consider a particle diffusing in a two - dimensional box with reflecting boundary conditions .we show a portion of a simulated trajectory of such a system in figure [ fig : rect_box ] .suppose that in the left half of the box the particle diffuses with coefficient and that in the right half of the box the particle diffuses with coefficient .we assume that there are no external forces acting on the particle .our question is : does the particle spend an equal fraction of time on each side of the box in the long run ?  one answer is based on statistical mechanics .the justification for this prediction is the principle of statistical mechanics which states that an isolated system in equilibrium is equally likely to be in any of its accessible states " ( reif 1965 , p. 54 ) .since all states in the box are accessible , and there are an equal number of states on each side of the box , the particle should spend an equal proportion of its time on each side of the box .another answer is based on the idea of rescaling time in one side of the box .the justification for this prediction is that , in the absence of any drift , faster diffusion is equivalent to time passing more quickly .this means that the periods of time the particle spends on the right side of the box will be shorter than those spent on the left side .hence the total time the particle spends on the right hand side of the box will be less .we show how this prediction is a straightforward consequence of interpreting the particle s motion as drift - free diffusion , where we interpret the state - dependent diffusion coefficient using the it convention ( gardiner , 2004 ) .we can write the equation for the particle s motion as where and is standard two - dimensional brownian motion .( equivalently we may write this equation as where is two - dimensional gaussian white noise . )we specify for and for . here , where is the corresponding diffusion coefficient .we enforce reflecting boundary conditions at the four walls of the box .the ( it-)fokker - planck equation for , the probability density of the particle s position at time , is ( gardiner 2004 , p. 118 ) =  \nabla \cdot [ \nabla \left (  d(x ) \rho(x , t ) \right ) ] .\ ] ] the equilibrium density satisfies reflecting boundary conditions for the diffusion correspond to neumann boundary conditions ( zero - flux ) for the fokker - planck equation . with these boundary conditions , the unique equilibrium density is for some constant .thus , since , the particle spends half as much time on the right side of the box as on the left .( we discuss the relation of our question to other interpretations of in section [ sec : conclusion ] . )   simulation of particle diffusing within a box with reflecting boundary conditions .points are generated by the euler - maruyama method , where , are independent standard two - dimensional gaussians , and is the state - dependent diffusion coefficient . here on the left side of the box and on the right side of the box . , width=491 ]  neither the statistical mechanics prediction nor the time - change prediction are definitive .the statistical mechanics prediction relies on the principle of equal probability of all accessible states , which needs to be independently justified for the mesoscopic level of description we are considering here .the time - change prediction is not definitive since it stochastic differential equations are themselves mesoscopic models whose use can only be rigorously justified by showing how they arise as the coarse - scale limit of microscopic dynamics .since the two predictions contradict each other , at least one of them must be wrong for any given physical system .the approach by which we resolve this apparent contradiction is to study a microscopic model of the box system we have described above and then see what proportion of the time the particle spends on each side in simulations of that system .if one prediction turned out to always be true for our model system , we could use our result as a baseline for investigating under which more general situations the prediction was still true .however , we will see that even for our simple system there are parameter choices that make the statistical mechanics prediction correct and parameter choices that make the time - change prediction correct .this demonstrates that there is no _ a priori _ reason to conclude that one prediction or another is correct , given only the diffusion coefficient for the system .  in section [ sec : model_sys ], we describe our model system , perform numerical experiments on it , and show that the result of the numerical experiments can be determined analytically from the properties of the model system .the system we consider is a deterministic hamiltonian billiard system : the random lorentz gas ( dettmann 2000 ) .the system has two free parameters : disc radius and free volume fraction .there is a one - parameter family of values of these parameters that can generate the same effective diffusion coefficient .we will show how to choose these parameters to obtain arbitrarily good approximations to diffusive motion for the particle on each side of the box .using the degree of freedom in the choice of parameters , we show that the equilibrium density of the particle is underdetermined by the diffusion coefficient on each side of the region . for a given and , exploiting the flexibility in the parameter choiceallows us to create systems in which either the statistical mechanics prediction or the time - change prediction is correct .our justification for studying a particular microscopic system is threefold .firstly , there is a solid analytical understanding of the random lorentz gas on which we can base our simulations .secondly , our purpose is not to conclude that a particular style of mesoscopic modelling is always the correct one , but to show that postulating a state - dependent diffusion rate is not enough to fully specify mesoscopic behaviour . for this objective ,it is enough to show that multiple mesoscopic behaviours are possible for a single class of simple models , as we do in this paper .finally , we have resorted to a purely deterministic microscopic model , rather than a stochastic microscopic model ( such as langevin dynamics or a random walk ) to avoid concerns that the method used to introduce randomness at the microscopic level somehow biases the results at the mesoscopic level .this last point distinguishes our work from similar discussions of van kampen ( 2007 , sec .xi.3 ) and othmer and stevens ( 1997 , sec . 2 ) .these authors show that spatially inhomogeneous random walks , with natural choices of parameters , can lead to either prediction holding true at the mesoscopic level .similarly , the conclusions of our section [ sec : model_sys ] closely parallel those of korabel and barkai ( 2011 ) , in which the same question is answered using a random - walk model on a one - dimensional lattice .there , following the earlier work of ovaskainen and cornell ( 2003 ) , they show that choosing how the random walk behaves at the interface of two regions of differing diffusion coefficient leads to different equilibrium probability densities for the system .korabel and barkai explain how to determine the correct interface behaviour of the random walk model using experimentally measurable quantities .our approach differs in that , in our model , the interface behaviour is determined indirectly through the microscopic dynamics that we describe .the results in section [ sec : model_sys ] show that choosing a diffusion coefficient is not enough to fully specify diffusive dynamics in the absence of other assumptions .in particular , our results show that fixing is not enough to specify , the equilibrium probability density for the position of the particle . in section [ sec : prop ] we propose a framework for modelling state - dependent diffusion that makes this fact explicit . rather than simply specifying a state - dependent diffusion coefficient ,we specify diffusion coefficient and an equilibrium density which together with a detailed balance assumption ( no - flux in equilibrium ) completely determine the dynamics .we then introduce a new method for the numerical simulation of diffusive dynamics which makes use of our framework : the numerical method is expressed in terms of and and makes no reference to a drift term .the method consists of euler - maruyama steps for a purely diffusive it stochastic differential equation together with metropolis rejections .the method is similar to the metropolis - adjusted langevin algorithm ( roberts and tweedie , 1996 ; bou - rabee and vanden eijnden , 2010 ) , except that there is no drift term in the euler - maruyama step and it is the metropolis rejections that induce any drift in the trajectories .the advantages of our method over the euler - maruyama method are that it samples space with the correct equilibrium density and performs well even when and have discontinuities .one area in which the proposed framework in section [ sec : prop ] could be used is cellular biology .although earlier models of chemistry in the cytoplasm of cells assumed that chemical species were ` well - mixed ' and thus ignored diffusion , more recent models have taken the geometry of the cell and the diffusion coefficient of various molecules into account ( turner , schnell , and burrage , 2004 ) .effective diffusion coefficients of a molecule in a cell differ from that in water due to the crowding effect of other molecules .one approach is to model the motion of a molecule as diffusion with constant coefficient , but then have the effective diffusion be modified by interaction with other particles which are also included in the model ( ridgway , et al . , 2008 ) .another is to not include the crowding particles in the model , but to model their effect with a modified diffusion coefficient ( hall and hoshino , 2010 ) . given the inhomogeneity of the cytoplasm , we can expect that this effective diffusion coefficient of the molecule ( and its equilibrium probability density ) will vary with location within the cell .our framework and numerical method are developed with this latter situation in mind .  in section [ sec : conclusion ] we conclude by explaining the relation between our results and three different interpretations of state - dependent diffusion : it , stratonovich , and the isothermal convention of lanon , batrouni , lobry , and ostrowski ( 2001 ) .in the introduction , we described a two - dimensional system consisting of a single particle diffusing inside a rectangular box and reflecting off the boundaries .the diffusion coefficient is twice as large on the right side of the box as the left . in this section ,we demonstrate how to construct a family of deterministic hamiltonian systems that approximates this behaviour on a coarse scale .  in subsection ( [ subsec : rlg ] ), we describe the random lorentz gas ( dettmann , 2000 ) , a deterministic system that when given a random initial condition yields constant - coefficient diffusion at a coarse scale . in subsection ( [ subsec : boxtwo ] ) , we show how to approximate the model system of the introduction by creating two adjacent domains of the random lorentz gas within a bounding box . in subsection ( [ subsec : numeric ] ) , we describe numerical experiments with the box system demonstrating that the fraction of time a particle spends on each side of the box can not be determined solely from the values of the diffusion coefficient . in subsection ( [ subsec : analdisc ] ) , we explain how the result of the numerical experiments in subsection ( [ subsec : numeric ] ) can be predicted from properties of the dynamics of the microscopic system .consider infinitely many discs with positions fixed in .the centres of the discs are distributed randomly with uniform density subject to the constraint that the discs do not overlap .we consider a single point particle interacting with the discs in the following manner . given an initial velocity and an initial position not on a disc , the particle moves with constant velocity until it meets a disc .then it undergoes an instantaneous elastic reflection with the boundary of the disc , with the angle of incidence equalling the angle of reflection .this model is the two - dimensional _ random lorentz gas _ ( dettmann , 2000 ) , a mathematical formulation of a model originally due to lorentz ( lorentz , 1905 ) .we will always consider the case when the particle has initial velocity of magnitude 1 .figure [ fig : lorentz ] shows trajectories of the random lorentz gas at two different scales .the random lorentz gas has two parameters : the radius of the discs , and the number of discs per unit area .the radius can take any positive value .the density has a maximum value corresponding to the close - packed hexagonal pattern of discs .we define to be the free volume fraction , the proportion of the area not occupied by discs .we have that .the free volume can take any value in where , regardless of the value of .the pair provides an alternative parametrization of the random lorentz gas , with the advantage that is dimensionless .portions of the trajectory of the random lorentz gas with at two levels of magnification , showing the discs and the trajectory of the moving particle.,title="fig : " ]  portions of the trajectory of the random lorentz gas with at two levels of magnification , showing the discs and the trajectory of the moving particle.,title="fig : " ]  for adjacent discs are touching and the particle remains in a small region of the plane for its entire trajectory .for the probability that two discs are touching anywhere in the plane is zero ( dettmann 2000 , p. 325 ) and motion of the particle is conjectured to be diffusive ( dettmann and cohen , 2000 ) .( this contrasts with the situation of overlapping discs for which there is a phase transition to anomalous transport for dense enough placement of discs ( hfling , munk , frey , and franosch , 2008 ) ) .specifically , if we start the particle at initial position not coincident with a disc and give it initial speed 1 and uniformly distributed direction in , then , the position of the particle at time is approximately distributed as a gaussian random vector with mean 0 and variance matrix . here is the identity matrix and denotes the diffusion coefficient .this conjecture is supported by analytical calculations ( dettmann , 2000 ; ernst and weyland , 1971 ) and numerical simulations ( dettmann and cohen , 2000 ; bruin , 1972 ) .  a stronger and more formal statement of the conjectureis stated in the language of weak convergence ( billingsley , 1999 ) .specifically , let denote the position of the moving particle at time .fix to be some point not coincident with a disc and let the initial velocity be chosen as above .it is conjectured that as , where is standard two - dimensional brownian motion ( meaning , ) , is the diffusion coefficient which depends on and , and denotes weak convergence in the space of continuous functions ( billingsley , 1999 ) .such a result holds for the certain periodic lorentz gasses ( bunimovich and sina , 1980/81 ; klages and dellago , 2000 ) , but remains open for the random lorentz gas .( we chose for our study not to use the standard periodic lorentz gas with discs centred on a hexagonal lattice since for large enough the particle undergoes superdiffusive motion in this case . )a scaling argument shows that , for fixed , is proportional to . to see this , letting denote the euclidean norm , observe that the coefficient can be obtained as , assuming .if we rescale space by a factor , we increase both the size of the discs and the distance the particle travels by a factor of , without changing .so increases by a factor of .to maintain the speed of the particle as , we also have to rescale time , increasing by a factor . the net effect on the ratio to increase it by a factor ( sanders 2005 , sec .so for some function of .figure [ fig : dvsphi ] shows the relation between and that was computed using the techniques similar to those described in ( dettmann and cohen , 2000 ) .it appears that the function is continuous on its domain , it is monotonically increasing , as and goes to infinity as .indeed , calculations from kinetic theory show that $ ] in the limit ( vanleeuwen and weijland , 1967 ; bruin , 1972 ) .in order to investigate the main question of this paper , we take a rectangular box and divide it into two equal regions . each regionis filled with randomly placed discs , with different and on each side .as in the lorentz gas , the centres of the discs are placed uniformly at random with the condition that they not overlap with each other .discs can intersect with the walls of the box but not the dividing line between the two sides of the box .figures [ fig : first_params ] and [ fig : second_params ] show two examples of such configurations of discs .the dynamics of the point particle are the same as in the infinite random lorentz gas , with the added condition that the particle reflects off the walls of the box .suppose we are given diffusion coefficients .we can choose and such that and .for small enough the dynamics of the particle in this system will be well - approximated by a particle that diffuses with coefficient on the left side of the box and diffuses with coefficient on the right side of the box . with appropriate choices of the parameters , we can investigate the question of the proportion of time the particle spends on each side of the box .we start the particle off at some position in the box not on a disc with velocity of magnitude 1 and randomly chosen direction .the motion of the particle is simulated with an event - driven simulation , computing a trajectory that is accurate up to the errors of floating point arithmetic ( dettmann and cohen , 2000 ) .periodically the position of the particle is recorded . at the end of a long trajectorywe compute the number of times the particle is on the right side of the box divided by the number of times the particle is on the left side of the box .we consider two choices of parameters on each side of the box , or set - ups , each of which leads to the effective diffusion coefficients satisfying .we have contrived the set - ups so that in set - up 1 , the statistical mechanics prediction is correct , and that in set - up 2 , the time - change prediction is correct .the parameters for each set - up are summarized in table [ tab : params ]  .[ tab : params ] the parameters used in each side of the box in the two set - ups , and the ratio between amount of time spent by the particle on the right side of the box and the left side of the box in each set - up . [ cols="^,^,^,^,^,^,^,^",options="header " , ]   the question we posed in the introduction may be rephrased as follows : in the absence of external forces , and given a diffusion , what is the correct choice of parameter and drift to model the motion of the particle ? a natural way to approachthe problem is to interpret the absence of external forces as meaning that . then the problem boils down to the choice of : the statistical - mechanics prediction follows from taking , and the time - change prediction follows from taking .the results in section [ sec : model_sys ] showed that neither answer is justified universally .  in section [ sec : prop ] we recommended a different approach .we fix and then choose to generate the desired equilibrium distribution . as we have explained here ,the choice of is not crucial once we allow a non - zero .accordingly , we have chosen , corresponding to it calculus .this is the main choice in the mathematics literature , and numerical methods such as the euler - maruyama method take a particularly simple form with it .once we have made this choice of , we are free to choose appropriately . in section[ sec : prop ] we chose to ensure a given equilibrium density .  beyond the particular needs of the present work , we believe the framework we describe in section [ sec : prop ] provides a natural and flexible way to model diffusive systems . in situations where a researcher is confident for physical reasons that the equilibrium probability is constant , then our framework takes a simple form .the numerical method we present functions even when the diffusion and equilibrium density are discontinuous .future work will study the convergence properties of the method , as well exploring applications of our general framework .c. p. dettmann .the lorentz gas : a paradigm for nonequilibrium stationary states . in _ hard ball systems and the lorentz gas _ , volume 101 of _ encyclopaedia math ._ , pages 315365 .springer , berlin , 2000 .d. ridgway , g. broderick , a. lopez - campistrous , m. ruaini , p. winter , m. hamilton , p. boulanger , a. kovalenko , and m. j. ellison .coarse - grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm ._ biophys . j. _, 94(10):37483759 , 2008 .

Abstract: consider a particle diffusing in a confined volume which is divided into two equal regions . in one region the diffusion coefficient is twice the value of the diffusion coefficient in the other region . will the particle spend equal proportions of time in the two regions in the long term ? statistical mechanics would suggest yes , since the number of accessible states in each region is presumably the same . however , another line of reasoning suggests that the particle should spend less time in the region with faster diffusion , since it will exit that region more quickly . we demonstrate with a simple microscopic model system that both predictions are consistent with the information given . thus , specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system , even assuming the absence of external forces . we propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller . we introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients .  p. f. tupper and xin yang + department of mathematics , simon fraser university , + burnaby bc , v5a 1s6 canada .


cryptography is the science of encrypting and decrypting messages for secure communication. it has been used for military purposes since ancient times .cryptography is widely used today , e.g. for secure internet communication .the protocols used are based on classical physics and the assumption of limited computing power , which may one day turn out to be wrong !the basic procedure of cryptography consists of a sender and a receiver , conventionally called alice and bob respectively , who wish to share a secret message despite the presence of a malicious eavesdropper , conventionally called eve. alice and bob can only communicate through public channels such as phone or internet .thus , they need to encrypt their message prior to sending it as an eavesdropper could listen in on their communication . one simple way of encrypting the message is if alice and bob share a secret key .such a key could be a random sequence of 0s and 1s , each of which is called a bit , only known to themselves . if the message is also in the form of a sequence of 0s and 1s , alice can encrypt her message using bitwise addition of her message and the secret key , using addition modulo 2 where 1 + 1=0 .for example , the message 11001 and the key 01001 give the encrypted message 10000 .this encrypted message is called a ciphertext .equipped with the same secret key , bob now performs a bitwise addition of the ciphertext and the key to retrieve the original message . in the example, bob performs a bitwise addition of the ciphertext 10000 and the key 01001 to retrieve the original message 11001 .the procedure just described is called a one - time pad , and is only secure if the key is truly secret and used only once .eve gains no information from intercepting the ciphertext , as for a ciphertext composed of bits there are equally likely messages that are compatible with the ciphertext .  how can alice and bob share the secret key needed for this procedure ?this is the formidable challenge of key distribution .classically , there is no way to ensure that the distribution of the key is absolutely secure .quantum mechanics makes such secure key distribution possible ! the fundamental principle of quantum mechanics that taking a measurement perturbs the system ( unless the measurement is compatible with the quantum state ) also applies to eve .thus , if alice and bob use appropriate protocols , eve can not gain information about alice s and bob s communication without introducing perturbations that reveal her presence . since alice and bob are using a quantum procedure to only create a key and not to send a message , they can simply disregard any key that has been compromised , keeping their communication secure .eve can not simply make a copy of the state that alice sends to perform measurements on the copy , as the so - called no - cloning theorem proves that in the quantum world it is not possible to make a perfect copy of an arbitrary quantum state.  the idea of quantum key distribution ( sometimes imprecisely called quantum cryptography ) was first developed by bennett and brassard in 1984 ( see section [ sectionbb84]). since then , the field has progressed enormously. for example , quantum key distribution ( qkd ) using photons has been demonstrated over large distances using optical fibres and in free space , and with systems that are stable over long periods of time .commercial qkd devices are already available .quantum mechanics courses often only discuss quantum information in advanced courses .we argue that aspects of quantum information can profitably be covered much earlier .quantum key distribution ( qkd ) is one of the simplest demonstrations of the emerging and rapidly growing field of quantum information technology . in the most common form ( see section [ sectionbb84 ] ) , it does not require complicated mathematics , and only requires a knowledge of basic quantum mechanics concepts such as superposition states and their measurement outcome probabilities , incompatible observables , the collapse of a quantum state on measurement , and a physical quantum system with two states as a realization of a qubit .it also links well with the spins - first or more generally two - level systems approach gaining favor in quantum mechanics instruction , where topics such as single photons or spin 1/2 particles that are physical realizations of qubits are covered early in the curriculum. while qkd relies on the no - cloning theorem , our experience is that qkd can be discussed at the introductory level without proving this theorem .qkd is a topic with direct everyday applications well known to students ( such as secure internet communication ) , and thus helps to make a subject that can often appear abstract and far - removed from everyday experience relevant to students lives .  despite distinct advantages of incorporating aspects of quantum information into quantum mechanics courses, many textbooks do not include this topic , and very limited numbers of multimedia resources and research - based materials exist compared with other quantum mechanics topics. this article gives an overview of interactive simulations developed as part of the quvis quantum mechanics visualization project to support the learning and teaching of the basic principles of quantum key distribution using three different protocols .the simulations use polarized single photons and spin 1/2 particles as physical realizations of qubits .simulations can be used both at the introductory and advanced levels , and are also aimed at instructors with other areas of expertise interested in learning the principles of qkd .simulations have been coded in html5/javascript , so that they run on a wide range of devices including tablets and pcs .all of the quvis simulations and accompanying activities are freely available for use online or download ( www.st-andrews.ac.uk/physics/quvis ) .instructors can download password - protected solutions to activities from the quvis website .instructors interested in obtaining the password for the solutions are requested to email quvismail-andrews.ac.uk .the simulations described here have a number of features that make them useful for learning about qkd .they allow students to experience how a raw key would be generated experimentally and to see the effect of an eavesdropper infiltrating the experiment .students can easily set up different configurations , such as using only a single basis or two bases and see the results of data - taking immediately without needing complex equipment or long periods of time .they allow students to collect data at different speeds i.e. via individual particles / photons or in fast - forward mode .they show simplified idealized systems , such as perfect sources and detectors and a simple intercept - resend eavesdropping attack , to focus on key ideas and reduce cognitive load .they help students make connections between physical representations of the experimental setup and mathematical representations , e.g. the key bits , the error rate and the error probability .  in what follows ,we describe three different qkd protocols and show how they are implemented in the simulations .we then describe how the design of the simulations was refined using feedback from individual student volunteer sessions and in - class trials to ensure their educational effectiveness .we conclude with an outlook for future work .the best known quantum key distribution protocol is the so - called bb84 protocol , published by bennett and brassard in 1984.  in this protocol , alice sends bob a sequence of photons / particles in one of four different states chosen at random from two conjugate bases . for each photon/ particle , bob chooses one of the bases at random and makes a measurement in this basis .once all measurements have been made , alice and bob exchange their bases ( not their measurement outcomes ! ) and discard those measurements where their bases did not coincide . in order to test for an eavesdropper, they compare a randomly selected subset of their measurement outcomes ( which they then discard , as these are not anymore secure ) and check for errors . if the error rate is below a certain threshold , the unshared outcomes form the raw key , which can be processed further classically .  the _ _ quantum key distribution ( bb84 protocol ) using polarized photons__ simulation ( see fig .1 ) uses single photons and photon polarization as a physical realization of the bb84 protocol . at startup, the simulation shows an introduction view , with introductory text and a start button .the introductory text includes the learning goals , e.g. `` to help alice and bob decide whether or not they have generated a secure key .how can they tell that an eavesdropper eve has infiltrated their experiment ? '' .this introduction view is similar across all the qkd simulations . for the simulations using single photons , pressingthe start button shows an animation depicting virtual reality goggles that allows students to `` see '' the photons in the simulation .the photon visualization was developed in an earlier study.    a screenshot of the controls view of the bb84 simulation is shown in fig . 1 .as shown in the top graphics panel , alice is equipped with a single photon source and a polarizer .she sends bob a sequence of photons randomly polarized along either the horizontal , vertical , or directions , i.e. each in one of the quantum states , , or .the horizontal and vertical directions constitute the basis , the and directions the basis .the two bases are chosen such that the two states in one basis are an equal superposition of the two states in the other basis , e.g. and .bob is equipped with a polarization analyzer and a single photon detector . for each measurement, bob chooses one of the two bases at random and and measures the polarization of the photon . in the simulation, it is assumed that alice tells bob each time she sends a photon .bob either registers a photon in his detector or not , determining his outcome . for example, if bob orients his analyzer horizontally but does not detect a photon , his outcome is `` vertical '' ; if he does detect a photon , his outcome is horizontal " . in real experiments , polarizing beam splitters and two detectors would be used , so that each orientation would have its own detector and bob would detect the photon in both cases .if by chance bob chooses the same basis as alice and no eavesdropper is present , his measurement outcome will be identical to what alice sent , as the simulation assumes perfect sources and detectors .if by chance bob chooses a different basis to alice , either of his outcomes are equally likely , and his measurement has no information on the state that alice sent .for example , if alice sends and bob measures in the basis , he will measure , but if he measures in the basis , both outcomes and are equally likely .thus , whenever alice and bob use the same basis , they know that their measurements agree without needing to communicate these outcomes ! the measurement outcomes are turned into bits by assigning the vertical and orientations the value 1 , and the horizontal and orientations the value 0 .  as can be seen in fig . 1, the simulation shows the basis and outcome for each individual measurement for the last five measurements with the current measurement highlighted ( middle right panel ) , the most recent key bits ( middle bottom panel ) and the number of key bits and errors for all measurements taken for a given configuration ( right bottom panel ) .the most recent key bits panel only includes those measurements for which alice and bob used the same basis , as only these measurements are used for the key .  in the simulation, eve uses a simple intercept - resend attack where she intercepts each photon sent to bob , measures its polarization , and passes on a photon to bob .if random bases are used , eve orients her polarizer at random , and then passes on a photon with the polarization state she measured to bob .if eve by chance uses the same basis as alice , she passes on alice s state unchanged .if however , she used the incorrect basis , her measurement changes the quantum state , so that bob has a 50% chance of measuring either outcome .this latter case can lead to errors , e.g. if alice sends and bob measures and thus their bit values disagree even though they used the same basis . as eve guesses the incorrect basis 50% of the time , and 50% of these cases lead to an error , the presence of eve leads to 25% of the key bits being incorrect on average .  in order to test for an eavesdropper ,alice and bob compare a randomly selected subset of their measurement outcomes .they can detect the presence of an eavesdropper by errors in their measurements . if they do not find errors , they discard the exchanged bits , with the remaining bits forming the raw key . via the most recent key bits panel ( middle bottom panel in fig . 1 ), students can let alice and bob carry out a test for errors by comparing 20 bits .in the view in fig . 1 , such a test has been carried out and the compared bits have then been removed ( seen as the gaps in the middle bottom panel ) as they are not anymore secure .the controls view of all the qkd simulations is designed with implicit scaffolding to help students progress from simpler to more complex situations . at startup ,the `` fixed bases '' button in the top graphics panel is selected , so that only a single basis is used , and no eavesdropper is inserted .the tickboxes are all unticked , so that only the top graphics window , the display controls and the main controls are visible .this leads to students typically first exploring the simpler situation of a single basis before progressing to two bases .students can see that when eve is eavesdropping and only a single basis is used , eve knows the entire key despite there being no errors in alice s and bob s measurements .if students let alice and bob compare 20 bits for errors , the simulation gives feedback `` no errors , but fixed bases , so not secure '' .thus , the one - basis case helps students recognize why two bases rather than a single basis are needed for secure qkd .the quvis collection includes a second simulation demonstrating the bb84 protocol ._ quantum key distribution ( bb84 protocol ) with spin 1/2 particles__ simulation uses spin 1/2 particles and the component of spin as determined with a stern - gerlach apparatus as a physical realization of the bb84 protocol .the four polarization states above correspond to the spin states spin - up and spin - down along the vertical ( denoted as and ) , and spin - up and spin - down along the horizontal ( denoted as and ) .in 1992 bennett demonstrated that four states are more than are actually needed for qkd. qkd requires at least two states , so that eve is unable to unambiguously distinguish between them without introducing some errors in the key .however , bennett demonstrated that two states are also sufficient if they are not orthogonal e.g. using photon polarization states and .  the _ _ quantum key distribution using two non - orthogonal states__ simulation ( fig . 2 ) demonstrates this two - state protocol using single photons and photon polarization for the physical realization .similarly to the bb84 simulation , alice sends polarized photons to bob , who is equipped with a polarization analyzer and single photon detector .however , alice now uses two non - orthogonal polarization states rather than four polarization states . in the simulation , alice randomly prepares each photon with either ( horizontal ) or polarization .the horizontal polarization is assigned a bit value of 0 , the polarization a bit value of 1 .for each measurement , bob randomly sets his analyzer to one of two directions orthogonal to alice s directions , so either (vertical ) or .alice informs bob whenever she sends a photon . if bob detects the photon , he knows with certainty the polarization and hence the bit value ( 0 or 1 ) sent by alice .for example , if bob detects a photon when measuring along , he knows that alice sent a photon with polarization ( it can not have been the polarization ) and thus with bit value 1 .bob can therefore assign detections with a bit value of 1 , and detections with a bit value of 0 .if bob does not detect the photon , he can not be certain which state alice sent .for example , if bob measures along and did not detect a photon , alice could have sent either or .thus , alice and bob keep only those measurements where bob detected a photon .for example , if alice sends , then in 50% of cases bob measures along and does not detect the photon , and in 50% of cases he measures along , with a 50% chance of detection .thus , key bits are produced in the simulation only in 25% of cases .alice and bob publicly communicate to determine which photons were detected .this sequence of 0s and 1s forms the raw key . as for the other qkd protocols, they then exchange a small number of their bit values ( which they then discard as they are not anymore secure ) to check for errors .  in the simulation ,eve uses an intercept - resend attack where she uses the same orientations as bob for her polarization measurement , and the same orientations as alice for her photon source .if eve detects the photon , she knows what state alice sent and sends the same state as alice on to bob . in this case, errors do not occur .if eve does not detect a photon , she does not know which state alice sent .eve then sends the state that is orthogonal to her polarizer as this is the more likely case .if eve chooses the incorrect state to pass on to bob , errors can occur .for example , if alice sends and if eve measures along and does not detect a photon and hence sends , then bob can detect the photon in state , leading to an error .the error probability is 25% , the same as for the bb84 protocol .excepting the _ quantum key distribution with entangled spin 1/2 particles _ simulation discussed in section [ sectionbbm92 ] , all the qkd simulations include a second challenges tab with multiple challenges aligned with the learning goals .challenges can be solved in any order , and include a score counter and green ticks showing which challenges have already been solved .2 shows one of the challenges for the _ quantum key distribution using two non - orthogonal states _ simulation .this challenge asks students what sequence of detections and non - detections would be compatible with the given polarizations of alice and bob .other challenges ask students what sequence of key bits could have been measured given bob s outcomes , whether eve detected a photon and what state she sent on to bob for a particular measurement where alice and bob find an error , and whether eve s analyzer orientation can be known for a case where alice and bob do not find an error .in 1991 , ekert suggested a qkd protocol using entangled particle pairs rather than individual particles and measurements along three axes. this scheme was simplified by bennett , brassard and mermin in 1992 to measurements along two orthogonal axes. this simplified scheme is implemented in the simulation described here .the _ _ quantum key distribution with entangled spin 1/2 particles__ simulation uses a source that emits entangled spin 1/2 particle pairs ( see fig . 3 ) . in the simulation ,the two particles in the pair are emitted back - to - back each with opposite spin components .the particle pair is in the maximally entangled quantum state , where and correspond to spin - up and spin - down states along the vertical and the indices a and b stand for alice and bob .  as can be seen in the top panel of fig . 3 , alice and bobare each equipped with a stern - gerlach apparatus , which uses two magnets ( one of which is pointed ) to create a region of non - uniform magnetic field aligned along a given axis . for spin 1/2 particles ,the particles separate into two discrete streams , one deflected in the positive direction ( in the simulation defined as measurement outcome 1 ) , one deflected in the negative direction ( measurement outcome 0 ) along this axis .alice and bob orient their stern - gerlach apparatuses independently of one another at random along two orthogonal axes , denoted x and z.  for each particle pair , alice and bob note the basis ( x or z ) and measurement outcome ( 0 or 1 ) . due to the maximally entangled state used , they know that their measurement outcomes are perfectly anticorrelated ( if alice measures 1 , bob measures 0 and vice versa ) for those measurements where both stern - gerlach apparatuses happened to be oriented along the same axis .in contrast to a classical anticorrelation , this perfect anticorrelation of the entangled state holds both along the x and the z bases .this can be seen mathematically using the transformations and , where and correspond to spin - up and spin - down states along the horizontal respectively .if alice and bob use different bases ( e.g. x and z ) , then their outcomes are completely uncorrelated .thus , as for the bb84 protocol , alice and bob publicly share the bases used for each measurement but not their measurement outcomes , and keep only those outcomes for which their bases were the same .alice and bob then exchange a small number of their actual measurement outcomes ( which they then discard from the key as they are not anymore secure ) to check for errors . as for all the qkd simulations , students can insert an eavesdropper eve who intercepts the particles sent to bob ( shown in fig . 3 ) .for the `` random orientations '' setting , eve measures the particle s spin component in the same way as bob using a stern - gerlach apparatus oriented at random along x or z. she then sends a particle on to bob with the spin state she measured , e.g. if eve measures spin - up along z , she sends a particle in the state to bob .errors occur in 50% of cases where eve by chance chooses the wrong basis , i.e. a different one to alice and bob . as alice and eve have used different bases , their outcomes are completely uncorrelated .eve s measurement changes the spin state of the particle .eve then passes on a particle to bob that has equal probabilities for outcomes 0 and 1 in his basis .as eve guesses the incorrect basis 50% of the time , and 50% of these cases lead to an error , the quantum bit error rate is 25% , the same as for the bb84 and two - state protocols .  fig .3 shows the setup with random orientations and eve inserted . in the middle right key panel, one can see measurements highlighted where eve has chosen the wrong basis .one can see that errors occur only sometimes , not always , if eve used a different basis to alice and bob ( compare lines 2 , 3 and 5 of this panel ) . in the middle bottom panel , a test for errors has just been carried out .the 20 bits that were compared are highlighted , with errors highlighted in red .the errors indicate the presence of an eavesdropper , so that the entire key needs to be discarded .we developed an initial version of the _ _ quantum key distribution with entangled spin 1/2 particle pairs__ simulation ( coded in adobe flash ) in 2013 .after development , we ran a small number of individual student volunteer sessions where students interacted with the simulation freely while thinking aloud and then worked on the associated activity .this led to minor revisions of the simulation such as the inclusion of a fast forward button , adding images of alice , bob and eve , and minor revisions to texts and layouts .after incorporation of these revisions , this version was used in 2014 at the university of st andrews in an introductory quantum physics course ( similar to a us modern physics course ) and in an advanced quantum mechanics course taken by students in their third or fourth year of study .table [ datatable ] summarizes data collection from 2014 and 2016 ( the latter described below ) .the 2014 version of the simulation had two differences to the current version described in section [ sectionbbm92 ] and used in the 2016 trial : instead of the middle bottom panel showing recent key bits and allowing a test to compare for errors ( see fig .3 ) , this simulation showed a graph of the error probability versus the number of key bits .this graph showed how the error probability converges to 0.25 in the limit of a large number of measurements , if random orientations are chosen and the eavesdropper is inserted .the other difference was in the key bits panel , which did not highlight measurements where eve chose the wrong basis ( the turquoise highlighting of the middle right panel in fig .in what follows , we motivate why these two changes were incorporated into all the qkd simulations described in sections [ sectionbb84 ] , [ sectionb92 ] and [ sectionbbm92 ] in light of the 2014 results ..evaluation carried out with two versions of the _ quantum key distribution with entangled spin 1/2 particle pairs _ simulation . [cols="<,^,^,^,^",options="header " , ]   [ outcometable ]  the post - test questions are given in the appendix .results for both years for the introductory level are shown in fig .the results show that the majority of students are achieving the learning outcomes in terms of understanding which measurements are used for the key and how errors come about in the presence of an eavesdropper .the results for questions 1 and 3 are the same within errors across both years .question 2 is answered significantly better in 2016 than in 2014 ( u=2127.5 , n1=82 , n2=60 , p=0.02 ) .the post - test results for the 2014 advanced level are not shown in fig .they were 100% , 92.9% and 85.7% respectively for the three questions ( n=14 ) , but students likely had substantial prior knowledge so that the outcomes do not relate to learning from the simulation alone .  in summary , these evaluation results point to the simulation being effective in helping students make sense of qkd at both the introductory and advanced levels .they point to the revisions incorporated into the qkd simulation and activity being successful in improving students understanding .qkd is a relevant and useful topic for inclusion in quantum mechanics courses at both the introductory and advanced levels , in terms of limited prior quantum mechanics knowledge needed , the rapidly growing field of quantum information technology and the link to motivating real - world applications .we have described four interactive simulations that support the learning and teaching of qkd .the simulations demonstrate the basic principles of qkd using three different protocols , and using either polarized single photons or spin 1/2 particles as physical realizations of qubits .the simulations have been shown in preliminary studies to be effective at helping students learn about qkd both at the introductory and the advanced undergraduate levels .the simulations described aim to only demonstrate the basic principles of qkd , and thus show idealized and simplified situations , e.g. assuming perfect sources and detectors so that no errors occur without the presence of an eavesdropper .the simulations also assume that eve uses a simple intercept - resend strategy and listens in on each bit .future work will aim to develop simulations that include imperfect detectors and other possible eavesdropping strategies , such as eve only intercepting a fraction of particles . due to instrumental imperfections, some errors would always be present .if the error rate is not too large , alice and bob can use classical post - processing to create a secure key. this consists of error - correction methods to reduce errors between their two raw keys and privacy amplification to reduce eve s information on alice s bits , both at the expense of reducing the length of the key . future work will aim to include in the simulations errors due to instrumental imperfections and classical post - processing .this will allow the simulations to demonstrate how a secure key can be generated even in the presence of some errors .consider secure key generation with entangled spin 1/2 particle pairs as shown in the simulation .the two particles in the pair are described by a single wavefunction .assume observers alice and bob each have a stern - gerlach apparatus which they can orient independently and randomly along two orthogonal axes , x and z.  alice and bob make measurements on 5 pairs in order to generate a secure key .bob uses the orientations x x z x z ( in this order ) for his measurements and obtains the results 01100 ( a deflection in the positive direction is 1 , and in the negative direction 0 ) .alice tells him he measured the first , second and last in the `` correct '' orientation ( so in the same orientation as alice ) .we thank the uk institute of physics and the university of st andrews for funding the simulation development .we thank martynas prokopas for coding the 2014 version of the quantum key distribution simulation .we thank holly scott - riddell and mark paetkau for work on the data analysis .we thank seth devore for suggestions for improvement of the _ quantum key distribution using two non - orthogonal states _ simulation .we thank charles baily and natalia korolkova for implementing the quantum key distribution simulation in their courses .we are grateful to all students taking part in evaluation studies and providing feedback on simulations .k. yoshino , t. ochi , m. fujiwara , m. sasaki and a. tajima , `` maintenance - free operation of wdm quantum key distribution system through a field fiber over 30 days , '' opt .express * 21 * ( 25 ) , 3139531401 ( 2013 ) .a. kohnle , i. bozhinova , d. browne , m. everitt , a. fomins , p. kok , g. kulaitis , m. prokopas , d. raine and e. swinbank , `` a new introductory quantum mechanics curriculum , '' eur . j. phys .* 35 * ( 1 ) , 015001 - 1 to 9 ( 2014 ) .b. mason et al ., `` report and recommendations on multimedia materials for teaching and learning quantum physics , '' 2014 girep - mptl proceedings , [ palermo , italy , july 7 - 12 , 2014 ] edited by c. fazio and r. m. s. mineo , 647655 .s. devore and c. singh , `` development of an interactive tutorial on quantum key distribution , '' 2014 perc proceedings , [ minneapolis , mn july 30 - 31 , 2014 ] edited by p. v. engelhardt , a. d. churukian and d. l. jones , 5962    a. kohnle , c. baily and s. ruby , `` investigating the influence of visualization on student understanding of quantum superposition , '' 2014 perc proceedings , [ minneapolis , mn july 30 - 31 , 2014 ] edited by p. v. engelhardt , a. d. churukian and d. l. jones , 139142  < http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/cryptography-bb84/quantum_cryptography.html>[<http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/cryptography-bb84/quantum_cryptography.html > ] .< http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/cryptography-b92/b92_photons.html > [ < http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/cryptography-b92/b92_photons.html > ] .< http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/cryptography/quantum_cryptography.html > [ < http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/cryptography/quantum_cryptography.html > ] .< http://www.st-andrews.ac.uk/physics/quvis/simulations_twolev/iop%20-%20quantum%20cryptography%20with%20spin-half%20particles%20v3%20rev.html>[<http://www.st-andrews.ac.uk/physics/quvis/simulations_twolev/iop%20-%20quantum%20cryptography%20with%20spin-half%20particles%20v3%20rev.html > ] .

Abstract: secure communication protocols are becoming increasingly important , e.g. for internet - based communication . quantum key distribution allows two parties , commonly called alice and bob , to generate a secret sequence of 0s and 1s called a key that is only known to themselves . classically , alice and bob could never be certain that their communication was not compromised by a malicious eavesdropper . quantum mechanics however makes secure communication possible . the fundamental principle of quantum mechanics that taking a measurement perturbs the system ( unless the measurement is compatible with the quantum state ) also applies to an eavesdropper . using appropriate protocols to create the key , alice and bob can detect the presence of an eavesdropper by errors in their measurements .  as part of the quvis quantum mechanics visualization project , we have developed a suite of four interactive simulations that demonstrate the basic principles of three different quantum key distribution protocols . the simulations use either polarized photons or spin 1/2 particles as physical realizations . the simulations and accompanying activities are freely available for use online or download , and run on a wide range of devices including tablets and pcs . evaluation with students over three years was used to refine the simulations and activities . preliminary studies show that the refined simulations and activities help students learn the basic principles of qkd at both the introductory and advanced undergraduate levels .


quantum key distribution ( qkd ) allows two remote parties , alice and bob , to generate a secret shared string of bits , that can be used for symmetric cryptography or other cryptographic protocols . unlike classical key distribution, whose security is based on the computational difficulty of solving certain classes of problems , the security of quantum key distribution comes from the impossibility for an eavesdropper ( eve ) to acquire information about an exchanged state without perturbing it .this goal is obtained by using non - orthogonal states , as in the first qkd protocol , introduced by bennett and brassard in 1984 ( bb84 ) , that uses four different states of two mutually unbiased bases . in 1992 , bennett showed that two non - orthogonal states are sufficient for qkd .this protocol , however , has the drawback that the secure key rate is strongly affected by losses : indeed , eve can extract information by increasing the losses and performing the so called unambiguous state discrimination ( usd ) attack  .the addition of a third state is sufficient to make the b92 protocol unconditionally secure independently from the noise in the quantum channel : however , rates comparable to the bb84 protocols can be obtained only when four states are detected at the receiver .  the optimal three - state qkd protocol , introduced in 2000 by phoenix - barnett - chefles ( pbc00 )  , uses states that form an equilateral triangle in the x - z plane of the bloch sphere .the symmetry of this protocol can indeed be exploited to obtain rates comparable with the bb84 protocol but requiring only three detectors instead of four .this protocol , however , still requires the public exchange of part of the sifted key in order to estimate the qber .an improvement of this protocol , introduced by renes in 2004 ( r04 )  , estimates the error rate from the number of inconclusive events , thus allowing to use all conclusive ones for key extraction .the unconditional security of the pbc00 has been demonstrated , in the asymptotic case , in 2005 , for a bit error rate of up to {\%} ] is collected into a polarization maintaining single mode fiber ( pm - smf ) for spatial mode filtering .the polarization is adjusted by a pbs followed by a hwp and the beam is focused by a {mm} ] periodically poled ktp crystal in a polarization - based sagnac interferometer .the source is pumped with a continuous wave ( cw ) laser at {nm} ] .the down - converted photons have a central wavelength of {nm} ] full width at half maximum ( fwhm ) , and are collected into single - mode fibers . in this configuration , the setup has a mean coincidence rate of {khz} ] is the coherence time of down - converted photons and {ns} ] for the vertical polarization , followed by a hwp at and a polarizing beam - splitter ( pbs ) .given an arbitrary input state , with , the ppbs routes it to detector with probability .the state at the transmitting output of the ppbs , , is transformed into by the hwp .thus , the probabilities of a click at detectors and are , respectively , and .it is easy to check that the above probabilities can be also written as , with : then , the above described apparatus implements the povm .photons are detected using silicon single photon counting modules ( spcm ) , characterized by a dead time of {ns} ] .a two hour continuous run of the apparatus has led to the exchange of about symbols within a coincidence window of {ns} ] , with a contribution of {db} ] each , with a mean number of sifted bits .the qber is estimated from inconclusive events , with poissonian error bars .the secret key rate is estimated for the case of infinitely long key , using equation ( [ eq : asymp_r ] ) . ]the qber is estimated as , where is the fraction of inconclusive results .the qber remains almost constant at a level below for all the acquisition , thus confirming the stability of both the source and the povm during the acquisition .the visibility of the source in two mutually unbiased bases at the exit of the sagnac interferometer , before fiber injection , has been measured to be between and : then the measured qber level can be attributed almost completely to the source . a small contribution to the qber is due to the small imbalances between the different channels of the povm .these values , estimated by computing the ratio between the raw counts at the different detectors , vary between and , in line with what observed in the previous implementation of the povm .  in a real scenario , the number of exchanged signals is always finite and the security analysis must take this fact into account .the finite key analysis of the r04 protocol is very similar to the one of the pbc00 , the only substantial difference lying in the estimation of the bit error rate .  .]  the pbc00 estimates it by comparing part of the sifted key through the public channel , while the r04 use the fraction of inconclusive results .the method is based on the fact that the choice between `` good '' conclusive and inconclusive results is given by a random event at alice s side after the exchange of all the qubits , therefore eve has no way of differentiate between the two and their number is approximately equal . defining the fraction of inconclusive results , the fraction of `` good '' conclusive ones can be written as . using the hoeffding bound , the inequality is valid with probability at least implies that , with the same probability , the qber is less than .the secret key fraction , in the case of collective attacks , is  - 7 \sqrt{\frac{1}{n } \log_2{\frac{2}{\bar{\epsilon } } } } \nonumber \\           & - \frac{1}{n}\log_2{\frac{1}{\epsilon_{ec } } } - \frac{1}{n}\log_2{\frac{2}{\epsilon_{pa } } } - f_{ec } h(q )         \label{eq : r_finite}\end{aligned}\ ] ] for exchanged signals , with security parameter .this result can be extended to the case of general attacks by exploiting the postselection technique , giving  with security parameter .figure [ fig : finite ] shows the secret key fraction of the r04 protocol in the finite key scenario . for each point , both the qber and the number of conclusive events is evaluated on the first exchanged symbols .the secret key fraction is calculated by using equations ( [ eq : r_finite ] ) and ( [ eq : r_finite_gen ] ) . for collective attacks ,the chosen security parameter is , with .the same value has been chosen for , therefore the term of equation ( [ eq : r_finite_gen ] ) is calculated using .the plots show that at least - signals are necessary to exchange a key , while already ( slightly more than half a minute at {khz}$ ] ) gives a reasonable key fraction .the difference between the key fraction for collective and general attacks is more marked for lower values of and tends to disappear for large number of exchanged symbols , where both approach the asymptotic key fraction .in this work , we demonstrated the experimental feasibility of the equiangular three state qkd protocol r04 .we showed that the scheme proposed by for the povm is suitable for applications in quantum key distribution .state preparation was simplified by using an entanglement - based version of the protocol , with the same povm at both alice s and bob s side .we also showed that the estimation of the bit error rate from inconclusive results is feasible in the finite key scenario .the implemented scheme was demonstrated to be stable and highly reliable , allowing a two hour data acquisition without any significative change in the qber value .the performance of the protocol is comparable with the bb84 , despite the less efficient parameter estimation , both in the asymptotic limit and for finite key .its simpler receiving apparatus , requiring only three single photon detectors , makes it a valid alternative to current implementations of qkd based on the bb84 protocol .finally , this work extends the experimental investigation of equiangular spherical codes to a still uncovered area of quantum information : quantum key distribution .we thank davide g. marangon and nicola laurenti for helpful discussions .this work has been carried out within the strategic - research - project quantumfuture of the university of padova and the strategic - research - project quintet of the department of information engineering , university of padova .m.s . acknowledges the ministry of research and the center of studies and activities for space ( cisas ) `` giuseppe colombo '' for financial support .

Abstract: quantum key distribution using three states in equiangular configuration combines a security threshold comparable with the one of the bennett - brassard 1984 protocol and a quantum bit error rate ( qber ) estimation that does not need to reveal part of the key . we implement an entanglement - based version of the renes 2004 protocol , using only passive optic elements in a linear scheme for the measurement povm , generating an asymptotic secure key rate of more than {kbit / s}$ ] , with a mean qber of . we then demonstrate its security in the case of finite key and evaluate the key rate for both collective and general attacks .


the duplication - divergence mechanism of network growth is traditionally used to model protein networks : a duplication of a node is a consequence of the duplication of the corresponding gene , and a divergence or loss of redundant links or functions is a consequence of gene mutations .general properties of the duplication - divergence growth have recently been studied for probably the simplest version of the duplication - divergence model which is the asymmetric divergence . yeteven this simplest model , where links are removed with a certain probability only from the replica node , turned out to have very rich phenomenology and to reproduce the degree distribution , observed in real protein - protein networks , surprisingly well .overall , when the link removal probability is small , the network growth is not self - averaging and an average vertex degree is increasing algebraically . for larger values of the link removal probability ,the growth is self - averaging , the average degree increases very slowly or tends to a constant , and a degree distribution has a power - law tail .a natural next step in exploring properties of the duplication - divergence networks is to consider their modular structure and distribution of various subgraphs or motifs .small subgraphs are often considered building blocks of network ; densities of particular subgraphs may tell if a network belongs to a certain superfamily or performs specific functions .abundances of triangles and loops have been studied in the internet , random and preferential attachment networks and regular scale - free graphs .densities of small motifs and cycles centered on a vertex were considered as a function of the vertex degree and clustering coefficient in . in protein - protein networks ,highly interconnected subgraphs were found to be well - conserved in evolution and to correspond to functional protein modules in living cells .an extreme case of highly interconnected motifs are cliques , or completely connected subgraphs .cliques have been found in higher than random abundances in protein - protein networks in yeast .  in the paperwe consider a generalization of duplication - divergence network growth mechanism , duplication - divergence - heterodimerization .the heterodimerization , or linking a certain number of the pairs of target and replica nodes , is essential for clustering and is observed in protein - protein networks .we show that duplication - divergence - heterodimerization produces the cliques in the number very similar to those observed in protein - protein networks .  as in our previous work , we again start with the simplest case of the completely asymmetric divergence . yet in real protein networks , apart from special cases of partially asymmetric divergence , the divergence is believed to be close to symmetric .it turns out that the asymmetric divergence results for the clique statistics as well as the previously obtained results for the network growth are qualitatively similar to those in the arbitrary divergence case , where links are removed with given probabilities both from the target and replica nodes .the paper consists of two principle parts : in the next section we derive the clique abundance distribution for the asymmetric case and compare it to the simulation and experimental results . in sec .iii we generalize these and previously obtained results for the network growth and degree distribution onto the arbitrary divergence case . a discussion and conclusion section completes the paper .protein - protein networks exhibit a distinct modular structure and contain densely linked neighborhoods or complexes ( and references therein ) .the extreme case of densely linked complexes are cliques or completely connected sub - graphs where each vertex is connected to all other subset members . cliques of the sizes of up to 14 vertices were found in much higher than `` random '' abundance in protein binding network of yeast . yet many large cliques observed in protein networks may be artifacts of specific experimental techniques or even of misinterpretation of the experimental data .for example , there is a strong evidence that all cliques of order higher than six in the yeast interaction network considered in result from the `` matrix '' recording of the experimental data from the mass - spectrometry experiments . in such experiments an immunoprecipitation is used to isolate stable protein complexes .usually a single protein is used as a target for the antibody ; binding of the antibody to this protein leads to an isolation of the the entire complex .however , the precise pairwise binding between proteins in the complexes strictly speaking remains undefined if a complex contains more than two proteins . yet in the matrix interpretation of the mass - spectrometry experiment all possible pairwise interactions between proteins in the complex are usually recorded .a well - known example of such erroneous recording is the anaphase - promoting complex .it is reported as a 11-node clique in three different mass - spectrometry high - throughput interaction surveys of yeast genome and in the mips database .the biggest reported clique in yeast network , saga / tfiid complex , is also the result of erroneous `` matrix '' recording of the data from a co - immunoprecipitation experiment described in .however , a virtually free of subjective interference two - hybrid method , used to determine the protein binding network of fruit fly , yields also higher than random number of cliques .specifically , the fly dataset contains 1405 triads , 35 4-cliques and one 5-clique , while a randomly re - wired graph of fly dataset contains only 1147 triads and 8 4-cliques . here and below , the lower - oder cliques that comprise the higher - order ones ( each clique with vertices or -clique consists of cliques with vertices which can be obtained by eliminating one of the vertices ) are counted along with the non - trivial cliques .the number of only non - trivial cliques is slightly lower ; the fly dataset contains 1297 non - trivial triads , 30 4-cliques and one 5-clique .is such high concentration of large cliques caused by an evolutionary pressure that specifically favors big cliques , or by some stochastic mechanism of network evolution ?evidently , a simple duplication - divergence network growth never produces even a single triad as new duplicates are never linked to their ancestors .random mutations , or re - wiring of some links will give rise to a certain number of cliques , yet their abundance will be much less than the experimentally observed one .however , in it was concluded that links between paralogs ( or recently duplicated pairs of proteins ) are significantly more common than if such links appeared by random mutations .most of these paralogous links are formed when a self - interacting protein or ( homodimer ) is duplicated , thus giving rise to a pair of interacting heterodimers . while after divergence certain pairs of heterodimers loose their ability to interact , some paralogs retain their propensity to heterodimerize . in the followingwe show that the simple duplication - divergence network growth complimented with heterodimerization of some pairs of duplicates does explain the observed abundance of cliques without invoking any evolutionary pressure .the duplication - divergence - heterodimerization process consists of duplication - divergence , previously introduced in ,  * * duplication*. a randomly chosen target node is duplicated , that is its replica is introduced and connected to each neighbor of the target node . ** divergence*. each link emanating from the replica is activated with probability ( this mimics link disappearance during divergence ) .  andheterodimerization ,  * * heterodimerization*. the target and replica nodes are linked with probability .it mimics the probability that the target node is a dimer and the propensity for dimerization is preserved during divergence .similarly to the pure duplication - divergence growth , the replica is preserved if at least one link is established ; otherwise the attempt is considered as a failure and the network does not change .let us first consider an evolution of population of triads , or 3-cliques .two processes that give rise to new triads are illustrated in figs .[ triada],[triadb ] . during the first process a target vertex 1 , initially linked to the vertices 2 and 3 ,is duplicated to produce a new vertex 4 .the resulting pair of duplicates 1 and 4 have a probability to be linked .in addition , links 4 - 2 and 4 - 3 are inherited with the probability each . as a result of this process , two new triads 1 - 4 - 2 and 1 - 4 - 3 are formed , each with probability .   a sketch of duplication event when a new triad is formed with a heterodimerization link .solid lines correspond to the existing links , dotted line is a heterodimerization link , established with the probability , and dashed lines denote the inherited with probability links . ]   a sketch of duplication event when a new triad is formed by duplicating the existing one .solid lines correspond to the existing links and dashed lines denote the links , each inherited with the probability . ]  in the second process ( fig .[ triadb ] ) a new triad is produced from the existing one when one of its vertices ( vertex 1 ) is duplicated .the new triad is formed only if both links , 4 - 2 and 4 - 3 survive divergence , which happens with the probability .correspondingly , a rate equation for the increase in the number of triads per duplication - divergence - heterodimerization step contains two terms , where and are the numbers of links and vertices in the network .the fraction in the first term is an average number of links picked up for a potential triad ( which is also equal to the average degree ) .the factor 3 in the second term indicates that each of the three vertices in the existing triad can be picked up as a target vertex for duplication .considering links as 2-cliques , the first term in eq .( [ tri ] ) can be interpreted as describing a creation of 3-clique from a lower - order 2-clique .it is easy to see that with such interpretation , the eq .( [ tri ] ) can be generalized to describe the evolution of population of cliques of an arbitrary order , here is an increment in the number of vertices per duplication step . in the following we focus on a biologically - relevant regime of where the average degree is constant or almost constant . in this regime , and assuming scaling for , , one obtains a recurrent relation for the rescaled -clique abundance , for large the second term in denominator becomes subdominant , it follows that the relative population of large cliques decays faster than exponentially .this rends large cliques highly improbable in networks of biologically relevant size of .  to check this analytical prediction and to see if the proposed duplication - divergence - heterodimerization model explains the observed population of cliques , we performed the following numerical simulation .as in , we fix so that the average degree is equal to that of the fly dataset , where for proteins . we select so that the number of triads in the simulated network is also similar to that in the fly dataset and count the number of 4- and 5-cliques in the resulting network .the theoretical are computed for the same and taking into account that .results of simulations averaged over 2000 network realizations , the computed , and the clique abundances in the fly dataset are shown in table [ tab_cliques ] ..[tab_cliques ] number of -cliques in networks with vertices and links for  fruit fly protein - protein binding network ,  simulations with and , and  eq .( [ cj ] ) prediction for the same and . [ cols="^,^,^,^",options="header " , ]   for symmetric divergence , we measure the number of components and the size of the largest component for the networks grown with various .the results for the networks of the size of fruit fly dataset , , are presented in table [ tab_comp ] .it follows that for the grown network consists of many fairly small components , while for there is usually one or few large components and several small ones .intuitively it is clear that if the average degree grows , even slowly , the probability to split the network into many parts becomes small .  a theoretical prediction for the size of the giant component exists only for the erds - rnyi random graph : when the average degree scales logarithmically with the number of vertices , i.e. , , the total number of vertices that do not belong to the giant component scales as for , while for the giant component engulfs the entire system .it turns out that for the same number of vertices and links , the completely random linking of the erds - rnyi graph keeps more vertices in a giant component than the corresponding duplication - symmetric divergence network . indeed , for the parameters corresponding to the fly dataset , , , , and , the number of vertices not belonging to the giant component is ( see table [ tab_comp ] ) .yet the erds - rnyi graph with the same number of vertices and links has only vertices outside of its giant component .this happens mainly because in our duplication - divergence growth model , once a component is split from the giant component , it never re - connects .if such separation happens at an early stage of the network growth , the separated component may grow to a significant size , thus leaving many vertices outside of the giant component . on contrary , at each step of the erds - rnyi growth , any two components can be united with a random link .this makes the co - existence of two or more large components very unprobable .in the previous sections the following conclusions on the clique abundances and growth laws of the duplication - divergence - heterodimerization networks have been made :  * we showed that the duplication - divergence network growth model , complimented with heterodimerization links between duplicates , correctly describes the statistics of cliques in biologically observed protein - protein networks .we derive an expression for clique population distribution that correctly describes the clique abundances in the duplication - divergence - heterodimerization networks .* generalizing the results obtained for the completely asymmetric divergence in , we demonstrated that similar regimes , such as presence and lack of self - averaging , growth and saturation of the average degree , scaling and fat tail in the degree distribution , exist in general duplication - divergence case as well .in addition , a clique density distribution is generalized onto the arbitrary divergence scenario .the heterodimerization links are not taken into account in our description of the network growth and degree distribution . despite their crucial role in the network topology and clique formation ,they constitute only about 1% of all links and do not contribute significantly to the degrees of the most of the vertices . for link inheritance and heterodimerization probabilities and , corresponding to the fly dataset , the resulting number of heterodimeric links in a network of the size of the fly dataset is .this is somewhat higher than the observed number of links between the pairs of recently duplicated ( paralogous ) proteins  .the main reason for this discrepancy is that in our simulation all heterodimeric links are counted , while in the real protein network one can reliably identify only the pairs of recently duplicated proteins .  in a case of not completely asymmetric divergence when links can disappear both from the target and replica nodes , a network may fragment into several components . yetthe biological protein networks are believed to be connected to ensure their functionality .hence during _ in vivo _ divergence the steps that lead to breaking the network into isolated components are excluded due to evolutionary pressure .our probabilistic network growth model does not take any evolutionary pressure into account .however , since for sufficiently high link retention probabilities the resulting network consists of one or very few large components , the number of link eliminations that have to be evolutionally overridden is small. hence most of the properties of the probabilistically grown graphs should be similar to those of the realistic evolutionary single - component networks .as the link inheritance probabilities decrease and the number of network components grow , the number of link removal steps that have to be evolutionally overridden becomes large .consequently , the probabilistic multi - component network becomes less similar to the real single - components one .  as we mentioned in the section ii , we selected the fly dataset as an example as being the most non - subjective one . other knowprotein - protein networks , such as for yeast , worm , and human , do contain parts of data that are results of the `` matrix '' recording of the experimental data from the immunoprecipitation experiments .these datasets contain a higher number of large cliques which can be attributed to this data interpretation . in principle, the clique population distribution derived here can be used to verify and filter the experimental datasets , removing the erroneously recordered large cliques .  in a recent publication , middendorfet al compared topological properties of the fly dataset to those of the networks grown by several mechanisms such as different versions of duplication - mutation model and preferential attachment .it was found that a duplication - mutation - complementation network provides the best fit to the fly dataset .the duplication - mutation - complementation network growth model is very close to the duplication - divergence - heterodimerization model studies here .complementation is equivalent to heterodimerization , the only difference between two models is in the way the links are deleted during divergence ( or mutation ) : unlike our model , in each neighbor remains connected to at least one of the two duplicates .thus we confirmed the conclusions made in that practically all considered properties of protein - protein networks are very well described by the duplication - divergence - heterodimerization model .  and finally a few words on the importance of heterodimerization links in clique formation .an alternative to heterodimerization way to connect paralogs is to link them randomly by mutation links . in this casethe probability to establish a heterodimeric link has to be replaced by a probability that a mutation link , emanating from a target node , selects the replica node out of network nodes .this probability is equal to where is the number of mutation links established at each duplication step . in the example of the fruit fly dataset where and , one needs random links at each step to form the correct number of triads and higher cliques .obviously , the mutation scenario which requires so many additional links is completely ruled out due to , for example , average degree constraint .we are thankful to s. maslov for interesting discussions and bringing the reference to our attention .this work was supported by 1 r01 gm068954 - 01 grant from nigms .                         a procedure , described in s. maslov and k. sneppen , _ science _ , * 296 * , 910 ( 2002 ) , randomizes a graph by re - wiring links without altering degrees of its verteces .when the fly data set is randomized according to this procudure , the number of cliques significantly decreases ; specifically , there are at average 1147 triads , 8 4-cliques and no higher cliques after randomiztion .s. janson , t. luczak , and a. rucinski , _ random graphs _( wiley , new york , 2000 ) . m. middendorf , e. ziv , and c. wiggins , q-bio.qm/0408010 .

Abstract: a population of complete subgraphs or cliques in a network evolving via duplication - divergence is considered . we find that a number of cliques of each size scales linearly with the size of the network . we also derive a clique population distribution that is in perfect agreement with both the simulation results and the clique statistic of the protein - protein binding network of the fruit fly . in addition , we show that such features as fat - tail degree distribution , various rates of average degree growth and non - averaging , revealed recently for only the particular case of a completely asymmetric divergence , are present in a general case of arbitrary divergence .