Research Article The Thermal Statistics of Quasi ...Research Article The Thermal Statistics of Quasi-Probabilities Analogs in Phase Space F.Pennini, 1,2 A.Plastino, 3 andM.C.Rocca

Research ArticleThe Thermal Statistics of Quasi-Probabilitiesrsquo Analogs inPhase Space

F Pennini12 A Plastino3 and M C Rocca3

1Departamento de Fısica Facultad de Ciencias Exactas y Naturales Universidad Nacional de La Pampa Avenida Peru 151Santa Rosa 6300 La Pampa Argentina2Departamento de Fısica Universidad Catolica del Norte Avenida Angamos 0610 1270709 Antofagasta Chile3Instituto de Fısica La Plata-CCT-CONICET Universidad Nacional de La Plata CC 727 1900 La Plata Argentina

Correspondence should be addressed to F Pennini fpenniniucncl

Received 14 May 2015 Revised 30 July 2015 Accepted 23 August 2015

Academic Editor Klaus Kirsten

Copyright copy 2015 F Pennini et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities (QP) in phase space for the importantcase of quadratic Hamiltonians We consider the three more important OPs Wignerrsquos 119875- and Husimirsquos We show that for allof them the ensuing semiclassical entropy is a function only of the fluctuation product Δ119909Δ119901 We ascertain that the semiclassicalanalog of119875-distribution seems to becomeunphysical at very low temperaturesThebehavior of several other information quantifiersreconfirms such an assertion inmanifold waysWe also examine the behavior of the statistical complexity and of thermal quantitieslike the specific heat

1 Introduction

A quasi-probability distribution is a mathematical construc-tion that resembles a probability distribution but does notnecessarily fulfill some of Kolmogorovrsquos axioms for prob-abilities [1] Quasi-probabilities exhibit general features ofordinary probabilities Most importantly they yield expec-tation values with respect to the weights of the distributionHowever they disobey the third probability postulate [1] inthe sense that regions integrated under them do not repre-sent probabilities of mutually exclusive states Some quasi-probability distributions exhibit zones of negative probabilitydensity This kind of distributions often arises in the studyof quantum mechanics when discussed in a phase spacerepresentation of frequent use in quantum optics time-frequency analysis and so forth

One usually considers a density operator 120588 defined withrespect to a complete orthonormal basis and shows that itcan always be written in a diagonal manner provided that anovercomplete basis is at hand [2] This is the case of coherentstates |120572⟩ [3] for which [2]

120588 = int1198892120572

120587119875 (120572 120572

lowast) |120572⟩ ⟨120572| (1)

We have 1198892120572120587 = 1198891199091198891199012120587ℏ with 119909 and 119901 being phase

space variables Coherent states right eigenstates of theannihilation operator 119886 serve as the overcomplete basis insuch a build-up [2 3]

There exists a family of different representations eachconnected to a different ordering of the creation and destruc-tion operators 119886 and 119886

dagger Historically the first of these isthe Wigner quasi-probability distribution 119882 [4] related tosymmetric operator ordering In quantum optics the particlenumber operator is naturally expressed in normal orderand in the pertinent scenario the associated representationof the phase space distribution is the Glauber-Sudarshan119875 one [3] In addition to 119882 and 119875 one may find manyother quasi-probability distributions emerging in alternativerepresentations of the phase space distribution [5] A quitepopular representation is the Husimi119876 one [6ndash9] used whenoperators are in antinormal order We emphasize that wework here with classical analogs of119882119875 and119876 As stated wewill specialize things to the three 119891-functions associated to aHarmonic Oscillator (HO) of angular frequency 120596 In such ascenario the three (classical analog) functions that we call forconvenience 119891

119875 119891119876 and 119891

119882are just GaussiansThe pertinent

treatment becomes wholly analytical

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 145684 8 pageshttpdxdoiorg1011552015145684

2 Advances in Mathematical Physics

11 Our Goal In this paper we wish to apply semiclassicalinformation theory tools associated with these analog 119875119876 and 119882 representations (for quadratic Hamiltonians) inorder to describe the associated thermal semiclassical featuresThe idea is to gain physical insight from the application ofdifferent information quantifiers to classical analogs of quasi-probability distributions It will be seen that useful insightsare in this way gained WE will discover that out of the threefunctions only 119891

119876and 119891

119882are sensible analogs while 119891

119875

exhibits problems if the temperature is low enoughWe insist that in this paper we will regard quasi-

probabilities as semiclassical distributions in phase spaceanalogs of the quantum quasi-probabilistic distributions andtry to ascertainwhat physical features they are able to describeat such semiclassical level One has [10 11]

119891119875= 120574119875119890minus120574119875|120572|

2

120574119875= 119890120573ℏ120596

minus 1 (119875-function)

119891119876= 120574119876119890minus120574119876|120572|

2

120574119876= 1 minus 119890

minus120573ℏ120596(119876-function)

119891119882

= 120574119882119890minus120574119882|120572|

2

120574119882

= 2 tanh(120573ℏ120596

2) (119882-function)

(2)

where 120573 = 1119896119861119879 119896119861is the Boltzmann constant and 119879 is

the temperature They will be used as semiclassical statisticalweight functions Since ours is not a quantum approach theordering of HO-creation and destruction operators 119886 and 119886

dagger

plays no role whatsoever

12 Historic Considerations and Organization The thermo-dynamics properties associated with coherent states havebeen the subject of much interest See for instance [12 13]Notice that HO is a really relevant system that yields usefulinsights of wide impact Indeed HO constitutes much morethan a mere example It is of special relevance for bosonicor fermionic atoms contained in magnetic traps [14ndash16] andfor system that exhibits an equidistant level spacing in thevicinity of the ground state like nuclei or Luttinger liquids

For thermal states the Gaussian HO-quantum phasespaces distributions are known in the literature for applica-tions in quantum optics

This paper is organized as follows Section 2 refers todifferent information quantifiers in a phase space represen-tation for Gaussian distributions In Section 3 we calculatethe classical analog Fano factor Features of the fluctuationsare analyzed in Section 4 Additionally we discuss the notionof linear entropy Finally some conclusions are drawn inSection 5

2 Semiclassical Information Quantifiers

Consider a general normalized gaussian distribution in phasespace

119891 (120572) = 120574119890minus120574|120572|2

(3)

whose normalized variance is 1120574 with 120574 taking values 120574119875

120574119876 and 120574

119882 We discuss next in these terms some important

information theory quantifiers

21 Shift-Invariant Fisherrsquos Information Measure The infor-mation quantifier Fisherrsquos information measure specializedfor families of shift-invariant distributions which do notchange shape under translations is [17 18]

119868 = int119889119909119891 (119909) (120597 ln119891 (119909)

120597119909)

2

(4)

and in phase space it adopts the appearance [19]

119868 =1

4int

1198892120572

120587119891 (120572) (

120597 ln119891 (120572)

120597 |120572|)

2

(5)

such that considering 119891(120572) given by (3) we get 119868 = 120574 whosespecific values are 120574

119875 120574119876 and 120574

119882for the three functions 119891

119875

119891119876 and 119891

119882 The behavior of these quantities is displayed

in Figure 1 The solid line is the case 119875 the dashed one isthe Wigner one and the dotted curve is assigned to theHusimi case Now it is known that in the present scenariothemaximumattainable value for 119868 equals 2 [19]The119875-resultviolates this restriction at low temperatures more precisely at

119879 lt 119879crit =(ℏ120596119896

119861)

ln 3asymp 091023

ℏ120596

119896119861

(6)

with 119879 being expressed in (ℏ120596119896119861)-units

22 Logarithmic Entropy 119878 The logarithmic Boltzmannrsquosinformation measure for the the probability distribution (3)is

119878 = minusint1198892120572

120587119891 (120572) ln119891 (120572) = 1 minus ln 120574 (7)

so that it acquires the particular values

119878119875= 1 minus ln (119890120573ℏ120596 minus 1) (8)

119878119876= 1 minus ln (1 minus 119890

minus120573ℏ120596) (9)

119878119882

= 1 minus ln(2 tanh(120573ℏ120596

2)) (10)

for respectively the distributions 119891119875 119891119876 and 119891

119882 These

entropies are plotted in Figure 2 Notice that 119878119875

lt 0 for119879 lt ℏ120596(119896

119861ln(1 + 119890)) asymp 076(ℏ120596119896

119861) lt 119879crit Negative

classical entropies are well known As an example one cancite [20]

Table 1 lists a set of critical temperatures 119879crit for typicalelectromagnetic (EM) waves

We see from Table 1 that serious anomalies are detectedfor 119875-distribution in the case of radio waves of high fre-quency 119875 becomes negative which is absurd for rather hightemperatures where one expects classical physics to reignAccordingly one concludes that quasi-probabilities do notexhibit a sensible classical limit in 119875-case contrary to whathappens in both119882 and 119876 ones

Advances in Mathematical Physics 3

Table 1 Critical temperatures 119879crit for typical radio waves with ℎ119896119861= 4799 10

minus11 Kelvin per second

Frequency (]) Critical temperatures (∘K)Extremely low frequency (ELF) 3ndash30Hz 14397 10minus10ndash14397 10minus9

Super low frequency (SLF) 30ndash300Hz 14397 10minus9ndash14397 10minus8

Ultra low frequency (ULF) 300ndash3000Hz 14397 10minus8ndash14397 10minus7

Very low frequency (VLF) 3ndash30 kHz 14397 10minus7ndash14397 10minus6

Low frequency (LF) 30ndash300 kHz 14397 10minus6ndash14397 10minus5

Medium frequency (MF) 300KHzndash3MHz 14397 10minus5ndash14397 10minus4

High frequency (HF) 3ndash30MHz 14397 10minus4ndash14397 10minus3

Very high frequency (VHF) 30ndash300MHz 14397 10minus3ndash14397 10minus2

Ultra high frequency (UHF) 300MHzndash3GHz 14397 10minus2ndash14397 10minus1

Super high frequency (SHF) 3ndash30GHz 14397 10minus1ndash14397Extremely high frequency (EHF) 30ndash300GHz 14397ndash14397Tremendously high frequency (THF) 300GHzndash3000GHz 14397ndash14397

0 1 2 3 4 50

1

2

3

4

5

T

Fish

er m

easu

res

Tcrit

Figure 1 Fisher measure versus temperature 119879 expressed in(ℏ120596119896

119861)-units The solid line is the case 119875 the dashed one is the

Wigner one and the dotted curve is assigned to theHusimi instanceThe vertical line represents the critical temperature 119879crit

23 Statistical Complexity The statistical complexity 119862according to Lopez-Ruiz et al [21] is a suitable product oftwo quantifiers such that 119862 becomes minimal at the extremesituations of perfect order or total randomness Instead ofusing the prescription of [21] but without violating its spiritwewill take one of these two quantifiers to be Fisherrsquosmeasureand take the other to be an entropic form since it is wellknown that the two behave in opposite manner [22] Thus

119862 = 119878119868 = 120574 (1 minus ln 120574) (11)

which vanishes for perfect order or total randomness Foreach particular case we explicitly have

119862119875= (119890120573ℏ120596

minus 1) [1 minus ln (119890120573ℏ120596 minus 1)] (12)

119862119876= (1 minus 119890

minus120573ℏ120596) [1 minus ln (1 minus 119890

minus120573ℏ120596)] (13)

119862119882

= 2 tanh(120573ℏ120596

2) [1 minus ln(2 tanh(

120573ℏ120596

2))] (14)

for respectively the distributions 119891119875 119891119876 and 119891

119882 The

maximum of the statistical complexity occurs when 120574 = 1

and the associated temperature values are

119890120573ℏ120596

minus 1 = 1 997904rArr 119879 =ℏ120596

119896119861

ln 2 gt 119879crit

for the 119891119875-function

1 minus 119890minus120573ℏ120596

= 1 997904rArr 119879 = 0 for the 119891119876-function

2 tanh(120573ℏ120596

2) = 1 997904rArr 119879 =

ℏ120596

2119896119861

arctan(12)

for the 119891119882-function

(15)

The statistical complexity 119862 is plotted in Figure 3

24 Linear Entropy Another interesting information quanti-fier is that of the Manfredi-Feix entropy [23] derived fromthe phase space Tsallis (119902 = 2) entropy [24] In quantuminformation this form is referred to as the linear entropy [25]It reads

119878119897= 1 minus int

1198892120572

1205871198912(120572) = 1 minusJ (16)

J = int1198892120572

1205871198912(120572) =

120574

2 (17)

Accordingly we have

119878119897= 1 minus

120574

2 0 le 119878

119897le 1 (18)

This is semiclassical result which is valid for small 120574 Inparticular

119878119897119875

= 1 minus120574119875

2= 1 minus

119890120573ℏ120596

minus 1

2

119878119897119876

= 1 minus120574119876

2= 1 minus

1 minus 119890minus120573ℏ120596

2

119878119897119882

= 1 minus120574119882

2= 1 minus

2 tanh (120573ℏ1205962)2

(19)

4 Advances in Mathematical Physics

0 2 4 6 8 1000

05

10

15

20

25

30

35

T

Loga

rithm

ic en

tropi

es

(a)

00 02 04 06 08 10minus05

00

05

10

15

T

Entro

pies

Scalss

(b)

Figure 2 Left logarithmic entropies 119878119875 119878119876 and 119878

119882 as a function of the temperature 119879 in (ℏ120596119896

119861)-units The solid line is the case 119875 the

dashed one is theWigner one and the dotted curve is assigned to the Husimi instance Right zoom of the logarithmic entropies as a functionof temperature 119879 in (ℏ120596119896

119861)-units Negative values of 119878

119875occur below 119879 = ℏ120596(119896

119861ln(1 + 119890)) with the critical temperature lt 119879crit Remaining

details are similar to those of left figure We have added the classical entropy of the harmonic oscillator 119878class = 1 minus ln(120573ℏ120596)

0 2 4 6 8 1000

02

04

06

08

10

T

Stat

istic

al co

mpl

exiti

es

Figure 3 Complexities 119862119875 119862119876 and 119862

119882versus the temperature 119879

in (ℏ120596119896119861)-units The solid line is the case 119875 the dashed one is the

Wigner one and the dotted curve is assigned to theHusimi instance

Note that in the 119875-instance the linear entropy becomesnegative once again for 119879 lt 119879crit Contrary to what happensfor the logarithmic entropy the linear one can vanish in 119882

representation The lineal entropies are plotted in Figure 4The ensuing statistical complexity that uses 119878

119897becomes

119862119897= 119878119897119868 = 119868 (1 minus

119868

2) = 120574 (1 minus

120574

2) (20)

vanishing for both 120574 = 0 and 120574 = 2 the extreme values of120574-physical range (we showed above that 120574 cannot exceed 2without violating uncertainty restrictions)

3 Fano Factorrsquos Classical Analog

In general the Fano factor is the coefficient of dispersion ofthe probability distribution 119901(119910) which is defined as [26]

F =Δ1199102

⟨119910⟩ (21)

where Δ1199102 = ⟨1199102⟩ minus ⟨119910⟩

2 is the variance and ⟨119909⟩ is the meanof a random process 119910

If 119901(119910) is a Poisson distribution then one sees that thepertinent Fano factor becomes unity (F = 1) [10 27] Weremind the reader of two situations

(1) ForF lt 1 sub-Poissonian processes occur(2) ForF gt 1 the process is super-Poissonian

For ourGaussian distribution (3) if one sets now119910 = |120572|2

one has the classical Fano analog

F =

⟨|120572|4⟩119891minus ⟨|120572|

2⟩2

119891

⟨|120572|2⟩119891

(22)

where the expectation value of the functionA(120572) is calculatedas

⟨A⟩119891 = int1198892120572

120587119891 (120572)A (120572) (23)

indicating that 119891(120572) is the statistical weight function Thusafter computing the mean values involved in (22) by takinginto account definition (23) the Fano factor becomes

F =1

120574=

1

119868 (24)

Advances in Mathematical Physics 5

0 2 4 6 8 1000

02

04

06

08

10

T

Line

ar en

tropi

es

(a)

0 2 4 6 8 10minus10

minus05

00

05

10

T

S lP

(b)

Figure 4 Left linear entropies 119878119897119875 119878119897119876 and 119878

119897119882versus the temperature 119879 in (ℏ120596119896

119861)-units The solid line is the case 119875 the dashed one is

the Wigner one and the dotted curve is assigned to the Husimi instance Right 119878119897119875

as a function of the temperature 119879 in (ℏ120596119896119861)-units We

note that the linear entropy 119878119897119875

is negative below the 119879 value 119905crit = 119879crit119896119861ℏ120596 = ln 3

which for a Gaussian distribution links the Fano factor tothe distributionrsquos width and to Fisherrsquos measure 119868 We arespeaking of processes that are of a quantumnature and cannottake place in a classical environment Thus with reference tothe critical temperature defined in (6) we have to deal with

F119875=

1

119890120573ℏ120596 minus 1(= 1 at 119879 =

ℏ120596

119896119861ln 2

gt 119879crit)

for the 119891119875function

F119876=

1

1 minus 119890minus120573ℏ120596(= 1 at 119879 = 0)

for the 119891119876function

F119882

=1

2 tanh (120573ℏ1205962)(= 1 at 119879 = 119879crit)

for the 119891119882

function

(25)

The 119891119876-case reaches the super- to sub-Poissonian transi-

tion only at 119879 = 0 while the other two cases reach it at finitetemperatures

4 Fluctuations

We start this section considering the classical Hamiltonian ofthe harmonic oscillator that reads

H (119909 119901) = ℏ120596 |120572|2 (26)

where 119909 and 119901 are phase space variables |120572|2 = 119909241205902

119909+

11990121205902

119901 1205902119909= ℏ2119898120596 and 120590

2

119901= ℏ1198981205962 [28]

Using the definition of the mean value (23) from (26) weimmediately find [29]

⟨1199092

21205902119909

⟩

119891

= ⟨1199012

21205902119901

⟩

119891

= ⟨|120572|2⟩119891 (27)

with

⟨|120572|2⟩119891= 120574int

1198892120572

120587119890minus120574|120572|2

|120572|2=

1

120574 (28)

where ⟨119909⟩119891

= ⟨119901⟩119891

= ⟨120572⟩119891

= 0 while 120574 takes therespective values 120574

119875 120574119876 and 120574

119882 The concomitant variances

are Δ1199092

= ⟨1199092⟩119891minus ⟨119909⟩

2

119891= 2120590

2

119901120574 and Δ119901

2= ⟨119901

2⟩119891minus

⟨119901⟩2

119891= 21205902

119901120574 Hence for our general Gaussian distribution

one easily establishes that

U = Δ119909Δ119901 =ℏ

120574 (29)

which shows that 120574 should be constrained by the restriction120574 le 2 (30)

if one wishes the inequality

Δ119909Δ119901 geℏ

2(31)

to holdSpecializing (29) for our three quasi-probability distribu-

tions yields

Δ119909Δ119901 =ℏ

119890120573ℏ120596 minus 1 for 119891

119875function

Δ119909Δ119901 =ℏ

1 minus 119890minus120573ℏ120596 for 119891

119876function

Δ119909Δ119901 =ℏ

2 tanh (120573ℏ1205962) for 119891

119882function

(32)

6 Advances in Mathematical Physics

These fluctuations are plotted in Figure 5The restriction (31)applied to the 119875-result entails that it holds if

119879 geℏ120596

ln 3119896119861

= 119879crit asymp 091023ℏ120596

119896119861

(33)

Thus the distribution119891119875seems again to becomeunphysical at

temperatures lower than 119879crit for which (31) is violated From(29) we have 120574 = ℏU Accordingly if we insert this into (7)the logarithmic entropy 119878 can be recast in U-terms via therelation

119878 = 1 minus ln( ℏ

Δ119909Δ119901) (34)

(also demonstrated in [30] to hold for the Wehrl entropy)which vanishes for

Δ119909Δ119901 =ℏ

119890 (35)

In the 119875-instance this happens at

119879 = 071463ℏ120596

119896119861

(36)

At this temperature Heisenbergrsquos-like condition (31) is vio-lated 119882 and 119876 distributions do not allow for such acircumstance Actually in the Wigner case which is exactthe minimum 119878-value is attained at 120573 = infin where

119878min = 1 minus ln 2 asymp 0306 (37)

The uncertainty restriction (31) seems to impede the phasespace entropy to vanish a sort of quasi-quantum effect It isclear then that in phase space the logarithmic entropy byitself is an uncertainty indicator in agreement with the workin other scenarios of several authors (see for instance [31]and references therein)

Define now the participation ratiorsquos analog as [32 33]

119898 =1

J=

2

120574 (38)

where J is given by (17) This is an important quantity thatmeasures the number of pure states entering the mixturedetermined by our general Gaussian probability distributionof amplitude 120574 [32 33] We again encounter troubles with the119875-distribution in this respect It is immediately realized bylooking at Figure 6 that for fulfilling the obvious condition119898 ge 1 one needs a temperature 119879 ge 119879crit

5 Conclusions

We have investigated here the thermal statistics of quasi-probabilitiesrsquo analogs 119891(120572) in phase space for the importantcase of quadratic Hamiltonians focusing attention on thethree more important instances that is those of Wigner 119875-and Husimi distributions

(i) We emphasized the fact that for all of them the semi-classical entropy is a function only of the fluctuation

0 1 2 3 4 50

1

2

3

4

T

Tcrit

ΔxΔp

Figure 5 Fluctuations versus the temperature 119879 in (ℏ120596119896119861)-units

The solid line is the case 119875 the dashed one is the Wigner one andthe dotted line is assigned to the Husimi instance

0 1 2 3 4 50

2

4

6

8

10

T

Part

icip

atio

n ra

tios

Figure 6 Participation ratio 119898 versus temperature 119879 in (ℏ120596119896119861)-

units The solid line is the case 119875 the dashed one is the Wigner oneand the dotted curve is assigned to the Husimi instance

product Δ119909Δ119901 This fact allows one to ascertain thatthe analog 119875-distribution seems to become unphysi-cal at low enough temperatures smaller than a criticalvalue 119879crit because in such an instance

(1) it would violate Heisenbergrsquos-like principle insuch a case The behavior of other informationquantifiers reconfirms such an assertion that is

(2) Fisherrsquos measure exceeds its permissible maxi-mum value 119868 = 2

(3) the participation ratio becomes lt 1 which isimpossible

(ii) It is also clear then that semiclassical entropy by itselfin phase space looks like a kind of ldquouncertaintyrdquoindicator

Advances in Mathematical Physics 7

(iii) We have determined the temperatures for which thestatistical complexity becomesmaximal as a signatureof the well-known transition between classical andnonclassical states of light whose signature is thetransition from super-Poissonian to sub-Poissoniandistributions [34]

We have seen that the 119875-distribution in the case of radiowaves of high frequency becomes negative which is absurdas the pertinent temperatures are rather high and thus oneexpects classical physics to reign Accordingly one concludesthat quasi-probabilities do not exhibit a sensible classical limitin the 119875-case contrary to what happens in both 119882 and 119876

ones

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors were supported by Consejo Nacional de Inves-tigaciones Cientıficas y Tecnicas (CONICET) ArgentinaUseful discussions with Professor R Piasecki of OpolersquosUniversity Poland are gratefully acknowledged

References

[1] J Von Plato ldquoGrundbegriffe der Wahrscheinlichkeitsrech-nungrdquo in Landmark Writings in Western Mathematics IGrattan-Guinness Ed pp 960ndash969 Elsevier Amsterdam TheNetherlands 2005

[2] E C Sudarshan ldquoEquivalence of semiclassical and quantummechanical descriptions of statistical light beamsrdquo PhysicalReview Letters vol 10 pp 277ndash279 1963

[3] R J Glauber ldquoCoherent and incoherent states of the radiationfieldrdquo Physical Review vol 131 pp 2766ndash2788 1963

[4] E P Wigner ldquoOn the quantum correction for thermodynamicequilibriumrdquo Physical Review vol 40 no 5 pp 749ndash759 1932

[5] F Pennini and A Plastino ldquoSmoothed Wigner-distributionsdecoherence and the temperature-dependence of the classical-quantum frontierrdquoThe European Physical Journal D vol 61 no1 pp 241ndash247 2011

[6] K Husimi ldquoSome formal properties of the density matrixrdquoProceedings of the Physico-Mathematical Society of Japan vol 22pp 264ndash283 1940

[7] S S Mizrahi ldquoQuantum mechanics in the Gaussian wave-packet phase space representationrdquo Physica A vol 127 no 1-2pp 241ndash264 1984

[8] S S Mizrahi ldquoQuantum mechanics in the Gaussian wave-packet phase space representation II Dynamicsrdquo Physica AStatistical and Theoretical Physics vol 135 no 1 pp 237ndash2501986

[9] S S Mizrahi ldquoQuantum mechanics in the Gaussian wave-packet phase space representation III from phase space prob-ability functions to wave-functionsrdquo Physica A vol 150 no 3pp 541ndash554 1988

[10] M O Scully and M S Zubairy Quantum Optics CambridgeUniversity Press New York NY USA 1997

[11] W P Scheleich Quantum Optics in Phase Space Wiley BerlinGermany 2001

[12] J-P Gazeau Coherent States in Quantum Physics Wiley-VCHWeinheim Germany 2009

[13] E H Lieb ldquoThe classical limit of quantum spin systemsrdquoCommunications in Mathematical Physics vol 31 pp 327ndash3401973

[14] M H Anderson J R Ensher M R Matthews C E Wiemanand E A Cornell ldquoObservation of Bose-Einstein condensationin a dilute atomic vaporrdquo Science vol 269 no 5221 pp 198ndash2011995

[15] K B Davis M-O Mewes M R Andrews et al ldquoBose-Einsteincondensation in a gas of sodium atomsrdquo Physical Review Lettersvol 75 no 22 pp 3969ndash3973 1995

[16] C C Bradley C A Sackett and R G Hulet ldquoBose-Einsteincondensation of lithium observation of limited condensatenumberrdquo Physical Review Letters vol 78 no 6 pp 985ndash9891997

[17] B R Frieden and B H Soffer ldquoLagrangians of physics and thegame of Fisher-information transferrdquo Physical Review E vol 52no 3 pp 2274ndash2286 1995

[18] M J W Hall ldquoQuantum properties of classical Fisher informa-tionrdquo Physical ReviewA vol 62 no 1 Article ID 012107 6 pages2000

[19] F Pennini and A Plastino ldquoFluctuations entropic quantifiersand classical-quantum transitionrdquo Entropy vol 16 no 3 pp1178ndash1190 2014

[20] E H Lieb ldquoProof of an entropy conjecture of Wehrlrdquo Commu-nications in Mathematical Physics vol 62 no 1 pp 35ndash41 1978

[21] R Lopez-Ruiz H L Mancini and X A Calbet ldquoA statisticalmeasure of complexityrdquo Physics Letters A vol 209 no 5-6 pp321ndash326 1995

[22] B R Frieden Science from Fisher Information CambridgeUniversity Press Cambridge UK 2nd edition 2008

[23] G Manfredi and M R Feix ldquoEntropy and Wigner functionsrdquoPhysical Review E vol 62 no 4 pp 4665ndash4674 2000

[24] C Tsallis Introduction to Nonextensive Statistical MechanicsSpringer New York NY USA 2008

[25] P Sadeghi S Khademi and A H Darooneh ldquoTsallis entropyin phase-space quantummechanicsrdquo Physical Review A vol 86no 1 Article ID 012119 2012

[26] U Fano ldquoIonization yield of radiations II The fluctuations ofthe number of ionsrdquo Physical Review vol 72 no 1 pp 26ndash291947

[27] C Brif and Y Ben-Aryeh ldquoSubcoherent P-representation fornon-classical photon statesrdquo Journal of the European OpticalSociety B Quantum Optics vol 6 no 5 pp 391ndash396 1994

[28] R K Pathria StatisticalMechanics PergamonPress Exeter UK1993

[29] F Pennini and A Plastino ldquoHeisenberg-Fisher thermal uncer-tainty measurerdquo Physical Review E vol 69 no 5 Article ID057101 2004

[30] A Anderson and J J Halliwell ldquoInformation-theoreticmeasureof uncertainty due to quantum and thermal fluctuationsrdquoPhysical Review D vol 48 no 6 pp 2753ndash2765 1993

[31] I Białynicki-Birula and J Mycielski ldquoUncertainty relations forinformation entropy in wave mechanicsrdquo Communications inMathematical Physics vol 44 no 2 pp 129ndash132 1975

[32] J Batle A R Plastino M Casas and A Plastino ldquoCon-ditional q-entropies and quantum separability a numericalexplorationrdquo Journal of Physics AMathematical and Generalvol 35 no 48 pp 10311ndash10324 2002

8 Advances in Mathematical Physics

[33] F Pennini A Plastino and G L Ferri ldquoSemiclassical informa-tion from deformed and escort information measuresrdquo PhysicaA vol 383 no 2 pp 782ndash796 2007

[34] X T Zou and L Mandel ldquoPhoton-antibunching and sub-Poissonian photon statisticsrdquo Physical Review A vol 41 no 1pp 475ndash476 1990

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