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Area of Science:

  • Quantum information science
  • Statistical estimation theory

Background:

  • Minimax estimators minimize worst-case error (risk).
  • Quantum state tomography (QST) aims to reconstruct an unknown quantum state.
  • Relative entropy is a common risk measure in QST.

Purpose of the Study:

  • To construct the first minimax estimators for quantum state tomography using relative entropy risk.
  • To analyze the scaling of minimax risk for nonadaptive tomography.

Main Methods:

  • Construction of minimax estimators for QST.
  • Analysis of risk scaling with the number of quantum state copies (N).
  • Identification of sampling mismatch as a key challenge.

Main Results:

  • Minimax risk for nonadaptive tomography scales as O(1/sqrt[N]).
  • This scaling is slower than classical probability estimation (O(1/N)).
  • Sampling mismatch leads to biased estimators.

Conclusions:

  • The developed minimax estimators for QST have a slower risk convergence than classical methods.
  • A computationally tractable alternative estimator is proposed, balancing worst-case performance with improved average accuracy.