We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.
 W. Pauli, Encyclopedia of Physics V , 17 (1958).
 J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018).
 R. Blume-Kohout, New Journal of Physics 12, 043034 (2010), arXiv:0611080 [quant-ph].
 E. L. Lehmann and G. Casella, Theory of Point Estimation, 2nd ed. (Springer-Verlag, New York, NY, USA, 1998).
 A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Elsevier Science Ltd, Amsterdam ; New York : New York, 1982).
 A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, and P. Perinotti, IEEE Journal of Selected Topics in Quantum Electronics 15, 1646 (2009), arXiv:1702.08751 [quant-ph].
 K. R. Parthasarathy, Probability measures on Metric Spaces, Probability and Mathematical Statistics (Academic Press, New York, 1967).
 M. Sion, Pacific J. Math. 8, 171 (1958).
 M. Derakhshani, Quantum t-design, Ph.D. thesis, University of Waterloo (2008).
 M. Hayashi, ``Group covariance and optimal information processing,'' in A Group Theoretic Approach to Quantum Information (Springer International Publishing, Cham, 2017) pp. 69-119.
 R. Kueng and C. Ferrie, New Journal of Physics 17, 123013 (2015), arXiv:1503.00677 [quant-ph].
 Trung Can, Narayanan Rengaswamy, Robert Calderbank, and Henry D. Pfister, "Kerdock Codes Determine Unitary 2-Designs", arXiv:1904.07842.
The above citations are from SAO/NASA ADS (last updated 2019-07-15 11:55:13). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref's cited-by service no data on citing works was found (last attempt 2019-07-15 11:55:11).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.