Quantuminformation theoretic properties of nuclei and trapped Bose gases
Abstract
Fermionic (atomic nuclei) and bosonic (correlated atoms in a trap) systems are studied from an informationtheoretic point of view. Shannon and Onicescu information measures are calculated for the above systems comparing correlated and uncorrelated cases as functions of the strength of short range correlations. Onebody and twobody density and momentum distributions are employed. Thus the effect of shortrange correlations on the information content is evaluated. The magnitude of distinguishability of the correlated and uncorrelated densities is also discussed employing suitable measures of distance of states i.e. the well known KullbackLeibler relative entropy and the recently proposed JensenShannon divergence entropy. It is seen that the same informationtheoretic properties hold for quantum manybody systems obeying different statistics (fermions and bosons).
1 Introduction
Informationtheoretic methods are used in recent years for the study of quantum mechanical systems. [1][17] The quantity of interest is Shannon’s information entropy for a probability distribution
(1) 
where .
An important step is the discovery of an entropic uncertainty relation (EUR),[2] which for a threedimensional system has the form
(2) 
where is the information entropy in positionspace of the density distribution of a quantum system
(3) 
and is the information entropy in momentumspace of the corresponding momentum distribution
(4) 
The density distributions and are normalized to one. Inequality (2), for the information entropy sum in conjugate spaces, is a joint measure of uncertainty of a quantum mechanical distribution, since a highly localized is associated with a diffuse , leading to low and high and viceversa. Expression (2) is an informationtheoretical relation stronger than Heisenberg’s. is measured in bits if the base of the logarithm is 2 and nats (natural units of information) if the logarithm is natural.
In previous work we proposed a universal property of for the density distributions of nuclei, electrons in atoms and valence electrons in atomic clusters.[5] This property has the form
(5) 
where is the number of particles of the system and the parameters depend on the system under consideration. It is noted that recently we have obtained the same form for systems of correlated bosons in a trap.[4] This concept was also found to be useful in a different context. Using the formalism in phasespace of Ghosh, Berkowitz and Parr,[9] we found that the larger the information entropy the better the quality of the nuclear density distribution.[10]
In previous work we employed onebody density distributions in the definition of . In the present paper we introduce twobody density distributions and the corresponding twobody momentum distributions . Our aim is to investigate the properties of at the twobody level for correlated densities. The correlated nucleon systems or the trapped Bose gas, in a good approximation, are studied using the lowest order approximation.[18, 19] Shortrange correlations (SRC) are taken into account employing the Jastrow correlation function.[20] Thus it is of interest to examine how is affected qualitatively and quantitatively by the same form of correlations in comparison with , in view of the fact that the quantities and carry more direct information for correlations than the quantities and which are only indirectly affected by correlations. The above procedure is repeated for an alternative measure of information i.e. Onicescu’s information energy .[21] So far, only the mathematical aspects of this concept have been developed, while the physical aspects have been neglected.
A well known measure of distance of two discrete probability distributions is the KullbackLeibler relative entropy [22]
(6) 
which for continuous probability distributions is defined as
(7) 
which can be easily extended for 3dimensional systems.
Our aim is to calculate the relative entropy (distance) between (correlated) and (uncorrelated) densities both at the one and the twobody levels in order to assess the influence of SRC (through the correlation parameter ) on the distance . It is noted that this is done for both systems under consideration: nuclei and trapped Bose gases. An alternative definition of distance of two probability distributions was introduced by Rao and Lin,[23, 24] i.e. a symmetrized version of , the JensenShannon divergence [25]
(8) 
where stands for Shannon’s entropy. We expect for strong SRC the amount of distinguishability of the correlated from the uncorrelated distributions is larger than the corresponding one with small SRC. We may also see the effect of SRC on the number of trials needed to distinguish and (in the sense described in [25]).
In addition to the above considerations, we connect and with fundamental quantities i.e. the root mean square radius and kinetic energy respectively. We also argue on the effect of SRC on EUR and we propose a universal relation for , by extending our formalism from the one and twobody level to the body level, which holds exactly for uncorrelated densities in trapped Bose gas, almost exactly for uncorrelated densities in nuclei (due to the additional exchange term compared to Bose gas) and it is conjectured to hold approximately for correlated densities both in nuclei and Bose gases.
The plan of the present paper is the following. In Sec. 2 we review the formulas of KullbackLeibler relative entropy entropy and JensenShannon divergence , while in Sec. 3 Onicescu’s information energy is described. In Sec. 4 we present the formalism of density distributions used in present work and their applications to Shannon’s and Onicescu’s entropies. In Sec. 5 we introduce SRC in nuclei. In Sec. 6 we apply the formulas of and in correlated distributions. In Sec. 7 we present our numerical results and discussion. Finally, Sec. 8 contains our main conclusions.
2 KullbackLeibler relative entropy and JensenShannon divergence
The KullbackLeibler relative information entropy for continuous distributions and is defined by relation (7). It measures the difference of form the reference (or apriori) distribution . It satisfies: for any distributions and . It is a measure which quantifies the distinguishability (or distance) of from , employing a wellknown concept in standard information theory. In other words it describes how close is to by carrying out observations or coin tossing, namely trials (in the sense described in [25]). We expect for strong SRC the amount of distinguishability of the correlated and the uncorrelated distributions is larger than the corresponding one with small SRC.
However, the distance does not satisfy the triangle inequality and in addition is i) not symmetric ii) unbounded and iii) not always well defined.[25] To avoid these difficulties Rao and Lin [23, 24] introduced a symmetrized version of (recently discused in [25]), the JensenShannon divergence defined by relation (8). is minimum for and maximum when and are two distinct distributions, when . In our case can be easily generalized for continuous density distributions. For minimum the two states represented by and are completely indistinguishable, while for maximum they are completely distinguishable. It is expected that for strong SRC the amount of distinguishability can be further examined by using Wooter’s criterion.[25] Two probability distributions and are distinguishable after trials if and only if .
The present work is a first step to examine the problem of comparison of probability distributions (for nuclei and bosonic systems) which is an area well developed in statistics, known as information geometry.[23]
3 Onicescu’s information energy
Onicescu tried to define a finer measure of dispersion distributions than that of Shannon’s information entropy.[21] Thus, he introduced the concept of information energy . For a discrete probability distribution the information energy is defined by
(9) 
which is extended for a continuous density distribution as
(10) 
The meaning of (10) can be seen by the following simple argument: For a Gaussian distribution of mean value , standard deviation and normalized density
(11) 
relation (10) gives
(12) 
is maximum if one of the ’s equals 1 and all the others are equal to zero i.e. , while is minimum when , hence (total disorder). The fact that becomes minimum for equal probabilities (total disorder), by analogy with thermodynamics, it has been called information energy, although it does not have the dimension of energy.[26]
It is seen from (12) that the greater the information energy, the more concentrated is the probability distribution, while the information content decreases. and information content are reciprocal, hence one can define the quantity
(13) 
as a measure of the information content of a quantum system corresponding to Onicescu’s information energy.
Relation (10) is extended for a 3dimensional spherically symmetric density distribution
(14) 
in position and momentum space respectively, where is the corresponding density distribution in momentum space.
has dimension of inverse volume, while of volume. Thus the product is dimensionless and can serve as a measure of concentration (or information content) of a quantum system. It is also seen from (12),(13) that increases as decreases (or concentration increases) and the information (or uncertainty) decreases. Thus and are reciprocal. In order to be able to compare with Shannon’s entropy , we redifine as
(15) 
as a measure of the information content of a quantum system in both position and momentum spaces, inspired by Onicescu’s definition.
4 Density Matrices and Information entropies
Let be the wave function that describes the nuclei or the trapped Bose gases and depends on 3A coordinates as well as on spin and isospin (in nuclei). The onebody density matrix is defined in [27]
(16) 
while the twobody density matrix by
(17) 
The above density matrices are related by
(18) 
where the integration is carried out over the radius vectors and summation over spin (or isospin) variables is implied. The corresponding definitions in momentum space are similar. The twobody density distribution which is a key quantity in the present work, is defined as the diagonal part of the twobody density matrix
(19) 
and expresses the joint probability of finding two nucleons or two atoms at the positions and , respectively. The density distribution is given by the diagonal part of the onebody density matrix, that is
(20) 
or by the equivalent integral
(21) 
The twobody momentum distribution is given by a particular Fourier transform of the , that is
(22) 
In the independent particle model, where the nucleons are considered to move independently in nuclei, the is a Slater determinant. In this case it is easy to show that the twobody density matrix is given by the relation
(23)  
where is the singleparticle wave function normalized to one and
In Bose gases the manybody groundstate wave function is a product of identical singleparticle groundstate wave functions i.e.
(24) 
where is the normalized to one groundstate singleparticle wave function describing bosonic atoms. The twobody density matrix in a Bose gas, is given by the relation
(25) 
where
(26) 
We consider that the atoms of the Bose gases are confined in an isotropic HO well, where .
As the mean field approach fails to incorporate the interparticle correlation which is necessary for the description of the correlated nuclei or trapped Bose gases, we introduce the repulsive interactions through the Jastrow correlation function [20]. The correlated nucleon systems or the Bose gases, in a good approximation, can be studied using the lowest order approximation,[18, 19] where the correlated twobody density matrices in nuclei and Bose gases have the following forms respectively
(27) 
(28) 
In the present work, in the case of nuclei and trapped Bose gas, the normalization factor , is calculated by the normalization condition
(29) 
The same holds for
(30) 
The Jastrow correlation function both in the case of nuclei and trapped Bose gas is taken to be of the form
(31) 
The uncorrelated case corresponds to , while SRC increase as decreases. The above ansatz has the advantage that it leads to analytical forms for the , , and .
The onebody Shannon information entropy both in position and momentum space are defined in (3) and (4), where the total sum is
(32) 
The twobody Shannon information entropy both in position and momentum space and in total are defined respectively [28, 29]
(33) 
(34) 
(35) 
The onebody Onicescu information entropy is already defined in (3) and (15), where the generalization to the twobody information entropy is straightforward and is given by
(36) 
where
(37) 
It is easy to prove that in the case of the uncorrelated trapped Bose gas
(38) 
and
(39) 
It is worth noting that the above relations hold only approximately in finite nuclei (see Table 1), due to the additional exchange term, originating from the antisymmetry of the nuclear wave function. There is an exception in the case of He, where it holds exactly due to the absence of the exchange term.
5 Introduction of SRC in nuclei
We consider that the single particle wave functions, which describe the nucleons is harmonic oscillator type. In order to incorporate the nucleonnucleon (or atomatom) correlations, as we mention in the previous section, we apply the lowest order approximation. In this case the twobody density distribution, for He, takes the following form
(40) 
The first term of the righthand side of Eq. (40) which represents the uncorrelated part of the twobody sensity distribution, has the form
(41) 
and the second term which represents the correlated part of the twobody density distribution, is written
(42)  
where .
In the above expression is the width of the HO potential and is the normalization constant which ensures that and has the form
(43) 
The density distribution can be written also in the form
(44) 
The twobody momentum distribution is given also by the formula
(45) 
where, as in the case of twobody density distribution, the uncorrelated part has the form
(46) 
and the correlated part is written as
where .
The momentum distribution is given also by the relation
(48) 
In the present work, we extend our calculations in nuclei heavier than (C, O and Ca) based on the fact that the highmomentum tails of are almost the same for all nuclei with .[11, 30] Inspired by previous work [31, 32] we suggest a practical method to calculate the one and twobody density and momentum distributions for nuclei heavier than He. The theoretical scheme of the method combines the meanfield predictions of the twobody density distributions and twobody momentum distributions of various nuclei with their correlated part of He. Specifically, in our treatment we consider the following forms
(49) 
(50) 
From the above expressions it is obvious that the uncorrelated part of the and originate from the independent particle model for every nucleus separately, where the correlated part in each nucleus is that coming from the nucleus He. The and have a similar form.
It should be emphasized that in the uncorrelated case the additional information which is contained in and in nuclei, compared to the trapped Bose gas is the statistical correlations which come from the antisymmetry character of the manybody wave function of nuclei. Moreover, in the correlated case the and contain additional information which originate from the character of the nuleonnucleon interaction, making our model more realistic and the description more complete. It is of interest to study how the correlations (both statistical and dynamical) affect quantitatively and qualitatively the various kinds of information entropy.
6 Application of the Formalism of Relative Entropy and JensenShannon divergence for Correlated Densities
The relative entropy is a measure of distinguishability or distance of two states. It is defined, generalizing (7), by
(51) 
In our case is the correlated case and the uncorrelated one. Thus
(52) 
where is the correlated onebody density and is the uncorrelated onebody density.
A corresponding formula holds in momentumspace
(53) 
where is the correlated onebody density and is the uncorrelated one.
For the twobody case we have
(54) 
where is the correlated twobody density in positionspace and is the uncorrelated one.
The generalization to momentum space is straightforward
(55) 
where is the correlated twobody density in momentumspace and is the uncorrelated one.
For the JensenShannon divergence we may write formulas for (onebody) and (twobody), employing definition (8) and putting the corresponding correlated and uncorrelated distributions in position and momentum spaces. We calculate and in position and momentum spaces, for nuclei and bosons.
7 Numerical results and discussion
For the sake of symmetry and simplicity we put the width of the HO potential . Actually for in the case of uncorrelated case it is easy to see that and also (the same holds for Onicescu entropy), while when there is a shift of the values of and by an additive factor . However, the value of does not affect directly the total information entropy (and also ). and are just functions of the correlation parameter .
Nucleus  

He  6.43418  12.86836  248.05  248.05 
C  7.50858  15.00784  922.60  921.15 
O  7.60692  15.20890  1057.25  1055.77 
Ca  8.43472  16.88498  2685.72  2711.75 
In Fig. 1 we present the Shannon information entropy using relation (32) and using relation (35) in nuclei and trapped Bose gas as functions of the correlation parameter . It is seen that and increase almost linearly with the strength of SRC i.e. in both systems. The relations and hold exactly for the uncorrelated densities in trapped Bose gas, while the above relations are almost exact for the uncorrelated densities in nuclei and in the case of correlated densities both in nuclei and trapped Bose gas. A similar behavior is seen for all nuclei considered in the present work (He, O, Ca).
Values of , , , for various nuclei in the uncorrelated case, are shown in Table 1. The relations (38) and (39) are satisfied exactly only in the case of He. However, for the other nuclei, due to the additional exchange term in the nuclear wave function, the relations (38) and (39) hold only approximately (the differences are of order for and for ).
In Fig. 2 we present the decomposition of in coordinate and momentum spaces, for the sake of comparison i.e. , , , for and trapped Bose gas employing (3), (4), (33), (34). The most striking feature concluded from the above Figures is the similar behavior between and and also and respectively.
In Fig. 3 we plot the Onicescu information entropy both onebody and twobody for nuclei and trapped Bose gas (relations (15), (36)). We conclude by noting once again the strong similarities of the behavior between one and twobody Onicescu entropy.
It is interesting to observe the correlation of the rms radii with as well as the corresponding behavior of the mean kinetic energy with , as functions of the strength of SRC for the nucleus and trapped Bose gas. This is done in Fig. 4 for and Fig. 5 for after apllying the suitable rescaling. The corresponding curves are similar for nuclei and trapped Bose gas.
A wellknown concept in information theory is the distance between the probability distributions and , in our case the correlated and the uncorrelated distributions respectively. A measure of distance is the KullbackLeibler relative entropy defined previously. The correlated and uncorrelated cases are compared for the onebody case in Fig. 6 and the the twobody case in Fig. 7 for nuclei and trapped Bose gas, decomposing in position and momentumspaces according to (52)(55). It is seen that , increase as the strength of SRC increases, while , have a maximum at a certain value of depending on the system under consideration.
Calculations are also carried out for the JensenShannon divergence for onebody density distribution ( entropy) as function of for nuclei and trapped Bose gas, decomposed in position and momentum spaces (Fig. 8). We observe again that increases with the strength of SRC in positionspace, while in most cases in momentumspace there is a maximum for a certain value of . It is verified that as expected theoretically.[25]
It is noted that the dependence of the various kinds of information entropy on the correlation parameter is studied up to the value , which is already unrealistic corresponding to strong SRC. In addition, lowest order approximation does not work well beyond that value. In this case threebody terms should be included but this prospect is out of the scope of the present work.
For very strong SRC the momentum distribution exhibits a similar behavior with the mean field . This is illustrated in Fig. 9, where we present for various values of . It is seen that for small and large SRC the tail of disappears. That is why for small and large SRC the relative entropy ( and ) is small, while in between shows a maximum (Fig. 6, 8). A similar trend of for large SRC explains also the maximum of the relative entropy in Fig. 7.
8 Conclusions and final comment
Our main conclusions are the following

Increasing the SRC (i.e. the parameter ) the information entropies , , and increase. A comparison leads to the conclusion that the correlated systems have larger values of entropies than the uncorrelated ones.

There is a similar behavior of the entropies as functions of correlations for both systems (nuclei and trapped Bose gas) although they obey different statistics (fermions and bosons).

There is a correlation of with and with in the sense that they have the same behavior as a function of the correlation parameter . These results can lead us to relate the theoretical quantities and with experimental ones like charge form factor, charge density distribution, and momentum distribution, radii, etc. A recent paper addressed in that problem.[33]

The relations and hold exactly for the uncorrelated densities in trapped Bose gas while the above relations are almost exact for the uncorrelated densities and in the case of correlated densities both in nuclei and trapped Bose gas. In previous work we proposed the universal relation where is the number of particles of the system either fermionic (nucleus, atom, atomic cluster) or bosonic (correlated atoms in a trap). Thus in our case
For 3body distributions and
and generalizing for the body distributions and
This is exact for the uncorrelated trapped Bose gas, almost exact in correlated nuclei and it is conjectured that it holds approximately for correlated systems (which has still to be proved for ).

The entropic uncertainty relation (EUR) is
It is wellknown that the lower bound is attained for a Gaussian distribution (i.e. the case of uncorrelated). In all cases studied in the present work EUR is verified.
A final comment seems appropriate. In general, the calculation of and is a problem very hard to be solved, especially in the case of nuclei, in the framework of short range correlations. Just a few works are addressed in that problem.[34, 35, 36] In the present work we tried to treat the problem in an approximate but selfconsistent way in the sense that the calculations of and are based in the same , which is the generating function of the above quantities. As a consequence the information entropy is derived also in a selfconsistent way and there is a direct link between and , as well as the other kinds of information entropies which are studied in the present work.
Acknowledgments
The work of Ch. C. Moustakidis was supported by the Greek State Grants Foundation (IKY) under contract (515/2005) while the work of K. Ch. Chatzisavvas by Herakleitos Research Scolarships (21866). One of the authors (Ch. C. M.) would like to thank Prof. Vergados for his hospitality in the University of Ioannina where the earlier part of this work was performed.
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