## Discriminating quantum states: the multiple Chernoff distance.(English)Zbl 1397.62611

Summary: 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 M. Nussbaum and A. Szkoła’s lower bound [ibid. 37, No. 2, 1040–1057 (2009; Zbl 1162.62100)]. 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 K. M. R. Audenaert and M. Mosonyi [J. Math. Phys. 55, No. 10, 102201, 39 p. (2014; Zbl 1309.81042)].

### MSC:

 62P35 Applications of statistics to physics 62G10 Nonparametric hypothesis testing 81P45 Quantum information, communication, networks (quantum-theoretic aspects)

### Citations:

Zbl 1162.62100; Zbl 1309.81042
Full Text:

### References:

 [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 . [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. · Zbl 1309.81042 [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. · Zbl 1175.81036 [4] Barnum, H. and Knill, E. (2002). Reversing quantum dynamics with near-optimal quantum and classical fidelity. J. Math. Phys. 43 2097-2106. · Zbl 1059.81027 [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. · Zbl 1092.94013 [6] Blahut, R. E. (1974). Hypothesis testing and information theory. IEEE Trans. Inform. Theory IT-20 405-417. · Zbl 0305.62017 [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. · Zbl 1190.81015 [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. · Zbl 0048.11804 [10] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory . Wiley, New York. · Zbl 0762.94001 [11] Csiszár, I. (1998). The method of types. IEEE Trans. Inform. Theory 44 2505-2523. · Zbl 0933.94012 [12] Csiszár, I. and Körner, J. (1981). Information Theory : Coding Theorems for Discrete Memoryless Systems . Academic Press, Inc., New York. · Zbl 0568.94012 [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. · Zbl 0232.94008 [14] Han, T. S. and Kobayashi, K. (1989). The strong converse theorem for hypothesis testing. IEEE Trans. Inform. Theory 35 178-180. · Zbl 0678.62011 [15] Hausladen, P. and Wootters, W. K. (1994). A “pretty good” measurement for distinguishing quantum states. J. Modern Opt. 41 2385-2390. · Zbl 0942.81516 [16] Helstrom, C. W. (1976). Quantum Detection and Estimation Theory . Academic Press, New York. · Zbl 1332.81011 [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. · Zbl 1153.81375 [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. · Zbl 0756.46043 [19] Hoeffding, W. (1965). Asymptotically optimal tests for multinomial distributions. Ann. Math. Stat. 36 369-408. · Zbl 0135.19706 [20] Holevo, A. S. (1973). Statistical decision theory for quantum systems. J. Multivariate Anal. 3 337-394. · Zbl 0275.62004 [21] Holevo, A. S. (1978). Asymptotically optimal hypotheses testing in quantum statistics. Theory Probab. Appl. 23 411-415. · Zbl 0426.62085 [22] König, R., Renner, R. and Schaffner, C. (2009). The operational meaning of min- and max-entropy. IEEE Trans. Inform. Theory 55 4337-4347. · Zbl 1367.81028 [23] Leang, C. C. and Johnson, D. H. (1997). On the asymptotics of $$M$$-hypothesis Bayesian detection. IEEE Trans. Inform. Theory 43 280-282. · Zbl 0868.94013 [24] Li, K. (2014). Second-order asymptotics for quantum hypothesis testing. Ann. Statist. 42 171-189. · Zbl 1321.62155 [25] Mosonyi, M. (2009). Hypothesis testing for Gaussian states on bosonic lattices. J. Math. Phys. 50 032105, 17. · Zbl 1187.82012 [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. · Zbl 1152.81567 [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. · Zbl 1308.81051 [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 . · Zbl 1162.62100 [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. · Zbl 1311.81069 [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. · Zbl 1310.81021 [32] Nussbaum, M. and Szkoła, A. (2011). An asymptotic error bound for testing multiple quantum hypotheses. Ann. Statist. 39 3211-3233. · Zbl 1246.62226 [33] 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 [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. · Zbl 1074.81536 [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. · Zbl 0307.62040 [37] Salikhov, N. P. (1998). On a generalization of Chernoff distance. Theory Probab. Appl. 43 239-255. · Zbl 0942.62005 [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. · Zbl 1364.94217 [39] Torgersen, E. N. (1981). Measures of information based on comparison with total information and with total ignorance. Ann. Statist. 9 638-657. · Zbl 0487.62006 [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. · Zbl 1187.81021 [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. · Zbl 0301.94001
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.