## Institute of Mathematical Statistics Collections

### On asymptotic quantum statistical inference

#### Abstract

We study asymptotically optimal statistical inference concerning the unknown state of $N$ identical quantum systems, using two complementary approaches: a “poor man’s approach” based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam’s theory of Local Asymptotic Normality.

#### Chapter information

Source
Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 105-127

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.imsc/1362751183

Digital Object Identifier
doi:10.1214/12-IMSCOLL909

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 62P35: Applications to physics

Rights

#### Citation

Gill, Richard D.; Guţă, Mădălin I. On asymptotic quantum statistical inference. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 105--127, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL909. https://projecteuclid.org/euclid.imsc/1362751183.

#### References

• [1] Barndorff-Nielsen, O. E., Gill, R. D. and Jupp, P. E. (2003). On quantum statistical inference. J. Roy. Statist. Soc. (B) 65 775–816. With discussion and reply by the authors. Available at arXiv:quant-ph/0307191.
• [2] Bowles, P., Guţă, M. and Adesso, G. (2011). Asymptotically optimal purification and dilution of mixed qubit and gaussian states. Phys. Rev. A 84 022320.
• [3] Fuchs, C. A. (1995). Distinguishability and accessible information in quantum theory. Ph.D. thesis, University of New Mexico. Available at arXiv:quant-ph/9601020.
• [4] Gill, R. D. (2001). Asymptotics in quantum statistics. In State of the Art in Probability and Statistics (Leiden, 1999) (A. W. van der Vaart, M. de Gunst and C. A. J. Klaassen, eds.). IMS Lecture notes Monogr. Ser. 36 255–285. Inst. Math. Statist., Beachwood, OH. Available at arXiv:math.ST/0405571.
• [5] Gill, R. D. (2005). Conciliation of Bayes and pointwise quantum state estimation: Asymptotic information bounds in quantum statistics. Available at arXiv:math/0512443.
• [6] Guţă, M. and Kahn, J. (2012). On asymptotic quantum statistical inference. Available at arXiv:abs/1112.2078.
• [7] Gill, R. D. and Levit, B. Y. (1995). Applications of the Van Trees inequality: A Bayesian Cramér-Rao bound. Bernoulli 1 59–79.
• [8] Gill, R. D. and Massar, S. (2000). State estimation for large ensembles. Phys. Rev. A 61 042312–042335. Available at arXiv:quant-ph/9902063.
• [9] Guţă, M. and Kahn, J. (2006). Local asymptotic normality for qubit states. Phys. Rev. A 73 052108. Available at arXiv:quant-ph/0512075.
• [10] Guţă, M. (2011). Fisher information and asymptotic normality in system identification for quantum markov chains. Phys. Rev. A 83 062324.
• [11] Guţă, M., Bowles, P. and Adesso, G. (2010). Quantum-teleportation benchmarks for independent and identically distributed spin states and displaced thermal states. Phys. Rev. A 82 042310.
• [12] Guţă, M., Janssens, B. and Kahn, J. (2008). Optimal estimation of qubit states with continuous time measurements. Commun. Math. Phys. 277 127–160.
• [13] Guţă, M. and Jençová, A. (2007). Local asymptotic normality in quantum statistics. Commun. Math. Phys. 276 341–379.
• [14] Guţă, M. and Kahn, J. (2006). Local asymptotic normality for qubit states. Phys. Rev. A 73 052108.
• [15] Guţă, M. and Kotlowski, W. (2010). Quantum learning: asymptotically optimal classification of qubit states. New J. Phys. 12 123032.
• [16] Kahn, J. and Guţă, M. (2009). Local asymptotic normality for finite dimensional quantum systems. Commun. Math. Phys. 289 597–652.
• [17] Hayashi, M. (1998). Asymptotic estimation theory for a finite dimensional pure state model. J. Phys. A 31 4633–4655. Available at arXiv:quant-ph/9704041.
• [18] Hayashi, M. (2003). Quantum estimation and quantum central limit theorem (in Japanese). Sugaku 55(4) 368–391. New, English translation: arXiv:quant-ph/0608198.
• [19] Hayashi, M. (ed.). (2005). Asymptotic Theory of Quantum Statistical Inference: Selected Papers. World Scientific, Singapore.
• [20] Hayashi, M. and Matsumoto, K. (2004). Asymptotic performance of optimal state estimation in quantum two level system. Available at arXiv:quant-ph/0411073.
• [21] Helstrom, C. W. (1976). Quantum Detection and Estimation Theory. Academic Press.
• [22] Holevo, A. S. (1982). Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam. First appeared, in Russian, 1980.
• [23] Leonhardt, U. (1997). Measuring the Quantum State of Light. Cambridge University Press.
• [24] Matsumoto, K. (2002). A new approach to the Cramér-Rao-type bound of the pure state model. J. Phys. A 35 3111–3123.
• [25] Nachtergaele, B., Scholz, V. B. and Werner, R. F. (2011). Local approximation of observables and commutator bounds. Available at arXiv:math-ph:1103.5663.
• [26] Nielsen, M. A. and Chuang, I. L. (2000). Quantum Computation and Quantum Information. Cambridge University Press.
• [27] Ohya, M. and Petz, D. (2004). Quantum Entropy and its Use. Springer Verlag, Berlin-Heidelberg.
• [28] Petz, D. and Jencova, A. (2006). Sufficiency in quantum statistical inference. Commun. Math. Phys. 263 259–276.
• [29] van Trees, H. (1968). Detection, Estimation and Modulation Theory, Part 1. Wiley, New York.