Finite-Sample Selected Covariance Spectra in Classical Shadows

@misc{hayashi2026finitesample,
  title = {{Finite-Sample Selected Covariance Spectra in Classical Shadows}},
  author = {Masahito Hayashi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.00527},
  eprint = {2606.00527},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.00527},
}

TY  - UNPB
TI  - Finite-Sample Selected Covariance Spectra in Classical Shadows
AU  - Masahito Hayashi
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2606.00527
UR  - https://arxiv.org/abs/2606.00527
AB  - Classical shadows use a single measurement snapshot to estimate many observables. These estimates are jointly distributed, and their covariance is not determined by their expectation values or individual variances. It also differs from physical correlations between observables because it depends on the measurement and reconstruction protocol. We study the covariance matrix for a fixed set of shadow estimates and its spectrum. For general shadow protocols, we give a finite-sample operator-norm bound for the empirical covariance computed using the sample mean. This bound also controls the eigenvalues and isolated spectral subspaces of the covariance matrix. For local product shadow protocols with fixed local dimension, observables supported on a bounded number of sites lead to bounds that do not depend on the total system size, provided that the number of selected observables and the relevant local reconstruction factors remain bounded. For biased local Pauli shadows, we obtain explicit finite-sample bounds and an exact covariance formula determined by Pauli compatibility and local basis probabilities. A comparison with global Clifford shadows shows that dimension-independent covariance estimation depends on the measurement protocol and does not follow from the general theory alone.
ER  -