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International Journal of Quantum Information

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

Humberto G. Laguna, Robin P. Sagar

Published Oct 1, 2010• 29 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749910006484

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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 local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
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.
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.
Quantum uncertainties in position, momentum and phase‐space are studied in the confined Harmonic Oscillator. Standard deviations and Shannon entropies are used to quantify these uncertainties and their behaviors are compared and contrasted. We observe a minimum in the momentum space Shannon entropy as the box length is increased, a feature that is not present in the momentum space standard deviation. The behaviors of the standard deviation product and the Shannon entropy sum, which form the basis of uncertainty relationships, are also analyzed. Maxima are observed in the product as the box length increases in sharp contrast to the entropy sum. The relationship between these behaviors and that of the Shannon entropy of the phase‐space Wigner function is analyzed and discussed. An analysis of the energetic components is also performed. The results reinforce the idea that the confined Harmonic Oscillator can be considered as an intermediate model which interpolates between the Particle in a Box Model and the Harmonic Oscillator and thus contains characteristics of both models. image
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.
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.