Quantum entanglement in (<i>d</i>−1)-spherium

Toranzo, I. V.; Plastino, A.; Sánchez-Moreno, P.; Dehesa, J. S.

doi:10.1088/1751-8113/48/47/475302

Cited by 9 publications

(11 citation statements)

References 48 publications

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“…Now it is straightforward to see that for three-dimensional hydrogenic and oscillator-like systems the integrals required for the determination of the variance and the angular Rényi entropy are of the type I 1 given by The extension of all these physical entropies from 3 to D, D > 3, dimensions is direct and then the parameter α of the involved orthogonal polynomials is directly proportional to D. The usefulness of high-and very high-dimensional quantum systems and phenomena has been amply shown in the the literature from general quantum mechanics and quantum field theory [2,3,4,5,7,8,6,9,10,11,12] to quantum information [14,13,15,16].…”

Section: )mentioning

confidence: 99%

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Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

Temme¹,

Toranzo²,

Dehesa³

2017

J. Phys. A: Math. Theor.

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The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre (L (α) m (x)) and Gegenbauer (C (α) m (x)) polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as α → ∞ of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.Keywords: asymptotic analysis of integrals, information theory of orthogonal polynomials, entropic functionals of Laguerre and Gegenbauer polynomials.

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Section: )mentioning

confidence: 99%

“…As explained for Proposition 3.2, we can expand them, rearrange the series in (4. 16), and obtain an expansion in negative powers of α. We summarize the above results as follows.…”

Section: The Case C = Dmentioning

confidence: 99%

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

Temme¹,

Toranzo²,

Dehesa³

2017

J. Phys. A: Math. Theor.

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“…As a final remark, there is a very recent work concerning a system of two Coulombically interacting particles confined to a D − 1 sphere, where the dependence of the entanglement measures on the radius of the system and the spatial dimensionality has been investigated [36]. Thus, as future perspectives we would like to study the effects of the dimensionality and the interaction strength on the entanglement of two confined particles which interact via a general potential, taking as a starting point the results obtained in the generalization to dimensions higher than two presented in the second section of the supporting information.…”

Section: Discussionmentioning

confidence: 99%

“…Hence Wigner localization is expected for low density systems or for large interaction strengths. The Calogero and Moshinsky models are the most salient examples of analytically solvable models of confined particles, including the exact computation of the entanglement entropies [3,4,[30][31][32][33][34][35], in this sense it can also be mentioned the spherium model [36], and the quasisolvable Hook model [37]. In particular, the Calogero model has been widely studied in condensed matter physics and has experienced several revivals [38,39], such as the discovery of an explicit relation of the Calogero model with the fractionary quantum hall effect [40] and fractional statistics [41].…”

Section: Introductionmentioning

confidence: 99%

Long- and short-range interaction footprints in entanglement entropies of two-particle Wigner molecules in 2D quantum traps

Cuestas¹,

Garagiola²,

Pont³

et al. 2017

Physics Letters A

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The occupancies and entropic entanglement measures for the ground state of two particles in a two-dimensional harmonic anisotropic trap are studied. We implement a method to study the large interaction strength limit for different short-and long-range interaction potentials that allows to obtain the exact entanglement spectrum and several entropies. We show that for long-range interactions, the von Neumann, min-entropy and the family of Rényi entropies remain finite for the anisotropic traps and diverge logarithmically for the isotropic traps. In the short-range interaction case the entanglement measures diverge for any anisotropic parameter due to the divergence of uncertainty in the momentum since for short-range interactions the relative position width vanishes.We also show that when the reduced density matrix has finite support the Rényi entropies present a non-analytical behaviour. *

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“…Nowadays, there is an increasing interest on the dimensional dependence of the entropic properties for the stationary states of the multidimensional quantum systems [12,13,[28][29][30][31][32][33][34][35][36][37][38] in order to contribute to the emergent informational representation of the quantum systems which extends and complements the standard energetic representation. This is basically because of the increasing relevance of entropic properties in numerous biological, chemical, and physical phenomena [39][40][41][42][43][44] as well as in electronic correlations [45][46][47][48] and chemical reactions. [49][50][51][52][53] The determination of these entropic properties requires huge numerical calculations which often obscure and hide the basic phenomena.…”

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The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Dehesa

Belega

Toranzo

et al. 2019

Int J of Quantum Chemistry

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There exists an increasing interest on the dimensionality dependence of the entropic properties for the stationary states of the multidimensional quantum systems in order to contribute to its emergent informational representation, which extends and complements the standard energetic representation. Nowadays, this is specially so for high-dimensional systems as they have been recently shown to be very useful in both scientific and technological fields. In this work, the Shannon entropy of the discrete stationary states of the high-dimensional harmonic (ie, oscillator-like) and hydrogenic systems is analytically determined in terms of the dimensionality, the potential strength, and the state's hyperquantum numbers. We have used an informationtheoretic methodology based on the asymptotics of some entropy-like integral functionals of the orthogonal polynomials and hyperspherical harmonics which control the wave functions of the quantum states, when the polynomial parameter is very large; this is basically because such a parameter is a linear function of the system's dimensionality. Finally, it is shown that the Shannon entropy of the D-dimensional harmonic and hydrogenic systems has a logarithmic growth rate of the type D log D when D ! ∞. K E Y W O R D S high-dimensional harmonic systems, high-dimensional hydrogenic systems, high-dimensional quantum systems, logarithmic integrals of orthogonal polynomials-asymptotics, Shannon entropy of high-dimensional quantum systems 1 | INTRODUCTION Multidimensional quantum systems consisting of many particles are a major challenge in Quantum Chemistry and Physics, as their behavior can be determined only with immense computational power. The physical solutions of their associated wave equations, and consequently all the chemical and physical properties of these systems, crucially depend on the space dimensionality D. Scientists are trying to discover elegant notions and techniques to simplify the problem (see, eg, ref. [1]). The 1986 Nobel Prize in Chemistry winner Dudley Robert Herschbach et al have developed [2][3][4][5] a very useful strategy to study the atomic and molecular systems, the D-dimensional scaling method, where the D is the basic variable. The pseudoclassical or highdimensional (D ! ∞) limit is the starting point of this strategy. Indeed, this method requires to solve a finite many-electron problem in the (D ! ∞)-limit, and then perturbation theory in 1D is used to have an approximate result for the standard dimension (D = 3), obtaining at times a quantitative accuracy comparable to the self-consistent Hartree-Fock calculations. Most important here is that the electronic structure for the (D ! ∞)-limit is beguilingly simple and exactly computable for any atom and molecule. For finite D but very large, the electrons are confined to harmonic oscillations about the fixed positions attained in the (D ! ∞)-limit. Indeed, at this high-dimensional limit, the electrons of a many-electron system assume fixed positions relative to

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Quantum entanglement in (d−1)-spherium

Cited by 9 publications

References 48 publications

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

Long- and short-range interaction footprints in entanglement entropies of two-particle Wigner molecules in 2D quantum traps

The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

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