# Minimax quantum state estimation under Bregman divergence

Maria Quadeer, Marco Tomamichel, and Christopher Ferrie

Centre for Quantum Software and Information, University of Technology Sydney, Ultimo NSW 2007, Australia

### Abstract

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.

### ► References

[1] U. Fano, Rev. Mod. Phys. 29, 74 (1957).
https:/​/​doi.org/​10.1103/​RevModPhys.29.74

[2] W. Pauli, Encyclopedia of Physics V , 17 (1958).

[3] J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018).

[4] Z. Hradil, Phys. Rev. A 55, R1561 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.R1561

[5] R. Blume-Kohout, New Journal of Physics 12, 043034 (2010), arXiv:0611080 [quant-ph].
https:/​/​doi.org/​10.1088/​1367-2630/​12/​4/​043034
arXiv:0611080

[6] C. Ferrie and R. Blume-Kohout, ArXiv e-prints (2018), arXiv:1808.01072 [quant-ph].
arXiv:1808.01072

[7] E. L. Lehmann and G. Casella, Theory of Point Estimation, 2nd ed. (Springer-Verlag, New York, NY, USA, 1998).

[8] F. Tanaka and F. Komaki, Phys. Rev. A 71, 052323 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.052323

[9] A. Banerjee, X. Guo, and H. Wang, IEEE Transactions on Information Theory 51, 2664 (2005).
https:/​/​doi.org/​10.1109/​TIT.2005.850145

[10] B. S. Clarke and A. R. Barron, Journal of Statistical Planning and Inference 41, 37 (1994).
https:/​/​doi.org/​10.1016/​0378-3758(94)90153-8

[11] N. Merhav and M. Feder, IEEE Transactions on Information Theory 44, 2124 (1998).
https:/​/​doi.org/​10.1109/​18.720534

[12] Q. Xie and A. R. Barron, IEEE Transactions on Information Theory 46, 431 (2000).
https:/​/​doi.org/​10.1109/​18.825803

[13] K. F., Journal of Statistical Planning and Inference 141, 3705 (2011).
https:/​/​doi.org/​10.1016/​j.jspi.2011.06.009

[14] T. Koyama, T. Matsuda, and F. Komaki, Entropy 19, 618 (2017).
https:/​/​doi.org/​10.3390/​e19110618

[15] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Elsevier Science Ltd, Amsterdam ; New York : New York, 1982).

[16] 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].
https:/​/​doi.org/​10.1109/​JSTQE.2009.2029243
arXiv:1702.08751

[17] K. R. Parthasarathy, Probability measures on Metric Spaces, Probability and Mathematical Statistics (Academic Press, New York, 1967).

[18] J. Pitrik and D. Virosztek, Letters in Mathematical Physics 105, 675 (2015).
https:/​/​doi.org/​10.1007/​s11005-015-0757-y

[19] A. Wehrl, Reviews of Modern Physics 50, 221 (1978).
https:/​/​doi.org/​10.1103/​RevModPhys.50.221

[20] M. Sion, Pacific J. Math. 8, 171 (1958).
https:/​/​projecteuclid.org:443/​euclid.pjm/​1103040253

[21] M. Derakhshani, Quantum t-design, Ph.D. thesis, University of Waterloo (2008).
https:/​/​pdfs.semanticscholar.org/​5492/​9ff60ecb9410f9f1e95896731453b5988981.pdf

[22] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Journal of Mathematical Physics 45, 2171 (2004), arXiv:0310075 [quant-ph].
https:/​/​doi.org/​10.1063/​1.1737053
arXiv:0310075

[23] N. Bogomolov, Theory of Probability & Its Applications 26, 787 (1982).
https:/​/​doi.org/​10.1137/​1126084

[24] M. Hayashi, Group covariance and optimal information processing,'' in A Group Theoretic Approach to Quantum Information (Springer International Publishing, Cham, 2017) pp. 69–119.
https:/​/​doi.org/​10.1007/​978-3-319-45241-8_4

[25] C. Ferrie and R. Blume-Kohout, ArXiv e-prints (2016), arXiv:1612.07946 [math.ST].
arXiv:1612.07946

[26] R. Kueng and C. Ferrie, New Journal of Physics 17, 123013 (2015), arXiv:1503.00677 [quant-ph].
https:/​/​doi.org/​10.1088/​1367-2630/​17/​12/​123013
arXiv:1503.00677

[27] H. B. Maynard, Transactions of the American Mathematical Society 173, 449 (1972).
http:/​/​www.jstor.org/​stable/​1996285

[28] D. Prato and C. Tsallis, Journal of Mathematical Physics 41, 3278 (2000), arXiv:9906173 [cond-mat].
https:/​/​doi.org/​10.1063/​1.533305
arXiv:9906173

### Cited by

[1] Olivia Di Matteo, John Gamble, Chris Granade, Kenneth Rudinger, and Nathan Wiebe, "Operational, gauge-free quantum tomography", Quantum 4, 364 (2020).

[2] Trung Can, Narayanan Rengaswamy, Robert Calderbank, and Henry D. Pfister, "Kerdock Codes Determine Unitary 2-Designs", arXiv:1904.07842, IEEE Transactions on Information Theory 66 10, 6104 (2020).

[3] Jun Suzuki, Yuxiang Yang, and Masahito Hayashi, "Quantum state estimation with nuisance parameters", Journal of Physics A: Mathematical and Theoretical 53 45, 453001 (2020).

The above citations are from Crossref's cited-by service (last updated successfully 2021-08-01 07:20:41) and SAO/NASA ADS (last updated successfully 2021-08-01 07:20:42). The list may be incomplete as not all publishers provide suitable and complete citation data.