Eur. Phys. J. B (2013) 86: 363DOI: 10.1140/epjb/e2013-40634-9

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

de Bruijn-type identity for systems with flux

Takuya Yamanoa

Department of Mathematics and Physics, Faculty of Science, Kanagawa University, 2946, 6-233 Tsuchiya, Hiratsuka,259-1293 Kanagawa, Japan

Received 30 June 2013 / Received in final form 17 July 2013Published online 28 August 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. We show that an information-theoretic relation called the de Bruijn-type identity can be refor-mulated in a physical context with probability currents. The time derivatives of relative entropies underthe continuity equation are presented, which shows that the conservation of distance between a pair of dis-tributions is generally not guaranteed. As an important implication of these results, we discuss and presenta possible conceptual framework for the classical no-cloning (deleting) theorem and qualitatively assertthat we can attribute the perfect performance of the operating machine to the openness (non-vanishingflow at boundaries between the processing machine and the system) during the process.

1 Introduction

Since the advent of information theory, it has played aprofound role in statistical physics. The investigation ofproven relations in information theory could facilitate fur-ther understanding of the laws of physics. In this study,we focus on the one of the salient identities that is widelycited in connection with entropy of random variables.

When an arbitrary random variable X with finitevariance is perturbed by another independent standardGaussian random variable Z with zero mean and unitvariance but scaled by

√t, the derivative of the associated

entropy H(X +√

tZ) with respect to t has an informationrelation:

d

dtH(X +

√tZ) =

12J(X +

√tZ), (1)

where J(X) denotes the Fisher information associatedwith the probability distribution f(x) for X , which is de-fined as J(X) =

∫f |∇ ln f |2dx. The entropy is of the

Shannon’s form H(X) = − ∫f ln fdx. This relation is

called the de Bruijn identity [1–3]. The Fisher informa-tion J(X) is obtained in the limiting case t → 0+ of theleft-hand side of equation (1). The factor 1/2 is includedby convention, and we will see later that it is absorbed intothe diffusion coefficient so that it is not essential in oursubsequent consideration. In the limiting case t → 0+,this identity is generalized to the non-Gaussian variablefor Z in reference [4].

Recently, Guo [5] extended this identity to the relativeversion, which is called the de Bruijn-type identity:

limt→0+

d

dtKL(Ttf‖Ttg) = −1

2I(f‖g), (2)

a e-mail: yamano@amy.hi-ho.ne.jp

where Tt denotes the perturbation operator acting on thedistribution. As a special case, Tt represents the heatsemigroup. The quantities KL(f‖g) and I(f‖g) are theKullback-Leibler distance [6,7] (or relative entropy) andthe relative Fisher information [8,9] between two distri-butions f and g, respectively. When Tt denotes a heatsemigroup, the distribution maintains the Gaussian formover time (Gaussian perturbation). We will not necessar-ily impose this restriction of the Gaussian perturbationfor Tt in the present paper.

In another recent work, in the context of the minimummean-square error in estimation theory, Verdu showedthat two equal-time Gaussian distributions satisfy thefollowing identity [10]:

d

dtKL(Ttf‖Ttg) = −1

2I(Ttf‖Ttg). (3)

Also, Hirata et al. [11] provided an elementary proof forthis identity by way of integration by parts. As an imme-diate consequence, an integral representation of the KLdistance in terms of the relative Fisher information fol-lows KL(f‖g) = 1/2

∫ ∞0 I(Ttf‖Ttg)dt by using the in-

distinguishability property, i.e., KL(Ttf‖Ttg) → 0 ast → ∞ [10,11]. These above-described intimate relationsbetween relative entropy and the relative Fisher informa-tion have not been fully investigated in a physics con-text. Since the original de Bruijn identity involves randomvariables, the dynamics behind the probability distribu-tion is by necessity the heat equation. However, thereare many probability density functions that do not gen-erally follow the heat equation. To deal with such cases(i.e., extensions of a de Bruijn-type identity) is our mo-tivation. We should also note that the time derivative ofa wider class of divergence measures (relative entropies)exactly vanishes for probability distributions that follow

Page 2 of 6 Eur. Phys. J. B (2013) 86: 363

the Liouville equation. This fact has been documented inreference [12].

In this paper, therefore, we consider the time changeof relative entropies between two probability distributions,both of which follow the same evolution dynamics. In thede Bruijn-type identity, the parameter t denotes the vari-ance of the distribution of

√tZ. In a diffusion process

described by the heat equation, the variance of the prob-ability distribution is proportional to time. Therefore, thederivative with respect to the variance refers to the timederivative. Hence, our first main aim is to see how thetime derivative of the distance measure between two per-turbed distributions can be expressed in terms of thosedistributions. To be precise, it is shown that when a sys-tem has a probability current, we have a correspondingexpression of the de Bruijn-type identity. We use the termde Bruijn-type identity in the sense of equation (3), i.e.,the time derivative without taking limit t → 0+. We notethat readers should distinguish clearly between the origi-nal de Bruijn and de Bruijn-type identities, since our focusis to present an extended form of the latter for nonequi-librium dynamics represented by the continuity equation.The continuity equation incorporates the probability cur-rent of the system so that our formulation can provide theexpressions in a general setting, i.e., the de Bruijn-typeidentity for systems with flux. This formulation can alsoprovide a physical interpretation to the de Bruijn-typeidentity that holds for a system, other than a heat equa-tion. These investigations strongly encourage us to furtherexplore the links between information-theoretic quantitiesand nonequilibrium physical laws.

We also discuss the potential implication of the coun-terintuitive and apparently contradictory conclusion thatin the literature about the existence of a classical counter-part to quantum-impossible processes (i.e., the no-cloningtheorem) [13–16]. Its validity has been based on the as-sumption that both distributions to be compared and theassociated flux vanish at the boundaries of the system.This premise might not be true in the copying processwhen the physical resources are allowed to interact withthe machine.

The organization of the present paper is as follows.We consider the time changes in relative entropies underseveral dynamics and introduce the relative Fisher infor-mation in Section 2. In this section, we also present thede Bruijn-type identities for systems with flux in termsof probability current. In Section 3, we discuss the possi-ble implications of the results to the classical analogue ofquantum-impossible processes. The final section providesthe concluding remarks.

2 Change of relative entropies in time

First, we briefly review the form of the relative Fisherinformation. It appears in the de Bruijn-type identityand can be defined for two probability distributionsP1 and P2 as:

I(P1‖P2) =∫

P1

∣∣∣∇ ln

P1

P2

∣∣∣2

dx. (4)

This form of definition can be seen in references [8,9], butto the best of our knowledge, its physical meaning and im-plication have not been fully explored. The identity equa-tion (3) indicates that the shrinking rate of the distancebetween two distribution functions in terms of the KL dis-tance is equivalent to this information. Recall that if wereplace P within the logarithm in the negative Shannonentropy with P1/P2, we obtain the KL distance from P1 toP2, i.e., KL(P1‖P2). We can obtain the relative Fisher in-formation in a similar manner, i.e., if we replace P withinthe logarithm in the Fisher information

∫ P|∇ lnP|2dxwith P1/P2, we obtain equation (4). In this sense, thedefinition of the form seems natural. The non-negativityof I(P1‖P2) is evident from the definition, and it can reachzero if and only if the two distributions are identical. Thequantity ∇ lnP is referred to as the score function. There-fore, I(P1‖P2) is regarded as the average of the squareddifference of two score functions. In application, this in-formation measure is used to understand the behaviors ofthe atomic density profiles [17,18].

Note that so far and hereafter, we omit the argumentof the coordinate and time variables as P = P(x, t) forsimplicity, and we assume that P1 is absolutely continuouswith respect to P2 for t > 0. Furthermore, the relativeprobability P1/P2 is expected to be bounded on R

n. Todeal with general nonequilibrium processes, we adopt thefollowing continuity equation:

∂P∂t

= −∇ · j. (5)

We now focus on the time derivative of relative distancesunder equation (5) and its particular instances.

– KL distance KL(P1‖P2)The time derivative is given as:

d

dtKL(P1‖P2) =

∫dx(∂tP1) ln

P1

P2

+∫

dxP2

((∂tP1)P2

− P1

P22

(∂tP2))

.

(6)

Substituting ∂tPi = −∇· ji for i = 1, 2 and performingintegration by parts under the assumption of vanishingsurface terms (denoted by the symbol s),

j1 lnP1

P2

∣∣∣s

= 0, j2P1

P2

∣∣∣s

= 0, (7)

we have the expression

d

dtKL(P1‖P2) =

∫P1∇

(

lnP1

P2

) (j1P1

− j2P2

)

dx.

(8)If we choose the probability current as ji = vPi fori = 1, 2 with common velocity v = dx/dt, which corre-sponds to the Liouville dynamics, then the time changeof the distance measure exactly vanishes. This fact isconsistent with the property of conservation of the KL

Eur. Phys. J. B (2013) 86: 363 Page 3 of 6

distance under the Liouville dynamics demonstratedclearly in reference [13].We note that the zero flux ji = 0 also provides theconstancy of the distance. However, we exclude sucha stationary circumstance in our consideration. In aparticular case, systems that follow from the Fick’slaw ji = −D∇Pi (i = 1, 2) with the diffusion con-stant D (i.e., the heat equation) recover the usual deBruijn-type identity:

d

dtKL(P1‖P2) = −D

∫P1

∣∣∣∇

(

lnP1

P2

) ∣∣∣2

dx

= −DI(P1‖P2). (9)

This is also true for systems governed by the linearFokker-Planck equation with a potential φ(x) and thecorresponding probability currents:

ji = −dφ(x)dx

Pi − D∂Pi

∂x, (i = 1, 2). (10)

Since the factor of the difference appearing inequation (8) reads as:

j1P1

− j1P1

= −D∂

∂x

(

lnP1

P2

)

, (11)

the de Bruijn-type identity immediately follows. Notethat the appended factor 1/2 in the original expressionequation (3) is replaced by the diffusion constant D .In this sense, equation (8) can be regarded as areformulation of the de Bruijn-type identity under thepresence of flow.

– Overlap distance Do(P1‖P2)As another class of distances, we consider the overlap,since it generates other Fisher-like relative informa-tion, as will be explained below. This quantity is re-ferred to as the fidelity measure and is also relevant forstudying the information-theoretic aspect of generalprobabilistic theories [19]. It is defined as:

Do(P1‖P2) :=∫ √

P1P2dx. (12)

The time derivative can be written by substitutingthe continuity equation and performing integration byparts under the boundary condition in equation (7),

d

dtDo(P1‖P2) =

12

∫dx

√P2

P1(∂tP1)

+12

∫dx

√P1

P2(∂tP2)

=12

∫j1∇

(√P2

P1

)

dx

+12

∫j2∇

(√P1

P2

)

dx. (13)

Rearranging the second line, we have the expression

d

dtDo(P1‖P2) = − 1

4

∫dx

√P1P2

×∇(

lnP1

P2

) (j1P1

− j2P2

)

. (14)

Similarly, in the case of the Liouvillian dynamics, wehave exactly the vanishing time derivative. However,for a system with the heat equations ji = −D∇Pi

(i = 1, 2), we have

d

dtDo(P1‖P2) = −D

4

∫ √P1P2

∣∣∣∇

(

lnP1

P2

) ∣∣∣2

dx.

(15)

This identity is a counterpart of the de Bruijn-typeidentity. The right-hand side is not the same asI(P1‖P2), but we refer it to a variant of the relativeFisher information here, denoting

Im(P1‖P2) =14

⟨√P2

P1

∣∣∣∣∇ ln

P1

P2

∣∣∣∣

2⟩

P1

. (16)

It is symmetric Im(P1‖P2) = Im(P2‖P1) under thechange of the argument, indicating its pertinence as apotential utility for a distance measure.

– A general distance G(P1‖P2)As a demonstration in the general setting, we con-sider a generic distance measure that can be speci-fied by a convex function (a.k.a. the Csiszar-Morimotodivergence [20,21]),

G(P1‖P2) :=∫

P1χ

(P1

P2

)

dx, (17)

where χ as a function of the relative density can betaken arbitrarily if the property χ(1) = 0 is satisfied.Then, the time derivative becomes

d

dtG(P1‖P2)=

∫dx(∂tP1)χ

(P1

P2

)

+∫

dxP1χ′∂t

(P1

P2

)

=∫

j1χ′∇(P1

P2

)

dx +∫

j1∇(P1

P2χ′

)

dx

−∫

j2∇[

χ′(P1

P2

)2]

dx. (18)

Similarly, we have performed integration by parts withvanishing terms at the boundary, as follows:

j1χ∣∣∣s

= 0, j1P1

P2χ′

∣∣∣s

= 0, j2

(P1

P2

)2

χ′∣∣∣s

= 0. (19)

After the operations of ∇ in the second and third termsin the last expression of the right-hand side and byrearranging terms, we have:

∫dx

(j1P1

− j2P2

)

P1

{

2χ′∇(P1

P2

)

+P1

P2∇χ′

}

.

(20)

Page 4 of 6 Eur. Phys. J. B (2013) 86: 363

Table 1. The de Bruijn-type identities for dynamics.

Liouville eq. Heat eq. Continuity eq.

dDKLdt

0 −DI(P1‖P2)⟨∇

(ln P1

P2

) (j1P1

− j2P2

)⟩

P1

dDOdt

0 −DIm(P1‖P2)⟨√P1P2

4∇

(ln P1

P2

) (j1P1

− j2P2

)⟩

P1

dGdt

0 −D

⟨

∇(ln P1

P2

)∇

[

χ′(

P1P2

)2]⟩

P2

⟨

∇[

χ′(

P1P2

)2] (

j1P1

− j2P2

)⟩

P2

Thus, we have the following final expression for thederivative:

d

dtG(P1‖P2) =

∫dxP2

(j1P1

− j2P2

)

∇[

χ′(P1

P2

)2]

.

(21)

We readily find that under the Liouville dynamicsvi = ji/Pi = v for (i = 1, 2), a wide class of distance mea-sure specified by the form of the function χ is time in-variant, which is consistent with the direct demonstrationpresented in reference [14]. In this sense, equation (21)provides the de Bruijn-type identity for systems withnonequilibrium state, which is widely applicable beyondheat phenomenon. The results obtained above are sum-marized in Table 1. Note that the time derivatives underthe continuity equation have a common factor of the dif-ference in the flux j1/P1 − j2/P2, which is the primarycause for the non-vanishing time change. We note furtherthat the de Bruijn-type identity equation (9) and a variantequation (15) are reminiscent of the H-theorems verifiedby the Fokker-Planck equations and master equations [22].Indeed, since the two relative Fisher information I and Im

are always positive, the distances between two probabil-ity distributions decrease in time, and it gives the rate atwhich the distances shrink.

3 On the relevance to the classical no-gotheorem

The derived de Bruijn-type identities indicate that dis-tance is not generally conserved under flow dynamics. Wediscuss here a significant implication of this fact to theclassical version of the fundamental property of quantummechanics, i.e., the impossibility of copying (or deleting)a source state into (or from) a target system [23,24] with-out perturbing it. We refer to both operations as broad-casting, but for the sake of simplicity, we hereafter onlyconsider copying (cloning). The no-cloning theorem hastraditionally been considered a genuine manifestation ofthe non-classical effect that appears in quantum informa-tion processing. It forbids us to make an identical copy ofthe original quantum state by a unitary operation with-out destroying the original [23,24] (for later advances, seee.g. [25]). In references [13,14], it is pointed out that casesof continuous variables under certain conditions also fol-low the same theorem. The counterpart to this quantumfeature in classical processing, if it exists, is an amaz-ing and counterintuitive idea. Therefore, a compromise

(coarse-grained) method to this assertion was presentedin reference [15], where input distributions with nonzeroresolution (see Appendix) under Liouville dynamics maybe broadcast with fidelity arbitrarily close to unity1. Theseconsiderations are significant for exploring the border be-tween classical and quantum physics, and it would be par-ticularly reasonable to address this issue here in view ofits high relevance to our derived de Bruijn-type identities.This resolution-based argument is, however, subtle, andwe must be circumspect about it in the following sense:

– the machine’s resolution εmachine is fixed before theoperation and is supposed to be smaller than the onedetermined by the input, i.e., ε[p] > εmachine is as-sumed. However, when the converse holds for some in-put distributions, we do not achieve high fidelity, asclaimed in reference [15];

– we normally do not know and cannot know the ini-tial source state a priori without observing it. Accord-ingly, we cannot prepare the resolution of the copy-ing machine corresponding to the inputs prior to theoperation. The resolution must be set before observa-tion (and subsequent copying) and not the other wayaround.

Without getting involved in the above incommensurateaspects of resolution, we will show that it is possible tointerpret the issue. We think that the non-conservationof distance measures demonstrated in the classical copy-ing processes [13,14,16] has the following interpretation.Recall that our argument presented so far has relied heav-ily on the premise of vanishing surface terms, i.e., prob-abilities and flow vectors disappearing at the system’sboundary. This assumption makes the integration by partsan effective step towards obtaining the time change of dis-tance measures and leads to the physical formulation ofthe de Bruijn-type identities.

The gist of the claim of the impossibility perfor-mance even in a classical macroscopic system lies in thefact that the extra inflow and outflow at the bound-ary are negligible. This allows that the relative entropyemployed remains constant before and after the trans-fer process. Therefore, a possible explanation is that thesource and the target systems become temporarily opensystems mediated by the machine in the course of the pro-cess. The two non-conservative initial probabilities conse-quently break the constancy of the distance measure, andthe probability does not evolve according to the Liouville

1 In this sense, the authors of references [13,14] only refer toinfinite resolution broadcasting or perfect accuracy.

Eur. Phys. J. B (2013) 86: 363 Page 5 of 6

equation. This may suggest a view that the machine worksas a probability “sink” or “bath” depending on the pro-cessing dynamics. A similar qualitative explanation canbe found in reference [26], where in the derivation of theupper bound for the entropy production, the authors men-tion that an open (quantum) system may entail a flowacross the boundaries. In general, the machine can havemuch a larger degree of freedom than both the system andthe target. Therefore, it works as a heat bath, and themachine’s source distribution itself will not be perturbedconsiderably during operation.

If we consider the above viewpoint, it makes sense thatthe time invariance of a wide class of relative metrics doesnot contradict the Liouville evolution of probability distri-butions. Once we consider that the classical cloning pro-cess does not follow the Liouville dynamics, there may beno need to introduce an artifice such as the resolutions ofinput distributions.

4 Conclusion

We have derived the time change of the distance measuresunder general dynamics. These identities can be regardedas extensions of the celebrated de Bruijn-type identity, inthat the original identity is obtained in a particular caseof the heat equation. We have emphasized that the invari-ance of relative entropies in time can always be achievedwhen a system follows the Liouvillian evolution. This factnaturally leads to a possible compromise interpretationof the classical impossibility processes that is originallyinherent in the quantumness of systems. The conserva-tion of distance between the input and output states inbroadcasting processes is violated except for the Liouvilledynamics in the sense that the time derivative of the rela-tive entropies does not vanish. The existence of a classicalversion of the no-go theorem can be interpreted as non-Liouvillian dynamics of the probability distribution. Asthe rate of change of the relative entropy can be inter-preted as information loss (or gain) across the boundariesin the form of a nonzero flux per unit time, the quantityis an indicator of the credibility of classical operations.However, the explanation presented is still intuitive, andto make the consideration more qualitative, we will need tointroduce a refinement of the comparative quantity, say,the partial success rate of the broadcast, instead of as-suming perfect performance of the machine. These inves-tigations will provide a deeper understanding of the func-tioning of machines in dynamical processes and provide abasis for the emergence of classicality from quantumness.The classical counterpart of the quantum no-cloning (andno-deleting) is not a phantom analogue.

5 Appendix

A particular type of resolution for probability distribu-tions was introduced to assist the success of the near-ideal copying of the classical distributions [15]. The shape

of the distribution to be copied is required to have well-posed features. The resolution to distinguish two probabil-ity distributions is called the ε-resolution and is defined as:

ε[p] := maxδ

2|δ|∫

dx|p(x) − p(x − δ)| ,

where δ is the displacement in position x. This resolutionhas the following features:

– it is essentially equivalent to the inverse of theLipschitz constant Lp(x) of the distribution functionon a given interval defined as:

Lp(x) := supδ

d(p(x), p(x − δ))δ

,

where the numerator denotes a metric to be measured.The factor 2 in the case of the ε-resolution comes fromthe choice of the metric d(p, q) as the trace norm de-fined as

∫dx|p − q|/2. Since the Lipschitz continu-

ity requires the distribution function to be a stronger(more demanding) condition for smoothness, distribu-tions that are not Lipschitz continuous need alterna-tive fine-grained procedures. Just as the Lipschitz con-stant depends on the function, the ε-resolution is alsocontingent on the source state;

– when the probability distributions are symmetric andmonotonically non-increasing away from the origin, itis given as the inverse of the height of the distribu-tion: ε[p] = 1/p(0). In many areas of science, wherewe cannot expect this well-posedness in distribution,skewed distributions such as the gamma, Weibull, andlog-normal distributions, appear. In these cases, wecalculate the resolution by the definition, since the dis-tributions have zero-height at the origin;

– when applied to the classical copying scenario, mis-matched resolutions i.e., ε[p] < εmachine can occur de-pending on the input distributions, since the initialmachine’s resolution is fixed. These situations couldoccur when we copy different types of input distribu-tions consecutively many times, as we usually do in ourlives. It is not a trivial matter to iterate equations (26)and (27) of reference [15]. As a consequence, it does notproduce as the desired high fidelity. The optimal tun-ing mechanism in the machine side corresponding toeach input distribution must be introduced2.

The author wishes to thank H. Yoshida at OchanomizuUniversity for valuable discussions on the de Bruijn identity.The part of this work was presented at the JPS (Japan Physi-cal Society) annual meeting held at the Hiroshima University,26 March 2013.

2 In this regard, we mention also that making an oper-ationally clear distinction between preparation procedures,transformation, and measurements is important for the re-cent ontic/epistemic discussion of the nature of the quantumstate (e.g., reference [27] and references therein). It has signifi-cance in the study of classical analogues of aspects of quantumphysics.

Page 6 of 6 Eur. Phys. J. B (2013) 86: 363

References

1. A.J. Stam, Inf. Contr. 2, 101 (1959)2. T. Cover, J. Thomas, Elements of Information Theory,

2nd edn. (Wiley-Interscience, New York, 2006)3. A. Dembo, T. Cover, J. Thomas, IEEE Trans. Inf. Theory

37, 1501 (1991)4. K.R. Narayanan, A.R. Srinivasa, arXiv/0711.1460v25. D. Guo, Proc. IEEE Int. Symp. Inf. Theory 3, 814 (2009)6. S. Kullback, R.A. Leibler, Ann. Math. Stat. 22, 79 (1951)7. S. Kullback, Information Theory and Statistics (Wiley,

New York, 1959)8. G. Blower, Random Matrices: High Dimensional

Phenomena, ser. London Mathematical Society LectureNotes (Cambridge University Press, Cambridge, 2009),Chap. 6, p. 211

9. C. Villani, Topics in Optimal Transportation, GraduateStudies in Mathematics (American Mathematical Society,2000), Vol. 58, p. 278

10. S. Verdu, IEEE Trans. Inf. Theor. 56, 3712 (2010)11. M. Hirata, A. Nemoto, H. Yoshida, Entropy 14, 1469

(2012)12. M.C. Mackey, Rev. Mod. Phys. 61, 981 (1989)

13. A. Daffertshofer, A.R. Plastino, A. Plastino, Phys. Rev.Lett. 88, 210601 (2002)

14. A.R. Plastino, A. Daffertshofer, Phys. Rev. Lett. 93,138701 (2004)

15. T.A. Walker, S. Braunstein, Phys. Rev. Lett. 98, 080501(2007)

16. T. Yamano, O. Iguchi, Europhys. Lett. 83, 50007 (2008)17. J. Antolın, J.C. Angulo, S. Lopez-Rosa, J. Chem. Phys.

130, 074110 (2009)18. P. Sanchez-Moreno, A. Zarzo, J.S. Dehesa, J. Phys. A 45,

125305 (2012)19. C. Zander, A.R. Plastino, Europhys. Lett. 86, 18004

(2009)20. T. Morimoto, J. Phys. Soc. Jpn 12, 328 (1963)21. I. Csiszar, Stud. Math. Hung. 2, 299 (1967)22. H. Risken, The Fokker-Planck Equation (Springer,

New York, 1989)23. W.K. Wootters, W.H. Zurek, Nature 299, 802 (1982)24. D. Dieks, Phys. Lett. A 92, 271 (1982)25. V. Scarani, S. Iblisdir, N. Gisin, A. Acın, Rev. Mod. Phys.

77, 1225 (2005)26. D. Brody, B. Meister, Phys. Lett. A 204, 93 (1995)27. P. Blasiak, Phys. Lett. A 377, 847 (2013)