Relative Entropies and Jensen Divergences in the Classical Limit

Metrics and distances in probability spaces have shown to be useful tools for physical purposes. Here we use this idea, with emphasis on Jensen Divergences and relative entropies, to investigate features of the road towards the classical limit. A well-known semiclassical model is used and recourse is made to numerical techniques, via the well-known Bandt and Pompe methodology, to extract probability distributions from the pertinent time-series associated with dynamical data.


Introduction
Many problems of quantum and statistical mechanics can be formulated in terms of a distance between probability distributions.A frequently used quantity to compare two probability distributions, which arose in information theory, is the Kullback-Leibler (KL) Divergence [1].Given two probability distributions  and , it provides an estimation of how much information  contains relative to  and measures the expected number of extra bits required to code samples from  when using a code based on , rather than using a code based on  [2].Some of them can also be regarded as entropic distances.
It is well-known that systems that are characterized by either long-range interactions, long-term memories, or multifractality are best described by a formalism [3] called Tsallis' -statistics.The basic quantity is the entropy [ ∈ R ( ̸ = 1)] with   being probabilities associated with the  different system-configurations.The entropic index (or deformation parameter)  describes the deviations of Tsallis entropy from the standard Boltzmann-Gibbs-Shannon (BGS) one.One has The BGS entropy works best for systems composed of either independent subsystems or interacting via short-range forces, whose subsystems can access all the available phase space [4][5][6][7].For systems exhibiting long-range correlations, memory, or fractal properties, Tsallis' entropy becomes the most convenient quantifier [4][5][6][7][8][9][10][11][12][13][14][15][16][17].Tsallis relative entropies, studied in [18], are NOT distances in probability space.An alternative tool is the Jensen Shannon Divergence, introduced by Lamberti et al. [2].How good are these quantifiers to statistically describe complex scenarios?To answer we will apply the abovementioned quantifiers to a well-known semiclassical system in its path towards the classical limit [19,20].The pertinent dynamics displays regular zones, chaotic ones, and other regions that, although not chaotic, possess complex features.The system has been investigated in detail from a purely dynamic viewpoint [20] and also from a statistical one [21][22][23].For this a prerequisite emerges: how to extract information from a time-series (TS) [24]?The data at our disposal always possess a stochastic component due to noise [25,26], so that different extraction-procedures attain distinct degrees of quality.We will employ Bandt and Pompe's approach [27] that determines the probability distribution associated to time-series on the basis of the nature of the underlying attractor (see [27] for the mathematical details).
In this paper we employ the normalized versions of Tsallis relative entropy [10,18,28], to which we add the Jensen Divergences associated to it.We will compare, via Jensen, (i) the probability distribution functions (PDFs) associated to the system's dynamic equation's solutions in their route towards the classical limit [20] with (ii) the PDF associated to the classical solutions.Our present Jensen way will be also compared to the -Kullback-Leibler analyzed in [18].
The relative entropies and Jensen Divergences mentioned above are discussed in Section 2, which briefly recapitulates notions concerning the Tsallis relative entropy and the Kullback-Leibler relative one.The Jensen Shannon and generalized Jensen Divergences are also discussed in this Section.A simple, but illustrative example is considered in Section 2.1.In Section 2.2, we consider normalized forms corresponding to the information quantifiers mentioned in Section 2.1.As a test-scenario, the semiclassical system and its classical limit are described in Section 3, and the concomitant results presented in Section 4. Finally, some conclusions are drawn in Section 5.

Kullback and Tsallis Relative
Entropies, Jensen Shannon, and Generalized Jensen Divergences The relative entropies (RE) quantify the difference between two probability distributions  and  [29].
The two entropies (3) and ( 4) provide an estimation of how much information  contains relative to  [29].They also can be regarded as entropic distances, alternative means for comparing the distribution  to .Our two entropies are not symmetric in  − .
So as to deal with the nonsymmetric nature of the KL Divergence, Lamberti et al. [2] proposed using the following quantity: as a measure of the distance between the probability distributions  and .(, ) = 0 if and only if  = .This quantity verifies (, ) = (, ).Moreover, its square root satisfies the triangle inequality [2].In terms of the Shannon entropy, expression (5) can be rewritten in the form We can obtain a generalization of Jensen's Divergence by using the relative entropy (4) instead of the KL Divergence; that is,
In real-life statistical problems  ≫ 2, on the basis of our simple but illustrative example, we expect quite different behaviors for   (, ) and   (, ) regarding the -dependence for large .-variations are much smoother for   (, ) than for   (, ).
In this work we consider (see Sections 3 and 4) a system representing the zeroth mode contribution of a strong external field to the production of charged meson pairs, more specifically, the roald lading to the classical limit.The ensuing dynamics is much richer and more complex.However, we will see that the pertinent difference in the -behaviors of our quantifier is the one of our simple example.The two quantities, (a) maximum  of Figure 1 and (b) the -value in Figure 3, were both chosen so as to coincide with our semiclassical system's number of states for this case, which facilitates comparison.

Normalized Quantities.
It is convenient to work with a normalized version for our information quantifiers, for the sake of a better comparison between different results [18].In this way the quantifier's values are restricted to the [0, 1] interval, via division by its maximum allowable value.Accordingly, the example of the preceding section can be useful as a reference.If we divide  KL (, ) by ln , expression (3) becomes  2, but for  = 720.The -value was chosen to coincide with the semiclassical system's number of states.We observe different behaviors for   (, ) and   (, ) regarding the -dependence for large .-variations are much smoother for   (, ) than for   (, ).

Classical-Quantum Transition: A Review
This is a really important issue.The classical limit of quantum mechanics (CLQM) can be viewed as one of the frontiers of physics [31][32][33][34][35], being the origin of exciting discussion (see, e.g., [31,32] and references therein).An interesting subissue is that of "quantum" chaotic motion.Recent advances made by distinct researchers are available in [36] and references therein.Another related interesting subissue is that of the generalized uncertainty principle (GUP) ( [37,38]).Zurek [33][34][35] and others investigated the emergence of the classical world from quantum mechanics.
We are interested here in a semiclassical perspective, for which several directions can be found: the historical WKB, Born-Oppenheimer approaches, and so forth.The two interacting systems, considered by Bonilla and Guinea [39], Cooper et al. [19], and Kowalski et al. [20,40], constitute composite models in which one system is classical and the other is quantal.This makes sense when the quantum effects of one of the two systems are negligible in comparison to those of the other one [20].Examples encompass Bloch equations, twolevel systems interacting with an electromagnetic field within a cavity, and collective nuclear motion.We are concerned below with a bipartite system representing the zeroth mode contribution of a strong external field to the production of charged meson pairs [19,20], whose Hamiltonian reads x and p above are quantum operators, while  and   are classical canonical conjugate variables.The term  2 =   2 +  2  2 is an interaction one introducing nonlinearity, with   a frequency.  and  cl are masses, corresponding to the quantum and classical systems, respectively.As shown in [40], when faced with (14), one has to deal with the autonomous system of nonlinear coupled equations: derived from Ehrenfest's relations for quantum variables and from canonical Hamilton's equations for classical ones [40].
To investigate the classical limit one needs also to consider the classical counterpart of the Hamiltonian (14), in which all variables are classical.In such case, Hamilton's equations lead to a classical version of (15).One can consult [40] for further details.The classical equations are identical in form to (15), after replacing quantum mean values by classical variables.The classical limit is obtained considering the limit of a "relative energy" [20]: where  is the total energy of the system and  is an invariant of the motion described by the system (15). relates to the uncertainty principle: To tackle this system one appeals to numerical solution.The pertinent analysis is affected by plotting quantities of interest against   that ranges in [1, ∞].For   = 1 the quantum system acquires all the energy  =  1/2   and the quantal and classical variables get located at the fixed point (⟨ x2 ⟩ =  1/2 /    , ⟨ p2 ⟩ =  1/2     , ⟨ L⟩ = 0,  = 0,   = 0) [40].Since  = 0, the two systems become uncoupled.For   ∼ 1 the system is of an almost quantal nature, with a quasiperiodic dynamics [20].
As   augments, quantum features are rapidly lost and a semiclassical region is entered.From a given value   cl , the morphology of the solutions to (15) begins to resemble that of classical curves [20].One indeed achieves convergence of (15)'s solutions to the classical ones.For very large   -values, the system thus becomes classical.Both types of solutions coincide.We regard as semiclassical the region 1 <   <   cl .Within such an interval we highlight the important value   =   P , at which chaos emerges [40].We associate to our physical problem a time-series given by the   -evolution of appropriate expectation values of the dynamical variables and use entropic relations so as to compare the PDF associated to the purely classical dynamic solution with the semiclassical ones, as these evolve towards the classical limit.
Remember that our "signal" represents the system's state at a given   .Sampling this signal we can extract several PDFs using the Bandt and Pompe technique [27], for which it is convenient to adopt the largest -value that verifies the condition ! ≪  [27].Such value is  = 6, because we will be concerned with vectors whose components comprise at least  = 5000 data-points for each orbit.For verification purposes, we also employed  = 5, without detecting appreciable changes.
In evaluating our relative entropies, our -distributions are extracted from the time-series for the different   's, while  is associated to the classical PDF.
Figure 4 represents the Kullback Divergence   KL (, ) and the generalized relative entropy    (, ) (i.e., the pseudodistances (psd) between PDFs) for  = 3.5 that were computed in [18].All curves show that (i) the maximal psd between the pertinent PDFs is encountered for   = 1, corresponding to the purely quantal situation, and (ii) that they grow smaller as   augments, tending to vanish for   → ∞, as they should.One sees that the results depend upon .We specially mark in our plots special   -values whose great dynamical significance was pointed above.These are   P (chaos begins),   cl (start of the transitional zone), and    , known to be endowed with maximum statistical complexity [41].As a general trend, when  grows, the size of the transition region diminishes and that of the classical zone grows.For the Kullback Divergence, the size of the transition region looks overestimated if one considers the location of   cl (Figure 4(a)).As found in [18], we can assert that our description is optimal for 1.5 <  < 2.5.The description instead lose quality for  ≥ 2.5 (Figure 4(b)).
In Figures 5 and 6, we plot the Jensen Shannon Divergence (, ) and the generalized Jensen Divergences   (, ), for different values of .All curves show that the maximal distance between the pertinent PDFs is encountered for   = 1 and that the distance grows smaller as   augments, save for oscillations in the transition zone.We note that the three regions are well described (which is not the case for   KL (, )) for any reasonable -value.Our probabilitydistance tends to vanish for   → ∞, but in slower fashion than for   KL (, ) and    (, ).The -dependence is quite different in the case of symmetric versus nonsymmetric relative entropies, as can be also seen in the simple example of Section 2.1, notwithstanding the fact the this difference is significantly reduced for small .As  grows,    (, ) changes in a much smoother fashion than    (, ), which casts doubts concerning the appropriateness of employing    (, ).As noted in [18], (i) the maximal distance (pseudodistance) between the pertinent PDFs is encountered for   = 1, corresponding to the purely quantal situation and (ii) that distance (pseudodistance) grows smaller as   augments, tending to vanish for   → ∞.For the Kullback Divergence description, the size of the transition region looks overestimated if one considers the location of   cl .In (b), the transition region almost disappears.We observe that the behavior is similar for both curves.As in previous figures, here the maximal distance is encountered for   = 1, and distance grows smaller as   augments.The description of the three dynamical regions is optimal.

Advances in Statistics
The maximum statistical complexity value [41] is attained at    .In Figures 4, 5, and 6 we detect shape-changes, before and after    .Figures 5 and 6 also display there a local minimum for the Jensen Divergence.

Conclusions
The physics involved is that of a special bipartite semiclassical system that represents the zeroth mode contribution of a strong external field to the production of charged meson pairs [19,20].The system is endowed with three-dynamical regions, as characterized by the values adopted by the parameter   .We have a purely quantal zone (  ≃ 1), a semiclassical one, and finally a classical sector.The two latter ones are separated by the value   cl (see Section 3).In evaluating (i) the normalized relative entropies (( 12), (13a) and (13b)) [18] and (ii) normalized versions of the Jensen Divergences given by ( 6) and ( 7), the -PDFs are extracted from time-series associated to different   -values, while  is always the PDF that describes the classical scenario.We are thus speaking of "distances" between (a) the PDFs extracted in the path towards the classical limit and (b) the classical PDF.
For the normalized Jensen Divergences, all curves show that the maximal distance between the pertinent PDFs is encountered for   = 1 and that the distance grows smaller as   augments, save for oscillation in the transition zone.
Our three regions are well described for any reasonable  (which is not the case with    (, )).The vanishing of the distance referred to above for   → ∞ acquires now a much slower pace than that of    (, ) (and   KL (, )).These results are to be expected, given the behavior of the concomitant statistical quantifiers in the example of Section 2.1.

Figure 1 :
Figure1: (a)   (, ) and (b)   (, ) versus , for the certainty versus the equiprobability cases (see Section 2.1) for  = 5.For large , when  → ∞, we ascertain that   (, ) → ∞.Instead, the Jensen Divergences attain an asymptotic value.The maximum  was chosen to coincide with the number of states  of the semiclassical system described in Figures4, 5, and 6.This fact facilitates comparison.

Figure 3 :
Figure 3: (a)   (, ) and (b)   (, ) versus  as in Figure2, but for  = 720.The -value was chosen to coincide with the semiclassical system's number of states.We observe different behaviors for   (, ) and   (, ) regarding the -dependence for large .-variations are much smoother for   (, ) than for   (, ).

Figure 4 :
Figure 4: (a) The normalized Kullback Divergence   KL (, ) is plotted versus   .(b) Normalized generalized relative entropy    (, ) versus   for  = 3.5.As noted in[18], (i) the maximal distance (pseudodistance) between the pertinent PDFs is encountered for   = 1, corresponding to the purely quantal situation and (ii) that distance (pseudodistance) grows smaller as   augments, tending to vanish for   → ∞.For the Kullback Divergence description, the size of the transition region looks overestimated if one considers the location of   cl .In (b), the transition region almost disappears.

Figure 5 :
Figure 5: We plot (a) the Jensen Shannon Divergence versus   and (b) the generalized Jensen Divergence versus   for  = 1.5.We observe that the behavior is similar for both curves.As in previous figures, here the maximal distance is encountered for   = 1, and distance grows smaller as   augments.The description of the three dynamical regions is optimal.

Figure 6 :
Figure 6: We plot (a) the Jensen Divergence and (b) the generalized Jensen Divergence, both versus   , for (i)  = 3 and (ii)  = 5.We observe that the behavior of both curves resembles that of Figure 5.The description of the three dynamical regions is optimal.