A generalized statistical complexity measure: Applications to quantum systems

Abstract

A two-parameter family of complexity measuresC (α,β) based on the Rényi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the LMC complexity, which is recovered for α = 1 and β = 2. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, α or β, 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. 89.75.Fb.

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

“…Recently, a generalized complexity measure has been introduced which extend these two previous measures: the biparametric LMC-Rényi [71][72][73][74], defined as…”

Statistical Complexity. Applications in Electronic Systems

“…Then, from this probability distribution, the different statistical magnitudes, Shannon entropy, disequilibrium, Fisher information, statistical complexity and Fisher-Shannon entropy, can be obtained. Other generalized magnitudes that have recently appeared in the literature could also be tested in this type of systems [63,64]. Here, we undertake the calculation of entropic products, a statistical measure of complexity C and the similar indicator [24,65,66] Figure 1.10.…”

Complexity measure and quantum shape-phase transitions in the two-dimensional limit of the vibron model

“…This definition leads to several appealing features. The complexity measure (9) verifies the mathematical properties of inversion symmetry, monotonicity, invariance under translations, rescaling transformations and replication, and near continuity (For an extensive description see [32]). The generalized complexity is an increasing function of β for a fixed α.…”

“…Recently, it has been shown that Shannon and Rényi entropies [25][26][27] Wehrl entropies [28,29], entanglement entropies [30] and Rényi-Fisher products [31] detect QPTs in several atomic and chemical quantum models. In this article we show that the generalized complexity mesures [32] are also excellent descriptors of QPT in the vibron model for molecules.…”