A generalized statistical complexity measure: Applications
to quantum systems
and J. Sañudo
DIIS and BIFI, Facultad de Ciencias, Universidad de Zaragoza, E-50009 Zaragoza,
Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, Hungary
Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de
Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
Departamento de Física, Facultad de Ciencias, Universidad de Extremadura, E-06071
Badajoz, Spain, and BIFI, Universidad de Zaragoza, E-50009 Zaragoza, Spain
͑Received 20 May 2009; accepted 19 November 2009; published online 31 December 2009
A two-parameter family of complexity measures C
based on the Rényi entro-
pies is introduced and characterized by a detailed study of its mathematical prop-
erties. This family is the generalization of a continuous version of the Lopez-Ruiz–
Mancini–Calbet complexity, which is recovered for
complexity measures are obtained by multiplying two quantities bringing global
information on the probability distribution deﬁning the system. When one of the
, goes to inﬁnity, one of the global factors becomes a local factor.
For this special case, the complexity is calculated on different quantum systems:
H-atom, harmonic oscillator, and square well. © 2009 American Institute of
The question concerning the quantiﬁcation of complexity
has been addressed in many dif-
ferent ﬁelds, from computer science to physics. Depending on the properties to be grasped differ-
ent answers are found in the literature.
The study of these statistical measures in physical
systems, and, in particular, in quantum systems, has a role of growing importance. So, information
entropies and statistical complexities have been calculated on different atomic systems.
ticular, the so-called Lopez-Ruiz–Mancini–Calbet ͑LMC͒ complexity
has been computed in
the position and momentum spaces for the density functions of the hydrogenlike atoms and the
quantum isotropic harmonic oscillator.
It has been found that the minimum values of that
statistical measure is taken on the quantum states with the highest orbital angular momentum, just
those wave functions that correspond to the Bohr-like orbits in the prequantum image.
Many LMC-like statistical complexities are deﬁned as a product of two factors, one of them
measuring the broadening of the distribution that deﬁnes the system and the other one quantifying
the narrowness of it. Both factors are global magnitudes that can be calculated by integrating over
the whole support of the distribution.
is an adequate indicator to grasp the spreading of a distribution and
thus it is employed as a basic ingredient of the ﬁrst factor of complexity measures. Concretely, it
plays an important role in the original LMC statistical complexity in which the second factor, the
was originally chosen to be the square distance to the equiprobability
distribution. Other functions can be used to deﬁne different families of LMC-like complexities.
When the simple power Shannon entropy is taken as the ﬁrst factor, it is a meaningful parameter
to characterize the shape of a distribution.
In this particular case, the log of the LMC complexity
coincides with the structural entropy introduced to study questions concerning quantum
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JOURNAL OF MATHEMATICAL PHYSICS 50, 123528 ͑2009͒
50, 123528-10022-2488/2009/50͑12͒/123528/10/$25.00 © 2009 American Institute of Physics