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

Physica Scripta

Particle number and interactions in the entropic uncertainty relations

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

Published Nov 13, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ad0c1a

Related papers

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

Entropic Uncertainty Relations and Mutual Information Correlation Sums in Two-level Superposition States of Coupled Oscillators

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

The effects of quantum interferences and interaction strength on the entropic uncertainty relations, and on mutual information correlation sums, are examined in two-level superposition states of two coupled oscillators. The presence of quantum interferences results in a movement of the entropy sums toward the uncertainty relation bound, for both attractive and repulsive interaction potentials. On the other hand, these interferences suppress the statistical correlations in the presence of an attractive potential, while the correlations increase for a repulsive one. In general, stronger interactions between particles move the entropy sums away from bound, with the result that the systems possess larger statistical correlations. However, there are superposition and attractive interaction regimes, where the entropy sum of an interacting system can actually lie closer to the bound, in comparison to the corresponding non-interacting one. In these cases, the statistical correlations between particles is lesser for the interacting systems, as compared to the non-interacting ones. These effects are not observed when repulsive potentials are present. Here, the non-interacting systems lower-bound both the entropy sums and correlation measures. These results offer insights into the nature of superposition or quantum interference effects in interacting quantum systems, and the behavior in terms of the entropic uncertainty relations, statistical correlations and interaction strength. Resumen. Se examinan los efectos que las interferencias cuánticas y la magnitud de la interacción tienen sobre las relaciones de incertidumbre entrópicas, así como sobre las sumas correlaciones me didas a través de la información mutua, en estados de superposición de dos niveles de dos osciladores acoplados. La presencia de interferencias cuánticas da como resultado un movimiento de las sumas entrópicas hacia la cota de la relación de incertidumbre, tanto para potenciales de interacción atractivos como repulsivos. Por otra parte, en presencia de un potencial atractivo, estas interferencias suprimen las correlaciones estadísticas, mientras que las correlaciones aumentan en presencia de uno repulsivo. En general, con interacciones más fuertes entre partículas, las sumas de entrópicas se alejan de la cota, dando como resultado mayores correlaciones estadísticas en los sistemas. Sin embargo, existen regímenes de superposición e interacción atractiva, en los cuales la suma entrópica de un sistema interactuante puede estar más cerca de la cota, en comparación con el sistema no interactuante correspondiente. En estos casos, las correlaciones estadísticas entre partículas son menores para los sistemas interactúantes que para los no interactuantes. Estos efectos no se observan en los potenciales repulsivos. En este caso, los sistemas no interactuantes establecen límites inferiores tanto para las sumas entrópicas como para las medidas de correlación. Estos resultados dan información sobre la naturaleza de los efectos de superposición o interferencia cuántica en sistemas cuánticos interactuantes, y su comportamiento en términos de relaciones de incertidumbre entrópica, correlaciones estadísticas y fuerza de interacción.

Journal of the Mexican Chemical Society•Sep 30, 2024•0 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

Quantum channels and the entropic uncertainty relation in two-level harmonic oscillator superposition states

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

Abstract The Shannon information entropy sum is examined as systems comprised of two-level harmonic oscillator states are passed through dephasing and depolarization quantum channels. We show how the entropy sum increases as the dephasing parameter is increased. The interpretation here is a loss of information in the system as the quantum coherences are removed. This result and interpretation also holds in the depolarization channel as the system transits to the maximally mixed state. The increase in the entropy sum is associated with a delocalization of both position and momentum densities. The behaviour of phase-space Wigner functions and their Shannon entropies is also addressed. Wigner negativity is seen to decrease and disappear with an increase of the dephasing parameter. Here, Wigner negativity is associated with the presence of quantum coherences. The imaginary component of the Wigner function entropy is shown to capture the changes in Wigner negativity. On the other hand, Wigner functions with diagonal input density matrices are non-negative in the depolarization channel. Their real-valued Shannon entropies increase with the depolarization parameter as the system approaches the maximally mixed state. The mutual information between the Wigner function and the product of position and momentum space densities is used to probe the statistical correlation between position and momentum variables. This measure decreases in both channels as the parameter increases. Thus, there is a loss of correlation as quantum coherences are eliminated in the dephasing case, and as the system transits to the maximally mixed case in the depolarizing channel.

Physica Scripta•Nov 1, 2025•0 citations

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

Sign-of-coupling dependence of von Neumann entanglement entropy in networks of quantum oscillators

Pei-Song He

In this paper, we investigated the sign-of-coupling dependence of the von Neumann Entanglement Entropy (EE) between system and environment in the ground state of networks of N coupled quantum oscillators. We find that whether the EE increases or decreases with the amplitude of coupling strengths between particles depends on the repulsive or attractive nature of the couplings, and it also depends on whether the coupling is between system-environment or particles both in the environment. In particular, the EE decreases with the amplitude of an attractive coupling between particles both in the environment. We also find that the EE varies nonmonotonically with number of particles directly interacting with the system with attractive couplings; besides, the position-momentum uncertainty of the particle in the system and the EE vary contrarily when the particle-number is small.

Modern Physics Letters B•Mar 13, 2024•0 citations
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 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.
The effects of quantum interferences and interaction strength on the entropic uncertainty relations, and on mutual information correlation sums, are examined in two-level superposition states of two coupled oscillators. The presence of quantum interferences results in a movement of the entropy sums toward the uncertainty relation bound, for both attractive and repulsive interaction potentials. On the other hand, these interferences suppress the statistical correlations in the presence of an attractive potential, while the correlations increase for a repulsive one. In general, stronger interactions between particles move the entropy sums away from bound, with the result that the systems possess larger statistical correlations. However, there are superposition and attractive interaction regimes, where the entropy sum of an interacting system can actually lie closer to the bound, in comparison to the corresponding non-interacting one. In these cases, the statistical correlations between particles is lesser for the interacting systems, as compared to the non-interacting ones. These effects are not observed when repulsive potentials are present. Here, the non-interacting systems lower-bound both the entropy sums and correlation measures. These results offer insights into the nature of superposition or quantum interference effects in interacting quantum systems, and the behavior in terms of the entropic uncertainty relations, statistical correlations and interaction strength. Resumen. Se examinan los efectos que las interferencias cuánticas y la magnitud de la interacción tienen sobre las relaciones de incertidumbre entrópicas, así como sobre las sumas correlaciones me didas a través de la información mutua, en estados de superposición de dos niveles de dos osciladores acoplados. La presencia de interferencias cuánticas da como resultado un movimiento de las sumas entrópicas hacia la cota de la relación de incertidumbre, tanto para potenciales de interacción atractivos como repulsivos. Por otra parte, en presencia de un potencial atractivo, estas interferencias suprimen las correlaciones estadísticas, mientras que las correlaciones aumentan en presencia de uno repulsivo. En general, con interacciones más fuertes entre partículas, las sumas de entrópicas se alejan de la cota, dando como resultado mayores correlaciones estadísticas en los sistemas. Sin embargo, existen regímenes de superposición e interacción atractiva, en los cuales la suma entrópica de un sistema interactuante puede estar más cerca de la cota, en comparación con el sistema no interactuante correspondiente. En estos casos, las correlaciones estadísticas entre partículas son menores para los sistemas interactúantes que para los no interactuantes. Estos efectos no se observan en los potenciales repulsivos. En este caso, los sistemas no interactuantes establecen límites inferiores tanto para las sumas entrópicas como para las medidas de correlación. Estos resultados dan información sobre la naturaleza de los efectos de superposición o interferencia cuántica en sistemas cuánticos interactuantes, y su comportamiento en términos de relaciones de incertidumbre entrópica, correlaciones estadísticas y fuerza de interacción.
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.
Abstract The Shannon information entropy sum is examined as systems comprised of two-level harmonic oscillator states are passed through dephasing and depolarization quantum channels. We show how the entropy sum increases as the dephasing parameter is increased. The interpretation here is a loss of information in the system as the quantum coherences are removed. This result and interpretation also holds in the depolarization channel as the system transits to the maximally mixed state. The increase in the entropy sum is associated with a delocalization of both position and momentum densities. The behaviour of phase-space Wigner functions and their Shannon entropies is also addressed. Wigner negativity is seen to decrease and disappear with an increase of the dephasing parameter. Here, Wigner negativity is associated with the presence of quantum coherences. The imaginary component of the Wigner function entropy is shown to capture the changes in Wigner negativity. On the other hand, Wigner functions with diagonal input density matrices are non-negative in the depolarization channel. Their real-valued Shannon entropies increase with the depolarization parameter as the system approaches the maximally mixed state. The mutual information between the Wigner function and the product of position and momentum space densities is used to probe the statistical correlation between position and momentum variables. This measure decreases in both channels as the parameter increases. Thus, there is a loss of correlation as quantum coherences are eliminated in the dephasing case, and as the system transits to the maximally mixed case in the depolarizing channel.
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.
In this paper, we investigated the sign-of-coupling dependence of the von Neumann Entanglement Entropy (EE) between system and environment in the ground state of networks of N coupled quantum oscillators. We find that whether the EE increases or decreases with the amplitude of coupling strengths between particles depends on the repulsive or attractive nature of the couplings, and it also depends on whether the coupling is between system-environment or particles both in the environment. In particular, the EE decreases with the amplitude of an attractive coupling between particles both in the environment. We also find that the EE varies nonmonotonically with number of particles directly interacting with the system with attractive couplings; besides, the position-momentum uncertainty of the particle in the system and the EE vary contrarily when the particle-number is small.