quantum state tomography Scharnhorst, Thilo Spilecki, Jack Wright, John Quantum Physics Computational Complexity Data Structures and Algorithms 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. |
| title | Optimal lower bounds for quantum state tomography |
| topic | Quantum Physics Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.07699 |