Minimax Quantum Tomography: Estimators and Relative Entropy Bounds
A minimax estimator has the minimum possible error (“risk”) in the worst case. Here we construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O (1/$$\sqrt{N}$$ ) —in contrast to that of classical probability estimation, which is O (1/N) —where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. Lastly, this makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.
 Authors:

^{[1]};
^{[2]}
 Univ. of New Mexico, Albuquerque, NM (United States). Center for Quantum Information and Control; Univ. of Sydney, NSW (Australia). Centre for Engineered Quantum Systems, School of Physics
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Publication Date:
 Report Number(s):
 SAND20151684J
Journal ID: ISSN 00319007; PRLTAO; 650745
 Grant/Contract Number:
 AC0494AL85000; PHY1212445; W911NF1410098; W911NF1410103
 Type:
 Accepted Manuscript
 Journal Name:
 Physical Review Letters
 Additional Journal Information:
 Journal Volume: 116; Journal Issue: 9; Journal ID: ISSN 00319007
 Publisher:
 American Physical Society (APS)
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org:
 USDOE National Nuclear Security Administration (NNSA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
 OSTI Identifier:
 1341781
 Alternate Identifier(s):
 OSTI ID: 1240449
Ferrie, Christopher, and BlumeKohout, Robin. Minimax Quantum Tomography: Estimators and Relative Entropy Bounds. United States: N. p.,
Web. doi:10.1103/PhysRevLett.116.090407.
Ferrie, Christopher, & BlumeKohout, Robin. Minimax Quantum Tomography: Estimators and Relative Entropy Bounds. United States. doi:10.1103/PhysRevLett.116.090407.
Ferrie, Christopher, and BlumeKohout, Robin. 2016.
"Minimax Quantum Tomography: Estimators and Relative Entropy Bounds". United States.
doi:10.1103/PhysRevLett.116.090407. https://www.osti.gov/servlets/purl/1341781.
@article{osti_1341781,
title = {Minimax Quantum Tomography: Estimators and Relative Entropy Bounds},
author = {Ferrie, Christopher and BlumeKohout, Robin},
abstractNote = {A minimax estimator has the minimum possible error (“risk”) in the worst case. Here we construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O (1/$\sqrt{N}$ ) —in contrast to that of classical probability estimation, which is O (1/N) —where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. Lastly, this makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.},
doi = {10.1103/PhysRevLett.116.090407},
journal = {Physical Review Letters},
number = 9,
volume = 116,
place = {United States},
year = {2016},
month = {3}
}