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Minimax Quantum Tomography: Estimators and Relative Entropy Bounds
Christopher Ferrie1,2, Robin Blume-Kohout3
1Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA.
Physical Review Letters
|March 19, 2016
Summary
We developed new minimax estimators for quantum state tomography, offering improved accuracy. These estimators address sampling mismatch issues inherent in classical probability estimation.
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.
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