pion multiplicity distribution and their associated fluctuation and correlations have been studied at the cern sps in the past , are being investigated at rhic presently , and in the future will be studied at the lhc . because of the very large number of pions and other produced particles in very high energy collisions , event - by - event studies can be carried out and are important tools for many reasons .one current reason for studying them resides in the hope that anomalous fluctuations will remain from a transition of a quark - gluon phase to hadronic phase , thus offering one of the signals of the formation of the qg phase .several models predict large fluctuations such as the disoriented chiral condensate and in density fluctuations from droplets arising in a first order phase transition .a well known procedure for studying correlations uses the bose - einstein symmetries associated with pions in a hanbury brown - twiss analysis .such an analysis gives information about the space time history of the collision through measurements of source parameters .if the density of pions becomes large , bose - einstein correlation may also lead to a strongly emitting system which has been called a pion laser .the pion laser model has been recently solved analytically by t. csrg and j. zimanyi .the importance of bose - einstein correlations has also been illustrated in the observation of a condensation of atoms in a harmonic oscillator or laser trap .previous interest in pionic distributions have centered around the possibility of intermittency behavior and fractal structure based on parallels with turbulent flow in fluids .a distribution widely used to discuss such features has been the negative binomial ( nb ) distribution with its associated clan structure and kno scaling feature .kno scaling properties have been interpreted in terms of a phase transition associated with a feynman - wilson gas .various other issues associated with pions include evidence for thermalization , critical point fluctuations , fluctuations from a first order phase transition , charge particle ratios and question of chemical equilibrium , the behavior of fluctuations in net charge in a qg plasma for transition .  for lower energy heavy ion collisions ,multifragmentation of nuclei takes place .the fragment distribution can also be described statistically by considering all the possible partition of nucleons into smaller clusters .this study gives a tool for the description of nuclear multifragmentation distributions , nuclear liquid - gas phase transition , critical exponent , intermittency , and chaotic behavior of nuclear multifragmentation .the same model can describe pionic distribution .this possibility arises in our approach which is based on feynman path integral methods where symmetrization of bosons or anti - symmetrization of fermions leads to a cycle class decomposition of the permutations associated with these symmetries .the correspondance comes from the identification of clusters of size and the cycles of length in a permutation as discussed below .  in next section a summary of the generalized statistical modelwill be given .various models of particle multiplicity distribution will then be developed .moreover , we derive a generalized model which can further be reduced to a geometric , negative binomial , and lorentz / catalan model . these various models used in describing pion and particle multiplicity distributionsthen are examined with the generalized model in sect .iii . in sect .iv , we further compare various statistical properties between different models within the generalized model .consider a system composed of different types of species or objects which could be the fragments in a fragmentation or in a cycle class description .any event of such a system can be associated with a vector or where the non - negative integer is the number of individuals of species . for example can be the number of clusters of size or the number of cycles of length in a given permutation of particles .the later is important for bose - einstein and fermi - dirac statistics . a general block picture of is shown in fig.[fig1]a .fig.[fig1]b shows how the various partition can be developed as an evolution from successively smaller systems .the multiplicity , i.e. , the total number of individuals is then the number of species can be infinity in general .( 25,8)(7.5,0.0 )  ( 0.5,0.5 )  ( 12.2,6.2)*a ) *  ( 8.0,0.25)* *  ( 8.01 , 1.51)(0,1)4.5 ( 8.0 , 5.5)(0.5,0)2 ( 8.0 , 5.0)(0.5,0)2 ( 8.0 , 4.5)(0.5,0)4 ( 8.0 , 4.0)(0.5,0)5 ( 8.0 , 3.5)(0.5,0)5 ( 8.0 , 3.0)(0.5,0)7 ( 8.0 , 2.5)(0.5,0)7 ( 8.0 , 2.0)(0.5,0)7 ( 8.0 , 1.5)(0.5,0)11 ( 8.99 , 5.0)(0,1)1.0 ( 9.99 , 4.5)(0,1)0.5 ( 10.49 , 3.5)(0,1)1.0 ( 11.49 , 2.0)(0,1)1.5 ( 13.49 , 1.5)(0,1)0.5 ( 8.01 , 1.51)(1,0)5.5 ( 11.51 , 2.01)(1,0)2.0 ( 10.51 , 3.51)(1,0)1.0 ( 10.01 , 4.51)(1,0)0.5 ( 9.01 , 5.01)(1,0)1.0 ( 8.01 , 6.01)(1,0)1.0 ( 7.5,6.5)(0,0.01)2(1,0)0.5 ( 9.5,6.5)(0,0.01)2(-1,0)0.5 ( 8.0,6.2)(1.0,0.6)* * ( 7.6,3.2)(0,0.01)2(0,-1)1.7 ( 7.6,4.3)(0,0.01)2(0,1)1.7 ( 7.4,3.5)(0.3,0.6)[r]* * ( 15.5,0 )  ( 10,8)(0,0 )  ( 0.5,0.5 )  ( 8,6.2)*b ) *  ( 3.5 , 7.0 ) ( 4.2,7.3) ( 3.6,6.9)(-3,-2)1.05 ( 3.9,6.9)(2,-3)0.85  ( 2.0 , 5)(0,0.5)2 ( 2.7,5.6) ( 2.05,4.9)(-2,-1)0.65 ( 2.4,4.9)(3,-2)1.1  ( 4.5 , 5.0)(0.5,0)2 ( 5.7,5.4) ( 4.52,4.9)(-1,-1)0.78 ( 5.45,4.9)(3,-4)0.95  ( 1 . , 3.0)(0,0.5)3 ( 1.7,4.0) ( 1.1,2.9)(-1,-1)0.75 ( 1.4,2.9)(1,-4)0.3  ( 3.5 , 3.0)(0,0.5)2 ( 4.0 , 3.0 ) ( 4.2,3.8) ( 3.6,2.9)(-4,-3)1.6 ( 3.9,2.9)(0,-1)1.75 ( 4.4,2.9)(3,-4)1.3  ( 6.0 , 3.0)(0.5,0)3 ( 7.1,3.7) ( 6.4,2.9)(-1,-3)0.6 ( 7.3,2.9)(1,-2)1.1  ( 0 . , 0.0)(0,0.5)4 ( 0.7,1.6)  ( 1.5 , 0.0)(0,0.5)3 ( 2.0 , 0.0 ) ( 2.15,1.1)  ( 3.5 , 0.0)(0,0.5)2 ( 4.0 , 0.0)(0,0.5)2 ( 4.7,0.7)  ( 5.5 , 0.0)(0.5,0)3 ( 5.5 , 0.5 ) ( 6.2,0.8)  ( 8.0 , 0.0)(0.5,0)4 ( 9.6,0.7)  various probability distributions related with this system can be developed by assigning an appropriate weight to each individual of type species .a weight is then given to each event and the type of weight that will be considered has the structure ;     \label{weight}\end{aligned}\ ] ] the are gibbs factors .the will be given below and contains various physical quantities .summing the weight over all the possible events of , the grand canonical partition function of the system is given as   = \exp\left[\sum_{k=1}^n x_k\right ]     \label{grandptf }  \end{aligned}\ ] ] the last equation holds due to the form of eq.([weight ] ) of the weight , i.e. , the factor is the -th order expansion term of .  introducing other quantities to each individual entity or group of type , , we can define a canonical partition function with a fixed as        \label{canoptf }  \end{aligned}\ ] ] with . here is the summation over all events with a fixed value of , i.e. , over a canonical ensemble of a fixed , and the is a summation over all the possible values of ; it becomes for the case of with positive integer .the case is encountered in fragmentation problems and permutation problems . if we take , then the canonical partition function is the dependent term of the grand canonical partition function .the physics of the canonical ensemble depends on the choice of the quantity  .there always is at least one event having , i.e. , the event where all s are zero , .thus if all are non zero positive since then there are no other possible events having . due to the form of the weight given by eq.([weight ] ) the canonical partition function satisfies a recurrence relation  this relation is nothing but the constraint eq.([asum ] ) in terms of the mean using eq.([factma ] ) .for non - zero positive , there is no case having , i.e. , for and thus the can be obtained by the recurence relation of eq.([recur ] ) starting from .  to study the statistical properties of a system where the number of individuals is independent of their species type, we can choose .for fragmentation , all the clusters are treated the same independent of their size or internal structure by choosing . in pion distribution , , , and treated the same if we choose independent of their charge . by choosing ,we can study jet distributions without considering any further process of hadronization .then and the canonical ensemble is a set of systems having the same number of individuals independent of their internal structure and the canonical partition function is the expansion of grand canonical partition function in terms of with a power of multiplicity and counts the total multiplicity of individuals in the system . as we will see later , this uniform treatment of all the species independent to their internal structure leads to a poisson distribution .if we take with the to be a positive integer representing the number of constituent particles ( such as nucleons in a nuclear fragment or cycle length of a permutation ) , then is the total number of the constituent particles in the system and term in is the canonical partition function with the total of constituent particles . for this casethe weight variable is the variable counting the number of constituent particles , i.e. , the fugacity of the constituent with and with the chemical potential of the constituent particle .this case is related to the nuclear multifragmentation model in refs . and various models of pion distribution discussed below . for , andif we also take then all the clusters are treated the same independent of the size .the weight is then the same for all partitions having the same multiplicity except for the gibb s factor .thus counts the multiplicity with being a fugacity of a cluster .if we choose then the weight is the same for all partitions having same .all the nucleons are treated the same independent of the cluster it belongs to thus counting the number of constituent nucleons with being a fugacity of the constituent .if we choose for a general value of then the weight is the same for all partitions having same .all the constituents per unit are treated the same independent of the cluster it belongs to thus counting as the corresponding total quantity of with being a fugacity of unit .  on the other hand , if we choose to be the energy of species or -th level , then is the total energy of the system and becomes the partition function of a canonical ensemble with a fixed total energy .if we choose to be the charge of species , then is the total charge of the system .most discussions in this section applies to a general value of .however we will concentrate more on the choice of here for simplicity in notation .in a canonical ensemble of fixed , we can define a probability distribution of a specific partition as with this probability , various mean values , fluctuations and correlations of the number of species can be evaluated as a ratio of canonical partition functions for two different values of such as  thus we have , which shows that the mean number of species in a canonical ensemble with fixed is proportional to the weight assigned to the species and the ratio of canonical partition function with different .the recurrence relation of eq.([recur ] ) then follows simply using the fact that .this distribution has been used in describing various fragment distributions in nuclear multifragmentation .  now knowing the partition functions and , we can associate a probability of the system to have a fixed value of in a grand canonical ensemble as {z=0 }          \label{pax }  \\z(\vec x , z ) & = & \sum_a z_a(\vec x ) z^a   = \exp\left[\sum_k x_k z^{\alpha_k}\right ]  \label{grandzxz}\end{aligned}\ ] ] the last step follows from the fact that the power term of is if we put .thus the generating function of can also be looked at as a grand canonical partition function with the weight replaced to be , where the variable counts explicitly and .if we consider then has two roles ; one as a weight which is assigned the same to each constituent and another as a generating parameter of the probability .for the case that , the void probability  is the inverse of the grand canonical partition function , i.e. , inversely the canonical partition function is the probability normalized or rescaled by the void probability , i.e. , {z=0 }   = \frac{p_a(\vec x)}{p_0(\vec x)}\end{aligned}\ ] ] another type of generating function of that is frequently used may be defined as \right ]       \label{genu }  \\p_a(\vec x ) & = & \frac{1}{\gamma(a+1 ) } \left[\left(-\frac{d}{d u}\right)^a     g(\vec x , u)\right]_{u=1}\end{aligned}\ ] ] we see that , and .once the probability is determined , various statistical quantities can be evaluated .also the grand canonical partition function is given once we know the thermodynamic grand potential this result can be used to study the statistical properties of a system .moreover various moments and cumulants , mean values and fluctuations may be obtained using the generating function .this subsection gives general expression for various quantities that will be used later when we discuss specific models .since the probability of a specific event in a grand canonical ensemble is given by the mean of a quantity in a grand canonical ensemble is related to the mean of in a canonical ensemble as thus once we obtain using the recurrence relation of eq.([recur ] ) , then using the mean in canonical ensemble given by eq.([factma ] ) , we can obtain mean in grand canonical ensemble . since, in a grand canonical enemble , can be infinity , i.e. , with for if all and if can be negative , we have for integer and . using this fact, we can see easily that this result shows that the weight factor in this model is the mean number in a grand canonical ensemble .the -th power moment of and its factorial moments are given simply by {z=1 }    \\\left<\frac{\gamma(a+1)}{\gamma(a - m+1)}\right>(\vec x )  & \equiv & \sum_{\vecn } \frac{\gamma(a+1)}{\gamma(a - m+1 ) } p(\vec x , \vec n )   = \frac{1}{z(\vec x ) } \left[\left(\frac{d}{d z}\right)^m      z(\vec x , z)\right]_{z=1 }    \end{aligned}\ ] ] similarly the -th cumulants , which is the power moments of , and the factoral cumulants are {z=1 }   = \sum_{k=1}^\infty \alpha_k^m x_k   \label{alpham }  \\f_m(\vec x ) & \equiv & \left <\sum_{k=1}^n      \frac{\gamma(\alpha_k+1)}{\gamma(\alpha_k - m+1 ) } n_k\right >   = \left[\left(\frac{d}{d z}\right)^m \ln z(\vec x , z)\right]_{z=1 }        \nonumber  \\  & = & \left[\left(- \frac{d}{d u}\right)^m \ln g(\vec x , u)\right]_{u=0 }   = \sum_{k=1}^n \frac{\gamma(\alpha_k+1)}{\gamma(\alpha_k - m+1 ) } x_k         \label{factfm } \end{aligned}\ ] ] we can see easily that , for the power moments of , the power moments of are directly related to the power moments of measured from the mean , i.e. , the cumulants are same with the central moments of . this simple relation does not hold for but we can evaluate them starting from using the recurrence relation similarly the -th factorial cumulants , which is the factorial moments of , can be found using recurrence relation starting from the reduced factorial cumulants defined in ref. corresponds to the factorial cumulants normalized with mean number as thus with and .  a taylor expansion of with a real number , w.r.t . for , is {p=0 }   = \sum_{n=0}^\infty \frac{\gamma(a+1)}{n ! \gamma(a - n+1 ) }     q^{a - n } p^n   = \sum_{n=0}^\infty { a \choose n } p^n q^{a - n }  \end{aligned}\ ] ] thus , from the factorial structure of and from eqs.([genu ] ) and ( [ factfm ] ) ,   = \exp\left[\sum_{m=1}^\infty \frac{(-u \bar a)^m}{m ! }      \kappa_m(\vec x)\right]\end{aligned}\ ] ] since the probability can be obtained by {u=1 }   = \frac{(-\bar a)^a}{\gamma(a+1 ) }    \left[\left(\frac{d}{d \bar a}\right)^a p_0(\vec x)\right]\end{aligned}\ ] ] when is independent of the mean as used in ref. . using the power moment of eq.([alpham ] ) , we also have \end{aligned}\ ] ] since ,    = \exp\left[\sum_{m=0}^\infty \frac{(\ln z)^m}{m ! }< \alpha^m>(\vec x)\right]\end{aligned}\ ] ] the generating function differs from the generating function only by an extra term of in their exponent .the relation between the s and the shows that the s are also the combinants of ref. . in turnthe combinants can be related to the factorial cumulants defined by the factorial cumulants are the -th order factorial moments of of eq.([factfm ] ) .thus for .the normalized factorial cumulants , i.e. , the reduced cumurant , is for a negative binomial ( nb ) distribution , this result of eq.([kappam ] ) shows that for nb has an hierarchical structure of a distribution at the reduced cumulant level which was realized for the nb distribution in ref. .this result will be generalized later .  also using the above power moments and factorial moments we can study voids and void scaling relation , hierarchical structure , combinant and cummulant properties which will be discussed below .   since our approachwas first used to discuss multifragmentation and then later extended to include multiparticle production , we briefly mention some of difference between multifragmentation and multiparticle production .  in nuclear multifragmentation , is the number of nucleons in a fragment and is the number of fragments of size .the mean values of the total number of fragments and the total number of nucleons are in nuclear multifragmentation , the total number of nucleons is usually fixed . in multiparticle productionthe is the total number of produced particles and is not fixed .thus in multifragmentation we consider canonical ensemble instead of a grand canonical ensemble .the mean multiplicity and in a canonical ensemble are related with grand canonical ensemble as the in a grand canonical ensemble .but the weight factor is different from the mean multiplicity of cluster size in a canonical ensemble . from eq.([factma ] ) , is the weight multiplied by the ratio of and , i.e. , . in multifragmentation studies we have used with the same weight for each cluster and the same weight for each nucleon .this choice of gives and thus as where the with the chemical potential . to determine the weight experimentally, we should use a grand canonical ensemble , i.e. , we need to consider various system with different values of .van hove and giovannini have introduced clan variables and to describe a general class of probability distributions , with most discussions of these variables centering around the negative binomial distribution .these variables are defined as where the mean number of clans is and the is the mean number of members per clan .the is the grand canonical generating function and thus where is the cycle class weight distribution .the is the mean number of total members ( particles ) .the here is the number of clans of size having members .the clan variable is also related to the void probability ; thus . an important function in voidanalysis is .thus the void parameters , void probability and void function  , are equivalent to the generalized clan parameters and with the equivalence given by : the is the normalized grand potential for the mean .  void analysis looks for scaling properties associated with ; specifically , is a function of the combination where is the coefficient of in the fluctuation . since ,the variance of in a grand canonical ensemble becomes \end{aligned}\ ] ] with , i.e. , the normalized factorial cumulant . since the variance for a poissonian distribution is the same as the mean , , the ; thus , the parameter represents a degree of departure from poissonian fluctuation normalized by mean of the distribution .a well known non - poissonian example is a nb distribution which has and this becomes plank distribution with . using the recurrence relation eq.([recfm ] ) for , we can show that      - 2 \frac{\xi(\vec x , z)}{<a > }  \end{aligned}\ ] ] a nb distribution has while the lorentz / catalan ( lc ) distribution discussed in ref . and below has .we will study in sect .[ voidscal ] , _ vs _ , i.e. , the void or clan variable _ vs _ the fluctuation for various choices of summarized in table [ tabl1 ] .the lc model was shown to be a useful model for discussing an underlying splitting dynamics when ancestral or evolutionary variables and were introduced into and as discussed in ref. .percolation or splitting dynamics with a branching probability and survival probability has a hierarchical topology as shown in fig.[lcfig ] .weighting each diagram by , the evolutionary or ancestral variables are related to the clan variables and . by taking to be the number of diagrams of size shown in fig.[lcfig ] ,the evolutionary dynamics is just that of the lc model .then with set equal to 1 , , , , , etc .the interpretation of this set of s reads as follows : has 1 surviving line without a branch ( ) and one diagram ( ) , has 1 branch ( ) leading to 2 surviving lines ( ) and one diagram ( ) , has 2 branch points ( ) leading 3 surviving lines ( ) and two diagrams ( ) , has 3 branch points( ) , 4 surviving lines ( ) and 5 diagrams ( ) , etc . in these evolutionary / ancestral variables the which determines is for all . for , is no longer a constant and is .since the clan variables are and , then , for the lc model with evolutionary or ancestral varables ,   \\ \beta & = & n_c \frac{n_c - ( 1 \pm 1)/2}{n_c - 1}\end{aligned}\ ] ] these can be reduced to , , and for while , , and for .since the branching probability varies in , the clan variable has with at and 1 and at .the behavior of above will be discussed in sec .[ sectlc ] where a will be introduced .the lc model thus connects the clan variable to the probability of branching in the evolutionary or ancestral picture of fig.[lcfig ] or in a percolation model . for poisson processes ( no branching ) , ( only unit cycles and no be correlations ) and ( one member in each clan in average ) .  (35,7)(1.2,4.0 ) ( 3,7)(0,1)4 ( 2,6.0) ( 1,4.5)unit cycles ( 8.5,9)(0,1)2 ( 8.0,9)(1,0)1 ( 8.5,9 ) ( 8.7,9.4) ( 8.0,7)(0,1)2 ( 9.0,7)(0,1)2 ( 7.0,6.0) ( 5.6,4.5)period 2 cycles ( 13.2,10)(0,1)1 ( 12.5,10)(1,0)1.5 ( 13.2,10 ) ( 13.4,10.4) ( 12.5,7)(0,1)3 ( 14.0,8.5)(0,1)1.5 ( 13.5,8.5)(1,0)1 ( 14.0,8.5 ) ( 14.2,8.9) ( 13.5,7)(0,1)1.5 ( 14.5,7.0)(0,1)1.5 ( 16.8,10)(0,1)1 ( 16.0,10)(1,0)1.5 ( 16.8,10 ) ( 17.0,10.4) ( 17.5,7)(0,1)3 ( 16.0,8.5)(0,1)1.5 ( 15.5,8.5)(1,0)1 ( 16.0,8.5 ) ( 16.2,8.9) ( 15.5,7)(0,1)1.5 ( 16.5,7.0)(0,1)1.5 ( 13.0,6.0) ( 12.2,4.5)period 3 cycles ( 20.9,10)(0,1)1 ( 20.2,10)(1,0)1.5 ( 20.9,10 ) ( 21.1,10.4) ( 20.2,7)(0,1)3 ( 21.7,9.0)(0,1)1.0 ( 21.2,9.0)(1,0)1 ( 21.7,9.0 ) ( 21.9,9.4) ( 21.2,7.0)(0,1)2.0 ( 22.2,8.0)(0,1)1.0 ( 21.7,8.0)(1,0)1 ( 22.2,8.0 ) ( 22.4,8.4) ( 21.7,7.0)(0,1)1.0 ( 22.7,7.0)(0,1)1.0 ( 24.1,10)(0,1)1 ( 23.4,10)(1,0)1.5 ( 24.1,10 ) ( 24.3,10.4) ( 23.4,7)(0,1)3 ( 24.9,9.0)(0,1)1.0 ( 24.4,9.0)(1,0)1 ( 24.9,9.0 ) ( 25.1,9.4) ( 24.4,8.0)(0,1)1.0 ( 25.4,7.0)(0,1)2.0 ( 23.9,8.0)(1,0)1 ( 24.4,8.0 ) ( 24.6,8.4) ( 23.9,7.0)(0,1)1.0 ( 24.9,7.0)(0,1)1.0 ( 27.4,10)(0,1)1 ( 26.7,10)(1,0)1.5 ( 27.4,10 ) ( 27.6,10.4) ( 26.7,8.5)(0,1)1.5 ( 28.2,8.5)(0,1)1.5 ( 26.2,8.5)(1,0)1 ( 26.7,8.5 ) ( 26.9,8.9) ( 26.2,7)(0,1)1.5 ( 27.2,7.0)(0,1)1.5 ( 27.7,8.5)(1,0)1 ( 28.2,8.5 ) ( 28.5,8.9) ( 27.7,7)(0,1)1.5 ( 28.7,7.0)(0,1)1.5 ( 30.8,10)(0,1)1 ( 30.0,10)(1,0)1.5 ( 30.8,10 ) ( 31.0,10.4) ( 31.5,7)(0,1)3 ( 30.0,9.0)(0,1)1.0 ( 29.5,9.0)(1,0)1 ( 30.0,9.0 ) ( 30.2,9.4) ( 29.5,7.0)(0,1)2.0 ( 30.5,8.0)(0,1)1.0 ( 30.0,8.0)(1,0)1 ( 30.5,8.0 ) ( 30.7,8.4) ( 30.0,7.0)(0,1)1.0 ( 31.0,7.0)(0,1)1.0 ( 34.1,10)(0,1)1 ( 33.3,10)(1,0)1.5 ( 34.1,10 ) ( 34.3,10.4) ( 34.8,7)(0,1)3 ( 33.3,9.0)(0,1)1.0 ( 32.8,9.0)(1,0)1 ( 33.3,9.0 ) ( 33.5,9.4) ( 32.8,8.0)(0,1)1.0 ( 33.8,7.0)(0,1)2.0 ( 32.3,8.0)(1,0)1 ( 32.8,8.0 ) ( 33.0,8.4) ( 32.3,7.0)(0,1)1.0 ( 33.3,7.0)(0,1)1.0 ( 25.5,6.0) ( 24.7,4.5)period 4 cycles   in describing either nuclear multifragmentation or multiparticle distributions , is taken to be a positive integer . in a nuclear fragment distribution , an initial number of nucleonsare clustered into small nuclei of nucleons .for multiparticle distributions , can be related to the length of the cycle in a cycle class representation of permutations for bosonic or fermionic symmetry .then is the total number of produced particles of a given type such as pions and is the number of cycle classes of cycle length for a permutation of identical particles .these cases have no situation with .if we consider a formation of jets which are followed by a pion creation process , then may be related to the number of pions from a jet . for this case , can also be included as a jet having no pion .once we identify an appropriate for a physical system , then we may use our general model to study the statistical behavior of the system .the various models used in pion distribution can be related to our general model with by choosing as a term in a gauss hypergeometric series ; [ b]_m}{[c]_m } \frac{z^m}{m ! }         \label{hyperf }   \\  { [ a]_m } = \frac{\gamma(a+m)}{\gamma(a ) }   & = & \frac{\gamma(a+n)}{\gamma(a ) } \frac{\gamma(a+n + m - n)}{\gamma(a+n ) }       \label{amfc }   = [ a]_n [ a+n]_{m - n }    \end{aligned}\ ] ] usefull values of ] , = n! ] . considering only positive , we choose {k-1 } [ b]_{k-1}}{[c]_{k-1 } } \frac{z^k}{(k-1 ) ! }    = x \frac{\gamma(a+k-1)}{\gamma(a ) } \frac{\gamma(c)}{\gamma(c+k-1 ) }      \frac{\gamma(b+k-1)}{\gamma(b ) } \frac{z^k}{(k-1)!}\end{aligned}\ ] ] for this case , the thermodynamic grand potential or the generating function is if we allow jets without a pion , then we may allow also .for such a case , {k } [ b]_{k}}{[c]_{k } } \frac{z^k}{k ! }    = x \frac{\gamma(a+k)}{\gamma(a ) } \frac{\gamma(c)}{\gamma(c+k ) }      \frac{\gamma(b+k)}{\gamma(b ) } \frac{z^k}{k ! }  \\  f_0(\vec x ) & = & f_0(x , z ) = \log z(x , z )   = - \omega(x , z ) = \sum_{k=0}^\infty x_k    = x f(a , b ; c ; z)\end{aligned}\ ] ] we can see the only difference of the generating functions between above two cases is the extra factor for the grand potential .we will mostly concentrate on the first case , i.e. , .  using eq .( [ factfm ] ) or the recurrence relation eq.([recfm ] ) and eqs.([hyperf ] ) ( [ amfc ] ) , {m } [ b]_{m}}{[c]_{m } } z^{m+1 }       f(a+m , b+m ; c+m ; z )         \nonumber  \\  & &    + x m \frac{[a]_{m-1 } [ b]_{m-1}}{[c]_{m-1 } } z^{m }       f(a+m-1 , b+m-1 ; c+m-1 ; z)\end{aligned}\ ] ] the second order normalized factorial cumulant and the void variable are then [ b]_2}{[c]_2 } z^2 f(a+2 , b+2 ; c+2 ; z )      + 2 \frac{a b}{c } z f(a+1 , b+1 ; c+1 ; z ) }     { \frac{a b}{c } z f(a+1 , b+1 ; c+1 ; z ) + f(a , b ; c ; z ) }   \\\chi(x , z ) & = & \frac{f_0(x , z)}{<a > }   = \frac{f_0(x , z)}{f_1(x , z ) }   \nonumber  \\   & = & \frac{f(a , b ; c ; z ) }      { \frac{a b}{c } z f(a+1 , b+1 ; c+1 ; z ) + f(a , b ; c ; z)}\end{aligned}\ ] ]  for some values of , , and , the hypergeometric function become a simple function ; various models of pion distributions can be related with these functions as listed in table [ tabl1 ] ..various models with specific choice of and in hypergeometric series of eq.([hyperf ] ) . here is not included , and thus . here with except for poisson which has a finite . [ cols="^,^,^,^,^,^",options="header " , ]   because of a unique role played by the poisson distribution and the form  ] . for in table [ tabl1 ] , the becomes the same probability with replaced by .as an example the hga with \right)$ ] can be decomposed as a sequential process consisting of a nb distribution of clusters with which is then followed by a breakup of clusters distributed with a hga distribution with but without voids , i.e. , with .thus this decomposition separates the parameter assigned to cluster from other parameters .compound distribution such as those considered in the previous section can also be understood using the fact that the laplace transform of is related to the poisson transform for the probability distribution  ; where is the generating function for poisson s distribution and for a poisson distribution , ^n}{n ! }e^{- \bar n }   = e^{-u \bar n}\end{aligned}\ ] ] by a laplace transform or poisson transform , where is the poissonian probability with a mean of .the poisson transform of is and the laplace transform of is .thus is an inverse poisson transform of or an inverse laplace transform of .however the probability which is a poisson transform of may also be considered as a laplace transform of instead of itself .the probability is a superposition of a poisson distribution with weight of .thus we may interprete the with a mean number as the probability in an ensemble of mixed systems with various values of the energy or temperature which is distributed with weight . each of the system with a fixed energy or temperature breakups to give a poisson distribution with a scaled mean number .  since the , we have for the case of andthus & = &   \int_0^\infty dy f(y ) e^{-y \bar n }    = \int_0^\infty dy f(y )       \left[\frac{1}{z^{\rm p}(y\bar n)}\right]\end{aligned}\ ] ] thus is also an inverse laplace transform of with , i.e. , . in general , the laplace transform and its inversion are some examples are      &  ( k > 0 )  \ \ \ \ \ \ { \rm lc } \\   \hline \end{array}\ ] ] for nb with and , thus is the poisson transform of which is the inverse laplace transform of with and and from eq.([laplac ] ) as shown in ref. . with ,the nb becomes bose - einstein ( be ) distribution with .for lc with and ,   \nonumber  \\  & = & e^{2x\sqrt{1-z } } \exp\left[-2\sqrt{x\sqrt{1-z } }       \sqrt{x\sqrt{1-z } +u \bar n}\right ]  \\  z^{-1}(x , z ) & = & \exp\left[-2x(1 - \sqrt{1-z})\right ]   = \exp\left[2x\sqrt{1-z }    \left(1 - \sqrt{\frac{{1 - z + z}}{{1-z}}}\right)\right ]   \nonumber  \\  & = & e^{2x\sqrt{1-z } } \exp\left[-2\sqrt{x\sqrt{1-z } }       \sqrt{x\sqrt{1-z } + \bar n}\right ]  = g(x , z,1)\end{aligned}\ ] ] thus is the poisson transform of \end{aligned}\ ] ] which is the inverse laplace transform of with and from eq.([laplac ] ) .similarly we can also decompose any distribution as a superposition of nb distribution instead of superposition of poisson distribution by replacing in laplace transform by the generating function of nb distribution .that is using mellin transform , with .thus a nb transform for the probability may be defined as a shifted mellin transform for the generator .further investigation of these transformation properties may give interesting features of various distributions .as a special case of mellin transform .event - by - event studies from ultrarelativistic heavy ion collisions such as done at rhic are being used to study the details of particle multiplicity distributions as , for example , those associated with pions .such studies not only give information about the mean number of particles produced , but also information about fluctuations and higher order moments of the probability distribution which are important tools for studying the underlying processes and mechanisms that operate .they are also useful in distinguishing various phenomenological models .issues associated with fluctuations play an important role in many areas of physics and departures from poisson statistics are of current interest .one purpose of this paper was an investigation of various models of particle multiplicity distributions that can be used in event - by - event analysis .this study was done using a generalized model based on a hypergeometric series ( hga ) and uses a grand canonical ensemble as its basic framework .this framework has its origin in a feynman path integral approach and involves a cycle class decomposition of the permutation symmetries that originate from the underlying wavefunctions associated with the produced particles .many of the existing distributions used in particle phenomenology are shown to be special cases of this more general hypergeometric model .several quantities are shown to be quite general ; such as heirarchical scaling relations ( eq.([kappan ] ) ) , initially discused in terms of a particular distribution such as the negative binomial distribution .various models and associated distribution can be developed in a unified way .these include the poisson distribution coming from coherent emission , chaotic emission producing a negative binomial distribution , combinations of coherent and chaotic process leading to signal / noise distributions and field emission from lorentzian line shapes producing the lorentz / catalan distribution which are all shown to be special cases of the hga model . the hga model and its associated special cases are used to explore a wide variety of phenomena .these include ; linked pair approximations leading to heirarchical scaling relations on the reduced cumulant level , generalized void scaling relations , clan variable descriptions and their connections with stochastic variables associated with branching processes , kno scaling behavior , enhanced non - poissonian fluctuations .our results show that even though various distributions have the same mean and fluctuation the distribution itself or the underlying mechanism could be very different .thus to find the correct distribution and underlying mechanism from the data more informations than just the mean and its fluctuation are necessary .the model used here , hga , has three parameters , , , and .thus with a given value of , there is no extra controlling parameter for a given mean and fluctuation .all the higher moments and cumulants are then determined by the model without any further controling parameter .further studies of general models having more parameters are needed where the higher moments can be used as extra conditions in comparing various distributions .we compared various pionic distribution within a generalized hypergeometric model ( hga ) which is a special case of much more general distribution in grand canonical ensemble . in our modelused here for a grand canonical ensemble , the weight factor for each species is the same as the mean number of the species in the grand canonical ensemble .thus we may determine the weight through experimental data .this result also shows that any power law behavior in is directly related to the power law of .it is explicitly shown that kno scaling works only for large fluctuations for all the distribution related to the hga model .comparison within the hga model also shows that just comparing void variables and is not enough to distinguish different models that describing pion data .thus new parameters have to be found which are quite different between different models . beside mean values and fluctuationshigher order reduced factorial cumulants need also to be evaluated .  in this paperwe have also generalized the poisson transformation and the compound distribution that arises from sequential process .specifically , the underlying sequential picture involves a two step process where the final distribution arises from the production of clusters followed by a subsequent decay of the clusters . for the hga model , the final distributionis obtained from compounding a poisson distribution of clusters with a nb distribution coming from the decay of each of the clusters .the hga may arise through a three step sequential process of poisson - poisson - logarithmic compound distribution .it is also shown that the hga can arise from a two step sequential process of a nb distribution followed by a new hga with a different mean value .na49 collaboration , g. roland et al , nucl .a638 * , 91c ( 1998 ) .na49 collaboration , h. appelhuser etal , phys .* b459 * , 679 ( 1999 ) . k. rajagopal and f. wilczek , nucl . phys .* b404 * , 557 ( 1993 ) . s. gavin and b. mller , phys . lett . * b329 * , 486 ( 1994 ) .g. bayer and h. heiselberg , phys .* b469 * , 7 ( 1999 ) . v. heinz and b.v .jacak , ann .nucl . part ., * 49 * , ( 1999 ) .s. pratt , phys .* b301 * , 159 ( 1993 ) .csrg and j. zimanyi , phys .lett . * 80 * , 916 ( 1998 ) .andrews , c.g .townsend , h.j .meisner , d.s .durfee , d.m .korn , w. ketterle , science * 275 * , 637 ( 1997 ) .a. bialas and r. peschanski , nucl . phys . *b272 * , 703 ( 1986 ) ; 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Abstract: various phenomenological models of particle multiplicity distributions are discussed using a general form of the grand canonical partition function . these phenomenological models include a wide range of varied processes such as coherent emission or poisson processes , chaotic emission resulting in a negative binomial distribution , combinations of coherent and chaotic processes called signal / noise distributions , and models based on field emission from lorentzian line shapes leading to lorentz / catalan distributions . these specific cases can be written as special cases of a more general distribution . using this grand canonical approach moments and cumulants , combinants , hierarchical structure , void scaling relations , kno scaling features , clan variables and branching laws associated with stochastic or ancestral variables are discussed . it is shown that just looking at the mean and fluctuation of data is not enough to distinguish these distributions or the underlying mechanism . a generalization of the poisson transform of a distribution and the poissonian decomposition of it into a compound or sequential process is also given .


in quantum metrology quantum properties such as squeezing and entanglement are employed to improve the precision with which physical quantities can be measured  .quantum metrological techniques have been very fruitful in developing new generation of quantum devices that can outperform their classical counterparts . in particular , the framework of quantum metrology is very useful for measurement of physical quantities that do not have an associated operator in quantum theory such as time , phase , temperature , acceleration , etc .the process of quantum parameter estimation consists of three stages ; the preparation of the probe state of the system , feeding the prepared state into the quantum channel which encodes the parameter of interest into the state of the system and finally decoding the parameter by performing a measurement on the system after it has gone through the channel . in order to reduce the statistical error in the estimation of a physical quantity, this whole process needs to be optimised .this is achieved by optimising over both the preparation of the initial state of the system and the measurement of the final state of the system . for a given prepared state ,the quantum cramr - rao bound provides us with the ultimate precision bound on the measurement of the physical quantities in quantum theory .+ holevo and helstrom laid the foundations of quantum metrology by phrasing the problem in the context of a communication protocol between two parties  . in this paradigm alicechooses a quantum system to encode a message which she then sends to bob .for example she might choose to use either a spin- system or a quantum harmonic oscillator as carrier of her message .the message can be encoded as a phase parameter in the state of the spin- particle or in the state of the quantum harmonic oscillator .bob then performs a measurement on the system in order to decode the message , i.e. the encoded phase parameter .the quality of this communication protocol can be improved by optimisation of both stages of the protocol , namely alice s encoding process and bob s decoding process .however , standard approaches to quantum communication , such as encoding qubits into polarisation degree of freedom of photons , require that all parties have knowledge of a shared reference frame .this means that in the absence of such knowledge , the involved parties need to initially establish aligned reference frames . despite the considerable amount of progress in the development of protocols for aligning reference frames such as clock synchronisation and cartesian frame alignment  , maintaining aligned reference framesis still a large obstacle in achieving such tasks .for instance when the parties are in relative motion with respect to each other , the relative orientation of their local reference frames can change in time  . however , quantum reference frames ( qrf ) enable us to circumvent this problem .let us briefly explain what qrfs are and in what way they differ from classical reference frames ( crf ) .+ aharonov and susskind in their seminal papers , showed that the concept of reference frame can be suitably accommodated in quantum theory . in recent years , such treatment of reference frames in quantum theory , i.e. as quantum objects has led to the formalism of `` _ _ quantum reference frames _ _ '' .a qrf is different from its classical counterpart in two ways .first , due to its quantum nature , it has an inherent uncertainty and the measurement results are only an approximation of what would be obtained using a classical reference frame .second , each time the qrf is used , it suffers a back - action , which causes the future measurements to be less accurate .phase measurement of single - rail qubits relative to a qrf has been investigated in  , while the degradation of a directional qrf has been analysed in  . in the past few years , the `` _ _ resource theory of quantum reference frames _ _ '' , also known as the `` _ _ resource theory of asymmetry _ _ '' , has been developed .this resource theory provides us with a very useful framework wherein the qrfs are the `` _ _ resource states _ _ ''they enable us to achieve quantum information processing tasks without first establishing a shared reference frame .in such schemes , a qrf stands in for the possibility of performing tasks in the absence of a common crf , in the same way that entangled states allow for the possibility of performing non - local quantum operations .+ in this paper , we bring to bear the powerful machinery of quantum metrology to study the ultimate precision bounds in measurement of physical parameters with respect to qrfs .first we explain the connection between the quantum mechanical treatment of reference frames and quantum parameter estimation in the presence of noise .then we investigate how the ultimate precision in measurement of a parameter decreases due to inaccessibility of a perfect crf . in order to do so , we analyse the decrease in quantum fisher information as a result of not having access to a perfect reference frame for the physical quantity of interest . in particular, we provide necessary and sufficient conditions for two extreme cases that can occur in quantum parameter estimation with imperfect frames of reference .the first case is when the absence of a perfect reference frame does not affect the precision with which one can measure the parameter and the second case is when measurement of the parameter of interest is no longer possible due to not having access to a crf .motivated by this analysis we split the problem into two separate cases .the first case is when the noise caused by the quantum nature of the rf commutes with the channel of interest and the second case is when the channel of interest and the noise do not commute .we show how the non - commutative noise allows for the estimation of the parameter even in the absence of dfss .moreover , we explain the connection between noisy quantum metrology and alignment - free quantum communication . in the end , we present three examples in order to further clarify different aspects of quantum metrology with imperfect frames of reference .the structure of this paper is as follows : in section [ qcr ] we briefly review some of the mathematical tools from quantum metrology , in particular quantum fisher information ( qfi ) and symmetric logarithmic derivative ( sld ) . in section [ afc ] we represent the general scheme of the alignment - free communication protocols . in section [ qmqrf ]we present the general framework for quantum parameter estimation in the absence of an ideal crf and we discuss its relation to the alignment - free communication protocols . in section [ ex ]we present three examples in which we explain different aspects of quantum parameter estimation in the absence of aligned crfs . finally in section[ do ] we discuss the results of this paper and we mention some of our research interests in quantum parameter estimation as possible future work .the main goal in quantum metrology is to estimate an unknown parameter of a completely positive trace - preserving ( cptp ) quantum channel , . in order to do so, first a probe state is prepared which is then fed into the channel of interest .finally a measurement on the final state of the probe will enable us to estimate , where is a physical quantity such as time , phase , temperature , acceleration , etc . given a measurement strategy , the conditional probability of obtaining outcome when the initial state is given by , where are elements of a complete positive - operator valued measure ( povm ) corresponding to the chosen measurement strategy ( see fig.[qcrb ] ) .the lower bound on how precise we can estimate is given by the `` _ _ classical cramr - rao bound _ _ '' , i.e. , where the classical fisher information is defined as  ^{2},\ ] ] and is the number of repeated measurements . in other words , classical fisher informations is an operational measure which tells us how much information we can gain about the unknown parameter by choosing a certain measurement strategy .+   of the channel , a probe state is fed into the channel of interest .then a measurement is performed on the final state of the system which makes it possible to extract information about the parameter of interest . ]braunstein and caves showed that optimisation over all the possible quantum measurements provides an even more stringent lower bound  , i.e. [ cramer - rao ] n()^2 , where is the quantum fisher information .this quantity is closely related to the symmetric logarithmic derivative which is defined by . in particular in the basis in which is diagonalised , i.e. , the sld and the qfi can be written as      with the relation   the summations above do not include the terms with . the optimal povm which achieves the quantum cramr - rao bound , can be constructed from the eigenstates of  .+ note that in this section the existence of ideal crfs was assumed .this treatment of reference frames can lead to great deal of confusion .for instance , in the role of an external phase reference frame in interferometric setups has been analysed . in this paperwe analyse the decrease in quantum fisher information as a result of not having access to a perfect reference frame for the physical quantity . in particular , we present necessary and sufficient conditions for two extreme cases ; the case where the qfi does not decrease when a crf is lacking and the case where the qfi vanishes due to imperfectness of the rf , i.e. one can no longer extract the parameter .as mentioned earlier , quantum parameter estimation can be phrased as a communication protocol between two parties . in this sectionwe briefly review how qrfs have been employed in order to achieve alignment - free communication protocols  .+ consider to be the group element that describes the passive transformation from alice s to bob s reference frame .furthermore , since bob is completely unaware of the relation between his local rf and alice s local rf , we can assume that the group element is completely unknown .it follows that if alice prepares a state relative to her local reference frame , then relative to bob s rf this state is seen as .we refer the readers for more details to . ]=\int { \mathrm{d } } g u(g ) \rho _ { a}u(g)^{\dag}.\ ] ]  therefore , lacking such a shared reference frame is equivalent to having a noisy completely positive trace - preserving map which is known as the `` _ _ g - twirling map _ _ '' , i.e. .however , despite the fact that alice has no information about the group element that relates her local rf to bob s local rf , she can still encode information in the so called `` _ _ decoherence - free subsystems ( dfs ) _ _ ''  .these subsystems are resilient to the decoherence caused due to the lack of knowledge about the relative direction of the local reference frames .the efficiency of this protocol depends on the dimensionality of the largest dfs , i.e. the subsystem which possesses the largest number of degenerate eigenstates .such communication scenarios in the absence of a shared cartesian reference frame have been analysed before  .the idea is to encode logical qubits into rotationally invariant states of multiple physical qubits . in this casethe number of logical qubits per number of physical qubits that can be transmitted scales as , where is the number of transmitted physical qubits .this remarkable result proves that in the limit of one logical qubit can be sent per one physical qubit . therefore in this limitthe efficiency of this scheme is the same as the scenario wherein the reference frames are aligned .this protocol has also been studied for the situation in which the parties do not share a common background phase reference frame  .as we explained in section [ qcr ] , the standard scenarios considered in quantum metrology normally presume the existence of perfect classical reference frames . in this sectionwe investigate how the ultimate precision in measurement of a parameter decreases due to lack of access to a perfect crf .+ let us first briefly explain the general picture of the estimation of a parameter in the absence of a perfect external rf .we consider the case where the parameter is encoded into the fiducial state via a unitary channel .after this encoding process , we need to choose the optimal measurement in order to extract the maximum amount of information about .we then need a suitable rf with respect to which we are able to perform the chosen measurement .for instance if we wish to measure time we need a clock or if we need to measure phase we will need a phase reference frame .the absence of such reference frames can be viewed as a noisy quantum channel , i.e.  =\lim_{t\rightarrow\infty } \frac{1}{t}\int_{0}^{t } dt\ u(t)\rho_{\lambda}u(t)^{\dag},\ ] ]  where , , and .note that and are determined by the type of rf that is lacking .for instance if we need to measure phase of a quantum harmonic oscillator then is a phase and , where is the total number operator  .+ assuming the spectral decomposition , where are the projectors into subspaces with eigenvalues and , one can easily check that the state  ] , this general noise reduces to the well - known `` _ _ collective dephasing noise _ _ '' with uniform prior probability  .in fact in the quantum information protocols considered in  and the references therein it is assumed that =0 ] .we call such noise non - commutative .the second is when when the noise is commutative , or more specifically , systems with mixed fiducial state .] , i.e. =0 ] and that the operator has a non - degenerate spectrum , i.e. all the projectors are rank- projections , then in the absence of a perfect crf all the information about will be lost ( for details see appendix [ alternative ] ) .this is no longer the case when the two operators do not commute .this means that , even though the decoherence - free subspaces are crucial for successful encoding of parameter in the commuting case , such subspaces are not necessary in the non - commuting case . as we will explain in section [ rafc ] , this fact stands in for the possibility of alignment - free communication whenever the spectrum of is non - degenerate .we will present examples of these two different cases , i.e. commutative and non - commutative noise in section [ ex ] .+ we can write the no - loss condition in a more intuitive way as   where the covariance of two observables and is defined as , and the variance can be written as .covariance is a measure of correlations between two observables and with respect to the state . multiplying this equation by the eigenvalues and summing over all the indices , we can write the necessary condition for not loosing any information about as this means that if operators and are correlated with respect to the pure initial state , i.e. , then some information is lost due to the imperfectness of rf , i.e. .it is worth emphasising that this condition is not sufficient .as an example consider the operators , , and the fiducial state . in this examplethe covariance between and is zero , nevertheless , since and commute and the fact that no non - degenerate subspace exists , we will not be able to extract any information about . +similar to the no - loss condition in , using eq .( [ maxloss ] ) the necessary condition for the extreme case of loosing all the information can be written as \rangle}_\rho=0.\ ] ] this means that if for a given initial state \rangle}_\rho ] is ) & = 4\mathrm{var}_{\rho_{\lambda}}(\hat{k})-4\sum_i p_i\left[\mathrm{cov}_{\rho_{\lambda}}\left(\frac{\hat{p}_i}{p_i},\hat{k}\right)\right]^2 , \end{split}\ ] ] where , and .  from eq .we deduce that the decrease in qfi is proportional to the mean of squared covariances between the normalized ] projectors and the encoding operator .this means that the more projectors are correlated with the encoding operator , the more precision is lost . roughly speaking , in order to lose the minimum amount of precision one should choose an encoding operator which is less correlated with the decoherence caused by the noisy channel . for an explicit examplewe refer the readers to the third example of section [ ex ] .+ for the special case of a commutative noise , i.e. when and commute , the expression can be further simplified . in this casewe have , where .this causes the qfi in the absence of a perfect rf to reduce to )=\sum_i p_i h_u(\rho_i) ] .as explained in the previous section this decoherence effect can be taken into account by analysing the efficiency of communication in the presence of noise , i.e. we can assume that alice and bob have access to a shared crf but they only have access to a noisy quantum channel as their means of communication , as depicted in figure .the type of noisy channel is dictated by the type of reference frame that the parties do not share .for instance , in the case of cartesian reference frame , the generators are the generators of the group so(3 ) .motivated by the analysis of the previous section , again we split the problem into two subproblems .+ the first case is the case of commutative noise , i.e. when and commute . in this casethe noisy quantum channel commutes with the unitary encoding process , i.e. =u_{\lambda}\g[|\psi_{\lambda}\rangle\langle\psi_{\lambda}|]u_{\lambda}^{\dag} ] . as an example consider the case where alice and bob do not share a phase reference frame .if alice encodes only using a single harmonic oscillator , then the states that she can prepare are of the general form and the operator is the number operator .it is easy to check that in this case these states get completely decohered from bob s point of view .in contrast if alice chooses two quantum harmonic oscillators as the carrier of her message and the operator , then the states of the form are invariant under the action of the g - twirling map . for an explicit example of this casewe refer the readers to the first example given in section [ ex ] .+ the second case is the case of non - commutative noise , i.e. when the operators and do not commute .this case is particularly interesting since estimation of the parameter is possible even in the absence of dfss as explained in section [ correlation ] . in the second and third example of section [ ex ] , we present two scenarios in which the absence of an ideal crf results in non - commutative noise . +   in the presence of a shared crf between alice and bob.,title="fig : " ]  in the presence of a shared crf between alice and bob.,title="fig : " ]   in the previous sections we analysed how qrfs modify our precision in measurement of physical parameters such as time , phase , direction in space , etc .we also explained how misalignment of local rfs is connected with commutative and non - commutative noise in quantum parameter estimation .we are now in place to present some explicit examples .   and for a squeezed , dispalced vacuum state , i.e. , as the initial state of the qrf .paramter denotes the fraction of mean energy due to displacing the vacuum , i.e. . ]   the scenario that we consider in this example is as follows .alice and bob do not have access to synchronised clocks , i.e. they do not share a common classical rf for time .alice prepares a state , where and is the operator which imprints the parameter into the fiducial state .since the local clocks of the parties are not synchronised , in bob s frame the state of the system is given by eq ., where and is the hamiltonian of the qubit and the .the operators are the projectors into subspaces with total energy .we analyse the quantum fisher information of the state  ] .note that this example is similar to the quantum communication scheme between two parties when they do not have a common phase reference frame as was considered in  .+ using eq ., it is straightforward to find the qfi in alice s frame as .note that alice s qfi is independent of the state of the qrf . on the other hand ,if we consider the state , then using either eq .  or eq ., we find the qfi in bob s frame as if alice chooses a uniform superposition of fock states , i.e. the state , then using  we can easily compute bob s qfi as . using eq . , we find the elusive sld for this case as   , and the optimal state . ]the sld provides us with the optimal observable to measure in order to minimise the statistical error in measurement of and saturate the quantum cramr - rao bound .this can be easily verified by checking that satisfies the condition .let us next consider a squeezed , displaced vacuum state  as the state of the qrf , i.e.   where is the hermite polynomial , is the squeezing parameter , is the displacement parameter and .the mean energy of this state is equal to .we define parameter as the fraction of initial mean energy due to displacing the vacuum , i.e. .note that with this definition , and represent a squeezed state and a coherent state respectively .in particular , noticing that in the fock basis a squeezed state is of the form together with eq . , we find that )=0 ] . as mentioned earlier whenever these two operators do not commute , even in the absence of degenerate subspaces of total energy, we may still be able to estimate the parameter .for simplicity we assume that frequency is not a fraction of the interaction strength , i.e.  this assumption ensures that the hamiltonian does not possess any degenerate eigenvalues . in order to make the computations easier , we change of the basis as  this change of basis allows us to write the hamiltonian as with the eigenvectors which can be written in terms of the fock basis as   that related his other two axes to alices.,scaledwidth=30.0% ]now let us consider that the qrf is initially prepared in the uniform superposition of fock states .using eq ., we derive the qfi of the averaged state as where , and is a floor function .+ in this example since and do not commute , ) ] in terms of for increasing values of the mean energy in the state of the qrf .the maximum and minimum of the qfi occurs at and respectively .note that even for very large qfi does not approach the ideal case .in other words , even in the limit of very large mean energy in the initial state of the quantum clock , we can not estimate the phase parameter as precise as we could if we had access to a classical clock .this can be proved using necessary conditions and .one can easily check that , which means that independent of and , the qfi is always smaller than one , i.e. )<1 ] , we can deduce that independent of for , the qfi is always positive .   via rotating the fiducial state around the unit vector .for the state of the qubit in bob s frame is and bob s qfi is the same as alice s . for , is .note that is the projection of onto the z - axis .also note that in the latter case bob s qfi is -dependent(see figure [ bobsqfiexample3 ] ) . ]   in the first example we observed how using a qrf enables alice and bob to perform an alignment - free communication protocol in the presence of commutative noise , while in the second example we analysed how the absence of a perfect crf can cause non - commutative noise which then reduces the precision with which a physical parameter can be estimated . here , we present an example in which the noise caused due to bob s lack of knowledge about alice s local reference frame is non - commutative .we analyse how precise bob can extract if alice does not send him a quantum sample of her local rf .+ let us start with the case where alice wishes to both encode and decode a parameter herself .she chooses a spin- particle as the physical system to encode a parameter and then she encodes this parameter using a unitary channel with the generator this is the generator of a general rotation in the bloch sphere around the axis , where and are real parameters . for simplicitywe choose the fiducial state to be the eigenstate of with eigenvalue , i.e. . using euler s formula for pauli matrices ], we can write alice s prepared state as   then using eq ., the qfi in alice s frame reads as note that for , the corresponding generator is which leaves the fiducial state invariant , i.e. . since the encoding process is not successful , the qfi , , vanishes which simply means that a different generator needs to be used at the preparation stage .the qfi takes its maximum value when when the parameter is encoded via a rotation around any vector in the -plane , i.e. when . +   and for general . ]now suppose that alice and bob only share their -axis , i.e. bob is completely unaware of the relative angle between his other two axes and alice s , as depicted in figure [ bobrotated ] . in this case , is the generator of rotations around -axis , i.e. . using eq ., the qfi in bob s frame can be written as   again note that for , the qfi is zero in bob s frame .this is expected , since bob lacks some information with respect to alice , therefore alice s inability in extracting information about means that bob will not be able to decode the message either , i.e. . on the other hand , as can be seen from , when the qfi is the same in alice s frame and bob s frame .figure [ blochspheres ] depicts the two cases of and . for the former case ,the efficiency of communication is -independent , whereas for the latter case it is -dependent , as can be seen in figure [ bobsqfiexample3 ] . in this figure , we have plotted bob s qfi in terms of and for general .we observe that as approaches the value , the qfi approaches its minimum value , i.e. .in other words , for the chosen encoding operator and the fiducial state , bob will not be able to distinguish form its neighbouring states , where is a very small change in . + also after some algebra and with the aid of eq ., we find the sld operator that achieves the qfi in  as + again the optimal povm can be constructed from the eigenvalues of this operator , i.e. .this simply means that the most informative measurement for bob is the measurement in the computational basis . in order to verify that this is in fact the case, we can either use the relation  or we can compute the classical fisher information using eq .  and show that it is equal to the qfi .in quantum metrological schemes the existence of a prefect classical reference frame is often assumed .here we have exploited the powerful mathematical tools from quantum metrology in order to analyse the modification of the ultimate precision limits due to the absence of such frames of reference .we considered the effects of commutative and non - commutative noise due to lack of a certain crf . in doing so , we showed that the more the encoding process and the nature of the noise resemble each other , the more precision is lost .we also presented necessary and sufficient conditions for two extreme cases .the first case is when the absence of an ideal rf does not reduce the accuracy of estimation and the second case is when the estimation of the parameter with respect to qrfs is no longer possible .moreover , by explaining the connection between noisy parameter estimation protocols and alignment - free communication schemes  , we shed light into different aspects of quantum communication in the absence of aligned reference frames .+ our future line of research includes incorporating other sources of noise in the alignment - free communication protocols .we are interested in the regimes where relativity starts to play a more significant role  .relativistic effects such as the decoherence caused due to non - uniform motion  or the effects of the gravitational field of the earth  are the possible sources of noise that yet need to be considered . recently in techniques for the optimal estimation of parameters which appear in quantum field theory in curved spacetime have been presented .this enables the estimation of parameters such as proper acceleration , proper time , relative distance , amplitude of gravitational waves  , as well as spacetime parameters of interest , such as the expansion rate of the universe or the mass of a black hole .finally , the reference frames of relativistic observers is inevitably misaligned with respect to non - relativistic observers , therefore it is crucial to consider the effect of relativistic noise on estimation of parameters of interest such as acceleration , time , phase , temperature , etc . + _acknowledgements_:we thank antony lee , katarzyna macieszczak and carlos sabn for useful discussions and comments .m. a. and i. f. acknowledge support from epsrc ( caf grant no .ep / g00496x/2 to i. f. ) .from eq . , we immediately observe that eigenvalues of transformed density matrix are with respective normalised eigenvectors .let be a set orthonormal eigenvectors of with respective eigenvalue .using eq .( [ qfi ] ) we have and together with the parseval identity , i.e. we can remove the dependence on states . then is after substituting  where for -independent basis .the sum in ( [ prefinal ] ) consists only of elements where , however , by differentiating and using schwarz inequality on we get . now summing over all and using the completeness relation we get _ symmetric logarithmic derivative _ can be derived analogously , where instead of parseval identity we use completeness relation .here we prove that and the equality conditions . follows immediately from definition .let us prove .looking at the expression for qfi loss , i.e. eq . , we need to prove that first , let us define .then using the fact that the state is normalised , i.e. , the schwarz inequality and that for any state , if and only if and therefore , together with the completeness relation , we have where for the last step we have used eq . . now , because schwarz inequality is saturated if and only if there exists a complex number such that , by re - writing the state as we find that eq .is saturated if and only if which together with orthogonality condition for the projectors leads to the no - loss condition , i.e. now let us derive the max - loss condition . from the definition of quantum fisher information loss and that , we can write therefore by comparing ( [ lossformula ] ) and ( [ paris ] ) we have similar to the previous case we can write in the complete orthonormal basis as where span the rest of the hilbert space which is not spanned by vectors . after multiplying by we get the parseval identity , i.e. comparing this with ( [ nolossformulaappendix ] ) we get condition for max - loss as  we introduce alternative an description of quantum fisher information using eigenvectors of the operator .let be a set of orthonormal eigenvectors of with respective eigenvalue .then the projection operator into the eigenspace corresponding to eigenvalue can be written as then we substitute this expression into and find the qfi in terms of the eigenvectors as in order to write this expression in a more compact way we define the unnormalised states ,   where denotes the dimension of the subspace corresponding to eigenvalue .this way we can write eq .  as where is a standard inner product on .+ let us here also prove that when and commute and the spectrum of is non - degenerate , then , i.e. all the information about is lost due to the noise .this can be done either starting from ( [ eigenformula ] ) or by using max - loss condition ( [ maxloss ] ) . herewe choose the latter . because has non - degenerate spectrum , the projectors are all rank - one projectors , i.e. . also since and commute, then all projectors also commute with , i.e. =0 $ ] , where is a complete basis .consider a subset of this basis that do not have any overlap with the state , i.e. the subset .this set replaces in the max - loss condition ( [ maxloss ] ) .now we have which makes the proof complete .   in terms of . from top to bottom for and . ]in the case that the two operators and commute and in the limit of large initial mean energy in the state of the qrf , the classical limit of a quantum reference frame should be recovered , i.e. , as was shown in the first example for the uniform superposition of fock states .here we analyse this asymptotic behaviour for a coherent state as the initial state of the qrf .we expect such that .as figure suggests , this is true for large enough mean photon number in the initial state of the coherent state .moreover , from this figure one can see that , for the mean photon number above a certain threshold , we can find constants and such that   re - arranging the expression above , we can find -dependent lower and upper bounds for the function as therefore the qfi in bob s reference frame is where .choosing and , we tighten the lower and upper bounds on , i.e. .                           p. gibilisco , d. imparato , and t. isola , http://scitation.aip.org/content/aip/journal/jmp/48/7/10.1063/1.2748210 [ j. math . phys . * 48 * , 072109 ( 2007 ) . ] s. luo , http://dx.doi.org/10.1007/s11232-005-0098-6 [ theor . math . phys . , * 143 * , 681688 , ( 2005 ) . ]

Abstract: quantum metrology studies quantum strategies which enable us to outperform their classical counterparts . in this framework , the existence of perfect classical reference frames is usually assumed . however , such ideal reference frames might not always be available . the reference frames required in metrology strategies can either degrade or become misaligned during the estimation process . we investigate how the imperfectness of reference frames leads to noise which in general affects the ultimate precision limits in measurement of physical parameters . moreover , since quantum parameter estimation can be phrased as a quantum communication protocol between two parties , our results provide deeper insight into quantum communication protocols with misaligned reference frames . our framework allows for the study of general noise on the efficiency of such schemes .