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Monotone measures of statistical complexity. (English) Zbl 1349.81065
Summary: We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the three main intrinsic measures of complexity (Crámer-Rao, Fisher-Shannon, LMC) and some of their generalizations.

MSC:
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A17 Measures of information, entropy
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[1] Lloyd, S.; Pagels, H., Complexity as thermodynamic depth, Ann. Phys., 188, 186, (1988)
[2] Gell-Mann, M.; Lloyd, S., Information measures, effective complexity, and total information, Complexity, 2, 44, (1996) · Zbl 1294.94011
[3] Badii, R.; Politi, A., Complexity: hierarchical structure and scaling in physics, (1997), Henry Holt New York · Zbl 1042.82500
[4] Shiner, J. S.; Davison, M.; Landsberg, P. T., Simple measure for complexity, Phys. Rev. E, 59, 1459, (1999)
[5] Catalán, R. G.; Garay, J.; López-Ruiz, R., Features of the extension of a statistical measure of complexity to continuous systems, Phys. Rev. E, 66, (2002)
[6] Yamano, T., A statistical complexity measure with nonextensive entropy and quasi-multiplicativity, J. Math. Phys., 45, 1974-1987, (2004) · Zbl 1071.94005
[7] Yamano, T., A statistical measure of complexity with nonextensive entropy, Physica A, 340, 131-137, (2004)
[8] Holland, J., Signals and boundaries: building blocks for complex adaptive systems, (2012), M.I.T. Press Cambridge, MA
[9] (Sen, K. D., Statistical Complexity, (2012), Springer Berlin)
[10] Sánchez-Moreno, P.; Angulo, J. C.; Dehesa, J. S., A generalized complexity measure based on Rényi entropy, Eur. Phys. J. D, 68, 212, (2014)
[11] Tan, R.; Terno, Daniel R.; Thompson, J.; Vedral, V.; Gu, M., Towards quantifying complexity with quantum mechanics, Eur. Phys. J. Plus, 129, 191, (2014)
[12] Vidal, G., J. Mod. Opt., 47, 355, (2000)
[13] Baumgratz, T.; Cramer, M.; Plenio, M. B., Quantifying coherence, Phys. Rev. Lett., 113, (2014)
[14] Rényi, A., Probability theory, (1970), North Holland Amsterdam · Zbl 0206.18002
[15] Shannon, C. E.; Weaver, W., The mathematical theory of communication, (1949), University of Illinois Press Urbana · Zbl 0041.25804
[16] Fisher, R. A., Theory of statistical estimation, (Bennet, J. H., Collected Papers of R.A. Fisher, (1972), University of Adelaide Press South Australia), 22, 15-40, (1925), Reprinted
[17] Frieden, B. R., Science from Fisher information, (2004), Cambridge University Press Cambridge · Zbl 1079.81013
[18] Cover, T. M.; Thomas, J. A., Elements of information theory, (2006), Wiley-Interscience New York · Zbl 1140.94001
[19] Costa, M. H., A new entropy power inequality, IEEE Trans. Inf. Theory, IT-31, 6, 751-760, (1985) · Zbl 0585.94006
[20] Dembo, A.; Cover, T. M.; Thomas, J. A., Information theoretic inequalities, IEEE Trans. Inf. Theory, 37, 1501-1528, (1991) · Zbl 0741.94001
[21] Guerrero, A.; Sánchez-Moreno, P.; Dehesa, J. S., Phys. Rev. A, 84, (2011)
[22] Dehesa, S.; Sánchez-Moreno, P.; Yáñez, R. J., Cramér-Rao information plane of orthogonal hypergeometric polynomials, J. Comput. Appl. Math., 186, 523-541, (2006) · Zbl 1086.33010
[23] Antolín, J.; Angulo, J. C., Complexity analysis of ionization processes and isoelectronic series, Int. J. Quant. Chem., 109, 586-593, (2009)
[24] López-Ruiz, R.; Nagy, Á.; Romera, E.; Sañudo, J., A generalized statistical complexity measure: applications to quantum systems, J. Math. Phys., 50, 123528, (2009) · Zbl 1373.81116
[25] López-Ruiz, R., Shannon information, LMC complexity and Rényi entropies: a straightforward approach, Biophys. Chem., 115, 215, (2005)
[26] Pipek, J.; Varga, I., Statistical electron densities, Int. J. Quant. Chem., 64, 85, (1997)
[27] Romera, E.; Nagy, A., Fisher-Rényi entropy product and information plane, Phys. Lett. A, 372, 6823, (2008) · Zbl 1227.81271
[28] Vignat, C.; Bercher, J.-F., Phys. Lett. A, 312, 27, (2003)
[29] Angulo, J. C.; Antolín, J.; Sen, K. D., Fisher-Shannon plane and statistical complexity of atoms, Phys. Lett. A, 372, 670, (2008) · Zbl 1217.81155
[30] Romera, E.; Dehesa, J., The Fisher-Shannon information plane, an electron correlation tool, J. Chem. Phys., 120, 8906-8912, (2004)
[31] Białynicki-Birula, I.; Mycielski, J., Uncertainty relations for information entropy in wave mechanics, Commun. Math. Phys., 44, 129, (1975)
[32] Heese, R.; Freyberger, M., Phys. Rev. A, 87, (2013)
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