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Advanced Theory and Simulations

Tuning Pair and Higher‐Order Mutual Information Measures in Oscillator Systems with N$N$‐Dependent Frequencies

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

Published Oct 17, 2024• 0 citations
Publisher

Abstract

Abstract Analytical expressions for pair and higher‐order mutual information measures in position and momentum space are examined for many‐particle () coupled oscillator systems in their ground states. These relations reveal that the magnitudes of the statistical correlations monotonically increase with the strength of the particle interaction for all particle numbers (). On the other hand, the magnitudes of the measures monotonically decrease as increases, when the intensities of the one‐ and two‐body potentials are fixed. However, extremal points appear in the measures as is varied, when the intensity of the one‐body potential is ‐dependent, and set at . Fractional values of lead to the appearance of extremal points. These results illustrate how the intensity of the one‐body potential can be used to tune the behavior of the statistical correlation measures in these systems.

DOI
10.1002/adts.202400777

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Tuning Pair and Higher‐Order Mutual Information Measures in Oscillator Systems with NNNN‐Dependent Frequencies
Abstract Analytical expressions for pair and higher‐order mutual information measures in position and momentum space are examined for many‐particle () coupled oscillator systems in their ground states. These relations reveal that the magnitudes of the statistical correlations monotonically increase with the strength of the particle interaction for all particle numbers (). On the other hand, the magnitudes of the measures monotonically decrease as increases, when the intensities of the one‐ and two‐body potentials are fixed. However, extremal points appear in the measures as is varied, when the intensity of the one‐body potential is ‐dependent, and set at . Fractional values of lead to the appearance of extremal points. These results illustrate how the intensity of the one‐body potential can be used to tune the behavior of the statistical correlation measures in these systems.
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.
Interaction information, a measure of higher-order statistical correlations, is examined in quantum systems consisting of three coupled and uncoupled oscillators, in position and in momentum space. The magnitudes of the interaction information are greater than zero, and equal in both spaces, in the uncoupled case. This is interpreted as a dominance of synergic effects due to the indistinguishability of particles, and is larger for symmetric wave functions compared to antisymmetric ones. In symmetric wave functions, the magnitude of interaction information increases with the coupling strength, and the inclusion of coupling reveals differences in behavior between the spaces. With attractive potentials, interaction information is positive in momentum space (synergic) and negative in position space (redundant). This is reversed with repulsive potentials. It is now position space which yields positive values, while momentum space gives negative ones. The inclusion of spin provides antisymmetric wave functions which yield that the overall tendencies are maintained. However, there are regions for both attractive and repulsive potentials where the sign of interaction information changes, with the interpretation that the balance changes from a dominance of synergic to that of redundant interactions, and vice versa. This suggests that the nature of the interactions in these systems can be tuned with the coupling strength.
Abstract The dependence of the entropic uncertainty relations on particle number and on attractive and repulsive interparticle potential strengths, is examined in the ground state of coupled harmonic oscillators. The reduced entropy sums depend on the normalization constants of the position and momentum densities, which in turn depend on particle number and interaction strength. These sums exhibit maxima as the particle number is varied, when the particles are in the presence of attractive potentials. On the other hand, no maxima are observed for systems with a repulsive potential. All reduced entropy sums increase and move away from the bound with greater interactions. Consistent behaviours are observed for both reduced one- and two-variable entropy sums. Further reductions of the N-particle densities induce that the respective entropy sums are now farther away from the corresponding bounds. The interpretation of these results is that reduction and increased interaction translates into a weakening of the strength of the uncertainty principle statement. The one- and two-variable entropy sums of these systems are shown to approximate the a + b ln N relation, when the intensity of the one-body harmonic potential is set at the number of particles.
The influence of wave function symmetry on statistical correlation is studied for the case of three non-interacting spin-free quantum particles in a unidimensional box, in position and in momentum space. Higher-order statistical correlations occurring among the particles in this three-particle quantum system is quantified via higher-order mutual information and compared to the correlation between pairs of variables in this model, and to the correlation in the two-particle system. The results for the higher-order mutual information show that in general symmetric wave functions are more correlated than the antisymmetric ones with same quantum numbers. This holds in position as well as in momentum space. This behavior is opposite to that observed for the correlation between pairs of variables in this model, and the two-particle system, where the antisymmetric wave functions are in general more correlated. The use of higher-order mutual information as a correlation measure, is monitored and examined by considering a superposition of states or systems with two Slater determinants.
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.