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Response to “Comment on “Entropy and complexity analysis of hydrogenic Rydberg atoms” [J. Math. Phys. 58, 104101 (2017)]. (English) Zbl 1373.81115
From the text: In their comment [J. Math. Phys. 58, No. 10, 104101, 4 p. (2017; Zbl 1373.81110)] on our paper [S. López-Rosa et al., J. Math. Phys. 54, No. 5, 052109, 18 p. (2013; Zbl 1282.81054)], L. G. Jiao and L. R. Zan argued that the expressions used for the Crámer-Rao complexity in both position and momentum spaces are incorrect due to the wrong definition of variance. Specifically they point out that: (1) the correct form of this complexity in position space is presented exactly in their work; (2) in momentum space, two different definitions of the variance are provided and corresponding Crámer-Rao complexities are discussed separately.
There are two underlying issues in the Jiao-Zan comment to our paper1 which are certainly controverted, but not yet fully solved. One is the extension to arbitrary dimensions of the notion of variance of a one-dimensional probability distribution \(\rho(x)\), corresponding to some random variable \(X\) (e.g., position, momentum, …) (see, e.g. M. J. W. Hall[Phys. Rev A 59, No. 4. 2602–2615 (1999: doi:10.1103/PhysRevA.59.2602)] and A. Demboet al. [IEEE Trans. Inf. Theory 37, No. 6, 1501–1518 (1991; Zbl 0741.94001)]), that is, to a probability distribution \(\rho(\vec{r})\) of a D-dimensional observable \(\vec{r}\). Another issue is the definition of the Crámer-Rao complexity so that it approaches as close as possible the intuitive notion of complexity [L. Rudnicki et al., Phys. Lett. A 380, No. 3, 377–380 (2016; Zbl 1349.81065)].
MSC:
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A17 Measures of information, entropy
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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[1] López-Rosa, S.; Toranzo, I. V.; Sánchez-Moreno, P.; Dehesa, J. S., Entropy and complexity analysis of hydrogenic Rydberg atoms, J. Math. Phys., 54, 052109, (2013) · Zbl 1282.81054
[2] Hall, M. J. W., Universal geometric approach to uncertainty, entropy, and information, Phys. Rev. A, 59, 2602, (1999)
[3] Dembo, A.; Cover, T. M.; Thomas, J. A., Information theoretic inequalities, IEEE Trans. Inf. Theory, 37, 1501, (1991) · Zbl 0741.94001
[4] Rudnicki, L.; Toranzo, I. V.; Sánchez-Moreno, P.; Dehesa, J. S., Monotone measures of statistical complexity, Phys. Lett. A, 380, 377, (2016) · Zbl 1349.81065
[5] Dehesa, J. S.; Plastino, A. R.; Sánchez-Moreno, P.; Vignat, C., Generalized Crámer-Rao relations for non-relativistic quantum systems, Appl. Math. Lett., 25, 1689, (2012) · Zbl 1254.81033
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