the spectral methods approximate the exact solution of partial differential equations by seeking an good " projection in the linear subspace spanned by various orthogonal systems of special functions .the resulting spectral accuracy is highly preferred than any other numerical method , especially when the solution is known to be globally smooth enough .therefore , they are very appropriate for the elliptic and parabolic equations , thanks to the regularization properties of the operators . when mentioning the nonlinear conservation laws , it is well known that the solution may develop spontaneous jump discontinuity , i.e. , shock waves .this irregularity of the solution destroys not only the accuracy of the spectral approximations at the point of discontinuity , but also that in the entire computational domain .it causes the oscillations throughout the domain , which is the so - called gibb s phenomenon .moreover , the instability is induced in the nonlinear case .it is shown in that the usual spectral approximate solution may not converge to the entropy solution , the physically relevant one .  despite all these deficiencies ,many mathematicians still pay their effort to deal with these problems .the problems caused by the irregularity has already been solved for piecewise smooth functions in bounded domain or periodic piecewise smooth function in unbounded domain by filter techniques or reconstruction methods such as the gegenbauer partial sum , see details in a series of papers , , , and references therein . and the instability of the usual spectral approximations can be avoided by introducing the vanishing viscosity , which was first established by e. tadmor .the main idea of the spectral viscosity method is the use of artificial diffusion to stabilize the spectral computation without sacrificing its spectral accuracy .the periodic spectral method has been further investigated in , and , etc .the nonperiodic legendre spectral viscosity method is first introduced by y. maday , et .al . .and h. ma proposed the nonperiodic chebyshev - legendre spectral viscosity method in , . for more literatures related to the spectral viscosity methods with various orthogonal basis in bounded domain, we refer the readers to , , and references therein .as we know , a large amount of physical problems are modeled in unbounded domain . during the past two decades, more and more attentions were paid to numerical solutions of differential equations defined in unbounded domains . among the existing literature ,the hermite and laguerre spectral methods are the most commonly used approaches based on orthogonal polynomials in infinite interval , referring to , .although the hermite polynomials appear to be a natural choice of orthogonal basis of , it is not as popular as fourier series and chebyshev polynomials , due to its poor resolution ( see ) and the lack of the analogue of fast fourier transformation ( fft ) , see .however , it is shown in that the poor resolution can be remedied by a suitable choice of scaling factor. some further investigations on the scaling factor can be found in and also in chapter 7 , .recently , a guideline of choosing the suitable scaling factors for gaussian / super - gaussian functions is summarized by the author and s. s .- t .yau in , where the hermite spectral method is used to resolve the conditional density function of the states in nonlinear filtering problems .the literatures on the spectral method in unbounded domains have already been not as rich as those in bounded domains , let alone the spectral viscosity method in unbounded domains .as far as we know , j. aguirre and j. rivas is the only paper that considered the spectral viscosity method based on the hermite functions .however , they defined the hermite functions in the weighted , where . and no scaling factor is introduced .this essentially causes their involved theoretical proof of the convergence rate of their proposed scheme . andit is more costly when they try to implement their scheme numerically .  in this paper, we shall revisit the nonlinear scalar conservation laws in : where is a smooth nonlinear function and .in general , the spontaneous jump discontinuity may be developed .therefore , we ca nt expect the classical solutions to this problem . moreover , we restrict ourselves to the physically relevant weak solution , the entropy solution , by imposing the entropy condition in the sense of distributions , for all entropy pairs , with convex and , see .we propose hermite spectral viscosity methods based on generalized hermite functions with two different viscosity terms .  1 .  with viscosity term : where is a positive parameter depending on tending to as tends to , and is a viscosity operator which modifies only the high modes of the fourier - hermite expansion , that is , with and is a positive integer that tends to with .2 .  with viscosity term : where is defined in , is a positive parameter depending on tending to as tends to .the convergences of both schemes have been rigorously shown under certain conditions .  1 .  with viscosity term : + [ thm - u ] let be a nonlinear function such that , and there exists a primitive function of , i.e. , . assume further that .let be the solution to the spectral approximation , which is uniformly bounded , i.e. , independent of , where and the assumption for some , holds .let , , with some .then converges ( strongly in , ) to the unique entropy solution , , of the problem , where  ] is an open and bounded subset .  compared to the scheme in , the schemes in our paper have at least two advantages :* the viscosity term stablized the numerical scheme , due to its symmetry and positivity , while the stability of the scheme in and the scheme with viscosity term in our paper are not guaranteed ; * our schemes can be implemented easily with the help of the scaling factor . the better resolution and fewer oscillationsare retained with much smaller truncation terms even without viscosity , see figure [ fig - withoutviscosity ] , compared with fig 6.1 - 6.3 , .we provide two alternative schemes to solve the nonlinear conservation laws in .the convergences are guaranteed under some reasonable condtions or .it is hard to tell whether these conditions are weaker or not than the one in , i.e. , , independent of , due to the unbounded domain .the paper is organized as follows . in section 2 ,we give the definition of the generalized hermite functions and their properties .the new hermite spectral viscosity methods are proposed in section 3 .the convergences of our schemes have been rigorously shown under some conditions by using the compensated compactness arguments .the reasonableness of the conditions have been verified numerically in section 4 for the inviscid burger s equation .in this section , we introduce the generalized hermite functions and derive some properties which are inherited from the hermite polynomials . to establish the convergence rate of the hermite spectral method , we shall also state the convergence rate of the orthogonal approximation .let be the lebesgue space , equipped with the norm and the scalar product . in the sequel, we shall follow the convection in the asymptotic analysis , means that there exists some generic constants such that ; means that there exists some generic constant such that .let be the physical hermite polynomials , i.e. , , .the three - term recurrence is handier in implementation .one of the well - known and useful fact of hermite polynomials is that they are mutually orthogonal with respect to the weight .we define the generalized hermite functions with the scaling factor as for .it is readily to derive the following properties for the generalized hermite functions ( [ new hermite ] ) :  1 .[ orthogonal of h_n^alpha ] the forms an orthonormal basis of , i.e. where is the kronecker function[ eigenvalue ] is the eigenfunction of the following strum - liouville problem with the corresponding eigenvalue for conciseness , let us denote the strum - liouville operator as 3 .  by convention , , for . for ,the three - term recurrence is inherited from the hermite polynomials : 4 .the derivative of with respect to  for convenience , let .then 5 .  the orthogonality " of follows immediately from , i.e. , where is the kronecker function .any function can be written in the form with where are the fourier - hermite coefficients .let us denote the linear subspace of spanned by the first hermite functions by   [ remark - equivalence of derivatives ] actually , we have the norms controlled by and , for any .let us consider where , for all or . recall that we arrive at similarly , we can get .\ ] ]we define the projection , given , for all .more precisely , where , , are the fourier - hermite coefficients defined in .  the error estimate of the orthogonal projection onto is readily shown in theorem 4.2 , for and it can be trivially extended for that  [ lemma - orthogonal error ] for any with ,   in the sequel , the superscript in will be dropped if no confusion will arise .it is well known that the entropy solution to can be obtained as the limit when the artificial introduced viscosity term vanishes . in this section, we shall introduce two appropriate viscosity terms and . with the viscosity term , the viscosity operator has been introduced in the spectral scheme as in , where only the high frequency terms appear in the artificial viscosity .the convergence of this scheme has been shown under the condition for some such that or .the reasonableness of the condition has been verified in table [ table-1 ] and the arguments in section 4 .the spectral scheme is introduced in .let us first understand the viscosity operator ( defined in ) .[ lemma - d_x < d_xq ] let be defined as in , .then and for all .let us show in detail only , and can be obtained by the similar argument .let , and , where is the identity operator .then we split in dyadic parts , where  . here and for . from the orthogonality relation ,one has we bound each summand above by using the fact that given a linear operator defined in such that where are real numbers .then for all , since , for , for the second summand on the right - hand side of , since , we have , for any , combine and , we obtain that substituting the above equation back to , we get the desired result . and can be obtained similarly , with the fact that   the apriori estimates on the approximate solution are obtained in the following lemma .[ lemma - apriori estimate ] let , and there exists a primitive function of , i.e. .let , , is given in , , and \times\r\rightarrow\mathcal{r}_n ] and .let us choose in and it yields that   it is clear that the first term on the right - hand side of is and the second term is zero .in fact , the second term gives \\   = & \int_{\mathbb{r}}p_{n+1}\partial_x(f(u_n))u_ndx   = \int_{\mathbb{r}}\partial_x(f(u_n))u_ndx   = \int_{\r}f'(u_n)u_n\p_xu_ndx,\end{aligned}\ ] ] where the first equality of follows from the fact that ,\ ] ] the second and third equalities of hold due to the orthogonality of generalized hermite function . if there exists a primitive function of , with the fact that for any , , we obtain that   next , we shall examine the last term on the right - hand side of . using integration by parts ,we have let us compute and term by term : since , and can be estimated as with , by young s inequality . therefore , equation can be estimated as integrating from to , we get hence , and are followed immediately .  we are now ready to show theorem [ thm - u ] .the uniform boundedness of in ) ] the solution of .let us assume that there is a positive constant , independent of and , such that then and   we multiply by and integrate with respect to : where the first and second term on the right - hand side of are given by the same argument as in lemma [ lemma - apriori estimate ] .the third term is followed from the fact that integrating on both sides of from to , we obtain that equation and follow immediately .[ thm - v ] let be a nonlinear function such that , and there exists a primitive function of .assume further that .let be the solution to the spectral approximation , which is uniformly bounded , i.e. holds .we assume further that let .then converges ( strongly in , ) to the unique entropy solution , , of the problem , where  ] guarantees that there exists a subsequence converges in the weak- * sense of , denote this subsequence also and the limit .we shall prove that is the unique entropy solution of , and the whole sequence tends to in , .we first show that is in a compact set of .\\\notag = & \e_n\p_x\d_xv_n-\e_n\a^2\p_x(xv_n)+\e_n\a^2v_n-\e_n\a^4x^2v_n+\p_x[(i - p_{n+1})f(v_n)]\\ = & \e_n\p_x^2v_n+\e_n\a^2v_n-\e_n\a^4x^2v_n+\p_x[(i - p_{n+1})f(v_n)].\end{aligned}\ ] ] let be a compact set .notice that and thus , can be written as a sum of four terms , two tend to in , one is bounded in and the other one tends to in .besides , is in for any , since and is uniformly bounded in . therefore , in view of the murat s lemma , is in a compact subset of .  next , we show that is also in a compact subset of , where is the entropy pair introduced in . notice the estimates in - and the fact that , , the first , third , fourth and fifth term on the right - hand side of can be handled similarly as before , i.e. the first and fifth term tend to in ; the third term tends to in ; and the fourth term is bounded in .the rest two terms on the right - hand side of are both bounded in .in fact , and therefore , as we argued before , in view of the murat s lemma , is in a compact subset of .we conclude that converges strongly in , to a weak solution , , of the conservation law .it remains to show that the entropy condition is satisfied in the weak sense .let us multiply a nonnegative test function on both sides of and integrate it with respect to both and : \phi dxdt\\\notag \overset{\eqref{eqn - thm - v - u}}=&-\e_n\int_0^t\int_\r u'(v_n)\p_xv_n\p_x\phi dxdt-\e_n\int_0^t\int_\r u''(v_n)(\p_xv_n)^2\phi dxdt\\\notag & + \e_n\int_0^t\int_\r u'(v_n)\a^2v_n\phi dxdt-\e_n\int_0^t\int_\r u'(v_n)\a^4x^2v_n\phi dxdt\\   & -\int_0^t\int_\r u'(v_n)\p_x(i - p_{n+1})f(v_n)\p_x\phi dxdt-\int_0^t\int_\r u''(v_n)\p_xv_n(i - p_{n+1})f(v_n)\phi dxdt.\end{aligned}\ ] ] estimates - and imply that all the terms except the second one on the right - hand side of tends to , as .meanwhile , the second term is nonpositive , due to the convexity of .therefore , the entropy condition is satisfied in the weak sense , i.e. for any nonnegative test function , \phi dxdt\leq0.\ ] ]  compare the viscosity term in and , the convergence analysis for is easier , but the price to pay is the condition is stronger than , since this section we use the spectral viscosity methods or to numerically solve the inviscid burger s equation in , with the initial condition .we compute the same problem in for the purpose of comparison .the exact solution is given implicitly by the method of characteristics , i.e. , with the initial condition . and the shock presents at time .all of the numerical results displayed below are at time .  in our numerical simulations, we implement the scheme or and overcome the difficulty of accurately approximating the fourier - hermite coefficients ( mentioned in section 6 ) with the help of the suitable scaling factor .the optimal choice of the scaling factor to accurately resolve the functions is still open , let alone the solution to some partial differential equations .but the suitable choice of the scaling factor is investigated in , , , , etc .it is known so far that the scaling factor should match the asymptotical behavior of the function / solution to be resolved . from , for any fixed time , it is easy to see that , as . according to the practical guidelines in , we choose . generally speaking, the asymptotical behavior of the solutions to evolution equations is time - dependent .a time - dependent scaling factor should improve the resolution further .the study on the time - dependent scaling factor of the hermite spectral method in solving evolution equations has been recently investigated in . here , we focus ourselves on the intrinsical problem by using constant scaling factor .  in both spectral schemes ,we let , , in or .the coefficients or , , are the solution of a nonlinear system of ordinary differential equation .it is solved by using the fourth order runge - kutta method with an adaptive time step ( _ ode45 _ in matlab ) .  ., and versus , respectively , are displayed , where is the solution obtained by the spectral viscosity method with . [ cols="^,^,^,^,^,^,^,^",options="header " , ]   figure [ fig - withviscosity_differentn ] displays the results of the spectral viscosity method with and various and , respectively .it reveals that the larger is , the better resolution at discontinuity is obtained .it is natural to ask how to tune . with fixed ,the results with and are plotted in figure [ fig - withviscosity_differentepsilon ] .we need to balance the resolution near the discontinuity and the oscillations away from the discontinuity in the choice of . with the same ,the smaller yields the better resolution of the discontinuity , yet the more oscillations away from the discontinuity . in table [ table-2 ] we display , and versus , respectively , where is the numerical solution to with .the norm is computed using the same rule as before .the -norm in space in is computed by trapezoid rule using equidistant grid .again , we obtain the growth rate of , and versus by using the command polyfit " in matlab , which are and , respectively . the condition has been numerically verified .in this paper , we propose two spectral viscosity methods based on the generalized hermite functions for the solution of nonlinear scalar conservation laws in the whole line .our scheme has been shown rigorously to converge to the unique entropy solution by using compensated compactness arguments .the numerical experiments illustrate the implementability of our schemes .thanks to the generalized hermite functions , the fewer oscillations and better resolutions compared with those in are obtained with much smaller truncation modes , even before adding the viscosity term .aaaa j. aguirre and j. rivas , _ a spectral viscosity method based on hermite functions for nonlinear conservations laws _ , siam j. numer ., 46(2 ) : 1060 - 1078 , 2008. j. boyd , _ the rate of convergence of hermite function series _ , math ., 35:1039 - 1316 , 1980. j. boyd , _ asymptotic coefficients of hermite function series _ , j. comput .phys . , 54:382 - 410 , 1984 . j. boyd , _ chebyshev and fourier spectral methods _. edition , dover , new york , 2001 g .- q .chen , _ the compensated compactness method and the system of isentropic gas dynamics _ , preprint msri-00527 - 91 , mathematical science research institute , berkeley , ca , 1990 .chen , q. du and e. tadmor , _ spectral viscosity approximations to multidimensional scalar conservation laws _ , math ., 61:629 - 643 , 1993 .d. funaro and o. kavian , _ approximation of some diffusion evolution equation in unbounded domains by hermite function _, math . comp ., 37:597 - 619 , 1991 .a. gelb and e. tadmor , _ enhanced spectral viscosity approximations for conservation laws _ , appl ., 33:3 - 21 , 2000 .d. gottlieb and s. orszag , _ numerical analysis of spectral methods : theory and applications _, soc . in . andappl . math . ,philadelphia , 1977 .d. gottlieb and c .- w .shu , _ on the gibbs phenomenon and its resolution _ , siam rev ., 39(4):644 - 668 , 1997 .d. gottlieb , c .- w .shu , a. solomonoff , and h. vandeven , _ on the gibbs phenomenon i : recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function _ , j. comput ., 43:81 - 92 , 1992 .d. gottlieb and e. tadmor , _ recovering pointwise values of discontinuous data within spectral accuracy _ , in progress and supercomputing in computational fluid dynamics , e. m. murman and s. s. abarbanel , eds . , birkhuser , boston , 357 - 275 , 1985 .guo , h. ma , and e. tadmor , _ spectral vanishing viscosity method for nonlinear conservation laws _ , siam j. numer ., 39(4):1254 - 1268 , 2001 .x. luo and s. s .- t .hermite spectral method to 1d forward kolmogorov equation and its application to nonlinear filtering problems _ , ieee trans .control , 58(10):2495 - 2507 , 2013 .x. luo , s .-yau and s. s .-t , yau , _ time - dependent hermite - galerkin spectral method and its applications _ , appl .comput . , 264:378 - 391 , 2015 . h. ma , _ chebyshev - legendre spectral viscosity method for nonlinear conservation laws _ , siam j. numer .anal . , 35(3):869 - 892 , 1998 . h. ma , _ chebyshev - legendre super spectral viscosity method for nonlinear conservation laws _ , siam j. numer ., 35(3):893 - 908 , 1998 . y. maday , s. m. ould kaber , and e. tadmor , _ legendre pseudospectral viscosity method for nonlinear conservation laws _ , siam j. numer ., 30:321 - 342 , 1993 . y. maday and e. tadmor , _ analysis of the spectral vanishing viscosity method for periodic conservation laws _ , siam j. numer ., 26:854 - 870 , 1989 . s. schochet , _ the rate of convergence of spectral - viscosity methods for periodic scalar conservation laws _ , siam j. numer ., 27:1142 - 1159 , 1990 . j. smoller , _ shock waves and reaction - diffusion equations _ , springer - verlag , new york , 1983. j. shen , t. tang and l .- l .wang , _ spectral methods : algorithm , analysis and application _ , springer , 2011 .j. shen and l .- l .wang , _ some recent advances on spectral methods fo unbounded domains _ , commun .phys . , 5:195 - 241 , 2009 .e. tadmor , _ convergence of spectral methods for nonlinear conservation laws _ , siam j. numer ., 26:30 - 44 , 1989 .e. tadmor , _ shock capturing by the spectral viscosity method _ ,methods appl .engrg . , 80:197 - 208 , 1990 . t. tang , _ the hermite spectral method for gaussian - type functions _siam j. sci ., 14:594 - 606 , 1993 . l. tartar , _ compensated compactness and applications to partial differential equations _ , in nonlinear analysis and mechanics : heriot - watt symposium , vol .notes n math .39:136 - 212 , j. knopps , ed . , pitman , boston , london , 1979 .h. vandeven , _ family of spectral filters for discontinuous problems _ , j. sci ., 8:159 - 192 , 1991 .x .- m . xiang and z .- q .wang , _ generalized hermite spectral method and its applications to problems in unbounded domains _ , siam j. numer ., 48(4):1231 - 1253 , 2010 .

Abstract: in this paper , we propose new spectral viscosity methods based on the generalized hermite functions for the solution of nonlinear scalar conservation laws in the whole line . it is shown rigorously that these schemes converge to the unique entropy solution by using compensated compactness arguments , under some conditions . the numerical experiments of the inviscid burger s equation support our result , and it verifies the reasonableness of the conditions .


the federal highway administration of the u.s .department of transportation has originated the next generation simulation community ( ngsim ) in order to `` improve the quality and performance of simulation tools , promote the use of simulation for research and applications , and achieve wider acceptance of validated simulation results ''  .as part of the program , a first data set has been collected at the berkeley highway laboratory ( bhl ) in emeryville by cambridge systematics and the california center for innovative transportation at uc berkeley . the bhl is a part of the i-80 at the east coast of the san francisco bay .six cameras have been mounted on top of the {m} ] length in a 30-minute period in december 2003 .the result has been published as the `` prototype dataset '' . as part of the california partners for advanced highways and transit ( path ) program ,the institute of transportation studies at uc berkeley further enhanced the data collection procedure  and in april 2005 , another trajectory dataset was recorded at the same location using seven cameras and capturing a total of 5648 vehicle trajectories in three 15-minute intervals on a road section of approximately {m} ] tall 10 universal city plaza next to the hollywood freeway us-101 . on a road section of {m} ] ( which are then reported as {m / s^2} ] .however , in the later datasets , the velocity distributions are very spiky , i.e. , velocities tend to snap to certain values .looking at the velocities of an example trajectory exhibits an unrealistic behavior : if taken for real , this would mean that drivers do not smoothly brake or accelerate but use the gas and brake pedal only occasionally but hard to quickly change between `` preferred velocities '' .also , to produce the spikes in the velocity distribution , all drivers must happen to `` like '' the same velocities .this is clearly unrealistic and therefore the velocity spikes must be an artefact of the measurement method .one may credit the velocity spikes to discretization errors ( time and space are discretized , thus velocity can only take certain discrete values as well ) , but two observations object to that : first , the spikes are not delta peaks , other velocities still do appear .second , given the time discretization {s} ] , this would mean that the spatial accuracy of the measurement method is just {m} ] while the later two use {s} ] .  however , this value is a suggestion for the _ acceleration _ smoothing width only. we will now show that it is not necessary to use such large smoothing widths for the positions and velocities .let be a random variable describing the positions of vehicle with expectation value and variance .the measured trajectory is a realization of and , assuming unbiased noise , the real trajectory is equal to .now , we define two new random variables describing the velocities and accelerations of vehicle in terms of symmetric difference quotients , since this is a linear combination of random variables , the expectation values of and will be the first and second derivative of , respectively ( the real velocities and the real accelerations ) .assuming uncorrelated noise , the variances of and are given by thus , the noise will be strongly amplified by the differentiation and therefore , the velocities must be weaker smoothed than the acceleration and the positions weaker than the velocities . in fig .[ fig : smoothing](b , c ) we plotted the lateral positions and longitudinal velocities and accelerations of a sample trajectory of the us-101 dataset as original data and for different smoothing widths .the position smoothing width is very critical , because the lane change duration is quite sensitive to it .as visible in the plot , a large will significantly smear out the trajectory leading to larger lane change durations . in order to resolve the issue with the `` preferred velocities ''the smoothing of the velocities should be strong enough so that the smoothed velocities no longer follow the trends of this `` semi - quantization '' .however , the smoothing should be as weak as possible because the velocity smoothing width also quantitatively influences some results .finally , we decided in favor of the smoothing times {s}\text { , } \ ,  t_v = \unit[1]{s}\text{,}\,\text { and } \ ,  t_a = \unit[4]{s}.\ ] ] the effects of this smoothing on the acceleration distribution of the prototype dataset and the velocity distribution of the two later datasets can be seen in fig .[ fig : smoothing](d - f ) .effects of the applied trajectory smoothing : ( a ) the dependency of the acceleration variances on the smoothing kernel width .( b ) acceleration distribution in the prototype dataset and ( c ) velocity distribution in the 7.50am8.05am datafile of the us-101 dataset . in the right columnthe lateral position ( d ) , longitudinal velocity ( e ) and acceleration ( f ) of a sample trajectory of the us-101 dataset with different applied smoothing kernel widths is shown.,title="fig : " ] effects of the applied trajectory smoothing : ( a ) the dependency of the acceleration variances on the smoothing kernel width .( b ) acceleration distribution in the prototype dataset and ( c ) velocity distribution in the 7.50am8.05am datafile of the us-101 dataset . in the right columnthe lateral position ( d ) , longitudinal velocity ( e ) and acceleration ( f ) of a sample trajectory of the us-101 dataset with different applied smoothing kernel widths is shown.,title="fig : " ] + effects of the applied trajectory smoothing : ( a ) the dependency of the acceleration variances on the smoothing kernel width .( b ) acceleration distribution in the prototype dataset and ( c ) velocity distribution in the 7.50am8.05am datafile of the us-101 dataset . in the right column the lateral position ( d ) ,longitudinal velocity ( e ) and acceleration ( f ) of a sample trajectory of the us-101 dataset with different applied smoothing kernel widths is shown.,title="fig : " ] effects of the applied trajectory smoothing : ( a ) the dependency of the acceleration variances on the smoothing kernel width .( b ) acceleration distribution in the prototype dataset and ( c ) velocity distribution in the 7.50am8.05am datafile of the us-101 dataset . in the right column the lateral position ( d ) , longitudinal velocity ( e ) and acceleration ( f ) of a sample trajectory of the us-101 dataset with different applied smoothing kernel widths is shown.,title="fig : " ] + effects of the applied trajectory smoothing : ( a ) the dependency of the acceleration variances on the smoothing kernel width .( b ) acceleration distribution in the prototype dataset and ( c ) velocity distribution in the 7.50am8.05am datafile of the us-101 dataset . in the right columnthe lateral position ( d ) , longitudinal velocity ( e ) and acceleration ( f ) of a sample trajectory of the us-101 dataset with different applied smoothing kernel widths is shown.,title="fig : " ] effects of the applied trajectory smoothing : ( a ) the dependency of the acceleration variances on the smoothing kernel width .( b ) acceleration distribution in the prototype dataset and ( c ) velocity distribution in the 7.50am8.05am datafile of the us-101 dataset . in the right columnthe lateral position ( d ) , longitudinal velocity ( e ) and acceleration ( f ) of a sample trajectory of the us-101 dataset with different applied smoothing kernel widths is shown.,title="fig : " ]most empirical traffic state data is gathered by stationary loop detectors that can measure quantities at different times , but at a single location only .these measurement devices are therefore capable of measuring temporal quantities , but not spatial quantities . however , since both spatial and temporal quantities are important in traffic science , it is common practice to derive the spatial quantities from temporal measurements by using some conservation assumptions ( e.g. , constant vehicle velocities within a certain time period ) .modern trajectory data like the ngsim recordings provide enough data to enable a validation of these practices .  in the following , we will describe the analysis process to obtain spatial information from temporal data , and vice versa , and check its accuracy for three examples : the microscopic fundamental diagram , and the distributions of the time gaps and times - to - collision .later , we will investigate lane changes in the ngsim data .all following analysis will use the smoothed datasets obtained by the smoothing method introduced and motivated above and all references to any `` ngsim dataset '' are to be understood as references to the smoothed datasets .the two measurement types we want to compare are the traditional stationary loop detector , which is singular in space but continuous in time , and an aerial photograph , which is continuous in space but singular in time .the basic idea of our analysis is to place virtual loop detectors into the trajectory data .these would correspond to lines parallel to the time axis in a space - time - plot , while lines parallel to the space axis correspond to momentary snapshots ( virtual photographs ) of the measurement area ( cf .[ fig : virtualdetectors ] ) . wherever those lines intersect , both stationary and momentary measurements are available for comparison . to maximize the amount of data available for comparison ,we applied the following algorithm to the data : for every tenth datapoint of each trajectory , the spatial leader and the temporal leader are determined .the spatial leader is the vehicle currently driving ahead of the vehicle and the temporal leader is the vehicle that most recently passed the actual position of vehicle ( for simplicity , we will denote the temporal leader with as well ) .the first information is only available to momentary measurements while the second is only available to stationary measurements .  [cols="^,^ " , ]   besides the ability to compare stationary and momentary measurements , the ngsim trajectory data sets also provide a good basis to investigate lane changes . in order to determine the lane change duration, we collected all lane changes in the ngsim data .however , from the processed video data supplied with the ngsim datasets it can be seen that sometimes the tracking algorithm accidentally misplaced a vehicle across the lane boundary and back after a few timesteps . also , sometimes drivers might have abort an already begun lane change or quickly crossed two lanes . since we just want to look at real and normal single - lane lane changes , we therefore filtered out all lane changes that were closer than a certain threshold to another lane change , which we chose to be {s} ] by looking at the curvature of the two mode values and .this procedure is similar to the approach done in ref . , where the lane change start and end time of each trajectory were determined by looking at the curvature of the lateral position . however , finding the correct point in the curvature might be somewhat arbitrary , thus we will in the following look at a more well - defined way to measure a lower bound of the lane change duration .the ngsim vehicle detection algorithm does not only detect the vehicle position but also its length and width .since the lane assignment algorithm works such that each datapoint is placed into the lane where its mid - point front - bumper position lies in , it is possible to determine the time where a lane - changing vehicle first intruded the destination lane and the time where it just completely left the source lane . given the lane - changing event time and the relative time and position as defined in eqs . , the relative start time and end time of the lane change may be defined as follows ( a higher lane index corresponds to a larger lateral position ) : then , of course , the lane change duration is obtained trivially from   in total , we have investigated 1231 lane changes , 1105 of which were suitable to calculate according to eq . .in the remaining 126 cases , either or were undefined because the corresponding condition was not fulfilled for any  ] ( mode value of the distribution ) , a value already found valid for german highways back in 1978  , which is , however , substantially different from the one obtained by rule of thumb from the conditional probability density .the mean and standard variation of the distribution are {s}.  \label{eq : ngsim_tlc}\ ] ] however , one should be aware that definition  measures the time span where the vehicle occupies two lanes , which can only be taken as a lower bound of the real lane change duration . including the preparation and possible post - processing of a lane change , a value of {s} ] .this indicates that drivers perceive a velocity advantage on the destination lane before performing the lane - changing maneuver and take anticipationary actions .   of finding a vehicle on lateral position relative to the lane boundary at a given time relative to the lane - changing event time .the lower left plot shows the distribution of lane change durations according to the definition  given in the text . the mean lane change duration is {s} ] .the lower right plot shows the conditional probability density of the velocity difference between the leader on the destination lane and the leader on the source lane for fixed times relative to the lane - changing event ( the white line shows the mean value).,title="fig : " ]  of finding a vehicle on lateral position relative to the lane boundary at a given time relative to the lane - changing event time .the lower left plot shows the distribution of lane change durations according to the definition  given in the text .the mean lane change duration is {s}$ ] .the lower right plot shows the conditional probability density of the velocity difference between the leader on the destination lane and the leader on the source lane for fixed times relative to the lane - changing event ( the white line shows the mean value).,title="fig : " ]the availability of the ngsim data sets spurred a considerable research activity , particularly with respect to lane changing , where larger - scale empirical investigations are now possible for the first time .to date , most researchers only used the positional information which allows , for example , to investigate the lane - changing rate , the duration of lane changes , the gap - acceptance behavior , or the propagation velocity of longitudinal density waves .the full potential of the data , i.e. , using the positional information together with that for velocity and acceleration , has hardly been tapped .a possible reason is that the velocity and acceleration information can not be used directly since the noise of the positional information is greatly increased by the necessary numerical differentiations . in this paper , we have developed a filter to extract more realistic velocity and acceleration information from the positional data . since the trajectories are comparatively short , we included the boundary regions in the filtered output by reducing the width of the necessary smoothing operations near the boundary .this implies determining the most efficient order of the smoothing and differentiation operations of the filter since they do no longer commute , and a wrong order may even lead to a systematic bias .it must be noticed that it is inherently difficult to determine the optimal filter parameters that eliminate most of the noise while retaining the real information .this is particularly crucial for mean - reverting quantities such as the accelerations , where large smoothing time intervals will eventually suppress the whole information .clearly , further research is necessary to develop more sophisticated , possibly nonlinear , filters .the velocity and acceleration information of the trajectories can be used in many ways . in this work ,we investigate the systematic errors in determining spatial quantities from temporal information , and vice versa .the background is that spatial quantities such as the gap to the leading vehicle , the density , or the times - to collision , are usually estimated by single - vehicle data from stationary detectors , i.e. , by using temporal information . using `` virtual stationary detectors '' that are fed with the trajectory data and simulating the estimation procedure, we could quantitatively determine the resulting estimation errors .besides the well - known underestimation of the real density of congested traffic , we found that the percentage of critical values of times - to - collision is underestimated by a factor of 2 and more when estimated from single - vehicle data .this clearly is relevant for safety - related applications .another application field are empirical tests and parameter calibrations for car - following and lane - changing models . in this work, we showed that , prior to a discretionary lane change , there is a noisy and small , but significant , velocity difference in favor of the target lane . from this , we conclude that lane - changing decisions are not only based on gaps and velocities , but also on velocity differences , and possibly , on accelerations as considered in ref . .more generally , the trajectory data allow , for the first time , to empirically investigate the strategical and tactical actions for preparing or facilitating a lane change  .apart from the actions of the lane - changing driver , this also includes the actions of the other drivers involved , such as cooperative actions of the follower on the target lane to allow zip - like merging .this is relevant for microscopic simulation software since it turned out to be notoriously difficult to model realistic lane changes , particularly in the case of mandatory changes in congested traffic .the acceleration information of the data can also be used to investigate to which extent the local traffic environment ( consisting , e.g. , of the next - nearest and further leading vehicles ) influences the longitudinal driving behavior  .for example , it has been proposed that the driving style is influenced by the local velocity variance as determined from few leading vehicles  .finally , the velocity and acceleration information can be used to determine the influence of traffic congestion on the fuel consumption and emissions  . sincereliable characteristic maps are available for the instantaneous fuel consumption and emission rates of various pollutants as a function of velocity and acceleration , these quantities can now be estimated , for real situations , with unprecedented accuracy .the authors would like to thank the federal highway administration for providing the ngsim trajectory data used in this study .m. treiber , a. kesting , and c. thiemann , `` how much do traffic congestion increase fuel consumption and emissions ? applying a fuel consumption model to the ngsim trajectory data , '' in _ trb annual meeting 2008 cd - rom _ , ( transportation research board of the national academies , washington , d.c . , 2008 ) .s. hirst and r. graham , `` the format and presentation of collision warnings , '' in _ ergonomics and safety of intelligent driver interfaces _ , y. noy , ed . , ( lawrence erblaum associates , new jersey , 1997 ) .m. treiber , a. kesting , and d. helbing , `` understanding widely scattered traffic flows , the capacity drop , and platoons as effects of variance - driven time gaps , '' physical review e * 74 , * 016123 ( 2006 ) .

Abstract: the ngsim trajectory data sets provide longitudinal and lateral positional information for all vehicles in certain spatiotemporal regions . velocity and acceleration information can not be extracted directly since the noise in the ngsim positional information is greatly increased by the necessary numerical differentiations . we propose a smoothing algorithm for positions , velocities and accelerations that can also be applied near the boundaries . the smoothing time interval is estimated based on velocity time series and the variance of the processed acceleration time series . the velocity information obtained in this way is then applied to calculate the density function of the two - dimensional distribution of velocity and inverse distance , and the density of the distribution corresponding to the `` microscopic '' fundamental diagram . furthermore , it is used to calculate the distributions of time gaps and times - to - collision , conditioned to several ranges of velocities and velocity differences . by simulating `` virtual stationary detectors '' we show that the probability for critical values of the times - to - collision is greatly underestimated when estimated from single - vehicle data of stationary detectors . finally , we investigate the lane - changing process and formulate a quantitative criterion for the duration of lane changes that is based on the trajectory density in normalized coordinates . remarkably , there is a very noisy but significant velocity advantage in favor of the targeted lane that decreases immediately before the change due to anticipatory accelerations .


entanglement has been considered as one of the most important resources for quantum information processing and quantum communication including quantum key distribution ( qkd ) and quantum secret sharing ( qss ) .entanglements can be obtained from several methods , one of which is to utilize a projective measurement on multi - particles . if more than two particles can be handled then a projective measurement on multi - particles such as a bell - state measurement allows the state of the particles to be entangled , although the particles have never been entangled with one another .we now consider the following four - particle state , here the subscript represents the index of each particle , each is an integer running from to , , and which is one of -dimensional bell states .the state in eq .( [ eq : entanglement_swapping ] ) has mutually entangled particles 1 and 2 , and mutually entangled particles 3 and 4 .there does not exist any entanglement between particles 1 and 4 , nor between particles 2 and 3 in the state .however , after the measurement on particles 1 and 4 ( or 2 and 3 ) by the observable with eigenstates [ or for , the state in eq .( [ eq : entanglement_swapping ] ) becomes a product state of the form , whose particles 1 and 4 , and particles 2 and 3 are mutually entangled , respectively .this interesting phenomenon is called the _ entanglement swapping _ ( es ) , which was originally proposed by ukowski_ et al . _ , was generalized to multipartite quantum systems by zeilinger _et al . _ and bose _et al . _ independently , and was experimentally realized by pan _et al . _furthermore , the generalizations of es for multipartite and arbitrary dimensional quantum systems were also presented .these generalizations possess the more significant meaning in view of the fact that quantum communications in higher dimensional quantum systems are more secure than in two qubits . on this account, we consider -dimensional quantum systems , called qudits , in this paper .recently , several practical applications of es , such as the purification protocols  and the cryptographic protocols  , have been proposed . in particular , the es - based two - party qkd protocol , presented by cabello  , permits the secure key distribution , whose structure is different from those of the previous qkd protocols  .in addition , the es - based multiparty quantum communication consisting of multiparty qkd and qss protocols  was suggested  .however , this multiparty quantum communication can be shown to be insecure by means of a variant of zhang , li , and guo s eavesdropping strategy  .we call this eavesdropping strategy the _ zhang - li - guo - type _ ( zlg - type ) attack . in this paper , we present the zlg - type attack applied to the multiparty quantum communication protocol , and show that the protocol is not secure against the attack . modifying the protocol in the method analogous to cabello s modification for the es - based two - party qkd  , we also show that there exists a multiparty quantum communication protocol which is secure against the zlg - type attack as well as the basic intercept - resend attack .this paper is organized as follows . in sec .[ sec : cabello_protocol ] we review the original cabello s protocol . in sec .[ sec : zlg ] we show that the protocol is not secure by employing the zlg - type attack . in sec .[ sec : modified_protocol ] we present the modified protocol which is secure against not only the basic intercept - resend attack but also the zlg - type attack . finally , in sec .[ sec : conclusions ] we summarize our results .cabello  extended the es - based qkd protocol between two parties  into multiparty quantum communication . in this section ,we review the extended protocol for the three - party quantum communication  .since the three - party quantum communication can easily be generalized to the multiparty case , we consider only the three - party case in this paper .  in the beginning, alice prepares one of -dimensional bell states and one of -dimensional greenberger - horne - zeilinger ( ghz ) states  , and bob ( carol ) prepares one of -dimensional bell states . in the protocol , we assume that the information on the initial state is publicly known .we now remark the following two formulas for swapping bell states and ghz states : and where , which is a ghz state .  by exploiting the above two formulas , the process can be described as follows .* alice makes a bell measurement on one qudit of the bell state and one qudit of the ghz state , and then alice sends one of the other qudits of the ghz state to bob ( carol ) . let be the given initial state . here, the subscripts represent the indices of the parties manipulating the state .if we let be the result of alice s measurement for some and , then we obtain from eq .( [ eq : es2 ] ) that the initial state becomes . * bob ( carol ) performs a bell measurement on one qudit of his ( her ) bell state and the qudit received from alice . after the measurements , bob ( carol ) sends alice his ( her ) qudit which is not measured .+ for some and , we let [ be the measurement result of bob ( carol ) . then it follows from eq .( [ eq : es3 ] ) that the resulting state is . * alice measures the ghz state , , which consists of her own qudit and the qudits received from bob and carol , and then publicly announces the result of the measurement. then the final state becomes the correlation in ( [ eq : original_final_state ] ) allows multiparty qkd and qss of classical information , simultaneously .for the secure communication , legal users check the eavesdropping by comparing random subsequences of their key strings with the original correlations .we note that the basic intercept - resend attacks analogous to the attack presented in  , may destroy the correlations among the keys , since any eve s attempt to acquire the keys may change the result of the public announcement in an unpredictable way. therefore , cabello s protocol is secure against such kind of attacks .zhang , li and guo  proposed a special strategy for eavesdropping , called the zlg - type attack , which shows that the es - based two - party qkd protocol presented by cabello  is not secure . in this section , we show that the security of cabello s protocol for the es - based multiparty quantum communication can also be threatened by the zlg - type attack .the eavesdropping strategy is illustrated in fig .[ fig : fi1 ] , and can be described as follows : initially , eve prepares two bell states , and .thus , the initial state is .* alice performs a bell measurement on one qudit of her bell state and one qudit of her ghz state as in the original cabello s protocol .+ then the state becomes for some and .* eve intercepts two qudits transmitted from alice and keeps them intact , and then sends one qudit of her bell state to bob ( carol ) as in fig .[ fig : fi1 ] ( b ) . then bob ( carol )makes a bell measurement on the received qudit and one qudit of his ( her ) bell state without being concerned of eavesdropping . +then the state becomes for some and .* after their measurements , eve intercepts the qudits transmitted from bob ( carol ) , and measures the bell state consisting of one of her qudit and the intercepted qudit from bob ( carol ) .eve performs the suitable local operation on the qudits which she intercepts from alice , according to the measurement result of the ghz state as in the following : if we let be the measurement result of the two bell states , then eve applies the local operation where and are the unitary operations defined by + using the following relation the resulting state can be described by , after applying the local operation in ( [ eq : local_operation ] ) . *eve returns the two qudits to alice after the above operations. then alice measures the ghz state , and publicly announces the result . +then the final state becomes +   alice , bob and carol now attempt checking the eavesdropping by comparing random subsequences of their measurement results .however , legal users can not verify the presence of eavesdropping since no errors can be introduced from this particular attack as seen in eq .( [ eq : original_final_state ] ) and eq .( [ eq : zlg_final_state ] ) .moreover , eve can perfectly obtain information on the keys , which can readily be shown from eq .( [ eq : zlg_final_state ] ) .therefore , we can prove that cabello s multiparty quantum communication protocol is insecure against the zlg - type attack .in the preceding section , we have shown that cabello s protocol for multiparty quantum communication is not secure against a particular eavesdropping attack , the zlg - type attack . in this sectionwe suggest a modified protocol , which is secure against not only the basic intercept - resend attack but also the zlg - type attack .   the basic idea of our modification results from cabello s modification for the es - based two - party qkd  .we now remark the following two formulas \nonumber\\   = { \frac{1}{d}}\sum_{k , l=0}^{d-1}\omega_{d}^{-kl } &   [ ( i\otimes i\otimes \mathcal{f})(u_1 -k , u_2,l)_{125}]\nonumber\\   & \otimes(k , u_3 -l)_{43 } ,  \label{eq : fourier01}\end{aligned}\ ] ]   \otimes [ ( \mathcal{f}\otimes i ) ( 0 , 0)_{45 } ] \nonumber\\   = { \frac{1}{d}}\sum_{k , l=0}^{d-1}&\omega_{d}^{-kl }   [ ( i\otimes \mathcal{f}\otimes \mathcal{f})(u_1 -k , u_2,l)_{125}]\nonumber\\   & \otimes(k , u_3 -l)_{43 } ,  \label{eq : fourier02}\end{aligned}\ ] ] which can be obtained from the straightforward calculations . here , is the -dimensional quantum fourier transform ( qft ) defined by .the modified protocol is illustrated in fig .[ fig : fi2 ] and is described as follows .  * for convenience ,legal users initially prepare the following state : . *alice sends one qudit of her ghz state to bob ( carol ) .bob ( carol ) also sends one qudit of his ( her ) bell state to alice , and then randomly applies either the qft or the identity operation to the other qudit of his ( her ) bell state .alice performs a bell measurement on her qudit of the ghz state and one qudit of her bell state , and bob ( carol ) also performs a bell measurement on the qudit received from alice and the other qudit of his ( her ) state .+ there exist three cases in this step : ( i ) both bob and carol apply the identity operation , ( ii ) only one of them applies the qft , and ( iii ) both of them apply the qft .the case ( i ) is the same as the case of the original cabello s protocol . in the case ( ii ) , if it is carol who applies the qft , and , and are the measurement results of alice , bob and carol respectively , then alice s three - particle state becomes by eq .( [ eq : fourier01 ] ) . in the case ( iii ) , if , and are the measurement results of alice , bob and carol respectively , then alice s three - particle state becomes by eq .( [ eq : fourier01 ] ) and eq .( [ eq : fourier02 ] ) . *bob ( carol ) announces his ( her ) operation chosen in the previous step .alice performs the inverse of his ( her ) operation to the qudit received from him ( her ) .alice now measures the ghz state which is automatically made from the legal user s measurements , and publicly announces the result . then the final state becomes   for the secure communication , legal users check the eavesdropping by comparing random subsequences of key strings .  as onecan see in eq .( [ eq : original_final_state ] ) and eq .( [ eq : modified_final_state ] ) , our modified protocol provides legal users with the same correlation as that in the original cabello s protocol .furthermore , legal users can detect any eavesdropping unless eve knows bob and carol s choices of their operations beforehand .we are now going to discuss the security of our modified protocol in detail .as mentioned in the previous section , it is impossible for eve to get all information unless she knows bob and carol s decisions beforehand .more precisely , legal users can detect eve s presence with the probability not less than ^ 2 ] , eve can be detected .eve can be detected with the probability , not less than we note that in the -party quantum communication the probability to detect eve in the zlg - type attack , is at least ^{n-1} ] .furthermore , we can obtain from these probabilities the fact that the higher the dimension of quantum system is , the more secure the communication becomes .   the two - party attack on our modified protocol is illustrated in fig .[ fig : fi4 ] and is described as follows .  * the initial states of legal users are the same as those in our modified protocol .eve prepares two bell states and one ghz state .* when alice sends one qudit of her ghz state to bob ( carol ) , and bob ( carol ) sends alice one qudit of his ( her ) bell state , eve intercepts the transmitted qudits , and sends her qudits to legal users instead ; she sends one qudit of each bell state to alice and one qudit of her ghz state to bob ( carol ) .bob ( carol ) randomly applies either the qft or the identity operation to the other qudit of his ( her ) bell state .alice , bob and carol perform bell measurements as in the our modified protocol .while they perform bell measurements , eve performs two bell measurements on her own qudits and the qudits which she intercepts from alice as in fig .[ fig : fi4 ] ( b ) . * after bob ( carol ) announces his ( her ) operation chosen in the previous step , eve performs the inverse of his ( her ) operation to the qudit received from him ( her ) , and then measures the ghz state containing the qudits .  in this eavesdropping strategy , eve can not forge the exact correlation on the keys for the three - party quantum communication without knowing bob s ( carol s ) decision to choose his ( her ) own operation .thus , legal users can detect eve s intervention by publicly comparing a sufficiently large random subset of their sequences of key strings .we can straightforwardly show that the probability with which eve can be detected in the three - party quantum communication , is at least for each testing key .thus our modified protocol is secure against the two - party attack since the probability for testing keys is not less than ^{m}$ ] .  in order that eve acquire the keys for the qss in the multiparty quantum communication , she should access the qudits transmitted to all legal users .thus , employing eq .( [ eq:3-party_2p_prob ] ) , one can obtain that the probability of detecting eve s presence in the -party quantum communication is not less than for each testing key .it follows from the above probability in eq .( [ eq : n - party_2p_prob ] ) that as the dimension of the system or the number of parties increases , the multiparty quantum communication including the multiparty qkd and qss is more secure against the two - party attack .in this paper , we presented the zlg - type attack applied to cabello s protocol for the multiparty quantum communication consisting of the multiparty qkd and the qss of classical information , and showed that the protocol is not secure against the attack . modifying the protocol, we also showed that there exists a multiparty quantum communication protocol which is secure against the zlg - type attack as well as the basic intercept - resend attack .cabello s es based protocol simultaneously makes the multi - party qkd and the qss of classical information possible .furthermore , it has a merit that no transmitted quantum data are rejected as mentioned in .our work shows that we can obtain the secure protocol with those advantages by exploiting an appropriate modification process .the authors acknowledge the kias quantum information group for useful discussions .is supported by a kias research fund ( no .02 - 0140 - 001 ) , j.k . by a korea research foundation grant ( krf-2002 - 070-c00029 ) , and s.d.o . by a korea science and engineering foundation grant ( r06 - 2002 - 007 - 01003 - 0 )

Abstract: extending the eavesdropping strategy devised by zhang , li and guo [ phys . rev . a * 63 * , 036301 ( 2001 ) ] , we show that the multiparty quantum communication protocol based on entanglement swapping , which was proposed by cabello [ quant - ph/0009025 ] , is not secure . we modify the protocol so that entanglement swapping can secure multiparty quantum communication , such as multiparty quantum key distribution and quantum secret sharing of classical information , and show that the modified protocol is secure against the zhang - li - guo s strategy for eavesdropping as well as the basic intercept - resend attack .


one of the most puzzling features of quantum theory is its nonlocal nature .separated observers on a shared entangled state may carry out measurements on such a way that the correlations they generate are outside the set of common cause correlations .in particular , such quantum correlations may find application in novel device - independent information tasks , which have no counterparts in the classical world .they enable perfect security , randomness generation and state tomography without the need to trust the internal working of the devices .the concept of bell inequalities is a particularly useful tool to detect nonlocal quantum correlations , as violation of a single bell inequality conclusively proves the nonlocal character of correlations .the standard scenario for a bipartite two - outcome bell test is as follows .two spacelike separated parties , alice and bob , both share copies of a quantum state of a given dimension .alice ( bob ) may choose between ( ) alternative measurements at random , where each measurement has two possible outcomes .in a single run of the experiment the correlations between the two -valued observables and can be represented by the product . in order to obtain an accurate estimation of the correlations for each pair , alice and bob repeat the experiment many times using a copy of the state in each round . averaging over many runs of the experimentyields the mean value .the chsh inequality is probably the most well - known and simplest example of a bell inequality consisting of two measurement settings ( ) both on alice s and on bob s part .any correlations in the framework of local classical theories obey the following chsh inequality , curiously , quantum theory allows for a violation of the chsh inequality , but the strength of nonlocal correlations is still limited , it obeys  where now the expectation values can be expressed by , , , with -valued observables , . according to tsirelson s theorem , the bound applies to any quantum correlations without making assumptions on the sort of measurements or the dimensionality of the states involved . though , the maximum value of can already be achieved with a maximally entangled two - qubit state .inequalities , whose bound are saturated with quantum correlations without relying on dimensionality such as in eq .( [ eq : tsirelson ] ) were coined as quantum bell inequalities . thereexist various methods in the literature , to cite just a few , which enable one to derive quantum bell inequalities .also , several explicit constructions exist in the literature ( e.g. , ) , including so - called irrelevant ones ( where the classical and quantum limits coincide ) .the method invented by navascus , pironio and acn ( npa ) is based on the solution of a hierarchy of semidefinite programming ( sdp ) relaxations and is particularly useful since it gives better and better upper bounds on the maximum violation of an arbitrary bell inequality by stepping to higher levels in the hierarchy .moreover , the series of upper bounds in the hierarchy keep to the exact quantum maximum in terms of commuting measurements . on the other hand , one can use heuristic algorithms to obtain nontrivial lower bounds in some finite dimensional hilbert spaces on bell inequalities , recovering the explicit form of the states and measurement operators as well .if the above computed upper and lower bounds coincide within numerical accuracy for a given bell inequality , then we may say that a quantum bell inequality has been obtained , which delimits the boundary of the quantum domain .  in particular , in our previous papers we computed lower bounds on the maximum quantum violations and determined the corresponding measurement operators and state vectors with two different methods , both involved some parametrization of the operators and applying a downhill simplex method to find the optimum parameters .the disadvantage of these methods is that the size of the hilbert space was very limited .we could actually handle systems of maximum eight dimensional real or six dimensional complex component spaces .different sizes of component spaces require different parametrizations for the operators , hence in order to extend these methods to a higher dimension would involve working out an appropriate parametrization .moreover , the choice of measurement operators were also limited .for example , in the case of eight dimensional component spaces only projection operators projecting onto four dimensional subspaces were allowed .  in spite of the limitations we could get the maximum quantum violation of the great majority of inequalities we considered ( the list comprises 241 bipartite bell inequalities with up to five settings per party collected from refs . and detailed results concerning their optimum violations are presented in the web page ) .however , there were still a few exceptions , where the upper bound value resulting from the npa method did not match the best lower bound result .the most interesting one was the case of .this is the smallest case we considered , and perhaps the simplest tight bell inequality after the chsh one , with only three measurement settings per party .it was introduced by froissart back in 1981 , and recently reinvented in refs .it reads in a local classical model we have the maximum value of 0 , while the largest violation one could get with qubits was 0.25 , which could already be achieved with a maximally entangled pair of qubits ( see e.g. ,  ) . on the other hand ,the best upper bounds are based on the npa method and at level three it yields the significantly higher upper bound ,  .we could even go above level three to an intermediate level in , and presently we have got the upper bound at level four . from the dependence of the bound on the level it was derived it seemed to be clear that it would not go much lower ( note also that the computational complexity of the sdp problem increases dramatically with higher levels of relaxations ) .in fact , it is the above observation , which motivated us to search for a quantum violation beyond the two - qubit value of .  in the present paper we set out to resolve the puzzling problem concerning the maximum quantum violation of the inequality .to this end , we introduce in sec .[ ital ] an efficient iterative algorithm which was applied to explore the largest quantum violation of for states up to local dimension 20 . sec .[ const ] contains the main results , presenting the explicit construction of states and measurement operators , which give by means of an iterative method the conjectured maximal quantum value of in function of the dimensionality .in particular , for dimensions very large we recover the upper bound computed with the npa method at level four within high numerical accuracy , thereby establishing a quantum bell inequality for .we note that the optimal quantum state in the infinite dimensional space is far from the maximally entangled state , thereby supporting the claim that entanglement and nonlocality are different resources . in sec .[ other ] we investigate the remaining 19 bell inequalities from a set of 241 inequalities ( plus two additional symmetric 4-setting inequalities from ref . ) , where the tightness of the quantum bound could not be proven previously , and in some of the remaining cases now we manage to close the gap . sec .[ conc ] summarizes the results achieved .as mentioned above , the drawback of our previous heuristic methods in refs . for computing lower bounds on the maximum quantum violation of bell inequalities is that the size of the local hilbert space we could handle is limited up to dimension eight . with the iterative methodwe introduce here for two - outcome bell inequalities we can go to higher dimensions , which is only limited by the computational difficulties due to the increasing complexity of the problem with increasing hilbert space dimensions . similarly to the previous methods , this algorithm does not guarantee to converge to the global optimum solution . in case of larger spaceswe may miss it even after tens of thousands of restarts with different initial values .nevertheless , it managed to find every optima we derived with our previous methods , and in almost all cases it has done it considerably faster .the problem to be solved consists of maximizing the quantum value of the bell expression with coefficients , which can be written as : where there are and measurement settings for alice and bob , respectively , ( ) and ( ) are the measurement operators of alice and bob , respectively , and are unit operators in the component spaces , and is the state vector. the maximum can always be reached by a pure state . in case of two - outcome measurementsit is enough to consider projection operators as the measurement operators .when there are only two parties , a further simplification is that when we take a matrix representation of the expression above , we may choose the bases in the component spaces such that they correspond to the schmidt decomposition of the state vector . if we confine ourselves to -dimensional component spaces we get : where and are components of matrices of and , respectively , and are the schmidt coefficients of the state vector . if is smaller than the dimensionality of the component spaces required for the maximum quantum violation , then .  to solve the problem firstwe choose appropriate random matrices and numbers for bob s measurement operators , and for . to get the matrices ,first we take a diagonal matrix with zero and one diagonal values chosen randomly , then we apply many random two dimensional unitary ( or if we confine ourselves to real matrices , orthogonal ) transformations . for take positive numbers between zero and one from a uniform distribution , then we normalize them . in the iterative algorithm the first step is to calculate the optimal measurement operators of alice , given bob s operators and the state vector .this can be done directly .equation ( [ eq : bellexmat ] ) can be rewritten as : where we can get the matrix of the optimum ( ) the following way . first we diagonalize the matrix of . then we create a diagonal matrix .we choose its diagonal matrix element one where the diagonalized matrix of contains a positive number , and zero otherwise .then we transform this matrix with the inverse of the transformation that diagonalized the matrix of . to show that this is the matrix of the optimal we note that trace is invariant to basis transformation , and it is easy to see that the optimum matrix of projector in the basis diagonalizing is the one we have chosen .the second step is to derive bob s optimal matrices while alice s matrices and the state vector are fixed .this can be done equivalently to the first step .these two steps are the same as the steps of the see - saw algorithm by werner and wolf , who used it to get maximum violation with fixed ( not necessarily pure ) states .a variant of the see - saw algorithm for bell inequalities with multiple outcomes was also devised  .  in the third stepthe best state vector is calculated for the measurement operators .it is done the same way as in our previous method .an algorithm applying this very third step has been also used by liang et al . to compute quantum optima for multiple - outcome bell expressions . from eq .( [ eq : bellexmat ] ) it is clear that the optimal corresponds to the eigenvector belonging to the largest eigenvalue of the matrix and the eigenvalue is just the value of the bell expression .it is not sure that the components of the largest eigenvector will all be positive real numbers .if they are not , they can not be called schmidt coefficients .however , this does not affect anything , it is best just to leave them as they are .if we want to see the final result for the state vector represented by its schmidt coefficients , it is enough to make the appropriate transformation at the end of the calculation .we simply have to multiply rows and columns of either alice s or bob s matrices with phase factors .the algorithm consists of repeating these three steps until the value of the bell expression reach convergency . as this number may only increase in each step, convergence is ensured .what is not ensured is that we will converge to the global optimum of the problem .therefore , we have to repeat the full procedure many times with different initial values .we may save computation time if we stop a run when it clearly goes towards an inferior solution .for example , if we are interested in a solution belonging to a larger hilbert space , we may stop whenever too many components of the state vector falls below some threshold .it is not difficult to extend the method to more than two participants . in that casethere is no schmidt decomposition , which would ensure that one can choose the bases in the component spaces such that the -dimensional state vector has only nonzero components .however , one can still use eq .( [ eq : aopti ] ) to calculate the optimum measurement operators for alice , and analogous equations for the operators of the other participants .the only difference is that the formula for the matrix of ( and the corresponding matrices for the other participants ) will be somewhat more complicated than eq .( [ eq : xmat ] ) , involving summations to more indices .the optimum state vector can also be calculated as the eigenvector belonging to the largest eigenvalue of a matrix , but this will be a more general matrix in the full -dimensional hilbert space . as the third step of the algorithm in this case involves the diagonalization of a much larger matrix than the first two steps , it is more efficient to do it less frequently , that is to repeat the first and the second steps several times before updating the state vector .from the optimality of formula ( [ eq : bellexmat ] ) it follows that the matrices of the optimum , , , , and are such that they satisfy the following eq .( [ eq : besta1 ] ) , eq .( [ eq : besta2 ] ) , eq .( [ eq : besta3 ] ) , eq .( [ eq : bestb1 ] ) , eq .( [ eq : bestb2 ] ) and eq .( [ eq : bestb3 ] ) , respectively : meanwhile , are the components of the eigenvector belonging to the maximum eigenvalue of the matrix the matrices of the measurement operators we got with the iterative method described in section [ ital ] as the best solutions with all differing from zero had very special forms .we found most of the matrix elements zero within expected numerical accuracy , and for odd number of dimensions it turned out that with some appropriate reordering of the bases we could also make all matrices blockdiagonal with one block having a single element , while the rest consisting of two by two blocks , with all matrix elements real . in the cases of , and , the block of size one is the first one , while in the cases of , and it is the last one .the actual form of the matrix of may be written as with are all positive .the matrix of tuns out to be the same , but the offdiagonal elements have a negative sign .the form of is the following : the matrix element may take the value of 1 and 0 , when and , respectively . for smaller dimensions we got the former value for all best solutions , but it turned out that from dimensions larger than the other value gives the better solution .the notation of will become clear later .the matrix of is the same as that of , but , and like in the case of , the offdiagonal matrix elements are negative .matrices and have even simpler forms : , while all elements of the two by two blocks are .it is easy to check that if all , then the matrices of and do satisfy the optimality conditions eq .( [ eq : besta3 ] ) and eq .( [ eq : bestb3 ] ) , respectively .the matrix of has the same block structure as , as it should , and the optimality condition can be checked block by block .the value of the single element block in the matrix of , that is is either 2 or 0 , depending on , in the first case the optimum is , while in the other case the chosen value is just as good as value would have been .the two by two blocks in the matrix of have zero diagonal elements , and positive nondiagonal elements . as all have the same sign, the number they will be multiplied with is also positive .therefore , the optimum two by two block of the matrix of is the symmetric two - dimensional matrix with eigenvalues and having the largest possible nondiagonal elements .this matrix is the one whose all four elements are .  from eq .( [ eq : besta2 ] ) we can get equations determining the parameters of the matrix of that is with even ( see eq .( [ eq : a2 ] ) ) in terms of the parameters of the matrices of bob s measurement operators , that is with odd , and the schmidt coefficients .as the matrix of is diagonal , the block structure of is the same as that of .the matrix of is :   the value of is determined by the sign of , the value is correct if , which turns out to be true .each , where is even , occurs only in one two by two block of the matrix of ( see eq .( [ eq : a2 ] ) ) .its optimum value depends on the corresponding block in eq .( [ eq : b1b2b3i ] ) , containing and .we denoted as such that the last block has the same form as the others .using eq .( [ eq : besta2 ] ) we arrive at the equation determining the optimum value of : multiplying the equation by four , dropping terms constant in and substituting we get : + 2\sqrt{1-c_i^2}\lambda_i\lambda_{i+1}\notag\\ & = \rm{max } , \label{eq : optci2}\end{aligned}\ ] ] from which it follows that where with chosen this way eq .( [ eq : besta1 ] ) , the optimality condition for is also satisfied . if we started from this condition instead of the one for , we would have arrived at exacly the same results .   from the upper bound of shown as function of the dimensionality for cases of and . ]  in the same way as above , from eq .( [ eq : bestb1 ] ) and eq .( [ eq : bestb2 ] ) we can derive formulae for the optimum values of the parameters of the matrices of and , that is for , where is odd .these formulae turn out to have exactly the same form as the ones derived for even values , that is eqs .( [ eq : optci1]-[eq : optci4 ] ) are valid for odd . to get , we have to introduce the notation .the value is the optimum value .either ( that is ) or ( that is ) is a possible choice .consistency with the optimality condition is fulfilled if in the first case , and in the second one .  with the specific forms of the measurement operators above , the matrix of eq .( [ eq : bestlambda ] ) , whose eigenvector corresponding to its largest eigenvalue gives the schmidt coefficients , is tridiagonal , and its nonzero matrix elements are : we may apply the iterative method with confining ourselves to the present specific forms of the operators , and get solutions for very large dimensions .this way we may increase the dimensionality of the component hilbert spaces even to a thousand , or more . the distance of the maximum value of the bell expression from the upper bound calculated at level four is shown in fig .[ convergence ] as a function of the dimensionality , both for and . for lower dimensionsthe former family of solutions is much better .still , even this gives a value larger than 0.25 , which one can obtain with a pair of real qubits only if .unfortunately , this family does not converge to the upper bound .the other family with is initially inferior , but above it overtakes the other one , and converges to , a value well consistent with the upper bound : it is slightly larger , but within the numerical accuracy of the bound . in fig .[ aszimcl ] and fig .[ szimcl ] we show and for with and , respectively . for small  is close to =0 , and tends to a constant value fast . in the case of stays basically constant up to the last few point , when it drops towards the zero value of fast .when and there are more than one solutions . in all solutions for small and for large .in the different solutions the change of sign happens between two neighbouring integers .for very large it always happens above and below . among all these solutions the best is the one in which changes sign exactly in the middle , that is between and . in fig .[ szimcl ] we show the result for this solution . in particular , the upper curve in fig . [ szimcl ] shows the behaviour of .the curve starts like in the case of , that is it drops from to a constant value fast , that stays there until it goes to minus one times the constant value in a relatively short interval , and then it stays almost constant again until it turns towards at the end .where is constant , behaves exponentially .   and are shown for with . ]  knowing the behaviour of , it is easy to choose initial values such that the iterative procedure will converge to the desired solution for the first time , there is no need for repeated runs .for example , we may initialize as for and for , with . then we can start with the third step instead of the first one , that is with the determination of .   and are shown for with . ]  the families of solutions may be extended to even values of .the matrices of the measurement operators are blockdiagonal , like for odd .the matrices of , and consist of two by two blocks .the blocks of , and are also two - dimensional except for the first and the last ones , which have single elements .the actual form of the matrices can be parametrized with analogously to the odd case , which will satisfy the same equations as optimality conditions , and the eigenvalue equation for will also have the same form .figure [ convergence ] does contain the results for even .the curve for in the case of the best solution with is not fully symmetric .it changes either very nearly half way between and or between and .if we make the change of size to happen at the middle , that is we enforce , for large the result will converge to the same suboptimal value as in the case of the family .probably the reason this family fails to converge to the true optimum is that must approach zero at an integer value , instead of midway between two integers .  in the matrices of the measurement operators we made very specific choices for the values of the single element blocks , leaving only as a single free parameter .we note that there are other choices consistent with the optimality conditions , but they lead to solutions equivalent with the ones discussed .0.2truecm  [ cols="<,^,<,<,^ , < " , ]   [ table : highlowdiff ]  in ref . we calculated both lower and upper bounds for all known tight binary bipartite bell inequalities with up to five measurement settings per party . for most cases we managed to find explicit solutions saturating the upper bound .there were 20 exceptions .the case of was discussed in the previous chapter .now we have applied the iterative algorithm with up to 25 dimensional component spaces for the remaining 19 problems . in 11 caseswe have found better solution than the ones we reported in ref .the solution for could have been found by our previous methods , but we missed it .all the other ones involve larger hilbert spaces than those methods could handle .we have also made the upper bound tighter by going to higher levels by using a computer with larger memory .as previously , the calculations have been done with borcher s code csdp for semidefinite programming  . for cases with five measurement settings per party we could do the calculation on level two plus plus . for the smaller cases we could afford to go up to level three , except for , where we did level four .the notion of levels and partial levels , and their notation is explained in ref . , and also in ref  .this way the two bounds met for .we sent at level three , and and at level three plus to brian borchers , as test cases for his new code which requires much less memory than csdp for very large cases .for this new upper bound does agree with the lower one .unfortunately , for there remained a difference of , significantly less than the level three value of , but more than numerical uncertainty . at an even higher level it would probably disappear , but it is not sure .therefore , we can not state that our solution with 12-dimensional component spaces is already the optimum one .the case at level three plus seems to be a difficult case for the new code , it could solve it only at a reduced accuracy , and the value it has given for the upper bound turned out to be lower than the lower bound .therefore , very probably the solution with a pair of real qubits we found the best so far is actually the optimum .if we added to level three just one quarter of the type terms , we could use csdp .in the cases of and this was already enough to reach the lower bound . by looking at the dependence of the upper bound on the level it has been calculated, we are almost sure that for , , , , , and the present lower bound will not be reached , therefore we do not yet have the optimum solution .probably is not the only inequality requiring infinite dimensional hilbert spaces for maximum violation . actually , the difference in the cases of and is very small , but it does not change much with increasing level . for and upper bound would more probably converge to our solution , while for the rest of the cases we can not tell from the present results .bancal et al . found recently two symmetric inequalities with four settings per party , which they called and .they were also included in our calculations .in fact , both of them could be saturated with measurements acting on qubits , we found the maximum violation to be and for and , respectively .the details of our best solutions , including schmidt coefficients and the matrices of the measurement operators are on our web site  .to summarize , we investigated the bell inequality and its maximal quantum violation .although is one of the simplest bell inequalities with just three measurements per party , surprisingly we found that the maximum value of 0.25 achievable with a pair of qubits could be only be overcome by using states of dimension at least .moreover , we found a family of measurement operators and states , which attains numerically in very large dimensional hilbert spaces ( and hence in the limit of infinite dimensions as well ) the largest possible value of the bell expression allowed by quantum theory . taking as a conjecture that our construction is optimal ( in the sense that no finite dimension suffices to achieve the true quantum maximum for ) , to our knowledge this constitutes the first example of a bell scenario concerning finite number of measurements and outcomes , where measuring finite dimensional quantum systems is not enough to obtain exactly all quantum correlations . in this respectwe wish to invoke the concept of dimension witnesses introduced recently in ref .  . in a broader sense ,the aim of this concept is to put a lower bound on the hilbert space dimension needed to reproduce the statistics arising from a bell experiment .dimension witnesses have been found or conjectured in many different scenarios , such as for the bipartite setting for the multipartite setting and even for the case of a single system .  in particular , for measurements with binary outcomesit was shown how to get dimension witnesses analytically for any dimensions .actually , these results entail that no finite dimension is sufficient to generate the whole set of quantum correlations provided an arbitrary number of two - outcome measurements is involved .consider now a scenario where the number of measurements ( each having a finite number of outcomes ) is finite .does there exist a bell inequality requiring infinitely large entangled states for obtaining the maximum possible quantum violation ?that is , are there quantum correlations corresponding to this scheme attainable exactly by measuring infinite dimensional quantum systems ? this is a question posed recently by navascus et al . .  indeed , in the present paper we suggested , supported by thorough numerical computations , that no finite dimension suffices to maximally violate the inequality .while we can not guarantee that for any finite dimension there exist no other construction beyond ours giving a larger quantum value , nevertheless we showed that our best conjectured values tend to the upper limit computed with the aid of the npa method .we pose it as a challenge to provide an analytical proof of our conjecture .also , in our view it would be interesting to find other , bipartite or multipartite , bell inequalities beyond ( but still involving finite number of settings and outcomes ) requiring infinitely large entangled states for their maximal quantum violation .t.v . has been supported by a jnos bolyai grant of the hungarian academy of sciences .we thank for the help of brian borchers , who used some of our problems as test cases for his code for semidefinite programming , and this way provided us with improved upper bounds for those cases .we would like to thank miguel navascus for illuminating discussions .n. linden , s. popescu , a.j .short , and a. winter , phys .* 99 * , 180502 ( 2007 ) ; k.f . pl and t. vrtesi , phys . rev .a * 80 * , 042114 ( 2009 ) ; m.l .almeida , j.d .bancal , n. brunner , a. acn , n. gisin , s. pironio , arxiv:1003.3844 ( 2010 ) .b. borchers , optimization methods and software * 11 * , 597 , ( 1999 ) ; b. borchers , optimization methods and software * 11 * , 613 , ( 1999 ) ; b. borchers and j.g .young , computational optimization and applications * 37 * , 355 ( 2007 ) ; ` https://projects.coin-or.org/csdp/ ` .

Abstract: the inequality is the simplest bipartite two - outcome bell inequality beyond the clauser - horne - shimony - holt ( chsh ) inequality , consisting of three two - outcome measurements per party . in case of the chsh inequality the maximal quantum violation can already be attained with local two - dimensional quantum systems , however , there is no such evidence for the inequality . in this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional hilbert space . further , it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum . we also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two - outcome bell inequality in any given hilbert space dimension . this algorithm played a key role to obtain our results for the inequality , and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two - outcome bell inequalities with up to five settings per party .