Finite Sample Analysis in Quantum Estimation pp 89-112 | Cite as

# Improvement of Estimation Precision by Adaptive Design of Experiments

## Abstract

As we have seen in Chap. 5, in quantum state tomography, an independently and identically prepared set of measurements are used throughout the entire experiment. However, during the experiment we can imagine changing the measurement at each trial according to the previously obtained data. This experimental design is referred to as adaptive. Clearly, adaptive experimental designs are a superset of the nonadaptive ones, and therefore can potentially achieve higher performance. In this chapter, we propose an adaptive experimental design whose expected losses are smaller than those of standard quantum state tomography. The update criterion is based on average-variance optimality (A-optimality) in classical statistics, and it has low computational cost for one-qubit state estimation. This chapter is structured as follows. In Sect. 6.1 we lay out the notation and terminology in adaptive experimental design. We also explain the A-optimality criterion in this section. In Sect. 6.2 we analyze the performance of the A-optimality criterion in one-qubit state estimation. We give the explicit form of the analytic solution of the A-optimal update criterion. This analytic solution makes it possible to reduce the computational cost of updating measurements. We compare several adaptive and non-adaptive estimation schemes numerically. The numerical results indicate that the A-optimality criterion gives more precise estimates than standard quantum state tomography. We also discuss the feasibility of implementing the proposed scheme experimentally. In Sect. 6.3 we review the history of adaptive design of experiments in quantum estimation. A summary appears in Sect. 6.4.

## Keywords

Adaptive design of experiments A-optimality criterion maximum-likelihood estimator## Supplementary material

## References

- 1.S. Watanabe, K. Hagiwara, S. Akaho, Y. Motomura, K. Fukumizu, M. Okada, M. Aoyagi,
*Theory and Implimentation of Learning Systems*(Morikita, Japan, 2005)Google Scholar - 2.F. Pukelsheim,
*Optimal Design of Experiments*. Classics in Applied Mathematics (SIAM, Philadelphia, 2006)Google Scholar - 3.P. Hall, C.C. Heyde,
*Martingale Limit Theory and Its Application*. Probability and mathematical statistics (Academic Press, New York, 1980)Google Scholar - 4.E. Bagan, M. Baig, R. Muñoz-Tapia, A. Rodriguez, Phys. Rev. A
**69**, 010304(R) (2004). doi: 10.1103/PhysRevA.69.010304 - 5.E. Bagan, M.A. Ballester, R.D. Gill, A. Monras, R. Muñoz-Tapia, Phys. Rev. A
**73**, 032301 (2006). doi: 10.1103/PhysRevA.73.032301 - 6.E. Bagan, M.A. Ballester, R.D. Gill, R. Muñoz-Tapia, O. Romero-Isart, Phys. Rev. Lett.
**97**, 130501 (2006). doi: 10.1103/PhysRevLett.97.130501 Google Scholar - 7.R. Blume-Kohout, New J. Phys.
**12**, 043034 (2010). doi: 10.1088/1367-2630/12/4/043034 - 8.F. Huszár, N.M.T. Houlsby, Phys. Rev. A
**85**, 052120 (2012). doi: 10.1103/PhysRevA.85.052120 - 9.H. Nagaoka, in
*Proceedings of 12th Symposium on Information Theory and Its Applications*, (1989), p. 577Google Scholar - 10.H. Nagaoka, in
*Asymptotic Theory of Quantum Statistical Inference: Selected Papers*, Chap. 10, World Scientific, 2005, ed. by M. HayashiGoogle Scholar - 11.A. Fujiwara, J. Phys. A: Math. Gen.
**39**, 12489 (2006). doi: 10.1088/0305-4470/39/40/014 - 12.D.G. Fischer, S.H. Kienle, M. Freyberger, Phys. Rev. A
**61**, 032306 (2000). doi: 10.1103/PhysRevA.61.032306 - 13.C.J. Happ, M. Freyberger, Phys. Rev. A
**78**, 064303 (2008). doi: 10.1103/PhysRevA.78.064303 - 14.C.J. Happ, M. Freyberger, Eur. Phys. J. D
**64**, 579 (2011). doi: 10.1140/epjd/e2011-20367-9 - 15.D.G. Fischer, M. Freyberger, Phys. Lett. A
**273**, 293 (2000). doi: 10.1016/S0375-9601(00)00513-2 - 16.T. Hannemann, D. Reiss, C. Balzer, W. Neuhauser, P.E. Toschek, C. Wunderlich, Phys. Rev. A
**65**, 050303 (2002). doi: 10.1103/PhysRevA.65.050303 - 17.R. Okamoto, M. Iefuji, S. Oyama, K. Yamagata, H. Imai, A. Fujiwara, S. Takeuchi, Phys. Rev. Lett.
**109**, 130404 (2012). doi: 10.1103/PhysRevLett.109.130404 Google Scholar - 18.M. Hayashi, K. Matsumoto, in
*RIMS Kôkyûroku*, vol. 1055 (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 1998), p. 96 (The original version is in Japanese. An English version is available in the arXiv.)Google Scholar - 19.M. Hayashi, K. Matsumoto, in
*Asymptotic Theory of Quantum Statistical Inference: Selected Papers*, Chap. 13, World Scientific, 2005, ed. by M. HayashiGoogle Scholar - 20.R.D. Gill, S. Massar, Phys. Rev. A
**61**, 042312 (2000). doi: 10.1103/PhysRevA.61.042312 - 21.J. Řeháček, B.G. Englert, D. Kaszlikowski, Phys. Rev. A
**70**, 052321 (2004). doi: 10.1103/PhysRevA.70.052321 - 22.D. Petz, K.M. Hangos, L. Ruppert, in
*Quantum bio-informatics, QP-PQ: Quantum Probability and White Noise Analysis*, vol. 21, ed. by L. Accardi, L. Accardi, M. Ohya (2007), p. 247Google Scholar - 23.R.A. Horn, C.R. Johnson,
*Matrix Analysis*(Cambridge University Press, New York, 1985)Google Scholar - 24.T. Sugiyama, P.S. Turner, M. Murao, Phys. Rev. A
**85**, 052107 (2012). doi: 10.1103/PhysRevA.85.052107