The Annals of Statistics

Second-order asymptotics for quantum hypothesis testing

Ke Li

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In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein’s lemma.

Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein’s lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory.

Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.

Article information

Ann. Statist., Volume 42, Number 1 (2014), 171-189.

First available in Project Euclid: 18 February 2014

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Zentralblatt MATH identifier

Primary: 62P35: Applications to physics 62G10: Hypothesis testing

Quantum hypothesis testing quantum Stein’s lemma second-order asymptotics finite sample size


Li, Ke. Second-order asymptotics for quantum hypothesis testing. Ann. Statist. 42 (2014), no. 1, 171--189. doi:10.1214/13-AOS1185.

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  • [1] Audenaert, K. M. R., Casamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, L., Acin, A. and Verstraete, F. (2007). Discriminating states: The quantum Chernoff bound. Phys. Rev. Lett. 98 160501.
  • [2] Audenaert, K. M. R., Mosonyi, M. and Verstraete, F. (2012). Quantum state discrimination bounds for finite sample size. J. Math. Phys. 53 122205.
  • [3] Audenaert, K. M. R., Nussbaum, M., Szkoła, A. and Verstraete, F. (2008). Asymptotic error rates in quantum hypothesis testing. Comm. Math. Phys. 279 251–283.
  • [4] Bjelaković, I., Deuschel, J.-D., Krüger, T., Seiler, R., Siegmund-Schultze, R. and Szkoła, A. (2005). A quantum version of Sanov’s theorem. Comm. Math. Phys. 260 659–671.
  • [5] Bjelaković, I., Deuschel, J.-D., Krüger, T., Seiler, R., Siegmund-Schultze, R. and Szkoła, A. (2008). Typical support and Sanov large deviations of correlated states. Comm. Math. Phys. 279 559–584.
  • [6] Bjelaković, I. and Siegmund-Schultze, R. (2004). An ergodic theorem for the quantum relative entropy. Comm. Math. Phys. 247 697–712.
  • [7] Blahut, R. E. (1974). Hypothesis testing and information theory. IEEE Trans. Inform. Theory 20 405–417.
  • [8] Brandão, F. G. S. L. and Plenio, M. B. (2010). A generalization of quantum Stein’s lemma. Comm. Math. Phys. 295 791–828.
  • [9] 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.
  • [10] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
  • [11] Csiszár, I. and Longo, G. (1971). On the error exponent for source coding and for testing simple statistical hypotheses. Studia Sci. Math. Hungar. 6 181–191.
  • [12] Han, T. S. (2003). Information-Spectrum Methods in Information Theory. Springer, Berlin.
  • [13] Han, T. S. and Kobayashi, K. (1989). The strong converse theorem for hypothesis testing. IEEE Trans. Inform. Theory 35 178–180.
  • [14] Hayashi, M. (2006). Quantum Information: An Introduction. Springer, Berlin.
  • [15] Hayashi, M. (2007). Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding. Phys. Rev. A (3) 76 062301.
  • [16] Hayashi, M. (2009). Information spectrum approach to second-order coding rate in channel coding. IEEE Trans. Inform. Theory 55 4947–4966.
  • [17] Hayashi, M. and Nagaoka, H. (2003). General formulas for capacity of classical-quantum channels. IEEE Trans. Inform. Theory 49 1753–1768.
  • [18] Helstrom, C. W. (1976). Quantum Detection and Estimation Theory. Academic Press, New York.
  • [19] Hiai, F. and Petz, D. (1991). The proper formula for relative entropy and its asymptotics in quantum probability. Comm. Math. Phys. 143 99–114.
  • [20] Hoeffding, W. (1965). Asymptotically optimal tests for multinomial distributions. Ann. Math. Statist. 36 369–408.
  • [21] Holevo, A. S. (1978). On asymptotically optimal hypothesis testing in quantum statistics. Theory Probab. Appl. 23 411–415.
  • [22] Korolev, V. and Shevtsova, I. (2012). An improvement of the Berry–Esseen inequality with applications to Poisson and mixed Poisson random sums. Scand. Actuar. J. 2 81–105.
  • [23] Mosonyi, M. and Datta, N. (2009). Generalized relative entropies and the capacity of classical-quantum channels. J. Math. Phys. 50 072104.
  • [24] Nagaoka, H. (2006). The converse part of the theorem for quantum Hoeffding bound. Preprint. Available at arXiv:quant-ph/0611289.
  • [25] Nagaoka, H. and Hayashi, M. (2007). An information-spectrum approach to classical and quantum hypothesis testing for simple hypotheses. IEEE Trans. Inform. Theory 53 534–549.
  • [26] Nussbaum, M. and Szkoła, A. (2009). The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Statist. 37 1040–1057.
  • [27] Nussbaum, M. and Szkoła, A. (2011). An asymptotic error bound for testing multiple quantum hypotheses. Ann. Statist. 39 3211–3233.
  • [28] Ogawa, T. and Nagaoka, H. (2000). Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE Trans. Inform. Theory 46 2428–2433.
  • [29] Polyanskiy, Y., Poor, H. V. and Verdú, S. (2010). Channel coding rate in the finite blocklength regime. IEEE Trans. Inform. Theory 56 2307–2359.
  • [30] Tomamichel, M. and Hayashi, M. (2012). A hierarchy of information quantities for finite block length analysis of quantum tasks. Preprint. Available at arXiv:1208.1478 [quant-ph].
  • [31] Verdú, S. and Han, T. S. (1994). A general formula for channel capacity. IEEE Trans. Inform. Theory 40 1147–1157.
  • [32] Wang, L. and Renner, R. (2012). One-shot classical-quantum capacity and hypothesis testing. Phys. Rev. Lett. 108 200501.