An asymptotic error bound for testing multiple quantum hypotheses. (English) Zbl 1246.62226

Summary: We consider the problem of detecting the true quantum state among \(r\) possible ones, based of measurements performed on \(n\) copies of a finite-dimensional quantum system. A special case is the problem of discriminating between \(r\) probability measures on a finite sample space, using \(n\) i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with the exponent given by the worst case binary Chernoff bound between any possible pair of the \(r\) probability measures, and define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of \(r\) pure states, and that it is unimprovable in general. We extend the attainability result to a larger class of \(r\)-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.


62P35 Applications of statistics to physics
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
Full Text: DOI arXiv Euclid


[1] Assalini, A., Cariolaro, G. and Pierobon, G. (2010). Efficient optimal minimum error discrimination of symmetric quantum states. Phys. Rev. A 81 012315.
[2] 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.
[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] Barnett, S. M. and Croke, S. (2009). On the conditions for discrimination between quantum states with minimum error. J. Phys. A 42 062001, 4. · Zbl 1156.81330
[5] 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
[6] Belavkin, V. P. (1975). Optimal multiple quantum statistical hypothesis testing. Stochastics 1 315-345. · Zbl 0353.62025
[7] Bergou, J. A., Herzog, U. and Hillery, M. (2004). Discrimination of quantum states. In Quantum State Estimation. Lecture Notes in Physics 649 417-465. Springer, Berlin.
[8] Calsamiglia, J., Munoz-Tapia, R., Masanes, L., Acin, A. and Bagan, E. (2008). The quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian states. Phys. Rev. A 77 032311.
[9] Chefles, A. (2000). Quantum state discrimination. Contemp. Phys. 41 401.
[10] Eldar, Y. C. (2003). von Neumann measurement is optimal for detecting linearly independent mixed quantum states. Phys. Rev. A (3) 68 052303, 4.
[11] Hayashi, M. (2006). Quantum Information. An introduction . Springer, Berlin. · Zbl 1195.81031
[12] Helstrom, C. W. (1976). Quantum Detection and Estimation Theory . Acadamic Press, New York. · Zbl 1332.81011
[13] 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
[14] Holevo, A. S. (1973). Statistical decision theory for quantum systems. J. Multivariate Anal. 3 337-394. · Zbl 0275.62004
[15] Holevo, A. S. (1974). Remarks on optimal quantum measurements. Probl. Inf. Transm. 10 51-55. · Zbl 0309.94001
[16] Holevo, A. S. (1978). On asymptotically optimal hypothesis testing in quantum statistics. Theory Probab. Appl. 23 411-415. · Zbl 0426.62085
[17] Hwang, W.-Y. and Bae, J. (2010). Minimum-error state discrimination constrained by the no-signaling principle. J. Math. Phys. 51 022202, 11. · Zbl 1309.81052
[18] Kimura, G., Miyadera, T. and Imai, H. (2008). Optimal state discrimination in general probabilistic theories. Phys. Rev. A 79 062306.
[19] 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
[20] Krob, J. and v. Weizsäcker, H. (1997). On the rate of information gain in experiments with a finite parameter set. Statist. Decisions 15 281-294. · Zbl 0928.62008
[21] Montanaro, A. (2007). On the distinguishability of random quantum states. Comm. Math. Phys. 273 619-636. · Zbl 1146.81019
[22] Nussbaum, M. and Szkoła, A. (2009). The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Statist. 37 1040-1057. · Zbl 1162.62100
[23] 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
[24] Nussbaum, M. and Szkoła, A. (2011). Asymptotically optimal discrimination between multiple pure quantum states. In Theory of Quantum Computation , Communication and Cryptography . 5 th Conference , TQC 2010, Leeds , UK. Revised Selected Papers (W. van Dam, V. M. Kendon and S. Severini, eds.). Lecture Notes in Computer Science 6519 1-8. Springer, Berlin. · Zbl 1310.81021
[25] Parthasarathy, K. R. (1999). Extremal decision rules in quantum hypothesis testing. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 557-568. · Zbl 1043.81512
[26] Parthasarathy, K. R. (2001). On consistency of the maximum likelihood method in testing multiple quantum hypotheses. In Stochastics in Finite and Infinite Dimensions 361-377. Birkhäuser, Boston, MA.
[27] Qiu, D. and Li, L. (2010). Minimum-error discrimination of quantum states: New bounds and comparisons. Phys. Rev. A 81 042329.
[28] 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
[29] Salikhov, N. P. (1998). On a generalization of Chernoff distance. Teor. Veroyatn. Primen. 43 294-314. Translation in Theory Probab. Appl. 43 (1999) 239-255. · Zbl 0942.62005
[30] Salikhov, N. P. (2002). Optimal sequences of tests for the discrimination of several multinomial schemes of trials. Teor. Veroyatn. Primen. 47 270-285. Translation in Theory Probab. Appl. 47 (2003) 286-298. · Zbl 1038.62020
[31] 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
[32] 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
[33] Tyson, J. (2010). Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates. J. Math. Phys. 51 092204, 35. · Zbl 1309.81023
[34] 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.