Particle number and interactions in the entropic uncertainty relations

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+blnN relation, when the intensity of the one-body harmonic potential is set at the number of particles.


Introduction
The uncertainty relation is one of the pillars of quantum mechanics and demonstrates the non commutativity of the position and momentum operators.It is usually quantified and presented in textbooks, as a product of standard deviations [1][2][3] There has been a movement in recent decades to formulate the uncertainty principle in terms of the entropic uncertainty relation [4][5][6][7][8], where the position and momentum space entropies are defined in terms of the respective one-variable densities as [9] ò ò

( ) [ ( )] ( ) [ ( )] ( )
For one-particle systems, the densities are ρ(x) = |ψ(x)| 2 and π(p) = |f(p)| 2 , defined in terms of the wave functions in each representation.However, the one-variable densities can also be reduced ones, obtained by integration of parent densities over one or several variables.For example, a N-particle or variable density, ρ N , may be reduced to a one variable one by integrating over N − 1 variables, x dx dx , , , . 4 The uncertainty relation above can be extended to two dimensions by [8] Here, the two-variable entropies are defined in terms of two-variable densities as [9] ò ò These one-and two-variable reduced entropy sums will be the principal focus of this work.
In fact, the uncertainty relations above can be extended to D-dimensional ones by [8] p + +  S S D 1 ln .7 The floor of the relationship is the ground state harmonic oscillator or Gaussian type function.Any movement away from this floor or bound can be interpreted as a weakening of the particular uncertainty statement [10].
The uncertainty relation is usually illustrated in text books with the use of one-particle systems.One question which we wish to address is how this changes, if one examines a N-particle system, by way of reduced or marginalized one-and two-variable densities.One would expect an increased uncertainty, not from the quantum mechanical formalism, but rather due to the reduction(s) of the N-particle density.In fact, many body approaches such as density functional theory, rely on the use of reduced densities.
Secondly, there is the presence of interaction potentials when one examines two or more particle systems.These interactions can be attractive or repulsive.What are the effects of these interactions on the entropic uncertainty relations, and how does this depend on the degree of reduction?These are the principal questions which we wish to address.
Thirdly, the reduced one-particle entropy sum, similar to that in equation (2), has been conjectured to exhibit a + a b N ln dependence for various types of systems [11, 38].This has been demonstrated by fitting procedures to numerical calculations.We will examine the pertinence of this relationship for the one-and twovariable entropy sums.
Another goal of the present work will be to obtain expressions for the entropy sum in ground-state coupled oscillator systems, to comment on the observed dependence.In fact, related expressions have been previously reported [39][40][41][42].A further aim here is to provide a detailed exposition of how these formulas can be obtained.The dependence of the entropy sums on the intensity of the one-body potential will also be given.Our formulation allows an appreciation of the dependence of the reduced entropy sums on the normalization constants of the reduced densities, and consequently their dependence on the one-and two-body potentials.Such a formulation could be applied to the study of excited states in future work.Note that for these oscillator systems, all integrals are defined over (− ∞ , ∞ ).

Coupled oscillators
The Hamiltonian of N-coupled oscillators in canonical coordinates in position space, considering atomic units where ω is the natural frequency of the oscillator and λ is the interaction potential, with the positive sign for the attractive case and the negative sign for the repulsive case.The value of λ is bounded for the repulsive case by l < w N in order to obtain a bound state.The Schrödinger equation can be solved exactly in the Jacobi coordinates [36,43,44].These are, for the center of mass coordinates, while the corresponding one for The N-particle function is 1 are the N-particle system quantum numbers while the ψ n (r) for the center of mass (R) and each relative coordinate (r i ) are, , .
Here, α 1 = ω 2 and α 2 = L = α N = ω 2 ± Nλ 2 , with H m (x), the Hermite polynomial of order-m.In this study, we confine ourselves to the study of ground states where all m are zero-valued.The corresponding density for N oscillators is where C N is a normalization constant.

Densities in matrix representation with canonical variables
In the following, we define the coupling potential ( a ) for a N-oscillator system as, 2 .We begin with the density of two oscillators, where we write the exponent of the exponential in quadratic form, This can be expressed in matrix representation as ( ) The density of three oscillators is written as which can be expressed in matrix form as , 20 x M x 1 2 3 2 3 where Thus, the general formula for the density function in position space is where Note that the matrix M N is symmetric and positive definite, and has diagonal elements which are equal and Ndependent.
In analogy, the expression for the N-oscillator momentum density in matrix representation is where The reader is referred to appendix A for the calculation of the C N and C N ˜normalization constants.

N-particle Shannon entropies
The position space entropy for N-oscillators is See appendix B for details of how these formulae are obtained.The corresponding entropic sum is This relation illustrates the dependence of the entropic sum on the product of the normalization constants in position and in momentum space.The bound is attained since the logarithmic argument is unity.

Reduced densities
We begin with the density of N-oscillators, where we now introduce a superscript to the normalization constant, C N k , where (N − k) conveys how many reductions or integrations have been performed on the parent density (k = N), while k establishes the dimension or variables of the reduced density.The first reduction or integration on the parent density in position space (x) is See details in appendix C for the calculation of the primed matrices, ¢ - M N 1 .The accompanying normalization constants can be calculated by inserting the primed matrices in appendix A. In position space, this results in and in momentum space,

Reduced Shannon entropies
The k-dimensional oscillator entropies that originate from the N-oscillator system, whose densities have been reduced or integrated (N − k) times, are ˜are the corresponding normalization constants and k = 1, 2, 3, L N.

Reduced Shannon entropy sums
The reduced entropic sum is defined as The reduced entropy sum will sit on the respective bound whenever the product in the logarithmic argument is unity.This shows the dependence of the entropy sums on the normalization constants of the reduced densities.The general expression S T k in the (N − k) − th reduction for N-oscillators is The first term is the entropic uncertainty relation bound, while the second term contains the information about the number of particles, the dimension of the densities, the number of reductions realized, and the magnitudes of the one-and two-body potentials.The denominator of the argument of this term is the product of the one and two-body potentials, where each factor has been weighted by the particle number.The numerator is the symmetric product of the sum of the potentials, where each term has been weighted by either the number of reductions (N − k) or by the dimension (k).One interpretation is that this represents a logarithmic difference between numerator and denominator, and thus contains the information on how the reductions or integrations modify the dependence of the entropy sum on the potentials.The symmetry in the relation shows that interchanging the values of the one-and two-body potentials will yield the same value for the entropy sum.With no coupling, λ = 0, Λ N = ω, and the numerator and denominator of the logarithmic term are equal to N 2 ω 2 .The entropy sum is then the bound k • S 0 for the respective uncertainty relation, where we define p = + S 1 ln 0 .Note also that with no reductions (k = N), the numerator and denominator of the logarithmic term are both equal to N 2 ωΛ N .The entropy sum of the N-particle system is now NS 0 , is independent of the potential strength, and also lies on the corresponding bound.It is only with reductions that the entropy sum moves away from the bound, when Λ N ≠ ω.This illustrates how the entropy sums of reduced densities are dependent on the potentials in the system.Equation (35) has a rather complex dependence on N, ω and λ, outside of the particular cases mentioned above.In the next sections, we first present an analysis varying λ, then subsequently varying N, and furthermore comment on the behaviour as a function of ω.This is done to facilitate the ease of analysis since equation (35) has dependencies on N, λ, ω and k.

Reduced Shannon entropy sums and coupling strength
It is reasonable to expect that with k, ω and λ fixed, the S T k values for larger N should be larger than those for smaller N, since the number of reductions, (N − k), is larger for greater N. The rationale here is that reducing (integrating the densities) results in an overall increased uncertainty.Figure 1 presents the reduced one-and two-variable entropy sums as a function of λ, for an attractive potential.Each curve represents the behaviour for a particular particle number.Note that the origin corresponds to the value of the bound or non-interacting system.The plots on the right hand side present the behaviour at smaller values of λ.All curves illustrate that the entropy sum increases with λ.The plots show that the relative ordering of the N-valued curves, in relation to the respective bound, is different at larger values of λ than at smaller ones.At smaller λ (right hand plot), the larger N curves are further away from the bound as compared to the smaller N curves.For greater λ (left hand plot), the larger N curves are now closer to the bound, and the relative ordering is inverted as compared to smaller λ.Thus, there must exist crossovers where the curves intersect.These are evident from the left hand side plot.It is possible to alter the ordering of the different N-valued curves with the intensity of the attractive potential.There is a further detail which deserves mention.In the case of the two-variable entropy sum, the inversion in the behaviour observed at small and larger λ is not complete, since the N = 3 (red) curve lies below the N = 15 one (blue), at all values of larger λ.
The respective behaviours are now presented for the repulsive potential in figure 2. In this case, there is no inversion of the ordering at small or larger λ, as observed for the attractive potential.The ordering is consistent with that observed for the attractive potential at smaller λ, where the larger N curves lie above those at smaller N.The curves are presented up to the value of λ where bound state solutions exist for N = 50.However, the relative ordering of the curves for other N is preserved at values of λ that are larger than those shown.
Analogous curves to figures 1 and 2, but now varying ω with fixed λ are not presented for brevity.These curves decay with larger ω, and move closer to the bound as the interaction is squeezed out of the system.This is independent of the value of N. With an attractive potential, different N-valued curves also intersect and change their ordering, similar to those in figure 1.These similarities with the λ-dependent curves can be expected, since it is the relation between the ω and λ values which determines the strength of the interaction between particles, as . At relatively larger ω, Λ N ≈ ω, and the system behaves as a set of uncoupled oscillators.We now address the question of how the use of reduced entropic sums contribute to uncertainties in the entropic uncertainty relations.Figure 3 presents the information of how the different orders of reduction influences the entropy sums, with regard to the particular bound, as a function of λ.One can see that more reductions (smaller k) induce a movement away from the respective bound, as more integrations on the parent density are performed.This observation is valid for both attractive and repulsive potentials, at all values of λ ≠ 0.Moreover, all reduced entropic sums display an increasing tendency with λ.That is, reduction or marginalization, and increasing interaction, weakens the strength of the particular uncertainty statement, as movement is away from the bound.On the other hand, with fixed N, ω and λ, the reduced entropy sum in equation (35) increases with larger k, toward the NS 0 bound.Note that there are pairs of (k, N − k) combinations which yield the same value in the logarithmic argument of the entropy sum in equation (35).Take for example N = 10.The first reduction, N − k = 9, provides the same value of the logarithmic argument as the one-variable entropy sum, N − k = 1.

Reduced Shannon entropy sums and particle number
Figure 4 shows the entropy sum as a function of N, for two different values of λ, in the presence of attractive and repulsive potentials.The non-interacting (λ = 0) case is a constant with N, whose value lies on the bound, and corresponds to the x-axis.The entropy sum for the attractive potential (top row) exhibits a maximum, whose position moves to smaller N, as the strength of the potential is increased.The entropy sum increases at smaller N, away from the bound, up to the maximum.After this, it begins to decrease towards the bound.This is in contrast to the entropy sum for the repulsive potential, where no maximum is observed.The observation in this case is that the appearance of the curve is more convex-like as the strength of the repulsive potential is increased.
These same types of observations are also valid for the two-variable reduced entropy sums presented in figure 5.The position of the maximum with the attractive potential occurs at larger N, as compared to the one-  variable sums presented in figure 4.Moreover, the presence of the maxima is not dependent on a particular value of ω.Maxima were observed for values of ω other than unity.It is the depth and location of the maxima that change with different values of ω.
These results can be interpreted from the perspective of the normalization constants.In the presence of an attractive two-body potential, the reduced entropy sums approach the respective bounds as N gets very large.Λ N behaves as » N for finite values of ω and λ, thus the logarithmic argument in equation (34) approaches unity in this limit.On the other hand, the logarithmic argument must deviate from unity in the presence of a repulsive potential, since the entropy sum moves away from the bound at larger values of N. In this case, for a fixed λ and N, w l > N , in order for Λ N to be real-valued.Thus ω must increase upon consideration of larger N, which is different from when an attractive interparticle potential is present.

Reduced Shannon entropy sums and + a b N ln behaviour
There has been considerable interest in the behaviour of the reduced one-particle entropic sum with N, over the past decades [11,21,38].It has been conjectured that the functional form resembles that of + a b N ln , where a and b are system dependent parameters.This conjecture has been supported with numerical calculations for a variety of different systems.It has also been reported that the value of b tends to be negative.Note that the -N ln term can be obtained from equation (35) by expanding the denominator of the logarithmic argument.
Figure 6 presents the curves corresponding to the fitted values of + a b N ln , for the repulsive potential.These can be compared and contrasted to the actual curves for S T 1 and S T 2 in these coupled oscillator systems.The curves for the attractive potential are not presented, since the functional form is unable to recover the maximum that was previously discussed.One can observe that the + a b N ln functional form is not a close representation of the actual values for S T 1 and S T 2 in these systems.This can be seen from the R 2 values reported in the figure caption.The agreement is only consistent in that both the fitted curves and the actual values, increase with N. Furthermore, the value of b is positive for S T 1 , in contrast to negative values that have been reported for other systems.It is also positive for S T 2 .On the other hand, note that the values of a are virtually identical to kS 0 , which one would expect from the constant term in equation (35).Deviations of the + a b N ln functional form from the actual data is due to a complicated dependence on N, ω and Λ N , as previously discussed.

Model potential
The value of ω is fixed at unity, for all systems in the analysis above.This is a marked difference from the studies of other systems.In the study of neutral atoms [11], for example, the one-body potential is not a constant, but depends on the nuclear charge (Z= N).One can model this behaviour by setting ω = N and in these oscillator systems.Plots of the entropy sums versus N with this potential are presented in figure 7. One can appreciate that the entropic sums approach the asymptotic values of S 0 and 2S 0 respectively, for larger N.This result is also reinforced from examination of equation (35) and taking the limit of large N.It is striking that the functional form now resembles the + a b N ln one, where b is negative valued.The calculated R 2 values are provided in table 1 below.Furthermore, one can observe in both cases how the value of the λ interaction influences the shape of the curve.These differences are more pronounced at smaller N.These results also illustrate the importance of the interparticle potential in obtaining the + a b N ln dependence.With no interaction, the entropy sums are constant.However, turning on the interparticle potential yields the + a b N ln form, with both attractive and repulsive potentials.This type of behavior was also observed for other values of λ.
Table 1 presents the values obtained from the fits of the entropy sums to the + a b N ln functional form, for both the attractive and repulsive interactions.All values of a are very close to the respective S 0 and 2S 0 values.The values of b are now negative, in contrast to when ω is a fixed value.This is in agreement with previously reported   calculations on different types of systems [11,22].The fits to the functional form are not excellent, since they do not take into account the presence of the term in equation (35), which depends on the one and two-body potentials.Considering such a term, or an approximation to it, could provide a means of obtaining information about the potentials in systems where analytical solutions are not available.

Conclusions
It is shown how entropy sums in the ground states of coupled oscillator systems depend on the normalization constants of the respective position and momentum space densities.This yields expressions for the reduced entropy sums which are dependent on the particle number, oscillator frequency and the interaction strength.
The behaviour of the one-and two-variable entropy sums, the basis of entropic uncertainty relations, is analysed as the particle number is varied, and as the strength of the attractive or repulsive interparticle potential is increased.Results show the existence of a maximum in the entropy sums as N is varied, when an attractive interparticle potential is present.The maximum moves towards smaller N as the interaction strength increases.
On the other hand, the entropy sums steadily increase with N with a repulsive interparticle potential, with the curve displaying more convex-like behaviour as the intensity of the potential is increased.No maximum is observed in this case.The behaviour of the entropy sums as a function of interaction strength, in the presence of an attractive potential, illustrates how the potential imposes an order in different N-particle systems.The entropy sums increase with greater interaction, moving away from the uncertainty bounds.At smaller intensities, the smaller N systems lie closer to the uncertainty bound, while at larger potential strength, it is the larger N systems which lie closer to the bound.With the repulsive potential, the ordering does not change with the potential strength, and the smaller N systems are closer to the bound.Results for both one-and two-variable entropy sums are shown to be consistent.The effects of marginalization or reduction of the densities used to compute the entropic sums results in a movement away from the corresponding bounds.The interpretation here is that reduction, and an increased interaction, results in a weakening of the uncertainty principle statement.We also show that the behaviour of the entropy sums with fixed ω, deviates from the + a b N ln functional form, previously conjectured for other types of systems.In contrast, this behaviour is exhibited when the magnitude of the one-body potential is not fixed, but rather depends on the number of particles.These results offer insights into how the entropy sums of the reduced densities depend on the particle number, and on the one-and two-body potentials in the system.The reduced entropy sums presented here can also be used as the basis to calculate statistical correlation measures (mutual information) at the pairwise or higher-order levels.It would be interesting to probe the relation of these measures with others of multipartite entanglement in mixed states.

Appendix B
We illustrate here the evaluation of the entropy of two oscillators with ω = 1.The first factor in the equation above is the reduced normalization constant, C 3 2 .Reductions in momentum space are carried out in an analogous manner using the M N ˜matrices and by integration over the p i variables.

Figure 6 . 4 ,
Figure 6.Left: Plot of the entropic sum S T 1 (red) and fitted curve (black) versus N. Right: Plot of the entropic sum S T 2 (red) and fitted curve (black) versus N. The strength of the repulsive potential is λ = 0.031.The value of ω is set at unity.The fitted equations are: + ´-N 2.143 1.74 10 ln 4

Table 1 . 2 2
Values of the parameters when fitting the entropy sums to functional form + a b N ln , where ω = N and l and m N 1000, for the attractive (λ + ) and repulsive (λ − ) potentials in figure 7. m = 2 for S T 1 and m = 3 for S T 2