adaptive optics ( ao ) systems may provide diffraction - limited imaging with ground - based telescopes , measuring the wavefront aberrations due to the atmosphere by means of a suitably bright reference star and compensating for these aberrations by means of a deformable mirror . in classical ao systems , based on a single natural reference star , such a star has to be located close to the target object , i.e. within the isoplanatic patch , so that the probability of finding it is quite low .this probability defines the fraction of the sky where an efficient correction can be achieved ( sky coverage ) , and for near infrared wavelengths it is few percent . in this contextthe potential power of laser guide star ( lgs ) ao , proposed by , is clear .the basic idea is to generate a bright guide star in the upper atmosphere by means of a laser .two alternative physical phenomena may be exploited for such a purpose : rayleigh scattering at low - medium altitude , up to 10 - 20 km , or excitation of the atoms in the sodium layer located at approximately 90 km , characterized by a thickness of 10 km .although low - altitude rayleigh scattering can provide much more backscattered light , the higher altitude of the mesospheric sodium layer reduces the cone effect , i.e. maximizes the atmospheric volume sampled by the artificial source .the use of lgs in principle solves the problem of the lack of bright reference sources , but other difficulties arise .apart from the above mentioned cone effect , that might be compensated using multiple reference sources , the problem analyzed in this paper is related to the perspective elongation of the lgs image , due to the vertical extension of the artificial source . referring , for instance , to a shack - hartmann ( sh ) wavefront sensor ( wfs ) , the subapertures located far from the laser launcher see an elongated lgs image , pointing towards the laser launcher .the angular elongation is given by ( see figure [ elongation ] ) since and are given by equation , in the small angles approximation , becomes where is the thickness of the sodium layer , is the mean altitude of the sodium layer and is the distance between the pupil subaperture and the lgs launcher ( figure [ elongation ] ) .this elongation can dramatically affect the capability to measure the wavefront , especially in the case of large diameter telescopes .( in this case at the edge of the telescope aperture ) from the launch site . represents the thickness of the sodium layer , is the mean altitude of the layer and is the angular elongation . ]  on a 30 - 40 m class telescope , like the tmt and the e - elt , the maximum elongation varies between few arcsec and 10 arcsec , depending on the actual telescope diameter , on the sodium layer properties and on the laser launcher position .  through numerical simulations , verified by means of analytical formulas , we compute the measurement error on the optical path difference ( opd ) per subaperture , considering the elongation pattern on a sh wfs for the 42 m e - elt , in the case of a laser guide star launched either from behind the secondary mirror ( i.e. from the center of the pupil ) or from the edge of the primary mirror . in section [sect : parameters ] we describe the simulations and the input parameters .the performance analysis of a specific algorithm ( weighted center of gravity ) is the subject of section [ sect : subnoise ] . in sections [ sect :propagation ] and [ sect : results2 ] we propagate the measurement error on the opd per subaperture to rms wavefront error over the whole pupil , hereafter referred to as wfe , assuming a modal least square wavefront reconstruction on a karhunen - loeve ( k - l ) basis .the resulting wfe is evaluated as a function of number of photons and read - out - noise ( ron ) .an estimation of the number of detected photons needed to achieve a pre - fixed wfe per lgs is given in section [ sect : results2 ] .in order to understand how the image of a lgs looks like as seen from each subaperture of a shack - hartmann wavefront sensor , it is important to choose a model of the sodium density profile .the density of the sodium layer can change by a factor of two or more depending on the telescope site and the time of the year .its mean altitude can change on a time scale of hours .we consider two different sodium profile models ( figure [ profile ] ) , with a shape similar to those presented in :  * single gaussian sodium density profile , peak at km and fwhm km ; * bi - modal sodium density profile given by the sum of two gaussians with the following characteristics : peak at km and fwhm km , peak at km and fwhm km .the described sodium profiles have been used to simulate the spots seen by a sh wfs .some examples of simulated spots are shown in figure [ spots ] .we consider two lgs launching schemes : from behind the secondary mirror ( central projection ) and from the edge of the primary mirror ( lateral projection ) .central projection gives a perspective elongation pattern with radial symmetry , while lateral projection leads to a non - symmetric pattern with maximum elongation two times larger than in the former option .the peak density of the sodium layer is imaged at the center of the field of view ( fov ) of each subaperture .the lgs spot imaged by each subaperture is modeled as follows :  * the starting point is a line source of length calculated accordingly to equation [ eq : na ] and with intensity distribution given by the adopted sodium density profile ; * the line source is convolved by a seeing - like disk of angular size arcsec , taking into account the round - trip propagation of the beam through the turbulent atmosphere ( seeing = 1 arcsec ) .this angular size corresponds to the width in the non - elongated axis .the subaperture diffraction ( arcsec ) is not considered in the spot model , as it is negligible with respect to the seeing effect , that always dominates over diffraction since the ao correction at the laser wavelength is quite poor .we generate sequences of instantaneous spots : each spot is characterized by a random jitter and is contaminated by photon noise and read - out noise ( ron ) .the random offset is extracted from a gaussian distribution , whose standard deviation may be tuned to simulate open or closed loop conditions .the photon noise is added to each pixel accordingly to a poisson distribution with a mean value equal to the pixel intensity ; the ron is extracted from a gaussian distribution of zero mean .different values of focal plane sampling and subaperture fov may be adopted in the generation of the sequence , in order to evaluate their impact on the wfs performance .depending on the chosen subaperture fov , the wings of the most elongated spots , especially for the the subapertures at the edge of the pupil , may fall outside the fov of the subapertures themselves . in a real system, this would translate into a light contamination of adjacent subapertures , a drawback that may be overcome using simple light baffling methods , for instance a diaphragm on the lgs image in an intermediate focal plane before the wfs .this baffling is always implicitly assumed here , so that the limited subaperture fov translates into a truncation of the spot wings , but not into a contamination of adjacent subapertures .the number of photons per subaperture referred to in the following includes also the photons blocked by the subaperture baffle .the sequence of instantaneous spots , typically formed by 500 realizations , is analyzed by a spot position measurement algorithm .the algorithm result is compared to the known true position , thus obtaining a sequence of position differences , the rms of which is the figure of merit for the algorithm performance evaluation .each instantaneous spot is tested for detectability . for this purposetwo checks are implemented : the maximum intensity value of the spot is requested to be at least twice the ron and the calculated position has to be within from the true position , where is the standard deviation of the best fit gaussian to the long - exposure spot , obtained by summing all the images in the sequence . when more than a certain fraction of images ( in our simulations 50% , following thomas et al .2006 ) are rejected , we consider the centroid measurements to have failed and not be reliable for those light conditions .  .simulation parameters .[ cols= " < , < " , ]the effect of the lgs perspective elongation on the performance of a shack - hartmann wfs for a large aperture telescope has been analyzed in this paper .the weighted center of gravity algorithm has been used for the spot position measurement in the subapertures . the opd measurement error in the subapertureshas been propagated to wfe ( rms wavefront error ) across the whole pupil , assuming a modal least square wavefront reconstruction ; a fast method to perform this error propagation has been proposed and validated , improving the efficiency of the calculation for a system with a very large number of subapertures . the method proposed hereis based on a single channel approach , i.e. it allows the computation of the propagated wavefront error per single lgs , starting from the first order properties of the wfs ( number of subapertures , number of detected photons per subaperture , ron , focal plane sampling , subaperture fov , calculation window ) .the analysis presented in this paper has been tuned for the specific case of a mcao module for the e - elt , assuming a target wfe per lgs of 200 nm .different sodium profiles have been considered : gaussian and bi - modal profiles .it has been found that , with a suitable optimization of the wcog algorithm , performance is quite similar in the two cases .two laser projection schemes have been considered : central and lateral projection . in this framework , assuming for instance a reasonable detector noise ron = 3 , it has been found that approximately detected photons per subaperture per frame are required for the central laser projection . for comparison , the photon return of the keck lgs system , properly adapted to the case of the mcao system for the e - elt considering all relevant parameters , would correspond to approximately ; therefore the number of detected photons required to achieve our target wfe is a factor larger . for the lateral laser projection, the required number of photons is , i.e. a factor larger that for the central projection .the choice between central and lateral projection depends not only on the number of required photons , but also on other aspects ( contamination due to the rayleigh scattering of the laser light in the lower atmosphere , size of the isoplanatic angle compared to the perspective elongation ) , that are beyond the scope of this paper .a notable reduction in the required number of detected photons ( ) might be obtained by canceling the perspective elongation by means of a dynamic refocus scheme combined with a pulsed laser ( baranec & dekany 2008 , beckers et al .2003 , georges et al .2003 ) , which is technologically challenging , or range gating , that however wastes a non negligible fraction of laser power .other approaches are under investigation to achieve similar results without fast moving parts .a detailed analysis of the influence of the ron has been performed . assuming a more conservative value ron = 5 , the required number of detected photons increases by a factor with respect to the case ron = 3 .another important aspect that has been analyzed is the number of pixels per subaperture , that determines the detector size . with typical sodium profiles , assuming nyquist sampling in the non elongated direction and central laser projection , the minimum number of pixels per subaperture is , so that 84 subapertures across the diameter ( table [ tab : parameters1 ] ) corresponds to a detector size of pixels , that doubles in the case of lateral laser projection .this detector format , especially combined with a ron = 3 or better looks a demanding requirement .  the analysis presented in this paper is based on the wcog algorithm .however it is worth comparing this method to other algorithms ( e.g. correlation maximization , quad - cell approach ) with the aim of relaxing the requirements on the number of photons and on the detector specifications .additional aspects to be considered in the evaluation of the algorithms are the variation of the sodium profile with time and the impact of the differential aberrations between science and lgs channel , that usually depend on the zenithal angle variation during the exposure .the sodium profile variations imply continuous updating of the algorithm reference ( the weighting function and the calibration curve in the case of the wcog ) ; the differential aberrations may translate into time dependent spot offsets in the subapertures , that need to be compensated or calibrated .the evaluation of different algorithms , considering these additional issues , is the subject of an additional work in preparation .the authors thank r. clare , n. hubin , t. fusco , m. le louarn , e. marchetti , c. petit , m. tallon for useful discussions .this work was supported by the european community ( framework programme 6 , elt design study , contract n. 011863 ; framework programme 7 , preparing for the construction of the european extremely large telescope , contract n. 211257 ) .baranec , c. and dekany , r. , 2008 , applied optics , 47 , 5155 beckers j. m. , 1993 , annu . rev .astron . astrophys . , 31 , 13 beckers , j. m. , owner - petersen , m. & andersen , t. , 2003 , spie proc , 5169 , 123 diolaiti e. et al , 2008 , spie proc . , 7015 , 70150u drummond j. , telle j. , denman c. , hillman p. , spinhirne j. , christou j. , et al ., 2004 , pasp , 116 , 952 foy r. , labeyrie a. , 1985 , a&a , 152 , 129 fugate r. q. , fried d. l. , ameer g. a. , boeke b. r. , browne s. l. , roberts p. h. , ruane r. e. , tyler g. a. , wopat l. m. , 1991 , nature , 353 , 144 fusco t. , , 2004 , spie proc ., 5490 , 33 gardner c. s. , welsh b. m. , thompson l. a. , 1990 , ieee proc . , 78 , 1721 georges j. a. , mallik p. , stalcup t. , angel j. r. p. , sarlot r. , 2003 , spie proc ., 4839 , 473 gilmozzi r. , spyromilio j. , 2008 , spie proc ., 7012 , 701219 happer w. , macdonald g. j. , max c. e. , dyson f. j. , 1994 , josa a , 11 , 263 hardy j. w. , 1998 , adaptive optics for astronomical telescopes , oxford university press martin , o. et al . , 2008 , spie proc , 7015 , 70154e nelson j. e. , sanders g. h. , 2006 , spie proc . , 6267 , 2006 osullivan , c. , et al . , 2000 , experimental astronomy , 10 , 147 ragazzoni , r. , esposito , s. , & riccardi , a. , 1998 , a&a , 128 , 617 ragazzoni , r. , marchetti , e. , & rigaut , f. , 1999 , a&a , 342 , l53 ribak , e. n. , & ragazzoni , r. , 2004 , optics letters , 29 , 12 rigaut f. , gendron e. , 1992 , a&a , 261 , 677 rousset g. , 2004 , in adaptive optics in astronomy , ed .roddier f. , cambridge university press schreiber l. , lombini m. , foppiani i. , diolaiti e. , conan j .- m ., marchetti e. , 2008 , spie proc ., 7015 , 70151o schreiber l. et al . , 2009 , in preparation thomas s. , adkins s. , gavel d. , fusco t. , michau v. , 2008 , mnras , 387 , 173 thomas s. , fusco t. , tokovinin a. , nicolle m. , michau v. , rousset g. , 2006 , mnras , 371 , 323 thompson l. a. , gardner c. s. , 1987 , nature , 328 , 229 tyson r. k. , 1998 , principles of adaptive optics , academic press wizinowich p. l. et al , 2006 , pasp , 118 , 297

Abstract: over the last few years increasing consideration has been given to the study of laser guide stars ( lgs ) for the measurement of the disturbance introduced by the atmosphere in optical and near - infrared astronomical observations from the ground . a possible method for the generation of a lgs is the excitation of the sodium layer in the upper atmosphere at approximately 90 km of altitude . since the sodium layer is approximately 10 km thick , the artificial reference source looks elongated , especially when observed from the edge of a large aperture . the spot elongation strongly limits the performance of the most common wavefront sensors . the centroiding accuracy in a shack - hartmann wavefront sensor , for instance , decreases proportionally to the elongation ( in a photon noise dominated regime ) . to compensate for this effect a straightforward solution is to increase the laser power , i.e. to increase the number of detected photons per subaperture . the scope of the work presented in this paper is twofold : an analysis of the performance of the weighted center of gravity algorithm for centroiding with elongated spots and the determination of the required number of photons to achieve a certain average wavefront error over the telescope aperture .   instrumentation : adaptive optics - atmospheric effects - telescope


a boolean network ( bn ) is a discrete dynamical system , which is , for example , used to study and model a variety of biochemical networks such as genetic regulatory networks .bns have been introduced in the late 1960s by kauffman who proposed to study random bns as models of gene regulatory networks .kauffman investigated their dynamical behavior and a phenomena called self - organization . aside from its original purpose , bns were also used to model ( small - scale ) genetic regulatory networks ; for example , in , it was demonstrated that bns are capable of reproducing the underlying biological processes ( i.e. , the cell cycle ) well .bns are also used to model large - scale networks , such as the _ escherichia coli _ regulatory network which is analyzed in section [ sec : ecoli ] .this network is , in contrast to kauffman s automata and the regulatory networks considered in , not an autonomous system , since the gene s states are determined by external factors .  in the literature addressing the analysis of bns , it is common to consider measures that quantify the effect of perturbations .whether a random bn operates in the so called ordered or disordered regime is determined by whether a single perturbation , i.e. , flipping the state of a node , is expected to spread or die out eventually .kauffman argues that biological networks must operate at the border of the ordered and disordered regime ; hence , they must be tolerant to perturbations to some extent .in contrast to measures of perturbations , determinative power in bns has not received much attention , even though there are several settings where such a notion is of interest .for example , given a feed forward network where the states of the nodes are controlled by the states of nodes in the input layer , we might ask whether a possibly small set of inputs suffices to determine most states , i.e. , reduces the uncertainty about the network s states significantly .this can be addressed by quantifying the determinative power of the input nodes .for example , in the _e. coli _ regulatory network , it turns out that a small set of metabolites and other inputs determine most genes that account for _ e. coli _ s metabolism ( see section [ sec : ecoli ] ) .  in this paper, we view the state of each node in the network as an independent random variable .this modeling assumption applies for networks with a tree - like topology , e.g. , a feed forward network , and is often applied when studying the effect of perturbations . for this setting , determinative power of nodes and perturbation - related measures are properties of single functions ; hence , the analysis of the bn reduces to the analysis of single functions .our main tool for the theoretical results is fourier analysis of boolean functions .fourier analytic techniques were first applied to bns by kesseli et al . . in , results related to derrida plots and convergence of trajectories in random bns were derived .ribeiro et al . considered the pairwise mutual information in time series of random bns , under a different setup that we use .specifically , in , the functions are random ; whereas here , the functions are deterministic , but the argument is random . finally , note that part of this paper was presented at the 2012 international workshop on computational systems biology .   mutual information between a set of inputs to a node andthe state of this node is a measure of the determinative power of this set of inputs , as mutual information quantifies mutual dependence of random variables . in order to understand the determinative power and mutual dependencies in boolean networks, we systematically study the mutual information of sets of inputs and the state of a node .we relate mutual information to a measure of perturbations and prove that ( maybe surprisingly ) a set of inputs that is highly sensitive to perturbations might not necessarily have determinative power .conversely , a set of inputs which has determinative power must be sensitive to perturbations . to prove those results ,we show that the concentration of weight in the fourier domain on certain sets of inputs characterizes a function in terms of tolerance to perturbations and determinative power of input nodes .furthermore , we generalize a result by xiao and massey , which gives a necessary and sufficient condition of statistical independence of a set of inputs and a function s output in terms of the fourier coefficients .this result can for instance be applied to decide for which classes of functions the algorithm presented in , which detects functional dependencies based on estimating mutual information , can succeed or fails . for _ unate _ functions , we show that any input and the function s output are statistically dependent and provide a direct relation between the mutual information and the influence of a variable .the class of unate functions is especially relevant for biological networks , as it includes all linear threshold functions and all nested canalizing functions , and describes functional dependencies in gene regulatory networks well .as an application of the theoretical results in this paper , we show that mutual information can be used to identify the determinative nodes in the large - scale model of the control network of _e. coli _ s metabolism .the paper is organized as follows .boolean networks and fourier analysis of boolean functions are reviewed in section [ sec : preliminaries ] . in section [ sec : inf ] , the influence and average sensitivity as measures of perturbations are reviewed , and their relation to the fourier spectrum is discussed . in section [ sec : mi ] , we study the mutual information of sets of inputs and the function s output .section [ sec : unate ] is devoted to unate functions .section [ sec : ecoli ] contains an analysis of the large - scale _ e. coli _ regulatory network , using the tools and ideas developed in previous sections .we start with a short introduction to boolean networks and fourier analysis of boolean functions , and introduce notation .a ( synchronous ) bn can be viewed as a collection of nodes with memory .the state of a node is described by a binary state at discrete time . choosing the alphabet to be rather than as more common in the literature on bns will turn out to be advantageous later . however , both choices are equivalent. the state of the network at time can be described by the vector \in \{-1,+1\}^{n} ] , .the assumption of independence holds for networks with tree - like topology , but is not feasible for networks with strong local dependencies and feedback loops .however , in many relevant settings , a bn has a tree - like topology , for instance the _ e. coli _ network analyzed in section [ sec : ecoli ] . for a network with few local dependencies, assuming independence will lead to a small modeling error .major results concerning the analysis of bns have been obtained under the assumptions as stated above , e.g. , the annealed approximation , an important result on the spread of perturbations in random bns .several important results on random bns , e.g. , , let the network size tend to infinity ; hence , there are no local dependencies .we use  ] . with , we mean the sum over all sets that are subsets of . throughout this paper , we use capital letters for random variables , e.g. , , and lower case letters for their realizations , e.g. , .boldface letters denote vectors , e.g. , is a random vector , and its realization . for a vector and a set ] with distribution } } , x_i \in \{-1,+1\} ] and expectations }{\ensuremath{\underset{{}}{\mathbb e}\left [ \cdot\right]}}} ] , the variance of by , its standard deviation by and finally }{\ensuremath{\underset{{}}{\mathbb e}\left [ x_i\right]}}} ] and , and denote .then ,   * orthonormality : * for  ] , over the function s output . as a measure of determinative power , we take the mutual information between and , since quantifies the statistical dependence between the random variable and .hence , this section is devoted to the study of .  before giving a formal definition of mutual information ,let us start with an example .consider the parity2 function and let its inputs be uniformly distributed . intuitively ,if has determinative power , knowledge about should provide us with information about .suppose we know the value of , say .since , we have with }}=1/2 ] .hence , knowledge of does not help to predict the value of .therefore , has no determinative power over .we indeed have .we next define mutual information .mutual information is the reduction of uncertainty of a random variable due to the knowledge of ; therefore , we need to define a measure of uncertainty first , which is entropy . as a reference for the following definitions ,see .the entropy of a discrete random variable with alphabet is defined as } } \log_2  { \ensuremath { { \mathrm{p}\!}\left[x = x\right]}}.\ ] ]  the conditional entropy of a pair of discrete and jointly distributed random variables is defined as } } h ( y | x = x ) .\ ] ]  the mutual information is the reduction of uncertainty of the random variable due to the knowledge of   for a binary random variable with alphabet and }} ] .hence , if is large , i.e. , close to 1 , we can be sure that has determinative power over since implies that for must be small then .   in order to study determinative power , its relation to measures of perturbations , and statistical dependencies , we start by characterizing the mutual information in terms of fourier coefficients .our results are based on the following novel characterization of entropy in terms of fourier coefficients .let be a boolean function , let be product distributed , and let be a fixed set of arguments , where  ] , according to . in figure[ fig2 ] , we depict for as a function of and .   as a function of and for .,scaledwidth=50.0% ]  it can be seen that , i.e. , and are statistically independent if and only if .that can be formalized as follows : is convex in .this can be proven by taking the second derivative of and observing that it is larger than zero for all pairs of values ( , ) for which is defined .next , from , we see that if ; hence , it follows that if and only if , which proves the following result :  let be a boolean function , and be product distributed . and are statistically independent if and only if .[ co : stat_indep_onev ]  corollary [ co : stat_indep_onev ] also follows immediately from a more general result , namely theorem [ le : statistical_independence_fourier_coeffs ] , which is presented later .recall that for parity2 , and ; hence , corollary [ co : stat_indep_onev ] comes at no surprise .  from figure [ fig2 ] , it can be seen that the larger , the larger becomes .formally , it follows from the convexity of and corollary [ co : stat_indep_onev ] that is increasing in .hence , has large determinative power , i.e. , is large , if and only if is large ( i.e. , close to one ) . is trivially maximized for the _ dictatorship function _ ,i.e. , for , or its negation , i.e. , .the output of the dictatorship function is fully determined by .next , let us consider the ( trivial ) case where  ] , i.e. , if the variance of is .in general , the closer to zero is , the larger the mutual information between a function s output and all its inputs becomes .let us finally relate the conditional entropy to the concentration of the fourier weight on the coefficients ,  ] .then , [ th : hlowerbound ]  see appendix [ proof_pr7 ] .theorem [ th : hlowerbound ] shows that can be approximated with .it further shows that is small if the fourier weight is concentrated on the variables in the set , i.e. , if is close to one .in contrast , as mentioned previously , for , it is relevant whether the fourier weight is concentrated on the coefficients with high degree .mutual information and average sensitivity are related as follows .  for any boolean function , for any product distributed , with [ pr : inf_bound_mi ]  see appendix [ proof_pr5 ] .note that the term is close to zero .specifically , for any we have , and for settings of interest , is very close to zero , as explained in more detail in the following .theorem [ pr : inf_bound_mi ] shows that if if large ( i.e. , close to one ) , must be sensitive to perturbations of the entries of .moreover , if is small ( i.e. , if is tolerant to perturbations of the entries of ) , then must be small ( i.e. , the entries of do not have determinative power ) .for the case that  ] be fixed , be a boolean function , and be product distributed .then , and the inputs are statistically independent if and only if [ le : statistical_independence_fourier_coeffs ]  see appendix [ proof_pr6 ] .for uniformly distributed , i.e. , } } = 1/2 ] , theorem [ le : statistical_independence_fourier_coeffs ] has been derived by xiao and massey .note that the proof provided here is also conceptually different from the proof for the uniform case in , as it does not rely on the xiao - massey lemma .theorem [ le : statistical_independence_fourier_coeffs ] shows that a function and small sets of its inputs are statistically independent if the spectrum is concentrated on the coefficients of high degree .the most prominent example is the parity function of variables , i.e. , : for uniformly distributed , each subset of or fewer arguments and are statistically independent .conversely , if a function is concentrated on the coefficients of low degree , which is the case for functions that are tolerant to perturbations , then small sets of inputs and the function s output are statistically dependent .  theorem [ le : statistical_independence_fourier_coeffs ] also has an important implication for algorithms that detect functional dependencies in a bn based on estimating the mutual information from observations of the network s states , such as the algorithm presented in .theorem [ le : statistical_independence_fourier_coeffs ] characterizes the classes of functions for which such an algorithm may succeed and for which it will fail .moreover , theorem [ le : statistical_independence_fourier_coeffs ] shows that in a boolean model of a genetic regulatory network , a functional dependency between a gene and a regulator can not be detected based on statistical dependence of a regulator and a gene s state , unless the regulatory functions are restricted to those for which holds for each relevant input .in this section , we discuss unate , i.e. , locally monotone functions .a boolean function is said to be unate in if for each and for some fixed , holds . is said to be unate if is unate in each variable ,  ] , is not necessarily unate . inserting into yields ,\ ] ] where the expectation in is over .based on , the discussion from section [ sec : reltov ] on applies by using and synonymously .hence , for unate functions , the mutual information is increasing in the influence .moreover , if is unate , and is a relevant variable , i.e. , a variable on which the functions actually depend on , then . from this fact and the same arguments as given in section [ sec : reltov ] follows :  let be unate . if and only if is a relevant variable , then .[ co : stat_dep_unate ]  in a boolean model of a biological regulatory network , this implies that if the functions in the network are unate , then a regulator and the target gene must be statistically dependent .in , the authors presented a complex computational model of the _ e. coli _ transcriptional regulatory network that controls central parts of the _ e. coli _ metabolism .the network consists of 798 nodes and 1160 edges . of the nodes ,636 represent genes and of the remaining 162 nodes , most ( 103 ) are external metabolites .the rest are stimuli , and others are state variables such as internal metabolites .the network has a layered feed - forward structure , i.e. , no feedback loops exist .the elements in the first layer can be viewed as the inputs of the system , and the elements in the following seven layers are interacting genes that represent the internal state of the system .our experiments revealed that all functions are unate ; therefore , the properties derived in section [ sec : unate ] apply .note that all functions being unate is a special property of the network , since if functions are chosen uniformly at random , it is unlikely to sample a unate function , in particular if the number of inputs is large .we first identify the input nodes that have large determinative power ( we will define what that means in a network setting shortly ) and then show that a small number thereof reduces the uncertainty of the network s state significantly .specifically , we show that on average , the entropy of the node s states conditioned on a small set of determinative input nodes , is small .  to put this result into perspective , we perform the same experiment for random networks with the same and different topology as the _ e. coli _ network .we denote by the set of inputs of the feed forward network and assume that the are independent and uniformly distributed .the remaining variables are denoted by and are a function of the inputs and the network s states , i.e. , . for our analysis , the distributions of the random variables need to be computed , since some of those variables are arguments to other functions .this can be circumvented by defining a collapsed network , i.e. , a network where each state of a node is given as a function of the input nodes only , i.e. , .the collapsed network is obtained by consecutively inserting functions into each other , until each function only depends on states of nodes in the input layer , i.e. , on .the collapsed network reveals the dependencies of each node on the input variables .interestingly , in the collapsed network , it is seen that the variables chol_xt , salicylate , 2ddglcn_xt , mnnh , altrh , and his - l_xt ( here , and in the following , we adopt the names from the original dataset ) , which appear to be inputs when considering the original _ e. coli _ network , turn out to be not .consider , for example , the node salicylate .the only node dependent on salicylate is mara = ( ( not arca or not fnr ) or oxyr or salicylate ) .however , arca = ( fnr and not oxyr ) , and it is easily seen that mara simplifies to mara = 1 .next , we identify the determinative nodes . as argued in section [ sec : mi ] , is a measure of the determinative power of over .this motivates the definition of the determinative power of input over the states in the network as note that a small value of implies that alone does not have large determinative power over the network s states , but may have large determinative power over the network states in conjunction with other variables . in principle be large for some  ] , such that , i.e. , orders the input nodes in descending order in their determinative power .we next consider as a function of to see whether knowledge of a small set of input nodes reduces the entropy of the overall network state significantly . has an interesting interpretation which arises as a consequence of the so called asymptotic equipartition property ( as discussed in greater detail in ) : consider a sequence of samples of the random variable .for and sufficiently large , there exists a set of typical sequences , such that and } } > 1-\epsilon,\ ] ] where denotes the cardinality of the set .this shows that the sequences obtained as samples of are likely to fall in a set of size determined by the uncertainty of .since the output layer consists of 653 nodes , the network s state space has maximal size . since is a function of , , where for the last equality , we assume uniformly distributed inputs .thus , without knowing the state of any input variable , the network s state is likely to be in a set of size roughly . given the knowledge about the states , the state of the network is likely to be in a set of size roughly .for a large network , however , is expensive to compute as by definition : } } \label{eq : adf } \cdot \sum_{{\ensuremath{\mathbf { y } } } }  { \ensuremath { { \mathrm{p}\!}\left[{\ensuremath{\mathbf { y } } } = { \ensuremath{\mathbf { y } } } | { \mathbf{x}_a } = { \mathbf{x}_a } \right ] } } \log_2  { \ensuremath { { \mathrm{p}\!}\left[{\ensuremath{\mathbf { y } } } = { \ensuremath{\mathbf { y } } } | { \mathbf{x}_a } = { \mathbf{x}_a } \right]}}.\ ] ] hence , the number of terms in the sum is exponential in and .an estimate of can be obtained by sampling uniformly at random over and .instead , we will consider the following upper bound which is computationally inexpensive to compute : with the bound above follows from the chain rule for entropy . is computationally inexpensive to compute , since depends on few variables only ( in the _e. coli _ network , on ) . for the _ e. coli _ network , is depicted in figure [ fig3 ] as a function of .figure [ fig3 ] shows that knowledge of the states of the most determinative nodes reduces the uncertainty about the network s states significantly .in fact , the upper bound is loose ; hence , we even expect to lie significantly below . also , note that when is small , must be small on average ; hence , }} ] for each function in the _e. coli _ network , again assuming that the are independent and uniformly distributed .we can see from figure [ fig4 ] that the average sensitivity of all functions is close to the lower bound on the average sensitivity .note that the functions with high in - degree ( i.e. , number of relevant input variables ) , which could have average sensitivity up to , also have small average sensitivity , because those functions are highly biased .we , therefore , can conclude that the functions have small average sensitivity either because they depend on few variables only or because they are highly biased .for other input distributions , i.e. , other values of } } , \forall i \in [ n] ] of each function in the _ e. coli _ network for different in - degrees and uniformly distributed . moreover , a lower bound on the average sensitivity , i.e. , poincare s inequality , is plotted.,scaledwidth=50.0% ]in a boolean network , tolerance to perturbations , determinative power , and statistical dependencies between nodes are properties of single functions in a probabilistic setting .hence , we analyzed single functions with product distributed argument .we used fourier analysis of boolean functions to study the mutual information between a function and a set of its inputs , as a measure of determinative power of over .we related the mutual information to the fourier spectrum and proved that the mutual information lower bounds the influence , a measure of perturbation .we also gave necessary and sufficient conditions for statistical independence of and .for the class of unate functions , which are particularly interesting for biological networks , we found that mutual information and influence are directly related ( not just via an inequality ) .we also found that for each relevant input , which , as an application , implies that in a unate regulatory network , a gene and its regulator must be statistically dependent . as an application of our results, we analyzed the large - scale regulatory network of _e. coli_. we identified the most determinative input nodes in the network and found that it is sufficient to know only a small subset of those in order to reduce the uncertainty of the overall network state significantly .this , in turn , reduces the size of the state space in which the network is likely to be found significantly .  a possible direction for future work is to provide an analysis similar to that of the _ e. coli _ regulatory network for other boolean models of biological networks , and see if similar conclusions as in section [ sec : ecoli ] can be reached .one of the main assumptions in our work is the independence among the input variables of the network .it would be interesting to provide methods that can be used beyond this setup .however , deriving such results is challenging because for dependent inputs , the basis functions do not factorize as in , and many results cited and derived in this paper make use of this particular form of the basis functions . in this paper ,we focused on generic properties of information - processing networks that may help identify possible principles that underly biological networks .assessing our findings from a biological perspective would be an interesting next step .for the proof of theorems [ th : relation_fourier_h ] and [ le : statistical_independence_fourier_coeffs ] , we will need the following lemma :  let be a boolean function , let be product distributed , and let $ ] and some fixed be given . then , }{\ensuremath{\underset{{}}{\mathbb e}\left [  f ( { \ensuremath{\mathbf { x } } } ) |  { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ] } } } = \sum_{s \subseteq a } { \hat f } ( s )  { \phi}_{s}({{\mathbf{x}_a } } ) .\label{lemma1:eq1}\ ] ] [ le : relation_fourier_evalue_general ]  inserting the fourier expansion of given by in the left - hand side of and utilizing the linearity of conditional expectation yields }{\ensuremath{\underset{{}}{\mathbb e}\left [  f ( { \ensuremath{\mathbf { x } } } ) |  { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ] } } } = \sum_{s\subseteq[n ] } \hat f(s ) ~ { \ifthenelse{\equal{{}}{}}{{\mathbb e } [ { \phi}_{s } ( { \ensuremath{\mathbf { x } } } ) |  { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } ] } { \ensuremath{\underset{{}}{\mathbb e}\left [  { \phi}_{s } ( { \ensuremath{\mathbf { x } } } ) |  { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right]}}}.\ ] ]  for , }{\ensuremath{\underset{{}}{\mathbb e}\left [  { \phi}_{s } ( { \ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ] } } } = { \phi}_{s } ( { { \mathbf{x}_a } } ) .\ ] ] conversely , for , }{\ensuremath{\underset{{}}{\mathbb e}\left [  { \phi}_{s } ( { \ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ] } } } = 0.\ ] ] to see this , assume without loss of generality that and . using the decomposition property of the basis function as given in section [ sec : fanalysisintro ] , }{\ensuremath{\underset{{}}{\mathbb e}\left [  { \phi}_{s } ( { \ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ] } } } & = \mathbb e \left [  \prod_{i\in s } \phi_{\{i\}}({\ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ]  \\ & = \prod_{i\in s } { \ifthenelse{\equal{{}}{}}{{\mathbb e } [ \phi_{\{i\}}({\ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } ] } { \ensuremath{\underset{{}}{\mathbb e}\left [  \phi_{\{i\}}({\ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right]}}}\end{aligned}\ ] ] which is equal to zero as }{\ensuremath{\underset{{}}{\mathbb e}\left [  \phi_{\{j\}}({\ensuremath{\mathbf { x } } } ) | { \ensuremath{\mathbf { x}}}_a = { { \mathbf{x}_a } } \right ] } } } =  { \ifthenelse{\equal{{}}{}}{{\mathbb e } [ \phi_{\{j\}}({\ensuremath{\mathbf { x}}})]}{\ensuremath{\underset{{}}{\mathbb e}\left [  \phi_{\{j\}}({\ensuremath{\mathbf { x}}})\right ] } } } = 0.\ ] ]we would like to thank sara al - sayed and dejan lazich for their helpful discussions and careful reading of the manuscript .j. saez - rodriguez , l. simeoni , j.a .lindquist , r. hemenway , u. bommhardt , b. arndt , u haus , r weismantel , e.d .gilles , s. klamt , b. schraven , a logical model provides insights into t cell receptor signaling .. biol . * 3*(8 ) , e163 ( 2007 )       r. heckel , s. schober , m. bossert , determinative power and tolerance to perturbations in boolean networks .paper presented at the 9th international workshop on computational systems biology , ulm , germany , 4 - 6 june 2012      r.r .bahadur , a representation of the joint distribution of responses to n dichotomous items . in _ studies in item analysis and prediction _ , ed . by h. solomon ( stanford university press , stanford , 1961 ) , pp.158168  m. ben - or , n. linial , collective coin flipping , robust voting schemes and minima of banzhaf values .paper presented at the 26th annual symposium on foundations of computer science , portland , oregon , usa , 2123 october 1985       b. rosell , l. hellerstein , s. ray , why skewing works : learning difficult boolean functions with greedy tree learners . paper presented at the 22nd international conference on machine learning , bonn , germany , 711 august 2005

Abstract: consider a large boolean network with a feed forward structure . given a probability distribution on the inputs , can one find , possibly small , collections of input nodes that determine the states of most other nodes in the network ? to answer this question , a notion that quantifies the _ determinative power _ of an input over the states of the nodes in the network is needed . we argue that the mutual information ( mi ) between a given subset of the inputs of some node and its associated function quantifies the determinative power of this set of inputs over node . we compare the determinative power of a set of inputs to the sensitivity to perturbations to these inputs , and find that , maybe surprisingly , an input that has large sensitivity to perturbations does not necessarily have large determinative power . however , for _ unate _ functions , which play an important role in genetic regulatory networks , we find a direct relation between mi and sensitivity to perturbations . as an application of our results , we analyze the large - scale regulatory network of _ escherichia coli_. we identify the most determinative nodes and show that a small subset of those reduces the overall uncertainty of the network state significantly . furthermore , the network is found to be tolerant to perturbations of its inputs .


it is now well known that information technology ( it ) that processes information according to quantum rules could have security and computational advantages over conventional it . over the last decadethere have been a great many proposals for the building blocks of such new technology how to realise qubits and do gates between them and the emergence of promising experimental results for many of these proposals .the next steps towards useful qit are further experimental investigations , and theoretical considerations of outstanding practical issues .the work we present here is of the latter variety , showing how to distribute entanglement and thus connect together small quantum dot ( qd ) registers , of realistic size , to enable a larger , useful , quantum processor architecture .  in the past few years, self - assembled semiconductor qds  have been considered as one of the most promising solid state hardware routes for implementing quantum computation ( qc )  .self - assembled qds are characterized by different properties , dependent upon whether they are _ stacked _ or _ in - plane_. in the stacked arrangement , the qds tend to be noticeably different in size due to the strain propagation , which makes them good candidates for energy - selective addressing of excitations in single qds  .in - plane qds can already be produced with fairly uniform size ( with just a few percent size variation ) , driven by the crucial uniformity requirements for qd - based lasing  .furthermore , 1d qd chains with controllable qd size and density have been recently grown and _ regular _ 2d and 3d qd arrays can be now produced , creating real optimism for future qd technology . for qc ,the key feature of qd hardware is the possibility of optically driven quantum evolution , which could enable useful quantum processing inside the relatively fast decoherence times typical of solid state systems . in order to implement such qc schemes , arrays of vertically stacked quantum dots  been proposed as quantum registers . however, for practical reasons it is clear that an individual stacked array can not be scaled to an arbitrarily large qubit number . for actual technology , an architectureis needed in which stacked registers of practical size are connected together , preferably in a way that is compatible with the manufacture of the registers themselves and this is what we propose in this paper .we exploit recent results showing the feasibility of quantum communication between separated quantum registers , based on quantum buses made from chains of in - plain quantum dots and the properties of relatively short spin chains  .our proposed architecture and how is used is described in the next section ; we discuss the fidelity of its operation in section [ fidelity ] , and conclude in secton [ conclusion ] .the key to our proposal is an entanglement distributor , an all - qd device for generating maximally entangled qubits on demand and distributing them to spatially separate regions .the system we envisage is shown schematically in in fig .registers of stacked qds are connected by entanglement distributors consisting of in - plane chains of qds .the entanglement distribution starts with the generation of a bell state in qda and qdb . for excitonic encoding ,the computational basis is the absence ( ) and presence ( ) of a ground state exciton in qd . due to strain propagation  qd energy spectra differ in the vertical ( stacked ) direction , i.e. between qda and qdb , between each dot in the buses ( baa and bab ) and its partner dot in the control array ( caa and cab ) , and along each of the quantum registers qra and qrb ( as shown in fig .[ fig1 ] ) .these energetically distinct dots may be addressed by laser pulses of different frequency .furthermore , since the excitons in vertically adjacent qds have different energies , resonant frster coupling  is strongly inhibited in this direction .the initial entangled state in qda and qdb is generated by a two - colour laser pulse sequence  ; this can be done on a picosecond time scale .the first pulse is a -pulse between the vacuum and ground state exciton in qda .this is equivalent to a hadamard operation applied to an empty qda , .( note , for later use , that . )the second laser pulse performs a c - not gate on qdb , flipping qdb conditional on the presence of an exciton in qda , due to the direct coulomb coupling  between these excitons .clearly the roles of qda and qdb can be interchanged .  as mentioned above , in - planeqds can be almost monodisperse , so that we can assume adjacent qds in each bus ( baa and bab ) and each control array ( caa and cab ) are very similar .this means that resonant frster energy transfer  takes place along the buses shown in fig .in addition , the relative stacked position of bus and control arrays allows us to address the control arrays caa and cab ( i ) separately and ( ii ) without exciting excitons also in the corresponding bus arrays .finally , the length of the bus arrays is made greater than the laser wavelength(s ) , so the central coupled qd pair ( qda and qdb ) and the two quantum registers qra and qrb can each be addressed with independent laser spots , as illustrated in fig .[ fig1 ] .  once generated , the entanglement needs to be distributed .the hamiltonian for a chain of qds , with zero or one exciton in each dot , is given by where is the frster coupling  .this hamiltonian governs the transfer of the entangled pair from qda(b ) to register qra(b ) . for uniform in - plane chains, we shall assume that is independent of and .the complete process of generating and distributing a maximally entangled qubit pair is shown in fig .[ scheme ] , panels ( 1 ) to ( 6 ) .    in the first step ( panel ( 1 ) ) the bus arrays baa and bab are set off resonance with respect to excitations in qda and qdb , by generating an exciton in each qd in the control arrays caa and cab .this multiple - excitation process , done with a single laser pulse , is not straightforward , and care must be taken when considering the system parameters . in our case in eq .[ hf ] and the hamiltonian has a series of eigenstate manifolds , each of which corresponds to a different total exciton number for the chain .the typical energy range covered by each manifold is on the order of .excitons are created in the dots by applying pulses centred on energies and that correspond to the for each chain .the coupling between laser and each individual dot is ( which determines the rabi frequency of the exciton creation process ) . for creation of an exciton on each dot ; in this case becomes the dominant energy in the system and the dots behave as though they are uncoupled ( to first order ) .we show how the population of the ground state of a 5-qd chain , and the population of the state with an exciton on all dots of a 5-qd chain varies as a function of time , and rabi frequency , in fig .[ control ] .notice that , for realistic frster couplings ( .2 mev  ) , it is possible to drive the desired control - array dynamics with very high accuracy on a sub - picosecond time scale .  ) and state with an exciton on every qd ( ) in a 5 qd chain , for various values of dot - laser coupling ..2 mev for all plots.,width=288,height=403 ]  the excitons created in the control bus couple to the excitonic transitions in baa and bab , causing a biexcitonic energy shift and effectively detuning them from resonance with respect to qda and qdb .this shift can be tailored up to several mev  , inhibiting frster processes between qda ( qdb ) and the qds forming the bus baa ( bab ) .this inhibited dynamics is shown in fig .[ forstblock ] . when bus baa is off resonance , due to the excitons in control array caa , an initial state of an exciton in qda ( with bus baa empty ) remains in that state with at least 99% probability as it evolves , for a ratio .this ratio could be achieved with a shift of 4 mev and a frster coupling of 0.2 mev , so very good confinement of an exciton is possible for practical parameters .  , at which such an overlap is minimum .this is plotted as a function of the ratio ( biexcitonic shift / frster coupling ) , the dimensionless shift in the excitonic transition of a qd in the bus baa due to the presence of an exciton in the corresponding qd in control array caa .inset : the time evolution of the overlap for a ratio of 20 , for the cases of five ( thin line ) and seven ( thick line ) qds in ( qda+baa+qdc ) .the difference in chain length only has an effect after several oscillations , and this is typical for a dimensionless shift .point ` a ' marks the minimum used in the main figure for calculating the overlap.,width=288,height=192 ]  the second step ( panel 2 ) is to generate a maximally entangled pair of excitons , one in qda and one in qdb .since qda and qdb are stacked qds , they can be individually addressed using sub - ps laser pulses of different color , and creation of the required state proceeds through hadamard and c - not gates as we discussed earlier .owing to the confinement discussed above , the entangled excitons do not propagate along the buses .  in the third step ( panel 3 ) , the bus arrays baa and bab are again set into resonance with the excitations in qda and qdb .this is done by recombining the excitons in each qd in the control arrays by the same laser pulses as those used in step 1 .now the entangled excitons in qda and qdb are free to travel along baa and bab , respectively ( panel 4 ) ; the time evolution follows the dynamics of relatively short chains described in ref . and governed by eq .( [ hf ] ) . in plane qdshave nearest - neighbor couplings that are typically _ un - modulated_. chains of such qds can be used for state transfer , and this avoids complexity and adds to the practicality of our scheme ; the dynamics of exciton transfer down these chains has been studied in ref ida .these dynamics are characterized by a series of resonances , corresponding to different fidelities of the transfer , the very first of which has a fidelity greater than 94% for the chains required for our entanglement distributor ( which is less that 10 qds long ) .the actual transfer time depends on the strength of the frster coupling between in - plane qds , and it is foreseen that such qd structures can be optimized to have a frster coupling of the order of a mev  and hence a time of transfer between qda(b ) and qdc(d ) of the order of a picosecond  . for the example discussed earlier ( where 0.2 mev ) the transfer time is still less than 10 ps for a bus with 9 qds .we underline that this transfer time is , on the one hand , much longer than the characteristic time corresponding to the control - array dynamics ( whose population / de - population can therefore be viewed as instantaneous ) , but on the other hand it is still much shorter than the relevant decoherence times ( of the order of a nanosecond ) , allowing for the possibility of further manipulations .  when the time elapsed is equal to the transfer time  , the two entangled excitons will be in qdc and qdd , i.e. at the bases of the two different quantum registers qra and qrb . using laser pulses to once again set the buses baa and bab out of resonance with respect to qda, b , c and d , will confine the two entangled excitons in qdc and qdd , ( panel 5 ) .thus the entangled pair has now been distributed to spatially separated regions , which can be addressed separately , e.g. by using laser pulses . using optical control ,quantum operations can be performed on such a pair ( panel 6 ) and the entanglement can be transferred to different degrees of freedom , e.g. other excitons , spatially distinguishable photons , or electronic spins .these may be characterized by much longer decoherence times than the original excitons .  as an example of this transfer ,we outline how a qubit encoded in the exciton basis in a qd can be swapped into the spin of an excess electron in an adjacent pauli - blockade qd . in terms of the spin ,the qubit computational basis is and . in a pauli - blockade qd a suitably polarized optical will create ( or remove ) an exciton _ conditional _ on the spin state of the excess electron , so , for example , a qubit with its ancillary exciton state is represented as and in such a qd after conditional creation .if such a qd is adjacent to another qd ( qdc or qdd in our scheme ) containing an excitonic qubit , a two - qubit gate equivalent to a controlled - phase gate ( which flips the sign of the computational basis amplitude and leaves the others alone ) can be achieved  using the biexcitonic shift , through conditional creation of an exciton for a chosen time .we denote this gate by where labels the dot c or d and labels the appropriate adjacent pauli - blockade qd containing the spin qubit .if this latter qubit is set to a known initial state , say , a swap ( denoted by ) between the two qubits can be achieved through two c - not gates , which decomposes to through this gate sequence an excitonic qubit that has been distributed to qdc or qdd can be swapped into a ( relatively ) long - lived spin qubit in an adjacent qd .we note that the two hadamard operations have to be performed on the spin qubit .however , fast optical techniques exist for performing such gates  , so in principle the whole swap operation can be done on a fast optical / excitonic timescale ( i.e. no more than a few ps ) , to mediate against the effects of decoherence acting on the excitons .from a practical quantum processing perspective , the swapping of the entangled qubits into spins , to create a long - lived entangled resource , is very important .the fidelity of our distribution scheme is set by a product of factors .first , the chain transfer fidelity of 94% , second the control chain blocking fidelity of around 99% , third , the swap gate fidelity which can also be greater than 99 %  and fourth , the natural decoherence of excitons due to spontaneous emission of photons , or exciton - phonon interactions  .the decoherence of excitons in qds consists of two main features : an initial sub - picosecond pure dephasing ( caused by the excitons interaction with acoustic phonons ) , followed by a decay of the remnant polarization that decays through exciton recombination .( see refs . ,  ,  and and for a comprehensive description of these effects and a comparison of experimental data with theory ) .however , for the purposes of our discussion , we will take the exciton recombination to be the principal source of dephasing .this is for two main reasons .first , the value of the residual polarization strongly depends on temperature and material parameters but is generally much larger than the lost polarization .second , the initial decay corresponds to the formation of exciton - phonon dressed states , and we could redefine our qubit to correspond to one of these .this would effectively eliminate the fast initial decay .we estimate that the total time for one entanglement distribution operation is no more than 20 ps , whereas the exciton decay time is about 1 ns  ( giving a ` natural ' fidelity of around 99% ) .thus , an estimate of the overall fidelity of distribution is 91% .such a good fidelity is certainly adequate for initial experimental investigations .it could be used to demonstrate teleportation and it might even be useful in certain few - qubit applications , which are run probabilistically with rather limited qubit resources .however , for practical quantum processing and a scalable processing architecture , very high fidelity entanglement is desirable .this could be achieved by purification  .a number of good fidelity pairs can be distilled to a smaller number of higher fidelity pairs using local qubit operations .this is feasible if distributed excitonic entanglement is mapped into long - lived spin entanglement prior to purification , as we propose .building up the entanglement resource _ off - line _ and using it to link together separate parts of the quantum device clearly has a practical advantage over trying to propagate a computation directly from one register to another , since it prevents errors caused by imperfect gate operations .we have presented a practical all - quantum - dot architecture for quantum information processing . the key ingredient to thisis an entanglement distributor , which provides a resource of distributed entangled states on demand .this enables the connection of stacked qd processors , of modest and practical size , to form a larger quantum processor .entangled exciton states are transported to different regions of a semiconductor based quantum information device using chains of in - plane qds , at which point they can , for example , be swapped into long - lived spin qubits , or used to create entangled photons .the fidelity of the distributed entanglement is good , and could be further increased through purification .such entanglement is a generic and flexible resource .for example , it could be used for teleportation of quantum states , to enable distributed quantum gates , or to build distributed cluster states  , so it can be utilised in a variety of approaches to quantum processing .our qd - based entanglement distributor should integrate with and thus enhance many of the current qd - based quantum information technology proposals .given the impressive progress with qd structures  , entanglement distribution forms a realistic experimental goal in the relatively short term and a route to scalable solid state qc in the long term .moritz a , et al . 1996 _ appl .* 69 * 212 ; su y k , et al .2004 _ semicond .technol . _* 19 * no 3 389 ; edwards p r , et al .2004 _ appl .lett . _ * 85 * 4281 ; gao q , et al .2005 _ electrochem .solid - 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Abstract: we propose a quantum dot architecture for enabling universal quantum information processing . quantum registers , consisting of arrays of vertically stacked self - assembled semiconductor quantum dots , are connected by chains of in - plane self - assembled dots . we propose an entanglement distributor , a device for producing and distributing maximally entangled qubits on demand , communicated through in - plane dot chains . this enables the transmission of entanglement to spatially separated register stacks , providing a resource for the realisation of a sizeable quantum processor built from coupled register stacks of practical size . our entanglement distributor could be integrated into many of the present proposals for self - assembled quantum dot - based quantum computation . our device exploits the properties of simple , relatively short , spin - chains and does not require microcavities . utilizing the properties of self - assembled quantum dots , after distribution the entanglement can be mapped into relatively long lived spin qubits and purified , providing a flexible , distributed , off - line resource .


during latest decades , we have seen large numbers of publications on the use of pareto - lev and other heavy - tailed distributions in finance ( see , for example , , and references there ) . one of the key authors that initiated this , was mandelbrot .he mentioned that gaussian distribution can not provide a solid explanation for observed large amount of ` outliers ' - in other words , the number of observations for which absolute value of deviation from empirical mean is bigger than , where is empirical variance .aparently , mandelbrot considered stable distributions as a unique alternative to gaussian family basing on the fact of classical limit theorem for independent identically distributed ( i.i.d . ) random variables , or , equivalently , on stability property . in the remainder of this paper, we show that stable distributions can not explain such a large number of outliers either .the connection to classical limit theorem for a non - random number of random summands seems to be questionable .a more natural perspective comes from the use of limit theorem for a random number of random variables .in this section we look at the probability of observing outliers in the case of distributions belonging to a domain of attraction of strictly stable distribution .specifically , suppose that is a sequence of i.i.d .random variables .denote by their empirical mean and empirical variance correspondingly .let be a fixed number .we are interested in the following probability our aim here is to prove the following theorem .[ th1 ] suppose that is a sequence of i.i.d .random variables belonging to a domain of attraction of strictly stable random variable with index of stability .then   since belong to the domain of attraction of strictly stable random variable with index , it is also true that belong to the domain of attraction of one - sided stable distribution with index .\1 ) consider at first the case . in this case ,{}{\longrightarrow } a={{\rm i}\kern-0.18em{\rm e}}x_1 ] .we have {}{\longrightarrow } 0.\ ] ]  \2 ) suppose now that . in this case, we have as . here is -stable random variable , and the sign is used for asymptotic equivalence .similarly , where has one - sided positive stable distribution with index .we have {}{\longrightarrow } 0.\ ] ]  \3 ) in the case we deal with cauchy distribution .the proof for this case is very similar to that in the case 2 ) .we omit the details .let us note that for the case of distributions having finite second moment and non - compact support , the probability ( [ eq1 ] ) has a positive limit as .indeed , if the second moment is finite then both and have finite limits and .the probability ( [ eq1 ] ) converges to as because of non - compactness of support of the distribution .therefore , it is clear that the explanation of presence a large number of outliers can not be provided by the heaviness of the tails .although the results of section [ s2 ] show that asymptotically the probability of presence of a large number of outliers is negligible , we may still have doubts for not too large samples .however , the samples in practice are not too small .so , basing on actual observed data , we can consider the sample size of order as typical one .we simulated samples of size ( is growing from 1,000 to 25,000 with step 2,000 ) and calculated the estimate of probability given by ( [ eq1 ] ) for .the behavior of this probability as a function of is reflected on figure [ fig1 ] .blue line corresponds to symmetric stable distribution with ; red line - to standard gaussian distribution .we see that the probability for -stable symmetric distribution becomes smaller that for gaussian case starting at about .therefore , we can not expect many outliers for such stable distribution for typical sample size .   for .blue line corresponds to symmetric stable distribution with ; red line - to standard gaussian distribution ]  \2 .again , we simulated samples of size ( is again growing from 1,000 to 25,000 with step 2,000 ) and calculated the estimate of probability given by ( [ eq1 ] ) for .the behavior of this probability as a function of is plotted on figure [ fig2 ] .blue line corresponds to symmetric stable distribution with ; red line - to standard gaussian distribution .we see that the probability for -stable symmetric distribution becomes smaller than that for gaussian case starting from about .the decrease of for the case is much slower .for example , for -stable symmetric distribution versus for gaussian distribution .in this situation , we can not say that stable distribution provides less outliers that gaussian law .but will the corresponding number of outliers be sufficient for the predicted figures to be in agreement with observed data ?unfortunately , there are only a few papers giving observed number of outliers , that is an estimate of probability for . some data of this kind may be found in , table 3 .corresponding estimates for probabilities given there vary from 0.009 to 0.013 . for symmetric stable distribution with , this probability is .we see that it is too small to explain the number of outliers in table 3 from .let us note that similar simulations for different values of parameters and sample size are given in .we shall not discuss them here .   for .blue line corresponds to symmetric stable distribution with ; red line - to standard gaussian distribution ]lately , _ tempered stable distributions _ have been growing increasingly popular .the idea of such distribution lies in altering the tails of stable laws by exponential tails starting from a certain point of distribution s support . of course, the probability in this case will not converge to zero anymore .however , one may expect that this probability will be small if the point of the tails alteration is far from the origin , since tempered distribution will be close to stable in this case .simulations support this opinion .let us consider only the case of symmetric tempered stable distributions having characteristic functions for , , .  on figure [ fig2a ] , plot of limit probability given for the distribution ( [ eqa ] ) as a function of parameters and for .this plot shows that the probabilities are too small to explain the presence of large number of outliers .   as a function of parameters and . ]here we discuss a way of constructing from a distribution another one having a higher probability to observe outliers .we call this procedure `` put tail down '' .let be a probability distribution function of random variable having finite second moment and such that for all . take a parameter andfix it . define a new function where for , and for .it is clear that is probability distribution function for any .of course , also has finite second moment , and .however , .let be a random variable with probability distribution function .then denoting rewrite previous equality in the form for to have more outliers than it is sufficient that there are many cases in which inequality ( [ eq4 ] ) is true for sufficiently large values of .let us mention two of them .  1.  random variable has exponential tail .more precisely , for some positive constants and .in this case , inequality ( [ eq4 ] ) is equivalent for sufficiently large to which is obviously true for large .2 .  has power tail , that is , where in view of existence of finite second moment .simple calculations show that ( [ eq4 ] ) is equivalent as to the last inequality is true for .let us note that the function has a jump at zero .however , one can obtain similar effect without such jump by using a smoothing procedure , that is by approximating by smooth functions .`` put tail down '' procedure allows us to obtain more outliers in view of two its elements .first element consists in changing the tail by smaller , but proportional to previous with coefficient .the second element consist in moving a part of mass into origin ( or into a small neighborhood of it ) , which reduces the variance .one of the arguments used to support the use of stable distributions in finance is that the observations may be considered as sums of a large number of random variables .however , to have the convergence to stable distribution , the summands themselves must have heavy tails , which seems to be unnatural .moreover , as we saw in sections [ s2 ] , [ s3 ] , the probability to observe large number of outliers is not in agreement with real data .however , alternatives to gaussian distribution are not restricted by stable distributions only .there are limit laws for the sums of i.i.d .random variables for _ random number of summands_. the theory of such limit distributions has been developed by robbins , dobrushin , gnedenko and others .the description of this theory can be found in the book .for some new results , see . as shown in these publications , the sums of a random number of i.i.d .random variables converges to so - called -normal distribution .the form of this distribution depends on the law for number of summands . in the case of geometric distribution for the number of summands the limit law is laplace distribution instead of gaussian . for the case oftransformed negative binomial distribution for the number of summands the role of gaussian law is played by symmetric gamma distribution .many other examples are given in .all such distributions have finite second moment .for many of them , the probability ( [ eq2 ] ) is essentially higher than for gaussian law .for example , laplace distribution gives the limit value of probability for equal to , versus for gaussian distribution .our suggestion is that such distributions are good alternatives to gaussian and stable laws .

Abstract: in the present paper , we discuss contra - arguments concerning the use of pareto - lev distributions for modeling in finance . it appears that such probability laws do not provide sufficient number of outliers observed in real data . connection with the classical limit theorem for heavy - tailed distributions with such type of models is also questionable . the idea of alternative modeling is given .   * keywords*:outliers ; financial indexes ; heavy tails ; stable distributions