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Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means

  • Yen-Huan Li
  • Wei-Fu Tseng
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Abstract

We study the problem of online learning of quantum states with logarithmic loss, which is a quantum generalization of online portfolio selection and has applications to quantum state tomography, the task of estimating an unknown quantum state from measurement outcomes. We first establish a regret lower bound, showing that the regret rates of Exponential Weights and the algorithm of Tseng et al. (2025) have optimal polynomial dependence on the dimension. Our analysis leads to two further results. First, our lower-bound proof leads to a quantum state tomography algorithm that is sample optimal up to polylogarithmic factors in the sample size. Second, we develop an algorithm for constructing confidence sequences for the mean of a matrix-valued random variable. A numerical experiment shows that the latter algorithm is competitive with existing methods.

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Cite

BibTeXRIS
BibTeX
@misc{li2026regret,
  title = {{Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means}},
  author = {Yen-Huan Li and Wei-Fu Tseng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21273},
  eprint = {2610.21273},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21273},
}
RIS
TY  - UNPB
TI  - Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means
AU  - Yen-Huan Li
AU  - Wei-Fu Tseng
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21273
UR  - https://arxiv.org/abs/2610.21273
AB  - We study the problem of online learning of quantum states with logarithmic loss, which is a quantum generalization of online portfolio selection and has applications to quantum state tomography, the task of estimating an unknown quantum state from measurement outcomes. We first establish a regret lower bound, showing that the regret rates of Exponential Weights and the algorithm of Tseng et al. (2025) have optimal polynomial dependence on the dimension. Our analysis leads to two further results. First, our lower-bound proof leads to a quantum state tomography algorithm that is sample optimal up to polylogarithmic factors in the sample size. Second, we develop an algorithm for constructing confidence sequences for the mean of a matrix-valued random variable. A numerical experiment shows that the latter algorithm is competitive with existing methods.
ER  -