Fano type quantum inequalities in terms of q-entropies

Quantum Information Processing - Tập 11 - Trang 1895-1910 - 2011
Alexey E. Rastegin1
1Department of Theoretical Physics, Irkutsk State University, Irkutsk, Russia

Tóm tắt

Generalizations of the quantum Fano inequality are considered. The notion of q-entropy exchange is introduced. This quantity is concave in each of its two arguments. For q ≥ 0, the inequality of Fano type with q-entropic functionals is established. The notion of coherent information and the perfect reversibility of a quantum operation are discussed in the context of q-entropies. By the monotonicity property, the lower bound of Pinsker type in terms of the trace norm distance is obtained for the Tsallis relative q-entropy of order q = 1/2. For 0 ≤ q ≤ 2, Fano type quantum inequalities with freely variable parameters are obtained.

Tài liệu tham khảo

Cover T.M., Thomas J.A. (1991) Elements of Information Theory. Wiley, New York Nielsen M.A., Chuang I.L. (2000) Quantum Computation and Quantum Information. Cambridge University Press, Cambridge Rényi, A.: On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–561. University of California Press, Berkeley–Los Angeles (1961) Erdogmus D., Principe J.C. (2004) Lower and upper bounds for misclassification probability based on Rényi’s information. J. VLSI Signal Process. 37: 305–317 Havrda J., Charvát F. (1967) Quantification methods of classification processes: concept of structural α-entropy. Kybernetika 3: 30–35 Tsallis C. (1988) Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52: 479–487 Gell-Mann, M., Tsallis, C. (eds) (2004) Nonextensive Entropy—Interdisciplinary Applications. Oxford University Press, Oxford Tsallis C. (1998) Generalized entropy-based criterion for consistent testing. Phys. Rev. E 58: 1442–1445 Dukkipati A., Narasimha Murty M., Bhatnagar S. (2006) Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization. Physica A 361: 124–138 Furuichi S. (2006) Information-theoretical properties of Tsallis entropies. J. Math. Phys. 47: 023302 Zozor S., Portesi M., Vignat C. (2008) Some extensions of the uncertainty principle. Physica A 387: 4800–4808 Rastegin A.E. (2010) Rényi formulation of the entropic uncertainty principle for POVMs. J. Phys. A: Math. Theor. 43: 155302 Majerník V., Majerníková E. (2001) The determination of bounds of the β-entropic sum of two noncommuting observables. Rep. Math. Phys. 47: 381–392 Rastegin A.E. (2011) Entropic uncertainty relations for extremal unravelings of super-operators. J. Phys. A: Math. Theor. 44: 095303 Rastegin A.E. (2011) Entropic formulation of the uncertainty principle for the number and annihilation operators. Phys. Scr. 84: 057001 Blahut R.E. (1976) Information bounds of the Fano-Kullback type. IEEE Trans. Inf. Theory 22: 410–421 Han T.S., Verdú S. (1994) Generalizing the Fano inequality. IEEE Trans. Inf. Theory 40: 1247–1251 Sharma N. (2008) Extensions of the quantum Fano inequality. Phys. Rev. A 78: 012322 Wehrl A. (1978) General properties of entropy. Rev. Mod. Phys. 50: 221–260 Ohya M., Petz D. (1993) Quantum Entropy and its Use. Springer, Heidelberg Borland L., Plastino A.R., Tsallis C. (1998) Information gain within nonextensive thermostatistics. J. Math. Phys. 39: 6490–6501 Borland L., Plastino A.R., Tsallis C. (1999) Erratum: Information gain within nonextensive thermostatistics. J. Math. Phys. 40: 2196 Csiszár I. (1963) Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Publ. Math. Inst. Hungar. Acad. Sci. 8: 85–107 Csiszár I. (1967) Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2: 299–318 Furuichi S., Yanagi K., Kuriyama K. (2004) Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45: 4868–4877 Abe S. (2003) Monotonic decrease of the quantum nonadditive divergence by projective measurements. Phys. Lett. A 312: 336–338 Abe S. (2004) Erratum: Monotonic decrease of the quantum nonadditive divergence by projective measurements. Phys. Lett. A 324: 507 Petz D. (1986) Quasi-entropies for finite quantum systems. Rep. Math. Phys. 21: 57–65 Petz D. (2010) From f-divergence to quantum quasi-entropies and their use. Entropy 12: 304–325 Jenčová A., Ruskai M.B. (2010) A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality. Rev. Math. Phys. 22: 1099–1121 Hiai F., Mosonyi M., Petz D., Bény C. (2011) Quantum f-divergences and error correction. Rev. Math. Phys. 23: 691–747 Sharma, N.: Equality conditions for the quantum f-relative entropy and generalized data processing inequalities. Quantum Inf. Process. (2011). doi:10.1007/s11128-011-0238-x Bhatia R. (1997) Matrix Analysis. Springer, New York Schumacher B. (1996) Sending entanglement through noisy quantum channels. Phys. Rev. A 54: 2614–2628 Raggio G.A. (1995) Properties of q-entropies. J. Math. Phys. 36: 4785–4791 Fuchs C.A., van de Graaf J. (1999) Cryptographic distinguishability measures for quantum mechanical states. IEEE Trans. Inf. Theory 45: 1216–1227 Biham E., Boyer M., Boykin P.O., Mor T., Roychowdhury V. (2006) A proof of the security of quantum key distribution. J. Cryptol. 19: 381–439 Rastegin A.E. (2011) Bounds on Shannon distinguishability in terms of partitioned measures. Quantum Inf. Process. 10: 123–138 Audenaert K. (2007) Subadditivity of q-entropies for q > 1. J. Math. Phys. 48: 083507 Rastegin A.E. (2011) Some general properties of unified entropies. J. Stat. Phys. 143: 1120–1135 Hu X., Ye Z. (2006) Generalised quantum entropies. J. Math. Phys. 47: 023502 Rastegin A.E. (2010) Partitioned trace distances. Quantum Inf. Process. 9: 61–73 Fannes M. (1973) A continuity property of entropy density for spin lattice systems. Commun. Math. Phys. 31: 291–294 Furuichi S., Yanagi K., Kuriyama K. (2007) A generalized Fannes’ inequality. J. Inequal. Pure Appl. Math. 8(1): 5 Zhang Z. (2007) Uniform estimates on the Tsallis entropies. Lett. Math. Phys. 80: 171–181 Rastegin A.E. (2010) Continuity and stability of partial entropic sums. Lett. Math. Phys. 94: 229–242 Audenaert K.M.R., Eisert J. (2005) Continuity bounds on the quantum relative entropy. J. Math. Phys. 26: 102104 Rastegin A.E. (2011) Upper continuity bounds on relative q-entropy for q > 1. J. Math. Phys. 52: 062203 Hiai F., Ohya M., Tsukada M. (1981) Sufficiency, KMS condition and relative entropy in von Neumann algebras. Pac. J. Math. 96: 99–109 Ruskai M.B., Stillinger F.M. (1990) Convexity inequalities for estimating free energy and relative entropy. J. Phys. A: Math. Gen. 23: 2421–2437 Fedotov A., Harremoës P., Topsøe F. (2003) Refinements of Pinsker inequality. IEEE Trans. Inf. Theory 49: 1491–1498