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Ann. Phys. (Berlin) 524, No. 8, 393–410 (2012) / DOI 10.1002/andp.201100299Review

Article

Event-by-event simulation of quantum phenomena

Hans De Raedt1,∗ and Kristel Michielsen2,3,∗∗

Received 9 November 2011, revised 7 July 2012, accepted 9 July 2012Published online 23 July 2012

A discrete-event simulation approach is reviewed that doesnot require the knowledge of the solution of the waveequation of the whole system, yet reproduces the statis-tical distributions of wave theory by generating detectionevents one-by-one. The simulation approach is illustratedby applications to a two-beam interference experimentand two Bell test experiments, an Einstein-Podolsky-Rosen-Bohm experiment with single photons employing post-selection for pair identification and a single-neutron Belltest interferometry experiment with nearly 100% detectionefficiency.

1 Introduction

1.1 Quantum theory and the observation of single events

Quantum theory has proven to be extraordinarily power-ful for describing the statistical properties of a vast num-ber of laboratory experiments. Starting from the axiomsof quantum theory it is, at least conceptually, straightfor-ward to calculate numbers that can be compared with ex-perimental data as long as these numbers refer to statis-tical averages. However, if an experiment records individ-ual clicks of a detector a fundamental problem appears.Although quantum theory provides a recipe to computethe frequencies for observing events it does not accountfor the observation of the individual events themselves, amanifestation of the quantum measurement problem [1,2]. A recent review of various approaches to the quantummeasurement problem and an explanation of it withinthe statistical interpretation is given in [3]. From the view-point of quantum theory, the central issue is how it canbe that experiments yield definite answers. As stated byLeggett [4], “In the final analysis, physics cannot forever

refuse to give an account of how it is that we obtain def-inite results whenever we do a particular measurement”.In fact, a common feature of all probabilistic models isthat they do not entail a procedure that specifies howparticular values of the random variables are being real-ized, but only contain a specification of the probabilitieswith which these values appear. As a probabilistic theory,quantum theory takes a special position in that it postu-lates that it is fundamentally impossible to go beyond thedescription in terms of probability distributions. At thepresent time, there is no scientific evidence that supportsthis assumption other than that it seems unsurmount-able to account for the observation of the individualevents within the context of quantum theory proper [1,2].This suggests that the search for a cause-and-effect de-scription of the observed phenomena should be doneoutside the realm of quantum theory.

1.2 From single events to probability distributions andnot vice versa

In general, the problem can be posed as follows: Given aprobability distribution of observing events, can we con-struct an algorithm which runs on a digital computerand produces events with frequencies that agree withthe given distribution without the algorithm referring, inany way, to the probability distribution itself. An affirma-tive answer to this primary question can give rise to sec-

∗ Corresponding author E-mail: [email protected],Phone: +31 50 3634950, Fax: +31 50 3634947

∗∗E-mail: [email protected] Department of Applied Physics, Zernike Institute for Advanced Ma-terials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The Netherlands

2 Institute for Advanced Simulation, Jülich Supercomputing Centre,Forschungszentrum Jülich, 52425 Jülich, Germany

3 RWTH Aachen University, 52056 Aachen, Germany

© 2012 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 393

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H. De Raedt and K. Michielsen: Event-by-event simulation of quantum phenomena

ondary questions such as “Can this be done efficiently?”,“Is the algorithm unique?”, and so on.

There are several elements in the statement-of-the-problem that deserve attention. The first is that we use“event” in the every-day meaning of the word. Thus, anevent can be a click of a detector, the experimenter push-ing a button, and so on. Second, a digital computer isnothing but a macroscopic physical device that evolvesin time by changing its state in a well-defined, cause-and-effect, discrete-event manner, as specified by the algo-rithm. Although not practical, it is possible to build a me-chanical apparatus that performs exactly the same func-tion as the digital computer. Thus, one can view a simu-lation on a digital computer as a perfectly controlled ex-periment on a macroscopic mechanical system, simulat-ing, in a cause-and-effect manner, the phenomenon thatis being observed in the laboratory. Third, and most im-portantly, the algorithm should not rely in any way onthe probability distribution of the events that it is sup-posed to generate. Otherwise it would be straightforwardto use pseudo-random numbers and generate events ac-cording to this distribution. However, this is not a solu-tion to the posed problem as it assumes that the proba-bility distribution of the quantum mechanical problemis known, which is exactly the knowledge that we wantto generate without making reference to quantum theory.Put differently, the algorithm should be capable of gener-ating events according to an unknown probability distri-bution.

In order to clarify our aim we draw an analogy [5]with the Metropolis Monte Carlo method for simulatingclassical statistical mechanics [6], a primary example ofa method that samples from an unknown probability dis-tribution. According to the theory of equilibrium statis-tical mechanics, the probability that a system is in thestate with label n is given by pn = e−βEn /

∑Nn=1 e−βEn ≡

e−βEn /Z , where N is the number of different states of thesystem, which usually is very large, En is the energy ofthe state, and β = 1/kB T where kB is Boltzmann’s con-stant and T is the temperature. Disregarding exceptionalcases such as the two-dimensional Ising model, for anontrivial many-body system the partition function Z isunknown. Hence, pn is not known. We can now pose thequestion "Can we construct a simulation algorithm thatgenerates states according to the unknown probabilitydistribution (p1, . . . , pN )?" As already mentioned, an affir-mative answer to this question was given a long time agoby Metropolis et al. [6]. The basic idea is to construct a dy-namical system, a Markov chain or master equation thatsamples the space of N states such that in the long run,the frequency with which this system visits the state n ap-proaches pn with probability one.

1.3 The event-based simulation approach

The basic ideas of the simulation approach that we re-view in the present paper are that (i) we stick to what weknow about the experiment, that is we consider the ex-perimental configuration, its outcome and its data anal-ysis procedure as input for constructing the simulationalgorithm; (ii) we try to invent a set of simple rules thatgenerates the same type of data as those recorded in anexperiment while reproducing the averages predicted byquantum theory; (iii) we keep compatibility with macro-scopic concepts. Our event-based simulation approachis unconventional in that it does not require knowledgeof the wave amplitudes obtained by first solving a wavemechanical problem. Evidently, as outlined below, ourevent-based approach requires a departure from the tra-ditional way of describing physical phenomena, namelyin terms of locally causal, modular, adaptive, classical(non-Hamiltonian) dynamical systems.

Our event-based approach has successfully beenused for discrete-event simulations of the single beamsplitter and Mach-Zehnder interferometer experiment ofGrangier et al. [7] (see [8–10]), Wheeler’s delayed choiceexperiment of Jacques et al. [11] (see [10, 12, 13]), thequantum eraser experiment of Schwindt et al. [14] (see[10, 15]), two-beam single-photon interference experi-ments and the single-photon interference experimentwith a Fresnel biprism of Jacques et al. [16] (see [10, 17]),quantum cryptography protocols (see [18]), the Han-bury Brown-Twiss experiment of Agafonov et al. [19] (see[10, 20]), universal quantum computation (see [21, 22]),Einstein-Podolsky-Rosen-Bohm (EPRB)-type of experi-ments of Aspect et al. [23, 24] and Weihs et al. [25] (see[5, 10, 26–30]), and the propagation of electromagneticplane waves through homogeneous thin films and strat-ified media (see [10, 31]). An extensive review of thesimulation method and its applications is given in [10].Interactive demonstration programs, including sourcecodes, are available for download [32–34]. A computerprogram to simulate single-photon EPRB experimentscan be found in [27]. So far the event-based simulationapproach has not been used to simulate the phenomenaof diffraction and evanescent waves.

Given the above list of successful simulations, clearlyan affirmative answer has been given to the primaryquestion posed in Sect. 1.2. A more detailed look to the al-gorithms that have been designed indicates that they arenot unique. Regarding the efficiency of the algorithmswe can say that (i) they are more efficient than their ex-perimental counterparts because idealized models of theexperimental equipment are considered and (ii) their ef-ficiency is limited by the computational power of exist-

394 www.ann-phys.org © 2012 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Ann. Phys. (Berlin) 524, No. 8 (2012)Review

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ing digital computers. Although the event-based simula-tion approach can be used to simulate a universal quan-tum computer [21,22], the so-called “quantum speed-up”cannot be obtained. This by itself is no surprise becausethe quantum speed-up is the result of a mathematicalconstruct in which each unitary operation on the stateof the quantum computer is counted as one operationand in which preparation and read-out of the quantumcomputer are excluded. Whether or not this mathemati-cal construct is realized in Nature is an open question.

1.4 Event-based modeling and detection efficiency

Applications in quantum information have increased theinterest in single-particle detectors. High detection effi-ciencies are essential in for example quantum cryptogra-phy and some Bell test experiments. Single-particle de-tectors are often complex devices with diverse proper-ties. In our event-based simulation approach we modelthe main characteristics of these devices by very simplerules. So far, we have designed two types of detectors,simple particle counters and adaptive threshold devices(see Sect. 2.1). The adaptive threshold detector can beemployed in the simulation of all single-photon experi-ments we have considered so far but is absolutely essen-tial in the simulation of for example the two-beam sin-gle photon experiment. The efficiency, which is the ra-tio of detected to emitted particles, of our model detec-tors is measured in an experiment with one single pointsource emitting single particles that is placed far awayfrom the detector. According to this definition, the sim-ple particle counter has an efficiency of 100% and theadaptive threshold detector has an efficiency of nearly100%. Hence, these detectors are highly idealized ver-sions of real single photon detectors. No absorption ef-fects, dead times, dark counts or other effects causingparticle miscounts are simulated.

In laboratory experiments, the single particle detec-tion process is often quantified in terms of the ratio ofdetected to emitted particles, the overall detection effi-ciency. Evidently, the efficiency of the detector plays animportant role in this overall detection efficiency but isnot the only determining factor. Also the experimentalconfiguration in which the detector is used plays an im-portant role. Therefore, the experimenter usually chosesthe best overall performing single-particle detector forher or his particular experiment. Also in the event-basedapproach the experimental configuration plays an im-portant role in the overall detection efficiency. Althoughthe adaptive threshold detectors are ideal and have a de-

tection efficiency of nearly 100%, the overall detection ef-ficiency can be much less than 100% depending on theexperimental configuration. For example, using adaptivethreshold detectors in a Mach-Zehnder interferometryexperiment leads to an overall detection efficiency ofnearly 100%, while using the same detectors in a single-photon two-beam experiment (see Sect. 2.1) leads to anoverall detection efficiency of about 15% [10, 17].

Also the data processing procedure which is appliedafter the data has been collected may have an influenceon the final detection efficiency. For example, in thesingle-photon EPRB experiment of Weihs et al. a post-selection procedure with a time-coincidence window isemployed to group photons, detected in two differentstations, into pairs [25]. Clearly, this procedure omits de-tected photons and therefore reduces the final ratio ofdetected to emitted photons. This is also the case in theevent-based simulation of this experiment (see Sect. 2.2).Although simple particle counters with a 100% detectionefficiency are used and thus all emitted photons are ac-counted for during the data collection process, the finaldetection efficiency is less than 100% because some de-tection events are omitted in the post-selection data pro-cedure using a time-coincidence window.

In conclusion, even if ideal detectors with a detectionefficiency of 100% would be commercially available, thenthe overall detection efficiency in a single-particle exper-iment could still be much less than 100% depending on(i) the experimental configuration in which the detectorsare employed and (ii) the data analysis procedure that isused after all data has been collected.

1.5 Event-based modeling and the interpretations ofquantum theory

This paper is not about interpretations or extensions ofquantum theory. The fact that there exist simulation al-gorithms that reproduce the statistical results of quan-tum theory has no direct implications for the founda-tions of quantum theory. The average properties of thedata may be in perfect agreement with quantum theorybut the algorithms that generate such data are outside ofthe scope of what quantum theory can describe. Never-theless, one could say that the event-based simulationapproach is in line with the ensemble-statistical inter-pretation of quantum theory but also goes beyond thisinterpretation since the method is able to give a logicalcause-and-effect description of how the ensemble is gen-erated event by event. This makes metaphysical interpre-

© 2012 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org 395

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tations [35,36] superfluous, at least for those experimentsthat can be simulated by the event-based approach.

Of course, providing a description that goes beyondthe statistical properties comes at a price. As it is well-known, “non-contextuality”, literally meaning “being in-dependent of the (experimental) measurement arrange-ment” is one of the properties that makes quantum(Maxwell’s) theory so widely applicable [1]. To go be-yond a statistical description unavoidably requires con-textuality [1]. Therefore, the discrete-event approach pro-vides a complete description of individual events butnon-contextuality is lost. Note that as long as no mea-surement is performed the event-based description isnon-contextual. For example, in the simulation of thesingle-photon EPRB experiment (see Sect. 2.2) the twophotons leaving the source have random but oppositepolarization. The photons have a well-defined predeter-mined (non-contextual) polarization which correspondsto a certain vector S. When an observer makes a mea-surement on one photon using a polarizing beam split-ter and two detectors, the observer gets a response +1or −1 depending on the angle of the polarizing beamsplitter. Hence, depending on the angle of the polarizingbeam splitter the observer measures +1 or −1 while thepolarization vector S of the photon has a well-definedpredetermined length and orientation. Thus, in this case,the contextuality (dependence on measurement arrang-ment) stems from the fact that the observer can only ob-tain partial information about the vector S when makinga measurement.

Finally, it should be noted that although the discrete-event algorithm can be given an interpretation as a re-alistic cause-and-effect description that is free of logicaldifficulties and reproduces the statistical results of quan-tum theory, at present the lack of relevant data makesit impossible to decide whether or not such algorithmsare realized by Nature. Only new, dedicated experimentsthat probe more than just the statistical properties canteach us more about this intriguing question. Proposalsfor such experiments are discussed in [17, 37].

2 Illustrative examples

As a detailed discussion of an extensive set of discrete-event rules cannot be fitted in this short review, we haveopted to present a detailed description of the algorithmsfor some fundamental experiments in quantum physics,namely interference of two coherent beams of parti-cles and two Bell test experiments, an Einstein-Podolsky-Rosen-Bohm (EPRB) experiment with single photons

[25] and a Bell’s inequality interference experiment withsingle neutrons [38]. Our motivation to select the two-beam experiment with single photons as an example isthat this experiment with minimal equipment shows in-terference in its purest form, that is without diffractionbeing involved. The experiment demonstrates that sin-gle particles coming from two coherent beams can grad-ually build up an interference pattern when the particlesarrive one by one at a detector screen. We choose theBell test experiments as another example because thequantum theoretical description of this type of experi-ments involves entanglement. The single-photon EPRBexperiment demonstrates that the two photons of a pair,post-selected by employing a time-coincidence window,can be in an entangled state. The neutron interferome-try experiment shows that it is possible to create correla-tions between the spatial and spin degree of freedom ofneutrons which, within quantum theory, cannot be de-scribed by a product state meaning that the spin- andphase-degree-of-freedom are entangled. In this experi-ment the neutrons are counted with a detector having avery high efficiency (≈ 99%), thereby not suffering fromthe so-called detection loophole.

Furthermore, we have chosen to present the close-to-simplest algorithms that reproduce the results of quan-tum theory, hoping that this will help the reader graspthe basic ideas. For an extensive review of an event-basedmodel for quantum optics experiments, see [10].

2.1 Two-beam interference

In 1924, de Broglie introduced the idea that also mat-ter, not just light, can exhibit wave-like properties [39].This idea has been confirmed in various double-slit ex-periments with massive objects such as electrons [40–43],neutrons [44, 45], atoms [46, 47] and molecules such asC60 and C70 [48, 49], all showing interference. In some ofthe double-slit experiments [16, 41, 42] the interferencepattern is built up by recording individual clicks of the de-tectors. According to Feynman, the observation that theinterference patterns are built up event-by-event is “im-possible, absolutely impossible to explain in any classicalway and has in it the heart of quantum mechanics” [50].

Reading “any classical way” as “any classical Hamilto-nian mechanics way”, Feynman’s statement is difficult todispute. However, taking a broader view by allowing fordynamical systems that are outside the realm of classi-cal Hamiltonian dynamics, it becomes possible to modelthe gradual appearance of interference patterns througha discrete-event simulation that does not make reference


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x

S0

D

X

y

S1 β(0, / 2)d

(0, / 2)d

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

-90 -60 -30 0 30 60 90

Cou

nts

θ [degrees]

Figure 1 (online color at: www.ann-phys.org) Left: schematicdiagram of a two-beam experiment with light sources S0

and S1 of width a, separated by a center-to-center dis-tance d . These sources emit coherent, monochromaticlight according to the light flux distribution J(x, y) =δ(x)

[Θ(a/2−|y −d/2|)+Θ(a/2−|y +d/2|)], where Θ(.)

denotes the unit step function and with a uniform angular dis-tribution, β denoting the angle. The light is recorded by detectorsD positioned on a semi-circle with radius X and center (0,0). The

angular position of a detector is denoted by θ. Right: detectorcounts (circles) as a function of θ as obtained from the event-basedmodel simulation of the two-beam interference experimentdepicted on the left. The solid line is a least-square fit of the simu-lation data to the prediction of wave theory, Eq. (5), with only onefitting parameter. Simulation parameters: on average, each of the181 detectors receives 104 particles,γ= 0.99, a = c/ f ,d = 5c/ f ,X = 100c/ f , where c denotes the velocity and f the frequencyof the particles.

to concepts of quantum theory. As a discrete-event sim-ulation is nothing but a sequence of instructions thatchanges the state of a macroscopic classical apparatus(most conveniently a digital computer) in a prescribedmanner, the demonstration that we give in this Sect. pro-vides a rational, logically consistent, common-sense ex-planation of how the detection of individual objects thatdo not interact with each other can give rise to the in-terference patterns that are being observed. As there areseveral breeds of such models [10, 17, 51] that reproducethe statistical distribution predicted by quantum theory,only a new kind of experiment, specifically addressingthis issue can provide a verdict about the applicability ofthese models to the problem at hand [17].

For the purpose of illustration, we consider the sim-ple experiment sketched in Fig. 1 (left). This two-beamexperiment can be viewed as a simplification of Young’sdouble-slit experiment in which the slits are regarded asthe virtual sources S0 and S1 [52]. In the two-beam experi-ment interference appears in its most pure form becausein contrast to the two-slit experiment the phenomenonof diffraction is absent. The event-based model for thisexperiment has two types of events only: (i) the creationof one particle at one of the sources and (ii) the detectionof that particle by one of the detectors forming the screen.

We assume that all these detectors are identical and can-not communicate among each other and do not allow fordirect communication between the particles, implyingthat this event-by-event model is locally causal by con-struction. Then, if it is indeed true that individual par-ticles build up the interference pattern one by one, justlooking at Fig. 1 (left) leads to the logically unescapableconclusion that the interference pattern can only be dueto the internal operation of the detector [53]. Detectorsthat simply count the incoming photons are not suffi-cient to explain the appearance of an interference pat-tern and apart from the detectors there is nothing elsethat can cause the interference pattern to appear. Mak-ing use of the statistical property of quantum theory onecould assume that if a detector is replaced by another oneas soon as it detects one photon, one obtains a similarinterference pattern if the detection events of all thesedifferent detectors are combined. However, since thereis no experimental evidence that confirms this assump-tion and since our event-based approach is based on lab-oratory experimental setups and observations we do notconsider this being a realistic option. Thus, logic dictatesthat a minimal event-based model for the two-beam ex-periment requires an algorithm for the detector that doesa little more than just counting particles.


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2.1.1 Event-based model

We now specify the model in sufficient detail such thatthe reader who is interested can reproduce our results (aMathematica implementation of a slightly more sophis-ticated algorithm [17] can be downloaded from the Wol-fram Demonstration Project web site [34]).– Source and particles: The particles leave the source

one by one, at positions y drawn randomly from a uni-form distribution over the interval [−d/2−a/2,−d/2+a/2]∪[+d/2−a/2,+d/2+a/2]. The particle is regardedas a messenger, traveling in the direction given by theangle β, being a uniform pseudo-random number be-tween −π/2 and π/2. Each messenger carries with it amessage e(t ) = (cos2π f t ,sin2π f t ) that is representedby a harmonic oscillator which vibrates with frequencyf (representing the “color” of the light). The internaloscillator is used as a clock to encode the time of flightt . When a messenger is created, its time of flight is setto zero. This pictorial model of a “photon” was usedby Feynman to explain quantum electrodynamics [54].The event-based approach goes one step further inthat it specifies in detail, in terms of a mechanical pro-cedure, how the “amplitudes” that appear in the quan-tum formalism get added together.The time of flight of the particles depends on thesource-detector distance. Here, we discuss as an ex-ample, the experimental setup with a semi-circular de-tection screen (see Fig. 1 (left)) but in principle anyother geometry for the detection screen can be con-sidered. The messenger leaving the source at (0, y)under an angle β will hit the detector screen of ra-dius X at a position determined by the angle θ givenby sinθ = (y cos2β + sinβ

√X 2 − y2 cos2β)/X , where

|y/X | < 1. The time of flight is then given by t =√X 2 −2y X sinθ+ y2/c , where c is the velocity of the

messenger. The messages e(t ) together with the ex-plicit expression for the time of flight are the only inputto the event-based algorithm.

– Detector: Microscopically, the detection of a particleinvolves very intricate dynamical processes [3]. In itssimplest form, a light detector consists of a materialthat can be ionized by light. This signal is then ampli-fied, usually electronically, or in the case of a photo-graphic plate by chemical processes. In Maxwell’s the-ory, the interaction between the incident electric fieldE and the material takes the form P ·E, where P is thepolarization vector of the material [52]. Assuming a lin-ear response, P(ω) = χ(ω)E(ω) for a monochromaticwave with frequency ω, it is clear that in the time do-main, this relation expresses the fact that the materialretains some memory about the incident field, χ(ω)

representing the memory kernel that is characteristicfor the material used.In line with the idea that an event-based approachshould use the simplest rules possible, we reason asfollows. In the event-based model, the kth messageek = (cos2π f tk ,sin 2π f tk) is taken to represent the ele-mentary unit of electric field E(t ). Likewise, the electricpolarization P(t ) of the material is represented by thevector pk = (p0,k , p1,k ). Upon receipt of the kth mes-sage this vector is updated according to the rule

pk = γpk−1 + (1−γ)ek , (1)

where 0< γ< 1 and k > 0. Obviously, ifγ> 0, a messageprocessor that operates according to the update ruleEq. (1) has memory, as required by Maxwell’s theory.It is not difficult to prove that as γ → 1−, the internalvector pk converges to the average of the time-series{e1,e2, . . .} [10, 17]. The parameter γ controls the preci-sion with which the machine defined by Eq. (1) learnsthe average of the sequence of messages e1,e2, . . . andalso controls the pace at which new messages affectthe internal state of the machine [8]. Moreover, in thecontinuum limit (meaning many events per unit oftime), the rule Eq. (1) translates into the constitutiveequation of the Debye model of a dielectric [17], amodel used in many applications of Maxwell’s theory[55]. The learning process of the detector, governedby the rule Eq. (1) can namely be viewed as a pro-cess that proceeds in discrete time steps τ, so thatpk = p(kτ) = p(t ) and ek = e(kτ) = e(t ). For small τ,pk−1 = p(t )−τ∂p(t )/∂t +O(τ2) and hence

∂p(t )

∂t≈−1−γ

τγp(t )+ 1−γ

τγe(t ). (2)

Letting the time step τ, that is the time between thearrival of successive messages, approach zero (τ→ 0),letting γ approach one (γ → 1−) and demanding thatthe resulting continuum equation makes sense, leadsto the following relation between τ and γ,

limτ→0

limγ→1−

1−γ

τγ= Γ, (3)

with 0 < Γ <∞. Hence, ∂p(t )/∂t = −Γp(t )+Γe(t ) or inFourier space P(ω) = χ(ω)E(ω), being the constitutiveequation of the Debye model of a dielectric, with Eq. (3)giving an explicit expression for the relaxation time 1/Γin terms of the parameters of the event-based model.After updating the vector pk , the processor uses theinformation stored in pk to decide whether or not togenerate a click. As a highly simplified model for the


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bistable character of the real photodetector or photo-graphic plate, we let the machine generate a binary out-put signal Sk according to

Sk =Θ(p2k − rk ), (4)

where Θ(.) is the unit step function and 0 ≤ rk < 1 is auniform pseudo-random number. Note that the use ofrandom numbers is convenient but not essential [10].Since in experiment it cannot be known whether a pho-ton has gone undetected, we discard the informationabout the Sk = 0 detection events and define the totaldetector count as N = ∑k

j=1 S j , where k is the numberof messages received. N is the number of clicks (one’s)generated by the processor.The efficiency of the detector model is determined bysimulating an experiment that measures the detectorefficiency, which for a single-photon detector is de-fined as the overall probability of registering a count ifa photon arrives at the detector [56]. In such an experi-ment a point source emitting single particles is placedfar away from a single detector. As all particles thatreach the detector have the same time of flight (to agood approximation), all the particles that arrive at thedetector will carry the same message which is encod-ing the time of flight. As a result pk (see Eq. (1)) rapidlyconverges to the vector corresponding to this identicalmessage, so that the detector clicks every time a pho-ton arrives. Thus, the detection efficiency, as definedfor real detectors [56], for our detector model is veryclose to 100%. Hence, the model is a highly simplifiedand idealized version of a single-photon detector. How-ever, although the detection efficiency of the detectoritself may be very close to 100%, the overall detectionefficiency, which is the ratio of detected to emitted pho-tons in the simulation of an experiment, can be muchless than one. This ratio depends on the experimentalsetup.

– Simulation procedure: Each of the detectors of the cir-cular screen has a predefined spatial window withinwhich it accepts messages. As a messenger hits a de-tector, this detector updates its internal state p, (the in-ternal states of all other detectors do not change) us-ing the message ek and then generates the event Sk . Inthe case Sk = 1, the total count of the particular detec-tor that was hit by the kth messenger is incrementedby one and the messenger itself is destroyed. Only af-ter the messenger has been destroyed, the source is al-lowed to send a new messenger. This rule ensures thatthe whole simulation complies with Einstein’s criterionof local causality. This process of creating and destroy-ing messengers is repeated many times, building upthe interference pattern event by event.

2.1.2 Simulation results

In Fig. 1 (right), we present simulation results for a rep-resentative case for which the analytical solution fromwave theory is known. Namely, in the Fraunhofer regime(d X ), the analytical expression for the light intensityat the detector on a circular screen with radius X is givenby [52]

I (θ)= A sin2 qa sinθ

2cos2 qd sinθ

2

/(qa sinθ

2

)2

, (5)

where A is a constant, q = 2π f /c denotes the wavenum-ber with f and c being the frequency and velocity of thelight, respectively, and θ denotes the angular position ofthe detector D on the circular screen, see Fig. 1 (left).Note that Eq. (5) is only used for comparison with thesimulation data and is by no means input to the model.From Fig. 1 (right) it is clear that the event-based modelreproduces the results of wave theory and this withouttaking recourse of the solution of a wave equation.

As the detection efficiency of the event-based detec-tor model is very close to 100%, the interference pat-terns generated by the event-based model cannot be at-tributed to inefficient detectors. It is therefore of interestto take a look at the ratio of detected to emitted photons,the overall detection efficiency, and compare the detec-tion counts, observed in the event-based model simu-lation of the two-beam interference experiment, withthose observed in a real experiment with single photons[16]. In the simulation that yields the results of Fig. 1,each of the 181 detectors making up the detection areais hit on average by ten thousand photons and the to-tal number of clicks generated by the detectors is 296444.Hence, the ratio of the total number of detected to emit-ted photons is of the order of 0.16, two orders of magni-tude larger than the ratio 0.5 × 10−3 observed in single-photon interference experiments [16].

2.1.3 What is the working principle?

In our event-based approach the simple particle coun-ters and the adaptive threshold detectors are ideal detec-tors with a detection efficiency of (nearly) 100%. In thecase of the two-beam interference experiment consid-ered in section 2.1, using simple particle counters wouldnot result in an interference pattern. These detectors sim-ply produce a click for each incoming photon and donothing with the information encoded in the messages ecarried by the particles. These messages contain informa-tion about the time of flight of the particles, that is aboutthe distance travelled by the particles from one of the


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two sources to one of the detectors constituting the cir-cular detection screen. It is precisely the difference in thetimes of flight (or the phase differences) which is impor-tant in the generation of an interference pattern. Since,in the single-photon two-beam experiment the detectorsare the only apparatuses available in the experiment thatcan process this information (there are no other appara-tuses present except for the source) we necessarily needto employ an algorithm for the detector that exploits thisinformation in order to produce the clicks that graduallybuild up the interference pattern. A collection of abouttwo hundred independent adaptive threshold detectorsdefined by Eq. (1) and Eq. (4) and each with a detec-tion efficiency of nearly 100% is capable of doing this. Aspointed out earlier, the reason why, in this particular ex-periment, this is possible is that not every particle thatimpinges on the detector yields a click.

Note that to simulate the interference pattern ob-served in single-photon Mach-Zehnder interferometryexperiments it is possible, but by no means necessary, touse the adaptive threshold detectors which do not nec-essarily produce a click for each incoming photon [10].Indeed, it suffices to use simple particle counters whichproduce a click for each incoming photon [8, 9]. In thesimulation of the Mach-Zehnder interferometry experi-ment the beam splitters process the information aboutthe time of flight of the particles [8–10]. Whether or notthe detectors also process this information has no influ-ence on the generation of the interference pattern [8–10].Hence, in this case the number of detected photons isequal to the number of emitted photons in the simula-tion.

2.2 Einstein-Podolsky-Rosen-Bohm (EPRB) experimentwith single photons

The EPRB experiment with photons, carried out by Weihset al. [25, 57], is taken as a concrete example to illustratehow to construct an event-based model that reproducesthe predictions of quantum theory for the single and two-particle averages for a quantum system of two spin-1/2particles in the singlet state and a product state [10, 30],without making reference to concepts of quantum theory.Recall that the quantum theoretical descriptions of theEPRB experiment with photons or with spin-1/2 particlesare identical.

In short, the experiment goes as follows. A sourceemits pairs of photons. Each photon of a pair travels toan observation station in which it is manipulated anddetected. The two stations are assumed to be identical.

They are separated spatially and temporally, preventingthe observation at station 1 (2) to have a causal effect onthe data registered at station 2 (1) [25]. As the photon ar-rives at station i = 1,2, it passes through an electro-opticmodulator (EOM) which rotates the polarization of thephoton by an angle depending on the voltage applied tothe modulator. These voltages are controlled by two in-dependent binary random number generators. A polar-izing beam splitter sends the photon to one of the twodetectors. The station’s clock assigns a time-tag to eachgenerated signal.

The firing of a detector is regarded as an event. Atthe nth event, the data recorded on a hard disk at sta-tion i = 1,2 consists of xn,i =±1, specifying which of thetwo detectors fired, the time tag tn,i indicating the time atwhich a detector fired, and the two-dimensional unit vec-tor αn,i that represents the rotation of the polarization bythe EOM at the time of detection. Hence, the set of datacollected at station i = 1,2 may be written as

Υi ={

xn,i =±1, tn,i ,αn,i |n = 1, . . . , Ni}

. (6)

In the experiment, the data {Υ1,Υ2} is analyzed after thedata of a particular run has been collected [25]. Adopt-ing the procedure employed by Weihs et al. [25, 57], weidentify coincidences by comparing the time differencestn,1 − tm,2 with a window W where n = 1, . . . , N1 and m =1, . . . , N2. By definition, for each pair of rotation angles aand b of the EOMs, the number of coincidences betweendetectors Dx,1 (x = ±1) at station 1 and detectors Dy,2

(y =±1) at station 2 is given by

Cx y =Cx y (a,b) (7)

=N1∑

n=1

N2∑m=1

δx,xn,1δy,xm,2δa,αn,1δb,αm,2Θ(W −|tn,1 − tm,2|),

where Θ(.) denotes the unit step function. In Eq. (7) thesum over all events has to be carried out such that eachevent (= one detected photon) contributes only once.Clearly, this constraint introduces some ambiguity in thecounting procedure as there is a priori, no clear-cut cri-terion to decide which events at stations i = 1 and i =2 should be paired. One obvious criterion might be tochoose the pairs such that Cx y is maximum [57] but, sucha criterion renders the data analysis procedure (not thedata production) acausal. It is trivial though to analysethe data generated by the experiment of Weihs et al. suchthat conclusions do not suffer from this artifact [58]. Thecorrelation of the two dichotomic variables x and y is de-fined as

E(a,b)= C++ +C−− −C+− −C−+C++ +C−− +C+− +C−+

, (8)


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where the denominator is the sum of all coincidences. Ingeneral, the two-particle averages E(a,b), and the totalnumber of the coincidences not only depend on the di-rections a and b but also on the time window W used toidentify the coincidences. For later use it is expedient tointroduce the function [59]

S ≡ S(a,b, a′,b′)

= E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′). (9)

Local-realistic treatments of the EPRB experimentusually assume that the expression for the correlation, asmeasured in the experiment, is given by [60]

C (∞)x y (a,b) =

N∑n=1

δx,xn,1δy,xn,2δa,αn,1δb,αn,2 , (10)

which is obtained from Eq. (7) (in which each photoncontributes only once) by assuming that N = N1 = N2,pairs are defined by n = m, and by taking the limit W →∞. However, the working hypothesis that the value of Wshould not matter because the time window only servesto identify pairs may not apply to real experiments. Theanalysis of the data of the experiment of Weihs et al.shows that the average time between pairs of photonsis of the order of 30 µs or more, much larger than thetypical values (of the order of a few nanoseconds) of thetime-window W used in the experiments [57]. In otherwords, in practice, the identification of photon pairs doesnot require the use of W ’s of the order of a few nanosec-onds. The small value of W is required to maximize |S|and is not really required for the data to violate Bell’s in-equality |S| ≤ 2. In other words, depending on the valueof W , chosen by the experimenter when analyzing thedata, the inequality |S| ≤ 2 may or may not be violated.Hence, also the conclusion about the state of the systemdepends on the value of W , which turns W into a so-called context parameter. Analysis of the data of the ex-periment by Weihs et al. shows that W can be as large as150 ns for the Bell inequality to be violated [58] and inthe time-stamping EPRB experiment of Agüero et al. [61]|S| ≤ 2 is clearly violated for W < 9 µs. Hence, the useof a time-coincidence window does not create a “loop-hole”. Nevertheless, very often it is mentioned that thesesingle-photon Bell test experiments suffer from the fairsampling loophole, being the result of the usage of a timewindow W to filter out coincident photons or being theresult of the usage of inefficient detectors [62]. The de-tection loophole was first closed in an experiment withtwo entangled trapped ions [63] and later in a single-neutron interferometry experiment [38] and in an exper-iment with two entangled qubits [64]. However, the lat-

ter three experiments are not Bell test experiments per-formed according to the CHSH protocol [59] because thetwo degrees of freedom are not manipulated and mea-sured independently.

The narrow time window W in the experiment byWeihs et al. mainly acts as a filter that selects pairs ofwhich the individual photons differ in their time tags bythe order of nanoseconds. The possibility that such a fil-tering mechanism can lead to correlations that are oftenthought to be a characteristic of quantum systems onlywas, to our knowledge, first pointed out by P. Pearle [65]and later by A. Fine [66], opening the route to a descrip-tion in terms of locally causal, classical models. A con-crete model of this kind was proposed by S. Pascaziowho showed that his model approximately reproducesthe correlation of the singlet state [67] with an accuracythat seems beyond what is experimentally achievable todate. Larson and Gill showed that Bell-like inequalitiesneed to be modified in the case that the coincidences aredetermined by a time-window filter [68], and models thatexactly reproduce the results of quantum theory for thesinglet and uncorrelated state were found [10, 26, 28, 30].Here, we closely follow [28, 30].


A minimal, discrete-event simulation model of the EPRBexperiment by Weihs et al. requires a specification of theinformation carried by the particles, of the algorithm thatsimulates the source and the observation stations, and ofthe procedure to analyze the data.– Source and particles: Each time, the source emits two

particles that carry a vector

Sn,i = (cos(ξn + (i −1)π/2),sin(ξn + (i −1)π/2)),

representing the polarization of the photons. This po-larization is completely characterized by the angle ξn

and the direction i = 1,2 to which the particle moves.A uniform pseudo-random number generator is usedto pick the angle 0 ≤ ξn < 2π. Clearly, the source emitstwo particles with a mutually orthogonal, hence corre-lated but otherwise random polarization. Note that forthe simulation of this experiment it is not necessarythat the particles carry information about the phase2π f ti ,n. In this case the time of flight ti ,n is determinedby the time-tag model (see below).

– Electro-optic modulator: The EOM in station i = 1,2rotates the polarization of the incoming particle by anangle αi , that is its polarization angle becomes ξ′n,i ≡EOMi (ξn + (i − 1)π/2,αi ) = ξn + (i − 1)π/2 − αi sym-bolically. Mimicking the experiment of Weihs et al. in


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which α1 can take the values a, a′ and α2 can take thevalues b,b′, we generate two binary uniform pseudo-random numbers Ai = 0,1 and use them to choose thevalue of the angles αi , that is α1 = a(1− A1)+a′A1 andα2 = b(1− A2)+b′A2.

– Polarizing beam splitter: The simulation model for apolarizing beam splitter is defined by the rule

xn,i ={+1 if rn ≤ cos2 ξ′n,i

−1 if rn > cos2 ξ′n,i

, (11)

where 0 < rn < 1 are uniform pseudo-random num-bers. It is easy to see that for fixed ξ′n,i = ξ′i , this rulegenerates events such that the distribution of eventscomplies with Malus law.

– Time-tag model: As is well-known, as light passesthrough an EOM (which is essentially a tuneable waveplate), it experiences a retardation depending on itsinitial polarization and the rotation by the EOM. How-ever, to our knowledge, time delays caused by retar-dation properties of waveplates, being components ofvarious optical apparatuses, have not yet been explic-itly measured for single photons. Therefore, in the caseof single-particle experiments, we hypothesize that foreach particle this delay is represented by the time tag[28, 30]

tn,i =λ(ξ′n,i )r ′n , (12)

that is, the time tag is distributed uniformly (0< r ′n < 1

is a uniform pseudo-random number) over the inter-val [0,λ(ξ′n,i )]. For λ(ξ′n,i ) = T0 sin4 2ξ′n,i this time-tagmodel, in combination with the model of the polar-izing beam splitter, rigorously reproduces the resultsof quantum theory of the EPRB experiments in thelimit W → 0 [28,30]. We therefore adopt the expressionλ(ξ′n,i ) = T0 sin4 2ξ′n,i leaving only T0 as an adjustableparameter.

– Detector: The detectors are ideal particle counters,meaning that they produce a click for each incomingparticle. Hence, we assume that the detectors have100% detection efficiency. Note that also adaptivethreshold detectors can be used (see Sect. 1.4) equallywell [10].

– Simulation procedure: The simulation algorithm gen-erates the data sets Υi , similar to the ones obtainedin the experiment (see Eq. (6)). In the simulation, itis easy to generate the events such that N1 = N2. Weanalyze these data sets in exactly the same manner asthe experimental data are analyzed, implying that weinclude the post-selection procedure to select photonpairs by a time-coincidence window W . The latter iscrucial for our simulation method to give results that

are very similar to those observed in a laboratory exper-iment. Although in the simulation the ratio of detectedto emitted photons is equal to one, the final detectionefficiency is reduced due to the time-coincidence post-selection procedure.

This algorithm fully complies with Einstein’s criterionof local causality on the ontological level. Once the par-ticles leave the source, an action at observation station 1(2) can, in no way, have a causal effect on the outcome ofthe measurement at observation station 2 (1).


In Fig. 2 we present some typical simulation results forthe function S, showing that the event-based model re-produces the predictions of quantum theory for the sin-glet state. The single-particle averages E1(a) and E2(b)(data not shown) are zero up to the usual statistical fluc-tuations and do not show any statistically relevant depen-dence on b or a, respectively. Additional results, mimick-ing quantum systems in the singlet and product state, aswell as a rigorous probabilistic treatment of the modelcan be found in [30]. For W = 50ns (data not shown),we find |S| = 2.62 which compares very well with the val-ues between 2 and 2.57 extracted from different data setsproduced by the experiment of Weihs et al. In all cases,the distribution of time-tag differences (data not shown)is sharply peaked and displays long tails, in qualitativeagreement with experiment [57].

From Fig. 2 (right), it follows that a violation of theBell inequality |S| ≤ 2 depends on the choice of W , a pa-rameter which is absent in the quantum theoretical de-scription of the EPRB thought experiment. There are twolimiting cases for which S become independent of W . IfW → ∞, it is impossible to let a digital computer vio-late the inequality |S| ≤ 2 without abandoning the rulesof Boolean logic or arithmetic [69]. For relatively smallW (W < 150 ns), the inequality |S| ≤ 2 may be violated.When W → 0 the discrete-event models which generatethe same type of data as real EPRB experiments, repro-duce exactly the single- and two-spin averages of the sin-glet state and therefore also violate the inequality |S| ≤ 2.Obviously, as the discrete-event model does not rely onany concept of quantum theory, a violation of the in-equality |S| ≤ 2 does not say anything about the “quan-tumness” of the system under observation [69–71]. Sim-ilarly, a violation of this inequality cannot say anythingabout locality and realism [69–72]. Clearly, the event-based model is contextual. The fact that the event-basedmodel reproduces, for instance, the correlations of thesinglet state without violating Einstein’s local causality


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-3

-2

-1

0

1

2

3

0 45 90 135 180

S

θ

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300

|S|

W [ns]

Figure 2 (online color at: www.ann-phys.org) Simulation resultsproduced by the locally causal, discrete-event simulation modelusing the time-tag model Eq. (12). Left: S as a function of θ witha = θ, a′ = π/4 + θ, b = π/8 and b′ = 3π/8 for a time win-dow W = 2ns. The solid line connecting the circles is the result−2

�2cos2θ predicted by quantum theory. The dashed lines rep-

resent the maximum value for a quantum system of two spin-1/2particles in a separable (product) state (|S| = 2) and in a singlet

state (|S| = 2�

2), respectively. Right: |S| as a function of W fora = 0, a′ = π/4, b = π/8 and b′ = 3π/8. For W ≈ 200ns, thesystem changes from “quantum-like” (|S| > 2) to “classical like”(|S| ≤ 2). Simulation parameters: for each value of θ, the num-ber of pairs generated is 3× 105 (roughly the same as in experi-ment [25]) andT0 = 2000ns, the adjustable parameter in the time-tag model Eq. (12).

criterion suggests that the data {xn,1, xn,2} generated bythe event-based model cannot be represented by a sin-gle Kolmogorov probability space. This complies withthe idea that contextual, non-Kolmogorov models canlead to violations of Bell’s inequality without appealingto nonlocality or nonobjectivism [73, 74].

2.2.3 Why is Bell’s inequality violated?

In [30], we have presented a probabilistic description ofour simulation model that (i) rigorously proves that forup to first order in W it exactly reproduces the single par-ticle averages and the two-particle correlations of quan-tum theory for the system under consideration; (ii) il-lustrates how the presence of the time-window W intro-duces correlations that cannot be described by the origi-nal Bell-like “hidden-variable” models [60].

The time-coincidence post-selection procedure withthe time-window W filters out the “coincident” photonsbased on the time-tags tn,i thereby reducing the final de-tection efficiency to less than 100%, although in the sim-ulation a measurement always returns a +1 or−1 for bothphotons in a pair (100% detection efficiency of the detec-tors). Hence, even in case of a perfect detection processthe data set that is finally retained consists only of a sub-

set of the entire ensemble of correlated photons that wasemitted by the source. In other words, the use of a time-coincidence window destroys the “fair-sampling hypoth-esis”.

We briefly elaborate on point (ii) (see [30] for a moreextensive discussion). Let us assume that there existsa probability P(x1, x2, t1, t2|α1,α2) to observe the data{x1, x2, t1, t2} conditional on {α1,α2}. The probabilityP(x1, x2, t1, t2|α1,α2) can always be expressed as an inte-gral over the mutually exclusive events ξ1, ξ2, represent-ing the polarization of the photons

P(x1, x2, t1, t2|α1,α2) = 1

4π2

∫2π

0

∫2π

0

×P(x1, x2, t1, t2|α1,α2,ξ1,ξ2)P(ξ1,ξ2|α1,α2)dξ1dξ2. (13)

According to Eq. (11) and Eq. (12), in the probabilistic ver-sion of our simulation model, for each event, (i) the val-ues of x1, x2, t1, and t2 are mutually independent randomvariables, (ii) the values of x1 and t1 (x2 and t2) are inde-pendent of α2 and ξ2 (α1 and ξ1), (iii) ξ1 and ξ2 are in-dependent of α1 or α2. With these assumptions Eq. (13)


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becomes

P(x1, x2, t1, t2|α1,α2)

(i)= 1

4π2

∫2π

0

∫2π

0P(x1, t1|α1,α2,ξ1,ξ2)P(x2, t2|α1,α2,ξ1,ξ2)

×P(ξ1,ξ2|α1,α2)dξ1dξ2

(ii)= 1

4π2

∫2π

0

∫2π

0P(x1, t1|α1,ξ1)P(x2, t2|α2,ξ2)


(i)= 1

4π2

∫2π

0

∫2π

0

×P(x1|α1,ξ1)P(t1|α1,ξ1)P(x2|α2,ξ2)P(t2|α2,ξ2)


(iii)= 1

4π2

∫2π

0

∫2π

0P(x1|α1,ξ1)P(t1|α1,ξ1)P(x2|α2,ξ2)

×P(t2|α2,ξ2)P(ξ1,ξ2)dξ1dξ2, (14)

which is the probabilistic description of our simulationmodel. The probabilistic model Eq. (14) is identical tothe local-realist model used by Larsson and Gill in theirderivation of a CHSH inequality with time-coincidencerestriction [68].

According to our simulation model, the probabilitydistributions that describe the polarizers are given byP(xi |αi ,ξi ) = [1 + xi cos2(αi − ξi )]/2 and those for thetime-delays ti that are distributed randomly over the in-terval [0,λ(ξi + (i −1)π/2−αi )] are given by P(ti |αi ,ξi ) =θ(ti )θ(λ(ξi + (i − 1)π/2−αi ) − ti )/λ(ξi + (i − 1)π/2−αi ).In the experiment [25] and therefore also in our simula-tion model, the events are selected using a time windowW that the experimenters try to make as small as possi-ble [57]. Accounting for the time window, that is multi-plying Eq. (14) by a step function and integrating over allt1 and t2, the expression for the probability for observingthe event (x1, x2) reads

P(x1, x2|α1,α2) (15)

=∫2π

0

∫2π

0P(x1|α1,ξ1)P(x2|α2,ξ2)ρ(ξ1,ξ2|α1,α2)dξ1dξ2,

where the probability density ρ(ξ1,ξ2|α1,α2) is given by

The simple fact that ρ(ξ1,ξ2|α1,α2) �= ρ(ξ1,ξ2) bringsthe derivation of the original Bell (CHSH) inequality to ahalt. Indeed, in these derivations it is assumed that theprobability distribution for ξ1 and ξ2 does not dependon the settings α1 or α2 [2, 60]. Larsson and Gill haveshown that due to the filtering by the time-coincidencewindow, Eq. (17) satisfies a modified CHSH inequality inwhich the upperbound of 2 is to be replaced by a numberthat depends on the ratio of the number of pairs satisfy-ing the time coincidence criterion and the total numberof pairs [68]. For finite W , this upperbound can be largerthan 2. Hence, it is to be expected that Eq. (17) can violatethe original CHSH inequality.

By making explicit use of the time-tag model (seeEq. (12)) it can be shown that [30] (i) if we ignore thetime-tag information (W > T0), the two-particle proba-bility takes the form of the hidden variable models con-sidered by Bell [60], and we cannot reproduce the resultsof quantum theory [60], (ii) if we focus on the case W → 0the single-particle averages are zero and the two-particleaverage E(α1,α2) =−cos(α1 −α2).

In summary, although our simulation model and itsprobabilistic version Eq. (14) involve local processes only,the filtering of the detection events by means of thetime-coincidence window W can produce correlationswhich violate Bell-type inequalities [66–68]. Moreover,for W → 0 our classical, local and causal model can pro-duce single-particle and two-particle averages which arethe same as those of the singlet state in quantum theory.

2.3 Violation of a Bell inequality in single-neutroninterferometry

The single-neutron interferometry experiment of Hase-gawa et al. [38], demonstrating that the correlation be-tween the spatial and spin degree of freedom of neutronsviolates a Bell-CHSH inequality, is taken to illustrate howto construct an event-based model that reproduces thiscorrelation by using detectors that count every neutronand without using any post-selection procedure.

A schematic picture of the single-neutron interferom-etry experiment is shown in Fig. 3. Incident neutrons arepassing through a magnetic-prism polarizer (not shown)that produces two spatially separated beams of neutronswith their magnetic moments aligned parallel (spin up),

ρ(ξ1,ξ2|α1,α2) =∫+∞−∞

∫+∞−∞ P (t1|α1,ξ1)P (t2|α2,ξ2)Θ(W −|t1 − t2|)P (ξ1,ξ2)d t1d t2∫2π

0

∫2π0

∫+∞−∞

∫+∞−∞ P (t1|α1,ξ1)P (t2|α2,ξ2)Θ(W −|t1 − t2|)P (ξ1,ξ2)dξ1dξ2d t1d t2

. (16)


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respectively anti-parallel (spin down) with respect tothe magnetic axis of the polarizer which is parallel tothe guiding field B. The spin-up neutrons impinge ona silicon-perfect-crystal interferometer [45]. On leavingthe first beam splitter, neutrons may or may not experi-ence refraction. A Mu-metal spin-turner changes the ori-entation of the magnetic moment from parallel to per-pendicular to the guiding field B. The result of passingthrough this spin-turner is that the magnetic moment ofthe neutrons is rotated by π/2 (−π/2) about the y axis, de-pending on the path followed. Before the two paths joinat the entrance plane of beam splitter BS3 a difference be-tween the time of flights (corresponding to a phase in thewave mechanical description) along the two paths canbe manipulated by a phase shifter. The neutrons whichexperience two refraction events when passing throughthe interferometer form the O-beam and are analyzed bysending them through a spin rotator and a Heusler spinanalyzer. If necessary, to induce an extra spin rotation ofπ, a spin flipper is placed between the interferometer andthe spin rotator. The neutrons that are selected by theHeusler spin analyzer are counted with a neutron detec-tor (not shown) that has a very high efficiency (≈ 99%).Note that neutrons which are not refracted by the centralplate of the Si single crystal (beam splitters BS1 and BS2)leave the interferometer without being detected.

The single-neutron interferometry experiment yieldsthe count rate N (α,χ) for the spin-rotation angle α andthe difference χ of the phase shifts of the two differentpaths in the interferometer [38]. The correlation E(α,χ)is defined by [38]

E(α,χ) (17)

= N (α,χ)+N (α+π,χ+π)−N (α+π,χ)−N (α,χ+π)

N (α,χ)+N (α+π,χ+π)+N (α+π,χ)+N (α,χ+π).

If quantum theory describes the experiment it is ex-pected that for the neutrons detected in the O-beam,E(α,χ) = cos(α+ χ) implying |S| = E(α,χ) − E(α,χ′) +E(α′,χ)+E(α′,χ′) > 2 for some values of α, α′, χ and χ′ sothat the state of the neutron cannot be written as a prod-uct of the state of the spin and the phase. Experimentsindeed show that |S| > 2 [38, 75].


A minimal, discrete event simulation model of the single-neutron interferometry experiment requires a specifica-tion of the information carried by the particles, of the al-gorithm that simulates the source and the interferometercomponents, and of the procedure to analyze the data.

– Source and particles: The particles (neutrons)leave the source one by one and carry a messagee = (eiψ1 cosθ/2,eiψ2 sinθ/2) where ψi = 2π f t +δi fori = 1,2. Here δ1 − δ2 and θ specify the magnetic mo-ment of the neutron and t specifies its time of flightwhere f is a frequency which is characteristic for a neu-tron that moves with a fixed velocity v . In the presenceof a magnetic field B = (Bx ,By ,Bz ), the magnetic mo-ment rotates about the direction of B according to theclassical equation of motion.

– Beam splitters BS0, . . . , BS3: The model for the beamsplitter uses the learning process governed by the ruleEq. (1). Exploiting the similarity between the magneticmoment of the neutron and the polarization of a pho-ton, we use the same model for the beam splitter as theone used in [10] for polarized photons. The only differ-ence is that we assume that neutrons with spin up andspin down have the same reflection and transmissionproperties, while photons with horizontal and vertical

Figure 3 (online color at: www.ann-phys.org) Dia-gram of the single-neutron interferometry exper-iment to test a Bell inequality violation (see alsoFig. 1 in [38]).


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polarization have different reflection and transmissionproperties [52].

– Mu metal spin turner: This component rotates themagnetic moment of a neutron that follows the H-beam (O-beam) by π/2 (−π/2) about the y axis.

– Spin-rotator and spin-flipper: The spin-rotator rotatesthe magnetic moment of a neutron by an angle α aboutthe x axis. The spin flipper is a spin rotator with α=π.

– Spin analyzer: This component selects neutrons withspin up, after which they are counted by a detector. Themodel of this component projects the magnetic mo-ment of the particle on the z axis and sends the particleto the detector if the projected value exceeds a pseudo-random number r .

– Detector: The detectors simply count each incomingparticle, meaning that we assume that the detectorshave 100% detection efficiency. This is an idealizationof the real neutron detectors which have a detector ef-ficiency of more than 99% [76].


In Fig. 4 (left) we present simulation results for the cor-relation E(α,χ), assuming that the experimental condi-tions are very close to ideal. For the ideal experimentquantum theory predicts that E(α,χ) = cos(α+ χ). Asshown by the markers in Fig. 4, disregarding the small sta-tistical fluctuations, there is close-to-perfect agreementbetween the event-based simulation data and quantumtheory. The laboratory experiment suffers from unavoid-able imperfections, leading to a reduction and distortionof the interference fringes [38]. In the event-based ap-

proach it is trivial to incorporate mechanisms for differ-ent sources of imperfections by modifying or adding up-date rules. However, to reproduce the available data itis sufficient to use the parameter γ to control the devi-ation from the quantum theoretical result. For instance,for γ = 0.55 the simulation (see Fig. 4) yields Smax ≡S(α = 0,χ = π/4,α′ = π/2,χ′ = π/4) = 2.05, in excellentagreement with the value 2.052±0.010 obtained in exper-iment [38]. Forγ= 0.67 the simulation yields Smax = 2.30,in excellent agreement with the value 2.291 ± 0.008 ob-tained in a similar, more recent experiment [75].

2.3.3 Working principle

From [10] we know that the event-based model for thebeam splitter produces results corresponding to thoseof classical wave or quantum theory when applied in in-terferometry experiments. Important for this outcome isthat the phase difference χ between the two paths in theinterferometer is constant for a relatively large numberof incoming particles. If, for each incoming neutron, wepick the angle χ randomly from the same set of predeter-mined values to produce Fig. 4, an event-based simula-tion with γ= 0.99 yields (within the usual statistical fluc-tuations) the correlation E(α,χ) ≈ [cos(α+χ)]/2, whichdoes not lead to a violation of the Bell-CHSH inequality(results not shown). Thus, if the neutron interferometryexperiment could be repeated with random choices forthe phase shifter χ for each incident neutron, and theexperimental results would show a significant violationof the Bell-CHSH inequality, then the event-based modelthat we have presented here would be ruled out.

0 90

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Figure 4 (online color at: www.ann-phys.org) Left: correlationE(α,χ) between spin and path degree of freedom as obtainedfrom an event-based simulation of the experiment depicted inFig. 3. Solid surface: E(α,χ) = cos(α+χ) predicted by quantumtheory; circles: simulation data. The lines connecting the markers

are guides to the eye only. Model parameters: reflection percent-age of BS0, . . . , BS3 is 20% and γ = 0.99. For each pair (α,χ),four times 10000 particles were used to determine the four countsN (α,χ),N (α+π,χ+π),N (α,χ+π) andN (α+π,χ+π). Right:same as figure on the left but γ= 0.55.


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3 Discussion and outlook

We have given a brief introduction to a new methodol-ogy for simulating, on the level of single events, whatare usually considered to be “quantum” phenomena. Inspirit, our approach is similar to cellular autonoma mod-eling advanced by Wolfram [77] or, for instance, latticeBoltzmann modeling of fluid dynamics [78]. The gen-eral idea is that simple rules, which are not necessar-ily derived from classical Hamiltonian dynamics, definediscrete-event processes which may lead to the (com-plicated) behavior that is observed in experiments. Thebasic strategy in designing these rules is to carefully ex-amine the experimental procedure and to devise rulessuch that they produce the same kind of data as thoserecorded in experiment, while avoiding the trap of simu-lating thought experiments that are difficult to realize inthe laboratory. The event-based model is entirely classi-cal in the sense that it uses concepts of the macroscopicworld and makes no reference to quantum theory but isnonclassical in the sense that some of the rules are notthose of classical Newtonian dynamics.

Depending on the experimental configuration, thesimulation of the interference and correlation phenom-ena, which are observed when individual photons andneutrons are detected one by one, can make use of oneor two of the four following detection processes:

– (1) The detectors are simple particle counters produc-ing a click for each incoming particle and no post-selection data procedure using a time-coincidencewindow is used. The final detection efficiency is 100%.

– (2) The detectors are simple particle counters anda post-selection data procedure using a time-coinci-dence window is used. Although the ratio of detectedto emitted particles is one, the final detection effi-ciency is less than 100%.

– (3) The detectors are adaptive threshold detectorsnot producing a click for each incoming particle andno post-selection data procedure using a time-coinci-dence window is used. The final detection efficiencycan be less than 100%, depending on the experimentalconfiguration.

– (4) The detectors are adaptive threshold detectorsand a post-selection data procedure using a time-coincidence window is used. The final detection effi-ciency is less than 100%.

In Table 1 we give an overview of which type of ex-periments our event-based approach can simulate witha particular detection process. From Table 1 it can be

clearly seen that discarding detection events in a post-selection data procedure using a time-coincidence win-dow or using detectors which do not produce a clickfor each incoming particle is not a characteristic of theevent-based simulation approach for simulating interfer-ence and quantum correlation phenomena.

Three fundamental experiments, two with photonsand one with neutrons, were used to illustrate our ap-proach. On purpose, we have kept the event-based modelas simple as possible, perhaps creating the impressionthat each experiment will require its own set of rules. Thisis not the case. One universal event-based model for theinteraction of photons with matter suffices to explain,without altering the rules, the interference and correla-tion phenomena that are observed when individual pho-tons are detected one by one [10]. This universal modelproduces the frequency distributions for observing manyphotons that are in full agreement with the predictionsof Maxwell’s theory and quantum theory [10]. The samemodel was used for simulating a single neutron interfer-ometry experiment. Similarly, we have constructed anevent-based model that simulates a universal quantumcomputer, reproducing the probability distributions ofthe quantum mechanical system without actually know-ing them [22].

An important question is whether an event-basedmodel leads to new predictions that may be tested exper-imentally. In the stationary state (after processing manyevents) the event-based model reproduces the statisti-cal distributions of quantum theory. Therefore, new pre-dictions can only appear when the event-based modelis operating in the transient regime, before the event-based model reaches its stationary state. Experimentswith a Mach-Zehnder interferometer consisting of twoindependent 50% reflective beam splitters and a phaseshifter (not an integrated Mach-Zehnder interferometer)and two-beam interference that may be able to addressthis issue have been discussed in [37] and [17], respec-tively. We hope that our simulation work will stimulatethe design of new experiments to test the applicabilityof our approach to event-based processes and to excludesome of our non-unique models.

Finally, it may be of interest to mention that thediscrete-event approach reviewed in this paper mayopen a route to rigorously include the effects of interfer-ence in ray-tracing software. For this purpose, it is neces-sary to extend the event-based model to include diffrac-tion, evanescent waves and the behavior of the Lorentzmodel for the response of material to the electromag-netic field [55]. We leave these extensions for future re-search.


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Table 1 Overview of the event-based simulation of photon (P) and neutron (N) experiments classified according to the detectionprocess: (1) Detectors are simple particle counters and no post-selection data procedure using a time-coincidence window; (2) de-tectors are simple particle counters and post-selection data procedure using a time-coincidence window; (3) detectors are adaptivethreshold detectors and no post-selection data procedure using a time-coincidence window; (4) detectors are adaptive thresholddetectors and post-selection data procedure using a time-coincidence window. Numbers in brackets are references to the papers inwhich simulation results are reported. The symbol “V” (“X”) indicates that it is possible (impossible) to obtain the results of quan-tum theory but that we did not publish the simulation results. The symbol “NA” denotes that the particular detection process isnot applicable to the specified experiment.

Experiment Particle (1) (2) (3) (4)

Single beam splitter experiment P [8, 9] NA [10] NA

Mach-Zehnder interferometry experiment P [8, 9] NA [10] NA

Wheeler’s delayed choice experiment P [12, 13] NA [10] NA

Quantum cryptography protocols P [18] NA V NA

Quantum eraser experiment P [15] NA [10] NA

Single photon tunneling P V NA [10] NA

Two-beam interference experiment P X NA [10, 17] NA

Reflection and refraction from an interface P [31] NA [10] NA

Multiple beam fringes with a plane-parallel plate P [31] NA [10] NA

Quantum computation P [21, 22] NA V NA

Einstein-Podolsky-Rosen-Bohm experiment P X [5, 26–30] X [10]

Classical correlations in HBT experiment P X X [20] X

Quantum correlations in HBT experiment P X V X [10]

Interferometry N Sect. 2.3 NA V NA

Bell-type experiment; path-spin correlations N Sect. 2.3 NA V NA

Interferometry; stochastic and deterministic absorption N V NA V NA

Interferometry; path-spin-energy entanglement N V NA V NA

Interferometry; time-dependent blocking of one path N V NA V NA

Hans De Raedt received the PhD from theUniversity of Antwerp (Belgium) for workon magnetism in one dimension in 1976.Since 1990 he is Professor of ComputationalPhysics at the Department of Physics, Uni-versity of Groningen (the Netherlands),where he leads the Computational Physicsgroup of the Zernike Institute for Advanced

Materials. His current research interests include computationalelectrodynamics, nanoscale magnetism, quantum statisticalphysics, and event-based simulationmethods of quantumphe-nomena.

Kristel Michielsen received her PhD fromthe University of Groningen, (the Nether-lands) for work on the simulation of strong-ly correlated electron systems in 1993. Since2009 she is group leader of the researchgroup Quantum Information Processing atthe Jülich Supercomputing Centre, For-

schungszentrum Jülich (Germany) and is also Professor ofQuan-tum Information Processing at RWTH Aachen University (Ger-many). Her current research interests include adiabatic quan-tum computation, quantum statistical physics, computationalelectrodynamics and event-based simulationmethods of quan-tum phenomena.


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Acknowledgements. We would like to thank K. De Raedt, F. Jin,and S. Miyashita for many thoughtful comments and contribu-tions to the work on which this review is based. This work is par-tially supported by NCF, the Netherlands.

Key words. Quantum mechanics, interference, EPR experiments,discrete event simulation.

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