a quite common method for the transportation of granular media is pneumatic conveying , where grains are driven through pipes by air flow .practical applications for pneumatic conveying can be found in food industry and in civil and chemical engineering .one distinguishes two modes of pneumatic conveying : dilute and dense phase conveying .dilute phase conveying has been studied in much detail  and is well understood .the grains are dispersed and dragged individually by the gas flow and the interaction between grains is small .this is not true for the dense phase conveying , where the particle interaction is important and where particle density waves can be observed .two modes of dense phase conveying can be distinguished , depending on the orientation of the pipe transporting the grains . in the horizontal transportthe granular medium separates into two layers , the moving bed at the bottom of the pipe and the irregular transport by traveling dunes or slugs  . in the vertical transport the granular medium forms strands at medium gas velocities  and plugs at low gas velocities with high mass loads  ; this mode is called plug conveying .currently plug conveying is gaining importance in industry , because it causes a lower product degradation and pipeline erosion than dilute phase conveying .unfortunately , current models  of plug conveying disagree even on the prediction of such basic quantities as the pressure drop and the total mass flow , and these quantities have a great impact in the industrial application .one of the reasons for the lack of valid models is that it is difficult to study plugs experimentally in a detailed way .usually experimental setups are limited to the measurement of the local pressure drop , the total mass flux and the velocity of plugs .most promising experimental studies have been performed by electrical capacity tomography  and stress detectors  .simulational studies are handicapped by the high computational cost for solving the gas flow and the particle - particle interaction , and are therefore mostly limited to two dimensions .for the dense phase regime simulations have been done for bubbling fluidized beds  , which show for high gas velocities first signs of pneumatic transport  , for the strand type of conveying  , and slugs in horizontal transport  .simulations of vertical transport which show plugs with considerable length are rare .there is only one simulation of a short pipe in two dimensions by tsuji et al . and a picture of a single plug from a simulation with 50 particles in three dimensions by ichiki et al . .the goal of this paper is to provide a detailed view of plugs , by using a discrete element simulation combined with a solver for the pressure drop .this approach provides access to important parameters like the porosity and velocity of the granulate and the shear stress on the wall at relatively low computational cost .contrary to the experiments , it is possible to access these parameters at high spatial resolution and without influencing the process of transportation at all . additionally to plug profiles ,characteristic curves of the pressure drop and the influence of simulation parameters are measured .the simulation results are compared with and verified by experiments .plug conveying is a special case of the two phase flow of grains and gas .it is therefore necessary to calculate the motion of both phases , as well as the interaction between them . in the following ,we explain how our algorithm treats each of these problems .   the model for the gas simulationwas first introduced by mcnamara and flekky and has been implemented for the two dimensional case to simulate the rising of bubbles within a fluidized bed . for the simulation of plug conveying we developed a three dimensional version of this algorithm .the algorithm is based on the mass conservation of the gas and the granular medium .conservation of grains implies that the density of the granular medium obeys where the specific density of the particle material is , the porosity of the medium is ( i.e. the fraction of the space available to the gas ) , and the velocity of the granulate is .the mass conservation equation for the gas is where is the density of the gas and its velocity .this equation can be transformed into a differential equation for the gas pressure using the ideal gas equation , together with the assumption of uniform temperature .the velocity of the gas is related to the granulate velocity through the darcy relation : where is the dynamic viscosity of the air and is the permeability of the granular medium .this relation was first given by darcy in 1856 .for the permeability the carman - kozeny relation was chosen , which provides a relation between the porosity , the particle diameter and the permeability of a granular medium of monodisperse spheres , after linearizing around the normal atmospheric pressure the resulting differential equation only depends on the relative pressure ( ) , the porosity and the granular velocity , which can by derived from the particle simulation , and some constants like the viscosity  : this differential equation can be interpreted as a diffusion equation with a diffusion constant .the equation is solved numerically , using a crank - nickelson approach for the discretization .each dimension is integrated separately .the boundary conditions are imposed by adding a term on the right hand side of equation ( [ eqn : dgl ] ) at the top and the bottom of the tube , where .this resembles a constant gas flux with velocity at a pressure into and out of the tube .the model for the granular medium simulates each grain individually using a discrete element simulation ( des ) . for the implementation of the discrete element simulation we used a version of the molecular dynamics method described by cundall .the particlesare approximated as monodisperse spheres , rotations in three dimensions are taken into account .the equation of motion for an individual particle is where is the mass of a particle , the gravitation constant and the sum over all contact forces .the last term , the drag force , is assumed to be a volume force given by the pressure drop and the local mass density of the granular medium .the interaction between two particles in contact is given by two force components : a normal and a tangential component with respect to the particle surface .the normal force is the sum of a repulsive elastic force ( hooke s law ) and a viscous damping .the tangential force is proportional to the normal force ( sliding coulomb friction ) or a viscous damping .the viscous damping is used when the relative surface velocity of the particles in contact is small .the same force laws are considered for the interaction between particles and the tube wall .the simulation method uses both a continuum and a discrete element approach .while the gas algorithm uses fields , which are discretized on a cubic grid , the granulate algorithm describes particles in a continuum .a mapping is needed for the algorithms to interact . for the mapping a tent function used : where is the grid constant used for the discretization of the gas simulation .  for the gas algorithmthe porosity and the granular velocity must be derived from the particle positions and velocities , where is the index of particle and is the index for the grid node . the tent function distributes the particle properties around the particle position smoothly on the grid : where is the position of the grid point and the sum is taken over all particles .  for the computation of the drag force on a particle the pressure drop and the porosity at the position of the particleare needed .these can be obtained by a linear interpolation of the fields and from the gas algorithm : where the sum is taken over all grid points .sketch of the experimental setup .gas is inserted at a constant mass rate at the bottom of the tube , the granulate enters from the left . s1 and s2 denotes pressure sensors .a ccd camera is used to record the transport at mid - height ., width=113 ]  in the experiment we study the vertical pneumatic transport of wax beads of diameter , density and a coulomb coefficient of 0.21 . the experimental transport channel is a vertical tube ( pmma ) of length and of internal diameter .these parameters were chosen to be able to compare to simulations with little computational effort possible .the air is injected at a constant mass flow rate at the bottom of the tube .the granular medium is injected from the side from a tube with a slope of .the wax beads , when leaving the vertical tube at the top , are gathered and weighted . from this data the mass flow rate of the granular medium can be obtained .the total pressure drop is detected through two pressure sensors , one at the bottom ( s1 ) and one at the top ( s2 ) .the velocity and length of the plugs can be obtained by a ccd - camera ( 30 frames per second ) positioned at the middle of the tube .  a series of photos showing a plug moving upwards .the height of the shown tube is , the frame rate is , from measurement series b. , width=377 ]  .[tab : experiment]experimental results for vertical plug conveying in a tube of length and diameter using wax beads ( ) as granular medium . [ cols="<,<,<,<,<",options="header " , ]   plug conveying is observed , as shown in figure [ fig : expplugs ] .the range of parameters for plug conveying in the experiment were rather limited .the plugs often get stuck in the tube , interrupting the plug transport completely . for long runs and repeated use of grains, electrostatic forces influenced the results .measurement series for three different air flow rates were made .the results for the total pressure drop and the granular flow are given in table [ tab : experiment ] , for more details see .the setup for the simulation is almost the same as for the experiment .the main difference is : particles are injected from the bottom at a constant mass flow rate . as default the mass flow rate of the granular medium is chosen equal to the measured flow rate in the experiment ( ) .default parameters for the particles are : diameter , density , coulomb coefficient and restitution coefficient .simulations are preformed for half tube length and tube diameter .the resulting gas volume has been discretized into 150x2x2 grid nodes , which corresponds to a grid constant of .the gas pressure is set to , simulations are preformed for gas viscosities from to and gas flows between and . the gas flow is usually given as the superficial gas velocity   , which is the equivalent gas velocity for an empty tube .spatio - temporal image of the porosity along the vertical tube .the dark regions correspond to low porosity , light regions to high porosity .the entire tube ( ) is displayed , the elapsed time is  .default parameters for the simulation were used , the superficial gas velocity is , the gas viscosity is , the coulomb coefficient is .particles are introduced at the bottom and on average 3600 particles are within the tube at a pressure drop of ., width=529 ]  the flow in the experiment is turbulent ( particle ) . in the simulation, an effective gas viscosity is used to account for the turbulence . for an effective gas viscosity , a gas flow and a coulomb coefficient plug conveyingis observed as shown in figure [ fig : rho - w0126 ] .the observed pressure drop , the plug velocity and the plug length 2 - 4 fit well with the experimental results .  as pointed out in the introduction ,only very limited simulations of vertical plug conveying have been published .vertical transport has been studied by tsuji  , but his simulation was limited to a short pipe in two dimensions ( ) . in his casethe tube contained about 300 particles , which is in our case about the number of particles within one plug .three dimensional plugs of about 50 particles have also been found by ichiki  , but no analysis of them was done .thus our simulations are the first to show full featured vertical plug conveying at the level of grains in three dimensions .the `` characteristic curves '' of a pneumatic transport system are plots of the pressure drop against the superficial gas velocity for different mass flows of the granulate .this kind of diagram is highly dependent on the material characteristics of the tube wall and the granulate and can be used to predict the overall transport performance for given parameter sets .such a diagram , from data from our simulation , is shown in figure [ fig : v - dp ] .total pressure drop against superficial gas velocity for different granular mass flows . plotted are characteristic curves from the simulation for the granular mass flows and , and the three data points from the experiment with granularmass flows . for mass flow plugs stick when the superficial gas velocity is below ., width=377 ]  the diagram provides the typical qualitative behavior for pneumatic transport . from topleft to bottom right with increasing superficial gas velocity , four regions can be distinguished .first , for small superficial gas velocities ( ) , there is bulk transport .the tube is completely filled with granulate , so the pressure drop is high .nevertheless the drag force on the bulk is too small to support the weight of the granular medium . in this casethe transport comes through the enforced granular mass flow at the bottom .spatio - temporal image of the porosity along the vertical tube .the dark regions correspond to low porosity , light regions to high porosity .the entire tube ( ) is displayed , the elapsed time is  .a lower superficial gas velocity ( ) has been used than in figure [ fig : rho - w0126 ] . on average 4900 particlesare within the tube at a pressure drop of . due to the low drag forcethe plugs quickly coalesce into very long plugs ., width=529 ]  for slightly higher gas velocities ( ) the first plugs appear .these plugs are often not able to compensate the force from material falling onto them .they usually collapse into big plugs sticking in the tube or their motion is reduced considerably ( fig .[ fig : rho - w0134 ] ) . in this regionthe pressure drop decreases rapidly with increasing gas velocity .  for moderate velocitiesplug conveying is observed ( fig .[ fig : rho - w0126 ] ) .the qualitative behavior of the granular medium in this regime is the same in the experiment and the simulation .the particles injected at the bottom organize into plugs .after a short acceleration at the bottom plugs move upwards with a constant velocity .when two plugs collide , they combine into a single plug .a plug always looses particles at the bottom and usually maintains its length by collecting particles at the top .a plug disintegrates when it gets too small .the porosity of the granular medium within a plug is close to the minimum porosity , the edges are sharp and the space between plugs is rather empty ( ) .as one can see in figure [ fig : v - dp ] the experimental data were measured for a superficial gas velocity at the lower limit for plug conveying . in this region the pressure drop is highly dependent on the material parameters ( ,, ) .spatio - temporal image of the porosity along the vertical tube .the dark regions correspond to low porosity , light regions to high porosity .the entire tube ( ) is displayed , the elapsed time is  .a higher superficial gas velocity ( 4 m/s ) has been used compared to the spatio - temporal image in figure [ fig : rho - w0126 ] ( ) . on average 850 particlesare within the tube at a pressure drop of . , width=529 ]  for high superficial gas velocities ( ) the tube is almost empty ( fig .[ fig : rho - w0136 ] ) , in the simulation the particles are pushed out as small plugs .their porosity increases with the superficial gas velocity . in this regionthe simulation method underestimates the pressure drop , because it does not consider the increasing drag force on single particles , which in the experiment dominates in this region .the regions as described above shift with changes of the simulation parameters .a nearly proportional relation is observed between the total pressure drop and the total number of particles in the tube .this can be explained by the observation that the amount of particles between the plugs is small and most particles are densely packed at a well defined porosity in the plugs . through the darcy s law , the total pressure drop depends linearly on the tube length filled with this porosity .the pressure drop on the granulate between the plugs is negligible and causes only a small deviation from the proportional relation . at high gas velocitiesthis is no longer true , because there is no dominating porosity for the plugs .  in the following the dependence of the pressure drop on the air viscosity , the atmospheric pressure and the coulomb coefficient discussed . for the parameter studiesthe superficial gas velocity has been fixed to .this resembles the gas velocities used for the experiment . for higher velocities the parameter dependence decreases .the gas algorithm can be influenced by changing the atmospheric pressure or the diffusion constant .an increase in background pressure combined with a proportional increase in superficial velocity leaves the pressure drop unchanged .this can be deduced directly from equation ( [ eqn : dgl ] ) by noting that both terms on the right hand side are proportional to .dependence of the pressure drop on the dynamic viscosity of the gas at superficial gas velocity .the data points can be fitted by a hyperbola ., width=377 ]  the diffusion constant can be changed through the particle diameter and the viscosity .therefore it is sufficient to analyze the parameter space for the viscosity at a constant diameter as shown in figure [ fig : vis - dp ] .the viscosity dependence of the pressure drop can be described by a hyperbola .the pressure drop converges for high viscosities to a finite value . in this casethe pressure drop supports the weight of the granulate embedded in the fluid and depends only on the velocity of the gas and the enforced mass flow of the granular medium . for small viscosities plug conveying starts to break down to bulk transport .the pressure drop diverges at the viscosity where the drag force can no longer balance the weight and friction of the bulk .the parameter of the particle simulation with most influence on the transport of the granular medium is the coulomb coefficient , i.e. an increase of the restitution coefficient from to leaves the pressure drop unchanged .dependence of the pressure drop on the coulomb coefficient , the superficial gas velocity is . for high coulomb coefficient ( ) a transition from plug conveying to bulk transport can be observed . ,width=377 ]  as one can see in figure [ fig : cc - dp ] the pressure drop increases with raising coulomb coefficients . for low coefficients , , plug conveyingis observed , for higher coulomb coefficients the plugs get sticky and coalesces into a big , slowly moving plug .these plugs are similar to the ones for low superficial gas velocities ( fig .[ fig : rho - w0134 ] ) .   the spatio - temporal image of the porosity along the vertical tube ( fig .[ fig : rho - w0126 ] ) provides a rough picture of plugs and their movement along the tube .a statistical approach is necessary to get more precise values .properties of interest are the porosity and the granular velocity within a plug , the plug length , and their dependence on the vertical position of the plug within the tube .to get some average values for the porosity and the granular velocity , the tube was segmented into horizontal slices of height . for each slicethe average porosity and granular velocity has been computed every .the contribution of a particle has been weighed by the volume occupied by that particle within a given slice .the resulting vertical porosity has been used to identify plugs .every region with a porosity lower than 0.6 is defined to belong to a plug .number of plugs per time as a function of height .the corresponding spatio - temporal image is shown in figure [ fig : rho - w0126 ] .the total tube height is , default parameters are used .the data is averaged over . , width=377 ]  figure [ fig : w0126_rate_his ] shows the number of plugs per time as a function of height . at the bottom of the tube the incoming granular medium forms into a lot of small plugs .even though their velocity is low , the resulting plug rate is high .as can be seen in figure [ fig : rho - w0126 ] , plugs collide along the tube and merge into bigger plugs . the merging of plugs reduces the plug rate along the tube .the smooth decrease of the plug rate indicates that there is no preferred height for the collision and merging of plugs .  for each plug the center of mass , the minimal porosity , the maximal granular velocity and the plug lengthhave been computed .minimal porosity found in plugs at height , corresponding to the plug rate shown in figure [ fig : w0126_rate_his ] .the bars denote the uncertainty of the average , and the dotted lines indicate the width of the distribution ; at each value of , half of the observed plugs have a porosity lying between the upper and the lower dotted lines .the plug porosity decreases at the bottom when the granulate enters the tube ( ) and increases at the top of the tube ( ) , when the preceding plug has been removed ., width=377 ]  figure [ fig : w0126_rho_his ] shows the minimum plug porosity as a function of the vertical position of a plug . at each value of ,the mean porosity and its uncertainty ( standard deviation divided by the square root of the number of plugs ) were calculated .these quantities are shown by the bars in figure 10 . to show the distribution of porosity about the mean , the two dotted lines were added . at each height ,half of the plugs have a porosity lying between these two lines .the same analysis was carried out for the data in figures 11 and 12 .as one can see in figure [ fig : w0126_rho_his ] at the left of the graph , the granular medium is inserted at the bottom of the tube with a porosity of 0.51 . from therethe porosity decreases quickly to about 0.47 at a height and then remains almost constant until . at the top of the tube ( ) the porosity increases until the grainsleave the simulation space at .maximal granular velocity found in plugs at height , corresponding to the plug rate shown in figure [ fig : w0126_rate_his ] .the bars and dotted lines have the same meaning as in figure [ fig : w0126_rho_his ] .the granular velocity saturates in the middle of the tube , the increase of the granular velocity at the bottom and at the top of the tube is due to the boundary effects described in figure [ fig : w0126_rho_his ] . , width=377 ]  figure [ fig : w0126_uxx_his ] shows the corresponding particle velocity .the change in porosity comes along with an increase of the granular velocity within the plugs .the granulate is inserted at the bottom with an initial velocity of .the granular velocity saturates to a final velocity for the granular velocity at a height of . at a height of about granular medium accelerates until the grains leave the tube .an explanation for the final constant plug velocity can be derived using the balance equation for the forces on a plug : where is the mass of a plug , is the gravity constant , is the force on the plug through the friction with the wall and the collisions of particles falling onto the plug .the last term is the drag force on the plug , which is proportional to the relative velocity between the particles and the gas within the plug . at the middle of the tube the porosity of the plug is constant . on small timescales the mass and the gas velocity can be assumed constant .only the drag force depends on the granular velocity through darcy s law . for a certain granular velocity the drag force balances the sum of gravitational and friction forces .if there is no acceleration of the plug , then remains constant .the solution is stable under small fluctuations in .mean plug length at tube height , corresponding to the plug rate shown in figure [ fig : w0126_rate_his ] .the bars and dotted lines have the same meaning as in figure [ fig : w0126_rho_his ] .the average plug length increases along the tube , consistent with the decrease in the plug rate ( fig .[ fig : w0126_rate_his ] ) . , width=377 ]  figure [ fig : w0126_l_his ] shows the mean plug length along the tube .the average plug length is increasing along the tube , due to merging of plugs . unlike the previous cases no equilibrium length is found .the increase of the plug length combined with the decrease of the number of plugs per time ( fig .[ fig : w0126_rate_his ] ) conserves the mass flux of the granulate .as can be seen in figure [ fig : rho - w0126 ] , the change in porosity , velocity and plug length at the top of the tube can be explained by the lack of grains falling onto the top of the uppermost plug .the diagrams in figures [ fig : w0126_rho_his ] to [ fig : w0126_l_his ] imply , that there is a typical porosity , granular velocity and length of plugs for a given height and a characteristic plug profile exists . inthe following averaged vertical and radial profiles of plugs at the height of are discussed .to get some sensible profiles the plugs were selected with length and granular velocity close to the mean values ( , ) .   while in experiments recognizing and measuring parameters for global plug conveyingare rather simple , the measurement of profiles for individual plugs remains a nearly impossible task .so one of the reasons for simulating plug conveying is to provide a detailed picture of what happens within a plug , and how parameters like porosity , granular velocity or shear stress change along the plug .a porosity profile of an averaged plug is shown in figure [ fig : rho_x ] .the provided profiles ( fig . [ fig : rho_x]-[fig : uxx_r ] ) average over seven plugs .these plugs were taken from the middle of the tube with granular velocity and plug length .the coordinate denotes the relative vertical position along the tube with respect to the center of mass .vertical porosity profile along an averaged plug , containing about 320 particles , positioned at the middle of the tube .the profile was averaged over seven plugs with granular velocity and plug length .the data belong to the simulation displayed in figure [ fig : rho - w0126 ] .the horizontal axis denotes the relative vertical position along the tube with respect to the center of mass .the vertical position is given in multiples of the particle diameter ., width=377 ]  at the top of the plug , on the right hand side of figure [ fig : rho_x ] , the porosity decreases quickly from 90% to . in the middlethe porosity of the plug remains almost constant , and increases slowly at the bottom .the minimum porosity is bigger than a random dense packing due to near wall effects .figure [ fig : uxx_x ] displays the velocity profile of the granular medium in the region of the plug shown in figure [ fig : rho_x ] .one can distinguish four different regions : above the plug ( ) the granulate particles are falling downwards , the porosity is high .velocity of the granular medium along the averaged plug of figure [ fig : rho_x ] . inside the plug ( )the velocity is constant , outside the granular medium accelerates downwards ., width=377 ]  these particles originate from the bottom of an upper plug .the drag force can not support their weight due to the high porosity . at the top of the plugthe particles collide with the low porosity region and decelerate rapidly ( ) .the granular medium gets denser as particles are added to the plug . in the following , this region at the top of the plug will be called collision region . inside the plug , where the porosity settles to a low value , the granulate velocity is almost constant ( ) . at the bottom ( ) of the plugthe granular medium starts to accelerate downwards again .thus the plug is always loosing material at the bottom .this region will be called disintegration region .forces acting on particles along the averaged plug in figure [ fig : rho_x ] .interparticle forces , gravity and friction are summed up and displayed as particle force .the drag force corresponds to the pressure drop of the gas . outside of the plug ( ) the particle force is , indicating that the particle are freely falling .the peak in the particle force corresponds to the collision region , where particles fall onto the plug ., width=377 ]  the trajectory of a single particle through the plug can be sketched by a snapshot of the vertical forces within a plug .figure [ fig : force_x ] displays the drag force and the sum over the interparticle , gravity and friction forces on a particle ( here called particle forces ) . these forces are averaged over the horizontal plane . above the plug , the particle accelerates constantly due to gravity .the drag force on a single particle can be neglected , as the particle velocity is far from the terminal velocity .the high peak in the particle force marks the collision region .the huge fluctuations in the particle force arise , when the falling particle collides with the top of the plug front . within the collision regionthe porosity decreases and therefore the drag force increases . below the collision region , within the plug , the drag force and the particle force balance each other . at the bottom of the plug the friction force drops to zero and gravity and drag forceare in equilibrium .below this point the disintegration region begins , the granular velocity within the plug decreases , the porosity increases .after a short distance the drag force is zero and the particles are falling freely again .stress between the wall and the granular medium along the averaged plug in figure [ fig : rho_x ] .the wall stress is limited to the plug , where the porosity is low ( ) . , width=377 ]  the normal wall stress corresponding to the plug in figure [ fig : rho_x ] is shown in figure [ fig : wallstress_x ] . due to the implementation of the simulation and the high dampingthe normal wall stress is proportional to the vertical shear stress ( coulomb friction ) .the wall stress is limited to the plug including the collision and disintegration region .the wall stress increases from zero within the collision region and then slowly decreases , reaching zero in the disintegration region .the maximal wall stress is of the same order of magnitude as the pressure imposed on the plug front by the falling granulate .granular temperature along the averaged plug in figure [ fig : rho_x ] .the granular temperature is only non - zero near the collision region between the plug and the falling particles at the top of the plug . due to the large damping ( )the granular temperature is dissipated quickly and remains on a low level within the plug ( ) ., width=377 ]  figure [ fig : eneryyz_x ] shows the granular temperature along a plug .the granular temperature is the average kinetic energy of particles minus the kinetic energy of the motion of their center of mass .appreciable granular temperatures occur in the collision region . due to the high damping ( ) these temperaturesare confined to a small region at the top front of the plug .there is a finite granular temperature within the plug , the temperature drops nearly to zero at the bottom of the plug .another point of view on the plug is given by the radial profiles .the volume used for averaging at a given radius is a cylindrical ring with a height of 1.5 particle diameters and a width of 0.25 particle diameters .the porosity at a given range is defined to be the volume fraction of the gas within the cylindrical ring .  radial porosity profile through an averaged plug at the height .the same plugs were used for averaging as for the vertical profiles in figure [ fig : rho_x ] .the radial distance from the middle of the tube is given in units of the particle diameter .the porosity minima correspond to three layers of particles moving upwards , parallel to the tube wall ., width=377 ]  as a result of this high radial resolution it is possible to resolve layers of granular particles .as can be seen in figure [ fig : rho_r ] , the plugs , averaged for the vertical profiles , are highly ordered in the radial direction .the radial profiles are taken from the middle of the averaged plugs , at half height of the tube ( ) .the porosity minima in figure [ fig : rho_r ] correspond to three layers of particles .this agrees well with the ratio between the tube and particle diameter , which implies that five particles fit into the tube horizontally .the minimum at corresponds to a vertical string of particles in the middle of the tube , the layer at touches the wall of the tube . for higher radii the porosity increases to one .the large error at the center of the tube is due to the smaller volume involved for averaging.  radial profile of the granular velocity for an averaged plug in figure [ fig : rho_r ] .the profile shows three layers of particles moving upwards with different velocities .the layer along the wall is the slowest due to the friction with the wall ., width=377 ]  the radial profile of the granular velocity ( fig .[ fig : uxx_r ] ) shows that the layers observed in the porosity profile move with different velocities .the layer at the wall , which has the largest impact on the total averaged granular velocity , moves slowest , due to the friction with the wall .in this paper a simple model  with coupled grain and gas flow has been applied to pneumatic transport .the implementation is three - dimensional ; rotation and coulomb friction are taken into account .the flux of gas and grains is set by the boundary conditions .plug conveying is observed .the simulation used for plug statistics and profiles contained on average 3600 particles in a tube of length and diameter .the simulation lasted ; during this time 90 different plugs were observed at the middle of tube , each with about 300 particles . additionally 24 simulations were preformed to obtain the characteristic curves .the behavior of the granular medium and the gas is consistent with experimental observations .the pressure drop , mass flux and superficial velocity can be obtained in quantitative agreement with experiment by introducing an effective viscosity and an effective friction .the effective viscosity reflects the increased momentum transport in the gas due to the turbulent flow around the grains .the effective friction reflects the complex interplay between sliding , rolling and static friction . by using the model ,large numbers of plugs could be studied , and their porosity , velocity , and size were measured as functions of height .the simulation results imply that the formation of the plugs at the bottom of the tube occurs spontaneously .there is a well defined preferred velocity of the plugs , which is independent of the plug size and the tube length .the average plug length increases along the tube due to the merging of plugs . to confirm experimentally the constant velocity and the increase of plug length ,statistics of plug length and velocity at different heights along the tube are needed .the results also show that boundary effects at the bottom and the top of the tube have to be taken into account . for the experimental parametersthe influence of the boundaries is limited to a distance of , which is about the distance between two plugs in the middle of the tube .plugs accelerate in these boundary regions .the acceleration at the upper boundary arises due to the lack of grains falling on the uppermost plug .this implies that the momentum transfer due to falling grains can not be neglected .a model of plug conveying must take into account that these falling grains reduce the plug velocity .also a detailed view into an average plug was presented .in contrast to experimental setups were are not limited to measure the plug properties only at few locations .experimental results usually provide the plug properties as a function of time at a given position along the tube , with the disadvantage that the plug profile is distorted by the relative motion of granulate along the plug .the vertical profiles were given for porosity , granular velocity , interparticle and drag forces , wall stress and granular temperature .in the experiment , these parameters usually are not accessible along the whole tube .additionally the radial profile is given at high resolution .the results show that the grains are ordered in radial direction .the grains arrange themselves in layers along the tube wall .these layers have different velocities ( up to 40% difference ) .experimental measurements of the granular velocity using a ccd - camera must take into account that they measure the velocity of the outer , slowest layer .  in conclusion ,the model presented here is a useful tool for investigating plug conveying .it is fast and flexible enough to make parameter studies on full featured plug conveying and provides at the same time access to the plug properties at the level of grains .it can obtain both characteristic curves and plug profiles .note also that the model can be easily used to describe horizontal transport , simply by rotating the gravitational acceleration .future optimizations and faster computers will permit this model to be applied to industrial - sized systems .we thank ludvig vinningland and eirik flekky for much help in the fluid solver , thomas ihle and adriano de oliveira sousa for their general help and advice .this research was supported by dfg ( german research community ) contract he 2732/2 - 1 and he 2732/2 - 3 .t. van den moortel , e. azario , r. santini , and louns tadrist .experimental analysis of the gas - 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Abstract: plug conveying along a vertical tube has been investigated through simulation , using a discrete element simulation approach for the granulate particles and a pressure field approach for the gas . the result is compared with an experiment . the dynamics of a plug are described by porosity , velocity and force profiles . their dependence on simulation parameters provides an overall picture of plug conveying .   plug conveying , dense phase , pneumatic transport , granular medium  47.55.kf , 45.70.mg


quantum cryptography , after its inception in 1984 , has been flourishing over the last decade .the prime reason is the possibility of unconditional security , a task unachievable in the domain of classical physics .this fact and already available marketable products based on quantum cryptography have motivated further research in this field . to name a few ,apart from the initial interest in quantum key distribution ( qkd ) , various schemes concerning direct communication ( secure communication circumventing the need of a prior shared key ) , quantum key agreement ( qka ) , quantum secret sharing , have been proposed in the recent past ( see for details ) .specifically , in the direct communication , the receiver may or may not require an additional classical information to decode the message sent by the sender ; depending upon this , the protocol falls under the category of deterministic secure quantum communication ( dsqc ) and quantum secure direct communication ( qsdc ) , respectively .there is another novel technique of direct communication , quantum dialogue ( qd ) , where both the users can send their information simultaneously , with no need of a prior shared key .all these schemes for secure direct quantum communication , provide us a vast potential for extension and modification to design protocols required in various real life scenarios .one such important facet of quantum cryptography provides solutions for maintaining the hierarchy in offices or government , in terms of the information accessible to each user .hierarchical quantum communication schemes are aimed to deal with these problems , when either only single sender holds all the information or it is distributed among two of them .we may consider another important scenario , where a controller supervises the communication among all the remaining users , and he can maintain his control by making sure that the communication is not accomplished without his consent .further , a scheme for quantum controlled communication based on a quantum cryptographic switch has been proposed recently , which allows the supervisor to control even the amount of information he wishes to share with the other users in a continuously varying degree .  it would be worth summarizing that the security achieved in all the cryptographic schemes is based on the principle of splitting the whole information in many pieces , and the whole information can only be extracted if all the pieces are available simultaneously .usually , one of the parties prepares an entangled state to be used as a quantum channel and shares it with all other parties in a secure way . by secure , we mean that a proper eavesdropping checking technique is employed , after inserting the decoy qubits with the entangled qubits to ensure the absence of eve . once this channel is shared the legitimate parties can securely share their secrets , either by teleportation or encoding their information using pauli operations and sending the qubits to the receiver again in a secure manner .an interesting observation , we would like to exploit here , is that if we start with a controlled quantum dialogue ( cqd ) scheme , we can reduce it to almost all the schemes of secure quantum communication .this point is discussed in detail in the forthcoming sections .the feasibility of implementation of various quantum communication schemes when subjected to noisy environment has been analyzed in the past .in particular , the schemes of qkd , qka , controlled dsqc ( cdsqc ) , qsdc , cqd , qd , asymmetric qd ( aqd ) , controlled bidirectional remote state preparation , among others , have been considered under the influence of both purely dephasing and damping noises .most of these investigations ( cf . ) were restricted to the domain of markovian environments , though , some attempts have been made to study the effects of non - markovian environments on quantum communication schemes , such as teleportation , densecoding and entanglement swapping .the security of a qkd protocol has also been analyzed over non - markovian depolarizing channel .all these attempts ( except ref . ) to examine the usefulness of entangled states under the influence of non - markovian environments were restricted to insecure quantum communication , where security is not required . however , in the secure quantum communication protocols , it becomes relevant to differentiate the disturbance caused due to eavesdropping and the effects of noise .this sets the motivation for this work , where we wish to analyze the effect of non - markovian noisy environment considering the scenario when no eavesdropping has been attempted .this would provide a threshold of error due to disturbance from a non - markovian environment ; errors exceeding this could be attributed to the presence of an eavesdropper . specifically , we would consider pure dephasing , damping and depolarizing interactions with a non - markovian reservoir .though entanglement can only be maintained for relatively longer time due to dephasing non - markovian interaction , it can show revival under dissipative interactions . as we are essentially using entangled states and entanglement revivalcould be an interesting feature to affect the feasibility of quantum cryptographic schemes , we would like to address the problem here .non - markovian noise has been attracting a lot of interest from both quantum optics and quantum information communities , theoretically as well as experimentally .a paradigm for studying non - markovian evolution is the quantum brownian motion .specifically , degradation of purity and nonclassicality of gaussian states have been studied under the effect of non - markovian channels .dynamics of entanglement has been discussed in both discrete and continuous variable channels .recently , dynamics of multipartite entanglement and its protection have been addressed .the additional problems due to non - markovian noise in quantum error correction and dynamical deocupling have also been discussed in the past .the non - markovianity was also characterized from an information theoretic approach in terms of quantum fisher information flow .a number of beautiful experiments depicting non - markovian nature of the system - reservoir interaction have been performed .first , the kraus operators of non - markovian dissipative and dephasing noise models are discussed in a concise manner ( in section [ sec : noise - models ] ) . in section [ sec :effect - of - noise ] , we introduce , briefly , a cqd scheme ( [ sub : cqd ] ) using bell states and based on the quantum cryptographic switch. then , we study the effect of non - markovian noise on the feasibility of the cqd scheme . to quantify the effect of noise ,a distance - based measure known as fidelity has been calculated .next , we reduce the scheme of cqd to design a cdsqc protocol ( in section [ sub : cdsqc ] ) , a qd protocol ( in section [ sub : qd ] ) , a dsqc and qsdc protocols ( in section [ sub : qsdc / dsqc ] ) , a qka protocol ( in section [ sub : qka ] ) , and finally , two well known qkd protocols ( in section [ sub : qkd ] ) . the qkd protocols discussed hereare well known as bb84 and bbm protocols .the feasibility of all these schemes under the action of non - markovian channels are also analyzed .finally , we conclude the paper in section [ sec : conclusion ] .we briefly discuss below , a few non - markovian models that are subsequently used to study the performance of various quantum cryptographic schemes .the dynamics of a system interacting with its surroundings can be expressed in terms of kraus operators as ( see for a review ) . here, we use this approach to describe the dissipative and purely dephasing interactions with non - markovian environments .the kraus operators for the damping noise under non - markovian effects are given by     where with .here , is the line width which depends on the reservoir correlation time ; and is the coupling strength related to qubit relaxation time . in the domain of large reservoir correlation time in comparison to qubit relaxation time ,memory effects come into play .the memory effects are characteristic of non - markovian nature of dissipation .interestingly , taking , the results obtained for amplitude damping noise under markovian regime can be deduced , with being the decoherence rate of amplitude damping channel .similarly , the kraus operators for purely dephasing non - markovian noise are     where  ] with for  .further , is the coupling strength of the system and is the noise bandwidth parameter .it should be noted that the markovian case can be deduced from the above by taking with for  .in what follows , we consider a set of quantum cryptographic protocols and analyze the feasibility of their implementation over the above discussed non - markovian channels .for all the one - way schemes for quantum cryptography that are discussed here , we consider alice as the sender and bob as the receiver , unless stated otherwise ; whereas , charlie is the third party supervising the protocol and referred to as the controller .however , for two - way schemes ( e.g. , qd , cqd ) , both alice and bob are considered to play dual roles of receiver and sender .let us start with a three party protocol for quantum cryptography , where two parties ( alice and bob ) wish to communicate simultaneously under the control of a third party ( charlie ) .in fact , all the controlled communication protocols work under an assumption that all the parties are semi - honest . otherwise , alice and bob can share a quantum state of their own and circumvent charlie s control . in literature , this is sometimes viewed as alice and bob lacking resources for state preparation , and consequently , they do not set up a quantum channel between them , rather they rely on charlie to prepare it for them .  to begin with, we consider a cqd scheme recently proposed by some of the present authors .charlie prepares copies of a bell state and makes two strings and of all the first and second qubits .subsequently , he sends both the strings to bob , only after permuting travel qubits , an equal number of decoy qubits are inserted randomly in the original sequence of the travel qubits , and subsequently , these decoy qubits are measured to check the existence of eavesdropper(s ) .various choices of decoy qubits and the corresponding principles of security are discussed in . ] .bob will encode his message by using pauli operations on the qubits in string .subsequently , bob sends to alice , who returns it to him after encoding her secret message as bob did .it is pre - decided that pauli operations ,  , and correspond to encoded bit values 00 , 01 , 10 , and 11 , respectively .finally , charlie discloses the permutation operator , and using this information bob performs a bell measurement on the partner qubits ( bell pairs ) . when bob announces the measurement outcome , both alice and bob can extract each other s message using their own encoding information and the knowledge of initial bell state prepared by charlie .if the choice of bell state prepared by charlie is made public , it leads to some leakage , which is often considered to be an inherent characteristic of schemes for qd and its variants .however , such leakage can be circumvented if charlie chooses the bell state randomly and sends his choice to alice and bob by using a scheme of dsqc or qsdc .in fact , the schemes of qd are the most efficient protocols without involving prior key generation .suppose charlie started with the initial state , where , and , .the transformed density matrix over the noisy channel would become where are the kraus operators for a specific kind of noise discussed in the previous section and are the pauli operations by alice and bob with for . here , we have used different values of corresponding to each operation of the kraus operator ( from eq .( [ eq : kraus - damping ] ) , ( [ eq : kraus - dephasing ] ) or the depolarizing channel ) on the initial quantum state as the coupling strength during various rounds of the quantum communication is assumed to be different .it may be noted that the second summation in the right hand side of eq .( [ eq : transformed - rho ] ) ensures that the map is positive while the first summation corresponds to the average over all the possible encoding operations that alice and bob are allowed to perform .thus , the fidelity that we are discussing here , and in the rest of the paper , is the average fidelity .further , here , we have assumed that the qubits not traveling through a quantum channel are not affected by noise .there are various distance - based measures to quantify the effect of noise on the quantum state , such as trace distance , fidelity , and the bures distance .the fidelity of the transformed density matrix with the quantum state in the ideal situation ( i.e. , in the absence of noise ) would be  where the expected quantum state .the fidelity of the quantum state transformed under the damping effect of non - markovian environment is ,\label{eq : damp - psi}\ ] ] when the initial quantum state prepared by charlie is . as the choice of initial state is solely a decision of charlie , an independent choice of the initial state , i.e. , , would lead to the following expression of fidelity .\label{eq : damp - phi}\ ] ]  if the state prepared by charlie were subjected to a non - markovian dephasing noise , the fidelity would be .\label{eq : deph}\ ] ] it is interesting to see that the obtained fidelity is independent of the choice of the initial bell state by charlie .this is also seen in analogous scenarios of markovian dephasing noise in and references therein .if we now consider that the system has evolved under the effect of a depolarizing channel , then following the above prescription , the analytical expression for fidelity can be obtained as .\label{eq : depolarizing - cqd}\ ] ]  it is interesting to observe the appearance of fourth order terms in all the non - markovian fidelities , a signature of four noisy channels acting on the , four , different rounds of quantum communication .it should be mentioned here that instead of sending both the strings to bob , charlie could have sent to alice and to bob .subsequently , alice would have sent to bob after encoding her message and bob would have encoded his message before performing the measurement .the obtained fidelity expressions in this case turns out to be the same as that of the cdsqc protocol , discussed in the next subsection .the effect of noise in the case discussed here is more than that in the case of cdsqc .making use of this observation , we analyze the scheme of cqd , described above , in detail as the results obtained in the following subsections can be reduced from it .now , we will discuss the fidelities for different scenarios depicted in eqs .( [ eq : damp - psi])-([eq : depolarizing - cqd ] ) , for both markovian and non - markovian noises .the case of the non - markovian damping / dephasing channels are also considered for strong and weak coupling regimes .specifically , we obtain results in the strong and weak coupling regimes over non - markovian damping channels and , whereas for very high values , such as , it is found to reduce to markovian case . in the following figures ,we have used the notation nm , m , and , which correspond to the non - markovian , markovian , and non - markovian ( under strong coupling strengths ) regimes of interactions , respectively .a comparative analysis of the effects of non - markovian ( for both strong and weak couplings ) and markovian noise can be seen from figs .[ fig : cqd - dam - deph]-[fig : cqd - transition - dam - dep ] . in this case , though four different coupling regimes are possible , one for each , we have restricted ourselves , for simplicity , to the scenario of charlie to bob quantum channel having the same coupling strength for both the travel qubits .similarly , bob to alice quantum channel has the same coupling strength as that for the other way round .we explicitly mention the two choices of regimes in fig .[ fig : cqd - dam - deph ] .specifically , fig .[ fig : cqd - dam - deph ] ( a ) and ( b ) show the effect of damping quantified by fidelity on the cqd scheme for different choices of initial bell states , i.e. , and , respectively .it is interesting to observe that when both the qubits undergo damping , either in markovian or in strong coupling non - markovian regimes , the choice of initial bell states becomes irrelevant ( see red ( dashed ) and orange ( large dashed ) curves in fig .[ fig : cqd - dam - deph ] ( a ) and ( b ) ) .however , this initial choice becomes considerably important for all the remaining cases , and states are seen to be preferable as these states are less affected by non - markovian damping noise than .further , it is seen that , due to non - markovian effects , the fidelity can be maintained for a relatively larger period of time ( i.e. , the quantum state decoheres slowly in non - markovian environments in comparison to the corresponding markovian environments ) , a feature that depends on the coupling strength ( cf .[ fig : cqd - dam - deph ] ( a ) and ( b ) ) .another interesting characteristic of this kind of non - markovian noise is periodicity and the kinks present in fig .[ fig : cqd - dam - deph ] ( a ) and ( b ) are its signature . in ref . it was shown that the dilapidating influence of decoherence , due to markovian damping , can be checked using squeezing . here , it is seen that the same task can also be achieved by exploiting non - markovianity .the effect of noisy environment is independent of the initial bell state over dephasing channel and the fidelity is observed to improve gradually with non - markovian effects and coupling strength ( cf .[ fig : cqd - dam - deph ] ( c ) ) .periodicity in the time variation of fidelity , when all interactions are ( strong ) non - markovian is not visible in the time scale of fig .[ fig : cqd - dam - deph ] ( a ) and ( b ) . for larger time scales ,this can be observed in fig .[ fig : cqd - dam - deph ] ( d ) , where fidelity over both the damping and dephasing non - markovian channels is shown together .it can be seen that the fidelity under the effect of the damping noise decays faster than that over dephasing channel . at times , the fidelity over damping channel is observed to be much larger than that over dephasing channels , which remains constant at 1/2 .  to analyze the effect of the coupling strength with varying time , we depict , in fig .[ fig : cqd-3d ] , a contour and a 3 dimensional plots .the ripple like plot ( cf .the blue - colored surface plot in fig .[ fig : cqd-3d ] ( b ) ) shows that with decreasing coupling strength the amplitudes of the revived fidelity gradually become smaller .the same fact is also illustrated through a contour plot shown in fig .[ fig : cqd-3d ] ( a ) , where we can see that the area of the light - colored region reduces as we move from bottom to top .physically , this corresponds to a transition from strong to weak coupling non - markovian regime and finally into markovian regime .  a similar analysis of the fidelity expression for the depolarizing channel is illustrated in fig .[ fig : cqd - depol ] . in fig .[ fig : cqd - depol ] ( a ) , homogeneous depolarizing noise is assumed , for which to ensure that the dynamical map is completely positive .interestingly , it can be  seen from fig .[ fig : cqd - depol ] ( a ) that the fidelity falls gradually with the parameter , which determines the fluctuation due to the depolarizing channel .however , it can be noted that for all the cases , the fidelity under non - markovian environment is always greater than that for the corresponding markovian case , till all the plots merge , with time , to a single value .further , for the case of inhomogeneous fluctuations , we observe revival in the fidelity in fig .[ fig : cqd - depol ] ( b ) . from fig .[ fig : cqd - depol ] , it can be summarized that non - markovian depolarizing channel affects the system less than the corresponding markovian channel .( color online ) the dependence of the obtained fidelity over the damping channel on the coupling strength and rescaled time is illustrated through a contour plot in ( a ) .( b ) depicts the variation of the fidelity for varying coupling strength and time for both purely dephasing and damping non - markovian channels in light ( yellow ) and dark ( blue ) colored surface plots , respectively . ]( color online ) ( a ) the effect of non - markovian depolarizing channel on the cqd scheme has been illustrated for different values of the dimensionless quantity indicated in the plot . (a ) shows the case of homogeneous non - markovian depolarizing channel ( i.e. , ) .( b ) illustrates a comparison between inhomogeneous case of non - markovian and markovian depolarizing channels . in ( a ) , the constant , which is the maximum value ensuring completely positive map for all times for the homogeneous case ; in ( b ) , the noise parameters are for . ]the change in coupling strength controls the transition from non - markovian to markovian regime for both damping and depolarizing channels .this dependence has been illustrated in fig .[ fig : cqd - transition - dam - dep ] .initial small changes in the value of coupling strength changes considerably the nature of the obtained fidelity , i.e. , the periodicity and maximum value of fidelity after revival show ample changes for even a small change of coupling strength. however , for small values of the coupling strength , this change becomes less sensitive as reflected in the dense black lines corresponding to smaller values of coupling strengths .a similar comparison of the effect of non - markovian and markovian depolarizing channels shows that the fidelity sustains for a longer period of time under the influence of a non - markovian depolarizing channel , and is more sensitive to small changes in the noise parameter , which controls the fluctuation . for higher values of noise parameter ,the variation due to small changes in noise parameters becomes negligible in both markovian and non - markovian depolarizing channels .the effect of a change in the coupling strength on the fidelity is illustrated here with a set of plots for damping and dephasing non - markovian noise in ( a ) and ( b ) , respectively .specifically , the parameter of the coupling strength varies from 0.001 to 0.03 in steps of 0.001 in both the plots . ]  in the following subsections , we will deduce corresponding results for the remaining cryptographic tasks from the results obtained in this section for the fidelity ( for the cqd scheme ) over the various non - markovian channels .   a protocol of cdsqc , based on quantum cryptographic switch , can be obtained from the cqd scheme discussed in the previous subsection , i.e. , when only a single party encodes and sends his / her message in a secure manner via the quantum channel , which is decoded by the other party . to be precise, charlie initially follows the same steps as in section [ sub : cqd ] but rather sends the two strings and to alice and bob , respectively .alice encodes her message as usual and sends the encoded qubit to bob , who decodes the secret by performing bell state measurement on partner pairs with the help of charlie .the cdsqc scheme and the effect of noise can be summarized as follows where all the parameters have the same meaning as in section [ sub : cqd ] .it is interesting to observe that the transformed density matrix in eq .( [ eq : transformed - rho - cdsqc ] ) can be obtained from eq .( [ eq : transformed - rho ] ) just by considering and .the fidelity can be calculated with the quantum state expected in the ideal situation , i.e. , .  due to this observation, the fidelity of the quantum states affected by the non - markovian noise , for the cdsqc scheme can be obtained from the corresponding cqd expressions by taking , in eqs .( [ eq : damp - psi])-([eq : deph ] ) .interestingly , for the case of the depolarizing channel , the fidelity can be shown to be ,\label{eq : depolarizing - cdsqc}\ ] ] where the presence of cubic terms manifests the fact that the number of rounds of quantum communication involved in this scheme is less than that for the scheme discussed in the previous subsection .specifically , the scheme for cdsqc requires three rounds of quantum communication , while the scheme for cqd requires four rounds .the qubit traveling through the noisy channel may have different coupling strength during each round of travel . here , we wish to emphasize this point with the help of three possible coupling strengths for three noisy channels acting on the travel qubits .the observations made above for the extreme cases , i.e. , the qubits traveling through either non - markovian channels with strong coupling or markovian channels all the time , remain valid here as well .nevertheless , it can not be conjectured that the more the number of non - markovian channels , the higher the fidelity . in particular ,the large dot - dashed ( purple ) curve in fig .[ fig : cdsqc - dam - deph ] ( a ) and ( b ) establishes that even lower fidelity is observed with lesser number of markovian channels acting on the travel qubits .in fact , fig . [fig : cdsqc - dam - deph ] ( b ) shows that the obtained fidelity for parity 1 bell states ( i.e. , ) is less for all the cases when various noise channels had different coupling strengths than that for the case of the travel qubits subjected to noisy channels with the same coupling strength .however , no such nature is visible in fig . [fig : cdsqc - dam - deph ] ( c ) for dephasing channels .it is worth stressing here that out of the three possible choices for different coupling regimes corresponding to each , we have emphasized only on the interesting cases and mentioned them accordingly in fig .[ fig : cdsqc - dam - deph ] .interestingly , the fidelity in the cdsqc protocol falls below the corresponding markovian value , under the influence of the non - markovian depolarizing channel , when all three noise parameters have different values ( cf .[ fig : cdsqc - dam - deph ] ( d ) ) . this nature can be attributed to the presence of cubic terms in the fidelity , eq .( [ eq : depolarizing - cdsqc ] ) .   a cqd scheme can be viewed as a qd scheme under the supervision of a controller .therefore , a qd scheme can be easily derived from the cqd scheme if we consider the scenario that one of the two communicating parties ( i.e. , either alice or bob ) prepares and measures the quantum state , while both the parties encode their secret on the same qubits .this qd scheme , which is obtained as a result of reduction from the cqd scheme described above , can be easily recognized to be equivalent to the first qd protocol proposed by ba an .the effect of noise on this scheme for qd can be obtained by considering in all the expressions of section [ sub : cqd ] .this would imply that the initial state is prepared by one of the communicating parties ( say , bob ) .then the transformed density matrix and the fidelity expressions over non - markovian channels can be deduced from eqs .( [ eq : transformed - rho])-([eq : deph ] ) . here, it is important to note that the effect of noise is independent of the choice of initial bell state by charlie / bob in all the schemes other than cqd and cdsqc .similarly , under the effect of depolarizing channels , the expression of fidelity turns out to be  ,\label{eq : depolarizing - qd}\ ] ]  due to two rounds of quantum communication of a travel qubit .   as mentioned beforehand in section [ sub : cdsqc ] , a cdsqc scheme can be viewed as a cqd scheme , where only one party is allowed to encode . in the same way , a qsdc scheme ( say , a ping pong protocol ) can be viewed as a scheme for qd , where one party ( say bob ) is restricted to encode identity only .therefore , all the expressions of the fidelity for a qsdc scheme are exactly the same as those for the scheme of qd.  a dsqc scheme can be reduced from the above mentioned protocols if bob incorporates information splitting in two quantum pieces and sends them one after the other in two different rounds of bob to alice communication .specifically , bob prepares two strings as in section [ sub : qd ] and sends the first string to alice .he subsequently sends the second string to alice only if the first quantum part is received by alice undisturbed .the effect of non - markovian noisy environment on this dsqc scheme can be obtained from the corresponding expressions for the cqd scheme obtained in sec .[ sub : cqd ] , if we consider and in eqs .( [ eq : damp - psi])-([eq : deph ] ) . here, is used to show the effect of noise on the second qubit traveling from bob to alice in the first round .in fact , it turns out to be exactly similar to what is obtained for the qsdc scheme .interestingly , in case of depolarizing noise , all the expressions for fidelity are found to be the same for qd , qsdc and dsqc schemes .for the convenience of discussion for the dsqc scheme , we have chosen bob ( alice ) as the sender ( receiver ) .so far , we have discussed quantum communication schemes where prior key generation is circumvented by proper use of quantum resources. we may now proceed to key generation schemes and investigate the effect of non - markovian environment on them .a qka scheme provides equal power to all the parties taking part in the key generation process , and does not allow members of a proper subset of the set of all users to solely decide the final key . here, we consider a completely orthogonal qka scheme proposed in . in this qka protocol, a party ( say alice ) sends her raw key to another party ( say bob ) by using a qsdc protocol , while the other party publicly announces his key .the security of the final key is achieved by the unconditional security of alice s transmission of raw key using quantum resources ( i.e. , from the security of the qsdc / dsqc scheme used by alice and bob for alice to bob communication ) .specifically , alice transmits a key to bob in a secure manner , whereas bob announces his key , publicly , and for all future communication they use a key , where denotes a bitwise xor operation .although , eve knows , she can not obtain any information about as she knows nothing about .thus , the security of depends on the security of . in other words , unconditional security of the qsdc scheme involved here would ensure the security of the protocol for qka .interestingly , in ref . , the present authors had already shown that the effect of noise on this scheme is identical to the qsdc scheme discussed in the previous section [ sub : qsdc / dsqc ] .since the observations made there remain valid here , we do not discuss it in further detail .any discussion on quantum cryptography remains incomplete without discussing a protocol that changed the course of cryptography by establishing the feasibility of unconditional security . in this section, we discuss two qkd protocols , which can be viewed as the variants of the same scheme , differing only in the measurement procedure .specifically , the bb84 and bbm qkd protocols are discussed here .before we proceed further , it would be apt to note that in contrast to the fidelity expressions obtained in the earlier sections ( which were average over all the encoding operations ) , for qkd protocols , the average fidelity is obtained over all possible equally probable measurement outcomes .  in the bbm protocol , alice prepares bell states and sends all the first qubits to bob , and both of them measure the qubits of the shared bell states randomly in computational ( ) and diagonal ( ) basis . using the outcome of these measurements, they finally obtain an unconditionally secure quantum key for those cases where both alice and bob perform measurement using the same basis .the bb84 protocol can also be viewed along the same lines , where alice first measures her qubit ( i.e. , second qubit ) of each bell state randomly either in computational or diagonal basis and then sends the other qubit to bob .finally , they can obtain a key by using the measurement outcomes of half of those cases , where they have chosen the same basis .the other half of the cases should be used for eavesdropping check .specifically , when alice and bob have performed measurement in the same basis , in the absence of eve , their measurement outcomes should match and a mismatch would indicate the presence of eve .interestingly , for the bbm protocol , the effect of noise can be considered by taking in eqs .( [ eq : damp - psi])-([eq : deph ] ) .similarly , the effect of depolarizing channel reduces the fidelity to  .\label{eq : depolarizing - qkd - bbm}\ ] ]  a similar study for the bb84 protocol results in the following fidelity over damping non - markovian channels  ,\label{eq : damp - qkd}\ ] ]  while , for the dephasing channel the fidelity is  .\label{eq : deph - qkd}\ ] ]  further , the fidelity when the travel qubit is subjected to a depolarizing channel is  .\label{eq : depolarizing - qkd - bb84}\ ] ]  additionally , the present results can also be used to deduce the fidelity for a few other quantum cryptographic schemes , which will reflect quantitatively the effect of non - markovian channels on the corresponding scheme . for example , the effect of noise on ekert s qkd protocol can also be deduced from the results in sec .[ sub : cqd ] , by taking as the source of entanglement is between both the parties , and both the entangled qubits travel to alice and bob from there .similarly , the feasibility of the b92 protocol can also be analyzed over the non - markovian channels in analogy with the study for bb84 protocol by only considering two of the four single qubit states ( one each chosen from computational and diagonal basis ) .( color online ) a comparative analysis of all the quantum cryptographic schemes discussed so far over the non - markovian channels .each line in all three plots corresponds to the different cryptographic scheme mentioned in the plot legend at the bottom of the figure .the light black lines in all three plots represent the corresponding markovian cases , and the black lines from bottom to top show the average fidelity for cqd , cdsqc , qd , bbm qkd , and bb84 qkd protocols .the fidelity obtained for qsdc , dsqc , and qka schemes is exactly the same as that of the qd protocol . ]  finally , we perform a comparative study for the fidelity obtained in each cryptographic scheme to reveal the general nature of the effect of non - markovian channels on all these schemes ( shown in fig . [fig : all - schemes ] ) .interestingly , the effect of noise depends on the number of rounds a qubit is required to travel through the noisy channel .this fact is consistent with the recent observations on a set of markovian channels .specifically , in the cqd scheme , one qubit travels from charlie to bob , while another qubit travels from charlie - bob - alice - bob .therefore , the maximum number of rounds of travels in the set of secure quantum communication schemes discussed here is four for cqd scheme , which decreases to three for cdsqc .it further reduces to two for qd , qsdc , dsqc , and qka schemes . the same fidelity for all these schemes further establish this point .finally , bbm and bb84 qkd protocols require only one round of quantum communication .in fact , bbm and bb84 protocols use entangled and single qubit states , respectively , to accomplish the same task . out of these two schemes ,the bb84 qkd scheme is least affected by noise as it uses single qubit states , which were shown to be less affected due to markovian channels in .  in fig .[ fig : all - schemes ] ( a ) and ( b ) , the fidelity variation over non - markovian channels due to the strong coupling of the travel qubits with the environment is depicted .similarly , the effect of different noise parameters corresponding to depolarizing channel is shown in fig .[ fig : all - schemes ] ( c ) . also shown is the effect of markovian environment on the fidelity in all three cases , depicted by thin smooth ( black ) lines .for all cases of markovian dynamics , the observation that the effect of noise depends on the rounds of quantum communication remains valid .  from fig .[ fig : all - schemes ] ( a ) , the revival in the fidelity over non - markovian damping channel is seen to decrease with an increase in the number of travel qubits .similarly , the fidelity falls with increasing rounds of quantum communication when subjected to dephasing non - markovian channel , as shown in fig .[ fig : all - schemes ] ( b ) . out of the set of fidelities , over the depolarizing channel ,those having odd power terms , such as for the cdsqc and qkd protocols , show fidelity less than that for the corresponding markovian case . otherwise , in all the remaining cases , the fidelity over non - markovian channels is more than that for the corresponding markovian channels ( cf .[ fig : all - schemes ] ( c ) ) .the present study on the effect of a set of non - markovian channels on various schemes for secure quantum communication tasks led to a number of interesting results .specifically , we have considered here a damping , a purely dephasing and a depolarizing non - markovian channel to analyze the feasibility of some quantum cryptographic schemes evolving under the influence of the non - markovian environments .we have started with a cqd scheme , based on a quantum cryptographic switch that uses bell states .later , this scheme is modified to deduce the results for other quantum cryptographic tasks , such as , cdsqc , qsdc , dsqc .apart from these direct communication schemes , the effect of non - markovian noise on some protocols of qkd and qka is also analyzed .it has been established that the effect of non - markovian noise depends on the number of rounds of the travel qubits .we have observed that the bb84 qkd scheme is least affected due to non - markovian channels , while the cqd scheme shows a maximum fall in the fidelity .in fact , from the results obtained here one can also show that the aqd scheme will have the same effect as that on the qd protocol if the number of travel qubits is kept unchanged .this fact is consistent with the results obtained here , that the fidelity for qsdc , dsqc , and qka schemes are exactly the same as that for the qd protocol . in the recent past, we have established that squeezing is a useful quantum resource for quantum cryptography as it can help to stop decoherence . here, we have shown that non - markovianity can also be used to accomplish a similar task .interestingly , the effect of noise on the cqd and cdsqc schemes is found to depend on chalie s initial choice of the bell state , while it is independent of this in all the remaining schemes .finally , our analysis has also revealed that the fidelity obtained in the case of damping and dephasing channels depends on the coupling strength .we hope these results would bring out the importance and utility of the non - markovian behavior in the understanding of quantum cryptographic protocols from the perspective of their practical implementation .* acknowledgment : * ap and kt thank defense research & development organization ( drdo ) , india for the support provided through the project number erip / er/1403163/m/01/1603 .sb acknowledges support by the project number 03(1369)/16/emr - ii funded by council of scientific and industrial research , new delhi .

Abstract: a set of schemes for secure quantum communication are analyzed under the influence of non - markovian channels . by comparing with the corresponding markovian cases , it is seen that the average fidelity in all these schemes can be maintained for relatively longer periods of time . the effects of non - markovian noise on a number of facets of quantum cryptography , such as quantum secure direct communication , deterministic secure quantum communication and their controlled counterparts , quantum dialogue , quantum key distribution , quantum key agreement , etc . , have been extensively investigated . specifically , a scheme for controlled quantum dialogue ( cqd ) is analyzed over damping , dephasing and depolarizing non - markovian channels , and subsequently , the effect of these non - markovian channels on the other schemes of secure quantum communication is deduced from the results obtained for cqd . the damped non - markovian channel causes , a periodic revival in the fidelity ; while fidelity is observed to be sustained under the influence of the dephasing non - markovian channel . the depolarizing channel , as well as the other non - markovian channels discussed here , show that the obtained average fidelity subjected to noisy environment depends on the strength of coupling between the quantum system with its surroundings and the number of rounds of quantum communication involved in a particular scheme .