Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Entropy

Statistical Correlations of the N-particle Moshinsky Model

Hsuan Peng, Y. K. Ho

Published Mar 31, 2015• 27 citations
Publisher

Abstract

We study the correlation of the ground state of an N-particle Moshinsky model by computing the Shannon entropy in both position and momentum spaces. We have derived the Shannon entropy and mutual information with analytical forms of such an N-particle Moshinsky model, and this helps us test the entropic uncertainty principle. The Shannon entropy in position space decreases as interaction strength increases. However, Shannon entropy in momentum space has the opposite trend. Shannon entropy of the whole system satisfies the equality of entropic uncertainty principle. Our results also indicate that, independent of the sizes of the two subsystems, the mutual information increases monotonically as the interaction strength increases.

DOI
10.3390/e17041882

Related papers

Statistical correlations in the Moshinsky atom

Humberto G. Laguna, Robin P. Sagar

We study the influence of the interparticle and confining potentials on statistical correlation via the correlation coefficient and mutual information in ground and some excited states of the Moshinsky atom in position and momentum space. The magnitude of the correlation between positions and between momenta is equal in the ground state. In excited states, the correlation between the momenta of the particles is greater than between their positions when they interact through an attractive potential whereas for repulsive interparticle potentials the opposite is true. Shannon entropies, and their sums (entropic formulations of the uncertainty principle), are also analyzed, showing that the one-particle entropy sum is dependent on the interparticle potential and thus able to detect the correlation between particles.

Physical Review A•Jul 8, 2011•35 citations

Position–momentum correlations in the Moshinsky atom

Humberto G. Laguna, Robin P. Sagar

Position–momentum correlation in the Moshinsky atom is studied by examining the mutual information between Wigner functions and position/momentum space densities, at the one- and two-particle levels. We show that this correlation is independent of the strength of the interparticle potential at the two-particle level. At the one-particle level, the correlation decreases as the potential is increased. Shannon entropies of one- and two-particle Wigner functions are also discussed as measures of the localization in phase-space distributions.

Journal of Physics A Mathematical and Theoretical•Dec 9, 2011•17 citations

Mutual information sums and relations with interaction energies in N -particle quantum systems

Saúl J. C. Salazar, Humberto G. Laguna, Robin P. Sagar

The sums of position- and momentum-space mutual information measures are used to examine the pairwise and higher-order statistical correlation in the ground states of N-particle coupled oscillators. Analytical expressions for these measures are shown to be related to the logarithmic interaction energies of these states, plus those of mirror states where the intensities of the one- and two-body potentials are interchanged, and the nature of the attractive or repulsive interaction is opposite to that in the original state. The measures separate the contributions from the interactions and those from the effective interactions due to marginalization into different terms. The pairwise mutual information sum is linearly related to the Shannon entropy sum in two particle systems, while the total correlation sum exhibits a similar relationship in three particle ones. In the latter instance, the interaction information sum can be related to entropy differences. This illustrates how entropy sums are connected to correlation measure sums in these systems. All measures approach zero with large N, when the magnitudes of the one- and two-body potentials are fixed. The pair mutual information and total correlation sums decay monotonically with N in the presence of an attractive potential and monotonically increase with a repulsive potential. On the other hand, the interaction information sum exhibits a minimum at small N, with an attractive potential. This is a consequence of the higher-order correlations governing behavior at smaller N while the pairwise ones dominate at larger N. Results are presented when the magnitude of the one-body potential is set to N.

Physical review. E•Sep 26, 2024•4 citations

Studying the Effect of Correlation on Two Electron Systems and Measurement of Information Entropies

Sudin Singh, Aparna Saha

In this paper, a three parameter (α, λ, μ) ‘Hartree and Ingman (1933)’ type correlated wave function has been used to derive an analytical model to quantify the two electron systems. The model is used to construct the expressions for the single particle wave functions [ 𝜓(𝑟) , 𝜙(𝑝)] in coordinate space and momentum space respectively with the subscripts marked as correlated (c) and uncorrelated (uc) ones. Further, the correlated and uncorrelated wave functions are used to form the single particle charge densities [𝜌(𝑟) , 𝛾(𝑝)] in coordinate space and momentum space. These wave functions and the single particle charge densities are used to compute the numerical values of the Shannon (S) and Fisher (I) information entropies for both the correlated and uncorrelated systems in coordinate space and momentum spaces and furthermore to examine their correlation effects in coordinate and momentum spaces. The uncorrelated and correlated values of Shannon information entropies indicate the reciprocity between the coordinate and momentum spaces such that low values of the Shannon information entropy in coordinate space ( 𝑆𝜌), are associated with the high values of Shannon information entropy in momentum space (𝑆𝛾) . The uncorrelated and correlated values of Shannon information entropies (S) also validate the entropic uncertainty relations Bialynicki-Birula and Mycielski (BBM) inequality which reads as (𝑆𝜌+𝑆𝛾)≥3(1+𝑙𝑛𝜋), as well as another stronger version of the uncertainty relation that is (𝑆𝜌𝑐+𝑆𝛾𝑐)>(𝑆𝜌𝑢𝑐+𝑆𝛾𝑢𝑐). Further, both the products of uncorrelated and correlated values of Fisher information entropies 𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 and 𝐼𝜌𝑐𝐼𝛾𝑐 satisfy the inequality condition 𝐼𝜌𝑐𝐼𝛾𝑐>𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 in coordinate and momentum spaces. At the same time these products also satisfy the Fisher based uncertainty relation, 𝐼𝜌𝐼𝛾≥36. The consistency of the results satisfying the uncertainty relations can be checked from the data demonstrated in the Table 1 and Table 2.

Journal of Emerging Technologies and Innovative Research•Oct 1, 2021•0 citations

SHANNON ENTROPY OF THE WIGNER FUNCTION AND POSITION-MOMENTUM CORRELATION IN MODEL SYSTEMS

Humberto G. Laguna, Robin P. Sagar

Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.

International Journal of Quantum Information•Oct 1, 2010•29 citations

Exploring the Shannon entropic uncertainty relation in 1D squeezed states

Angel García-Chung, Saúl J. C. Salazar, Humberto G. Laguna +1 more

The time-dependent behaviour of harmonic oscillator squeezed states is examined from the viewpoint of position and momentum space Shannon entropies and their sum. Analytical expressions for these entropies are obtained from a symplectic group approach. The entropy sum returns to the entropic uncertainty bound at different time intervals, and is characterized by the presence of either a maximum (minimum) in the position space Shannon entropy and a minimum (maximum) in the momentum space one. The maximal value of the entropy sum is characterized by equality of position and momentum Shannon entropies. The transition of the entropy sum from the bound to its maximal value is studied, and relations between position and momentum space Shannon entropies are probed. The relation between the position and momentum space entropies is deformed as the value of the squeezing parameter increases. An analysis of the dependence of the Shannon entropies on the squeezing parameter shows different behaviours at different times. Both position and momentum space Shannon entropies exhibit a linear dependence on the parameter when their sum lies on the uncertainty bound. Here, the entropy sum is independent of the parameter. At other times, the relation between the entropies and squeezing parameter is non-linear, and extremal points are present. The entropy sum here is seen to be an increasing function of the parameter. Kullback–Leibler distances between squeezed and unsqueezed densities illustrate that the position or momentum space densities of the squeezed system can revert back to the unsqueezed ones during the time evolution. The phase-space Wigner function, entropy and mutual information are also examined and discussed. The statistical correlation as measured by mutual information is the entropy sum minus its value at the uncertainty bound.

Discover Quantum Science•Aug 17, 2026•0 citations
We study the correlation of the ground state of an N-particle Moshinsky model by computing the Shannon entropy in both position and momentum spaces. We have derived the Shannon entropy and mutual information with analytical forms of such an N-particle Moshinsky model, and this helps us test the entropic uncertainty principle. The Shannon entropy in position space decreases as interaction strength increases. However, Shannon entropy in momentum space has the opposite trend. Shannon entropy of the whole system satisfies the equality of entropic uncertainty principle. Our results also indicate that, independent of the sizes of the two subsystems, the mutual information increases monotonically as the interaction strength increases.
We study the influence of the interparticle and confining potentials on statistical correlation via the correlation coefficient and mutual information in ground and some excited states of the Moshinsky atom in position and momentum space. The magnitude of the correlation between positions and between momenta is equal in the ground state. In excited states, the correlation between the momenta of the particles is greater than between their positions when they interact through an attractive potential whereas for repulsive interparticle potentials the opposite is true. Shannon entropies, and their sums (entropic formulations of the uncertainty principle), are also analyzed, showing that the one-particle entropy sum is dependent on the interparticle potential and thus able to detect the correlation between particles.
Position–momentum correlation in the Moshinsky atom is studied by examining the mutual information between Wigner functions and position/momentum space densities, at the one- and two-particle levels. We show that this correlation is independent of the strength of the interparticle potential at the two-particle level. At the one-particle level, the correlation decreases as the potential is increased. Shannon entropies of one- and two-particle Wigner functions are also discussed as measures of the localization in phase-space distributions.
The sums of position- and momentum-space mutual information measures are used to examine the pairwise and higher-order statistical correlation in the ground states of N-particle coupled oscillators. Analytical expressions for these measures are shown to be related to the logarithmic interaction energies of these states, plus those of mirror states where the intensities of the one- and two-body potentials are interchanged, and the nature of the attractive or repulsive interaction is opposite to that in the original state. The measures separate the contributions from the interactions and those from the effective interactions due to marginalization into different terms. The pairwise mutual information sum is linearly related to the Shannon entropy sum in two particle systems, while the total correlation sum exhibits a similar relationship in three particle ones. In the latter instance, the interaction information sum can be related to entropy differences. This illustrates how entropy sums are connected to correlation measure sums in these systems. All measures approach zero with large N, when the magnitudes of the one- and two-body potentials are fixed. The pair mutual information and total correlation sums decay monotonically with N in the presence of an attractive potential and monotonically increase with a repulsive potential. On the other hand, the interaction information sum exhibits a minimum at small N, with an attractive potential. This is a consequence of the higher-order correlations governing behavior at smaller N while the pairwise ones dominate at larger N. Results are presented when the magnitude of the one-body potential is set to N.
In this paper, a three parameter (α, λ, μ) ‘Hartree and Ingman (1933)’ type correlated wave function has been used to derive an analytical model to quantify the two electron systems. The model is used to construct the expressions for the single particle wave functions [ 𝜓(𝑟) , 𝜙(𝑝)] in coordinate space and momentum space respectively with the subscripts marked as correlated (c) and uncorrelated (uc) ones. Further, the correlated and uncorrelated wave functions are used to form the single particle charge densities [𝜌(𝑟) , 𝛾(𝑝)] in coordinate space and momentum space. These wave functions and the single particle charge densities are used to compute the numerical values of the Shannon (S) and Fisher (I) information entropies for both the correlated and uncorrelated systems in coordinate space and momentum spaces and furthermore to examine their correlation effects in coordinate and momentum spaces. The uncorrelated and correlated values of Shannon information entropies indicate the reciprocity between the coordinate and momentum spaces such that low values of the Shannon information entropy in coordinate space ( 𝑆𝜌), are associated with the high values of Shannon information entropy in momentum space (𝑆𝛾) . The uncorrelated and correlated values of Shannon information entropies (S) also validate the entropic uncertainty relations Bialynicki-Birula and Mycielski (BBM) inequality which reads as (𝑆𝜌+𝑆𝛾)≥3(1+𝑙𝑛𝜋), as well as another stronger version of the uncertainty relation that is (𝑆𝜌𝑐+𝑆𝛾𝑐)>(𝑆𝜌𝑢𝑐+𝑆𝛾𝑢𝑐). Further, both the products of uncorrelated and correlated values of Fisher information entropies 𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 and 𝐼𝜌𝑐𝐼𝛾𝑐 satisfy the inequality condition 𝐼𝜌𝑐𝐼𝛾𝑐>𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 in coordinate and momentum spaces. At the same time these products also satisfy the Fisher based uncertainty relation, 𝐼𝜌𝐼𝛾≥36. The consistency of the results satisfying the uncertainty relations can be checked from the data demonstrated in the Table 1 and Table 2.
Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.
The time-dependent behaviour of harmonic oscillator squeezed states is examined from the viewpoint of position and momentum space Shannon entropies and their sum. Analytical expressions for these entropies are obtained from a symplectic group approach. The entropy sum returns to the entropic uncertainty bound at different time intervals, and is characterized by the presence of either a maximum (minimum) in the position space Shannon entropy and a minimum (maximum) in the momentum space one. The maximal value of the entropy sum is characterized by equality of position and momentum Shannon entropies. The transition of the entropy sum from the bound to its maximal value is studied, and relations between position and momentum space Shannon entropies are probed. The relation between the position and momentum space entropies is deformed as the value of the squeezing parameter increases. An analysis of the dependence of the Shannon entropies on the squeezing parameter shows different behaviours at different times. Both position and momentum space Shannon entropies exhibit a linear dependence on the parameter when their sum lies on the uncertainty bound. Here, the entropy sum is independent of the parameter. At other times, the relation between the entropies and squeezing parameter is non-linear, and extremal points are present. The entropy sum here is seen to be an increasing function of the parameter. Kullback–Leibler distances between squeezed and unsqueezed densities illustrate that the position or momentum space densities of the squeezed system can revert back to the unsqueezed ones during the time evolution. The phase-space Wigner function, entropy and mutual information are also examined and discussed. The statistical correlation as measured by mutual information is the entropy sum minus its value at the uncertainty bound.