On the Chernoff bound for efficiency of quantum hypothesis testing. (English) Zbl 1070.62117

Summary: The paper estimates the Chernoff rate for the efficiency of quantum hypothesis testing. For both joint and separate measurements, approximate bounds for the rate are given if both states are mixed, and exact expressions are derived if at least one of the states is pure. The efficiencies of tests with separate and joint measurements are compared. The results are illustrated by a test of quantum entanglement.


62P35 Applications of statistics to physics
81P15 Quantum measurement theory, state operations, state preparations
62G10 Nonparametric hypothesis testing
Full Text: DOI arXiv


[1] Ballester, M. A. (2004). Estimation of unitary quantum operations. Phys. Rev. A 69 022303.
[2] Barndorff-Nielsen, O. E., Gill, R. D. and Jupp, P. E. (2003). On quantum statistical inference (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 65 775–816. · Zbl 1059.62128
[3] Bennett, C. H., DiVincenzo, D. P., Fuchs, C. A., Mor, T., Rains, E., Shor, P. W., Smolin, J. A. and Wootters, W. K. (1999). Quantum nonlocality without entanglement. Phys. Rev. A 59 1070–1091.
[4] Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493–507. · Zbl 0048.11804
[5] Cirac, J. I. and Zoller, P. (1995). Quantum computations with cold trapped ions. Phys. Rev. Lett. 74 4091–4094.
[6] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory . Wiley, New York. · Zbl 0762.94001
[7] Fuchs, C. A. and Caves, C. M. (1995). Mathematical techniques for quantum communication theory. Open Systems and Information Dynamics 3 345–356. Available at http://arxiv.org/abs/quant-ph/9604001. · Zbl 1020.81530
[8] Fuchs, C. A. and van de Graaf, J. (1999). Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inform. Theory 45 1216–1227. · Zbl 0959.94020
[9] Gill, R. D. (2001). Asymptotics in quantum statistics. In State of the Art in Probability and Statistics 255–285. IMS, Beachwood, OH. · Zbl 1371.81193
[10] Gill, R. D. and Massar, S. (2000). State estimation for large ensembles. Phys. Rev. A 61 042312.
[11] Helstrom, C. W. (1976). Quantum Detection and Estimation Theory . Academic Press, New York. · Zbl 0278.94003
[12] Hoeffding, W. (1965). Asymptotically optimal tests for multinomial distributions. Ann. Math. Statist. 36 369–401. · Zbl 0135.19706
[13] Holevo, A. S. (1976). Investigation of a general theory of statistical decisions. Trudy Mat. Inst. Steklov. 124 . [English translation in Proc. Steklov Inst. Math. 3 (1978) Amer. Math. Soc., Providence, RI.] · Zbl 0497.62010
[14] Holevo, A. S. (2001). Statistical Structure of Quantum Theory . Springer, Berlin. · Zbl 0999.81001
[15] Keyl, M. and Werner, R. F. (2001). Estimating the spectrum of a density operator. Phys. Rev. A 64 052311. · Zbl 1071.81013
[16] Nielsen, M. A. and Chuang, I. L. (2000). Quantum Computation and Quantum Information . Cambridge Univ. Press. · Zbl 1049.81015
[17] Ogawa, T. and Hayashi, M. (2002). On error exponents in quantum hypothesis testing. Available at http://arxiv.org/abs/quant-ph/0206151. · Zbl 1303.81070
[18] Ogawa, T. and Nagaoka, H. (2000). Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE Trans. Inform. Theory 46 2428–2433. · Zbl 1003.94011
[19] Parthasarathy, K. R. (2001). On consistency of the maximum likelihood method in testing multiple quantum hypotheses. In Stochastics in Finite and Infinite Dimensions (T. Hida et al., eds.) 361–377. Birkhäuser, Boston. · Zbl 1074.81536
[20] Peres, A. (1995). Quantum Theory : Concepts and Methods . Kluwer, Dordrecht. · Zbl 0867.00010
[21] Sanov, I. N. (1957). On the probability of large deviations of random variables. Mat. Sb. N. S. 42 11–44.
[22] Santaló, L. A. (1976). Integral Geometry and Geometric Probability . Addison–Wesley, Reading, MA. · Zbl 0342.53049
[23] Turchette, Q. A., Wood, C. S., King, B. E., Myatt, C. J., Leibfried, D., Itano, W. M., Monroe, C. and Wineland, D. J. (1998). Deterministic entanglement of two trapped ions. Phys. Rev. Lett. 81 3631–3634.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.