The Annals of Statistics

Discriminating quantum states: The multiple Chernoff distance

Ke Li

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We consider the problem of testing multiple quantum hypotheses $\{\rho_{1}^{\otimes n},\ldots,\rho_{r}^{\otimes n}\}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the minimal average error probability $P_{e}$ decays exponentially to zero, that is, $P_{e}=\exp\{-\xi n+o(n)\}$. However, this error exponent $\xi$ is generally unknown, except for the case that $r=2$.

In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkoła’s conjecture that $\xi=\min_{i\neq j}C(\rho_{i},\rho_{j})$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C(\rho_{i},\rho_{j}):=\max_{0\leq s\leq1}\{-\log\operatorname{Tr}\rho_{i}^{s}\rho_{j}^{1-s}\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\rho_{i}^{\otimes n}$ versus $\rho_{j}^{\otimes n}$.

The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkoła’s lower bound. Specialized to the case $r=2$, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.

Article information

Ann. Statist., Volume 44, Number 4 (2016), 1661-1679.

Received: September 2015
Revised: January 2016
First available in Project Euclid: 7 July 2016

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Zentralblatt MATH identifier

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

Quantum state discrimination quantum hypothesis testing error exponent quantum Chernoff distance multiple hypotheses


Li, Ke. Discriminating quantum states: The multiple Chernoff distance. Ann. Statist. 44 (2016), no. 4, 1661--1679. doi:10.1214/16-AOS1436.

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  • [1] Audenaert, K. M. R., Casamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, Ll., Acin, A. and Verstraete, F. (2007). Discriminating states: The quantum Chernoff bound. Phys. Rev. Lett. 98 160501. Available also at arXiv:quant-ph/0610027.
  • [2] Audenaert, K. M. R. and Mosonyi, M. (2014). Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination. J. Math. Phys. 55 102201, 39.
  • [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] Barnum, H. and Knill, E. (2002). Reversing quantum dynamics with near-optimal quantum and classical fidelity. J. Math. Phys. 43 2097–2106.
  • [5] Bjelaković, I., Deuschel, J., Krüger, T., Seiler, R., Siegmund-Schultze, R. and Szkoła, A. (2005). A quantum version of Sanov’s theorem. Comm. Math. Phys. 260 659–671.
  • [6] Blahut, R. E. (1974). Hypothesis testing and information theory. IEEE Trans. Inform. Theory IT-20 405–417.
  • [7] Brandão, F. G. S. L., Harrow, A. W., Oppenheim, J. and Strelchuk, S. (2015). Quantum conditional mutual information, reconstructed states, and state redistribution. Phys. Rev. Lett. 115 050501.
  • [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. Stat. 23 493–507.
  • [10] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
  • [11] Csiszár, I. (1998). The method of types. IEEE Trans. Inform. Theory 44 2505–2523.
  • [12] Csiszár, I. and Körner, J. (1981). Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic Press, Inc., New York.
  • [13] 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.
  • [14] Han, T. S. and Kobayashi, K. (1989). The strong converse theorem for hypothesis testing. IEEE Trans. Inform. Theory 35 178–180.
  • [15] Hausladen, P. and Wootters, W. K. (1994). A “pretty good” measurement for distinguishing quantum states. J. Modern Opt. 41 2385–2390.
  • [16] Helstrom, C. W. (1976). Quantum Detection and Estimation Theory. Academic Press, New York.
  • [17] Hiai, F., Mosonyi, M. and Ogawa, T. (2007). Large deviations and Chernoff bound for certain correlated states on a spin chain. J. Math. Phys. 48 123301, 19.
  • [18] Hiai, F. and Petz, D. (1991). The proper formula for relative entropy and its asymptotics in quantum probability. Comm. Math. Phys. 143 99–114.
  • [19] Hoeffding, W. (1965). Asymptotically optimal tests for multinomial distributions. Ann. Math. Stat. 36 369–408.
  • [20] Holevo, A. S. (1973). Statistical decision theory for quantum systems. J. Multivariate Anal. 3 337–394.
  • [21] Holevo, A. S. (1978). Asymptotically optimal hypotheses testing in quantum statistics. Theory Probab. Appl. 23 411–415.
  • [22] König, R., Renner, R. and Schaffner, C. (2009). The operational meaning of min- and max-entropy. IEEE Trans. Inform. Theory 55 4337–4347.
  • [23] Leang, C. C. and Johnson, D. H. (1997). On the asymptotics of $M$-hypothesis Bayesian detection. IEEE Trans. Inform. Theory 43 280–282.
  • [24] Li, K. (2014). Second-order asymptotics for quantum hypothesis testing. Ann. Statist. 42 171–189.
  • [25] Mosonyi, M. (2009). Hypothesis testing for Gaussian states on bosonic lattices. J. Math. Phys. 50 032105, 17.
  • [26] Mosonyi, M., Hiai, F., Ogawa, T. and Fannes, M. (2008). Asymptotic distinguishability measures for shift-invariant quasifree states of fermionic lattice systems. J. Math. Phys. 49 072104, 11.
  • [27] Mosonyi, M. and Ogawa, T. (2015). Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Comm. Math. Phys. 334 1617–1648.
  • [28] Nussbaum, M. (2013). Attainment of the multiple quantum Chernoff bound for certain ensembles of mixed states. In Proceedings of the First International Workshop on Entangled Coherent States and Its Application to Quantum Information Science 77–81. Tamagawa Univ., Tokyo, Japan.
  • [29] Nussbaum, M. and Szkoła, A. (2009). The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Statist. 37 1040–1057. Available also at arXiv:quant-ph/0607216.
  • [30] Nussbaum, M. and Szkoła, A. (2010). Exponential error rates in multiple state discrimination on a quantum spin chain. J. Math. Phys. 51 072203, 11.
  • [31] Nussbaum, M. and Szkoła, A. (2011). Asymptotically optimal discrimination between pure quantum states. In Theory of Quantum Computation, Communication, and Cryptography. Lecture Notes in Computer Science 6519 1–8. Springer, Berlin.
  • [32] Nussbaum, M. and Szkoła, A. (2011). An asymptotic error bound for testing multiple quantum hypotheses. Ann. Statist. 39 3211–3233.
  • [33] Ogawa, T. and Nagaoka, H. (2000). Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE Trans. Inform. Theory 46 2428–2433.
  • [34] Parthasarathy, K. R. (2001). On consistency of the maximum likelihood method in testing multiple quantum hypotheses. In Stochastics in Finite and Infinite Dimensions. Trends Math. 361–377. Birkhäuser, Boston, MA.
  • [35] Qiu, D. W. (2008). Minimum-error discrimination between mixed quantum states. Phys. Rev. A 77 012328.
  • [36] Salihov, N. P. (1973). Asymptotic properties of error probabilities of tests for distinguishing between several multinomial testing schemes. Dokl. Akad. Nauk SSSR 209 54–57.
  • [37] Salikhov, N. P. (1998). On a generalization of Chernoff distance. Theory Probab. Appl. 43 239–255.
  • [38] Tomamichel, M. and Hayashi, M. (2013). A hierarchy of information quantities for finite block length analysis of quantum tasks. IEEE Trans. Inform. Theory 59 7693–7710.
  • [39] Torgersen, E. N. (1981). Measures of information based on comparison with total information and with total ignorance. Ann. Statist. 9 638–657.
  • [40] Tyson, J. (2009). Two-sided estimates for minimum-error distinguishability of mixed quantum states via generalized Holevo–Curlander bounds. J. Math. Phys. 50 032106, 10.
  • [41] Yuen, H. P., Kennedy, R. S. and Lax, M. (1975). Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inform. Theory 21 125–134.