Open Phys. 2017; 15:713–733

Research Article Open Access

Hans De Raedt*, Kristel Michielsen, and Karl Hess The photon identication loophole in EPRB experiments: computer models with single-wing selection https://doi.org/10.1515/phys-2017-0085 1 Introduction Received August 1, 2017; accepted October 5, 2017

Abstract: Recent Einstein-Podolsky-Rosen-Bohm experi- The debate between Einstein and Bohr, about the founda- ments [M. Giustina et al. Phys. Rev. Lett. 115, 250401 (2015); tions of quantum mechanics, resulted in a Gedanken- ex- L. K. Shalm et al. Phys. Rev. Lett. 115, 250402 (2015)] periment suggested by Einstein-Podolsky-Rosen (EPR) [28] that claim to be loophole free are scrutinized. The com- that was later modied by Bohm (EPRB) [7]. The schemat- bination of a digital computer and discrete-event simula- ics of this experiment is shown in Figure 1 and involves two tion is used to construct a minimal but faithful model of wings and two measurement stations. EPR used the quan- the most perfected realization of these laboratory experi- tum mechanical predictions for possible outcomes of this ments. In contrast to prior simulations, all photon selec- experiment to show that quantum mechanics was incom- tions are strictly made, as they are in the actual experi- plete. ments, at the local station and no other “post-selection” is Many years later, John S. Bell derived an inequality for involved. The simulation results demonstrate that a man- the possible outcomes of EPRB experiments that he per- ifestly non-quantum model that identies photons in the ceived to be based only on the physics of Einstein’s rel- same local manner as in these experiments can produce ativity [6]. Bell’s inequality, as it was henceforth called, correlations that are in excellent agreement with those of seemed to contradict the quantum predictions for EPRB the quantum theoretical description of the corresponding experimental outcomes altogether [6]. thought experiment, in conict with Bell’s theorem which Experimental investigations, following Bell’s theoreti- states that this is impossible. The failure of Bell’s theo- cal suggestions, provided a large number of data that were rem is possible because of our recognition of the photon violating Bell-type inequalities and climaxed in the suspi- identication loophole. Such identication measurement- cion of a failure of Einstein’s physics and his basic under- procedures are necessarily included in all actual experi- standing of space and time [3, 12, 15, 57, 89]. ments but are not included in the theory of Bell and his Central to these discussions and questions are the cor- followers. relations of space-like separated detection events, some of which are interpreted as the observation of a pair of enti- Keywords: Einstein-Podolsky-Rosen-Bohm experiments, ties such as photons. The problem of classifying events as Bell inequalities, discrete-event simulation, foundations the observation of a “photon” or of something else is not of quantum mechanics as simple as in the case of say, billiard balls. The particle PACS: 03.65.Ud, 03.65.Ta, 02.70.-c identication problem is, in fact, key for the understand- ing of the epistemology of correlations between events. What do we know about such correlations of space- like separated events? Popular presentations of Bell’s work typically involve two isolated agents (Alice, Bob) at sepa- *Corresponding Author: Hans De Raedt: Zernike Institute for rated measurement stations (Tenerife, La Palma), who just Advanced Materials, University of Groningen, Nijenborgh 4, NL- collect data of local measurements. But how does Alice 9747AG Groningen, The Netherlands, E-mail: [email protected] know that she is dealing with a particle of a pair of which Kristel Michielsen: Institute for Advanced Simulation, Jülich Su- Bob investigates the other particle? She is supposed to be percomputing Centre, Forschungszentrum Jülich, D-52425 Jülich RWTH Aachen University, D-52056 Aachen, Germany, E-mail: totally isolated from Bob’s wing of the experiment in order [email protected] to fulll Einstein’s separation and locality principle! The Karl Hess: Center for Advanced Study, University of Illinois, Ur- answer is that neither Alice nor Bob know they deal with bana, Illinois, USA Email: [email protected]

Open Access. © 2017 H. De Raedt et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. 714 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

a a D 1 2 D +,1 +,2 S D D -,1 -,2

Figure 1: Diagram of the EPRB thought experiment. The source S, activated at times labeled by n = 1, 2, ... , N, sends a photon to the observation station R1 and another photon to the observation station R2. Depending on the settings of these observation stations repre- sented by unit vectors a1 and a2, the signal going to the left (right) triggers the counters D+,1 or D−,1 (D+,2 or D−,2).

correlated pairs if their stations are completely separated of the physical reality of the experiments in both wings. It from each other and have no space-time knowledge of the is important to note that these data must involve measure- other wing ever. ments in both stations and are necessarily inuencing the In his papers on the EPRB experiment, Bell did not ad- possible knowledge of the correlations of the single mea- dress this fundamental question but considered correlated surements in these stations. In the case of atomic or sub- pairs as given, without any trace of the tools of measure- atomic measurements the measurement equipment does ment and of space-time concepts that are both necessary to not only inuence the single outcomes as Bohr has taught accomplish the identication of events. He then claimed to us, but correlated measurement equipment (such as syn- have discovered a conict between his theoretical descrip- chronized clocks or instruments that determine thresh- tion and the quantum theoretical description of the EPRB olds) also inuence the knowledge of correlations of these thought experiment [6]. As a consequence of this discov- single outcomes. It is this extension of the Copenhagen ery, much research was devoted to view that leads to a loophole in Bell’s Theorem, the pho- ton identication loophole. The violations of Bell-type in- (i) the actual derivation of Bell-type inequalities from equalities described in this paper are based on this loop- Einstein’s framework of physics (particularly his sep- hole. aration principle that derives from the speed of light Bell and followers envisage that the correlations may in vacuum (c being the limit of all speeds) and Kol- be measured in the laboratory in complete separation and, mogorov’s probability theory [59], therefore, physical models of the Bell opponents must only (ii) designing and performing laboratory experiments use the measurements of two completely separated wings that provide data that are in conict with the Bell- operated by Alice and Bob who know nothing of each type inequalities, other. As just explained, correlations of spatially separated (iii) constructing mathematical-physical models, at times events can only be conceived by involving the human- supported by computer simulations that entirely invented space-time system in order to demonstrate the comply with Einstein’s and Einstein’s separation pairing, the knowledge that measurements of particles be- principle, that do not rely on concepts of quantum long together. Therefore, correlations can only be deter- theory, and are nevertheless in conict with the Bell’s mined in a consistent way under the umbrella of a given theorem. space-time system that encompasses the two or more mea- surement stations. This space-time system that all experi- menters subscribe to enables us to ban spooky inuences Bell’s theory, and the theories of all his followers, includ- out of science, particularly for the EPR experiments that ing Wigner, do not deal with the identication of the corre- Einstein constructed for the purpose to show this fact. lated particles and assume that they are known automat- We maintain that it is highly non-trivial to identify ically, so to speak per at. But the knowledge of pairing (correlated) photons by experimental methods and that requires additional data or additional channels of infor- this identication involves, at least in some way, a space- mation. These additional data may be measurement times, time system, a system developed by the human mind and certain thresholds for detection and many other elements The photon identication loophole Ë 715 agreed upon by all experimenters and evaluators of the Section 4 discusses the relevance of counterfactual Bell-type experiments. In fact, the identication of particle deniteness (CFD) in relation to the derivation of Bell-type pairs requires certain knowledge of the space-time prop- inequalities and hence also for computer simulation mod- erties of all the experimental equipment involved. This els that yield data that are in conict with these inequali- knowledge must necessarily extend to measurement sta- ties. In Section 5, we introduce a CFD-compliant computer tions far from each other and is, therefore, “non-local”. Of simulation model of the recent EPRB experiments [34, 85]. course, this non-local knowledge does not imply that there We discuss the correspondence of the essential elements are non-local physical inuences on the data measured in of the latter with those of the simulation model but refrain the two wings. In EPRB experiments [34, 38, 85, 89], great from plunging into the details of the algorithm itself. care is taken to rule out the possibility that the observed Section 6 gives a simple, rigorous proof that oper- correlations are due to physical inuences that travel with ationally but not conceptually, photon identication in velocities not exceeding the speed of light in vacuum. But each wing by a local voltage threshold [34], photon identi- without that non-local knowledge, only spooky inuences cation in each wing by a local time window, and photon are left as a possibility for connecting events in the two ex- identication by time-coincidence counting are all math- perimental wings. A space-time knowledge of all involved ematically equivalent. In Section 7, we discuss the con- equipment is nevertheless required to apply the scientic sequences of excluding from a model, at least one fea- method. ture that is essential for an experiment to yield useful data. We argue that Bell-type models, which are believed to be relevant for the description of recent EPRB experi- 2 Aim and structure of the paper ments [34, 85], suer from the photon identication loop- hole in a dramatic manner. Section 8 is devoted to a simple proof that the sim- As explained above, it is essential that the identication of ulation model introduced in Section 5 is CFD-compliant. photons is included into any meaningful theoretical model In Section 9, we give a simple derivation of the Bell [5], of an EPRB experiment because otherwise, the model is Clauser-Horn-Shimony-Holt (CHSH) [14], Eberhard [27], too simple to describe this experiment. Failing to do so, and Clauser-Horn (CH) [13] inequalities for real data, not only spooky inuences can explain the observed pair cor- for the imagined data produced by probabilistic models relations. Specically, omitting the inclusion of data which (which are discussed in the Appendix) and in Section 10 select the photons and/or pairs opens an Einstein-local we present a more general inequality that accounts for loophole, which we call henceforth the photon identica- the local photon identication procedure, employed in the tion loophole. By design, Bell-type models for the recent laboratory experiments. experiments that claim to be loophole free [34, 85] suer In section 11, we specify the computer simulation al- from this loophole. gorithm and the simulation procedure in full detail. A rep- The main aim of this paper is to show that by exploit- resentative collection of simulation results is presented in ing this loophole, or formulated more positively, by con- section 12. The main conclusion from these simulations is structing a model that captures the essence of these re- that a non-quantum model that employs the same pho- cent laboratory experiments [34, 85], a manifestly non- ton identication method in each wing of the EPRB experi- quantum computer simulation of an Einstein-local model ments as the one used in recent EPRB experiments [34, 85], that employs the same local photon identication method reproduces the results of quantum theory of the EPRB in each wing of the EPRB experiments as in recent EPRB ex- thought experiment. periments [34, 85], yields the photon pair correlation of a In section 13, we argue that all EPRB laboratory ex- pair of photons in the singlet state (of their polarizations), periments with photons can be viewed as a tool to char- in blatant contradiction with Bell’s theorem. acterize the response of the observation stations, leading The paper is structured as follows. Section 3 argues to the conclusion that this response, in particular the lo- that a simulation on a digital computer is a perfect lab- cal photon identication rate, depends on the settings of oratory experiment with a physically existing device and these stations, consistent with the assumptions made in can therefore be used as a metaphor for other laboratory constructing the simulation model and debunking the hy- experiments. The material in this section forms the con- pothesis that the observed pair correlations can be ex- ceptual basis for developing computer simulation models plained by non-local inuences only. The paper ends with of the recent EPRB experiments [34, 85]. section 14 which contains our conclusions. 716 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

Figure 2: Diagram indicating the direction of the modeling process adopted in this paper. An algorithm executing on a digital computer is an instance of an experiment on a physically existing device, a metaphor for a perfect laboratory experiment in which there are no unknown influences.

3 Metaphor for a perfect laboratory In the analysis of laboratory EPRB experiments, it is essential that all the important degrees of freedom that af- experiment fect the data analysis are identied and included, other- wise the conclusions drawn from an incomplete analysis An important, characteristic feature of digital computers may be wrong [36]. Computer simulation puts us in the po- is that their logical operation does not depend on the tech- sition to perform experiments under the same mathemati- nology that is used to construct the machine. These days, cal conditions for which e.g. Bell-type inequalities can be the rst thing that comes to mind when talking about dig- derived, simply because we can carry out real, perfect ex- ital computers are the electronic machines based on semi- periments that are void of any unknown elements that may conductor technology but it is a fact that, although not aect the results and analysis. cost-eective nor particularly useful in practice, digital A digital computer is a physical (electronic or me- computers can also be built from mechanical parts, e.g. chanical) device that changes its physical state (by ip- LegoTM elements. ping bits) according to well-dened rules (the algorithm). In former times, one could read o the state of the Therefore, assuming that the machine is operating aw- computer’s internal registers from a LED display. Although less for the time period of interest (a very reasonable as- not practical at all, in principle one could use a huge LED sumption these days), executing an algorithm on a digi- display to show the internal state of the whole computer. tal computer is a physics experiment in which there are no This is only to say that there is a one-to-one mapping from unknown elements of physical reality that might aect the the state of the computer to sense impressions (e.g. light outcome. In this sense, the “digital computer + algorithm” on/o). Therefore, the metaphor also oers unique possi- can be viewed as a metaphor for a perfected laboratory ex- bilities to confront man-made concepts and theories with periment, a discrete-event simulation that represents the actual facts, i.e. real perfect experiments, because it guar- so called “loophole-free” EPRB experiments [34, 85]. antees that we have a well-dened, precise representation Starting with Bell’s work [5], most theoretical work of the concepts and algorithms (both in terms of bits) in- on the subject matter is based on probability theory. This volved that directly translate into sense impressions. mathematical framework contains conceptual elements (probability measures and innitesimals) that are outside The photon identication loophole Ë 717 the domain of our sensory experiences and have no coun- Mathematically, we may express this dependence as y = terpart in our physical world. Therefore, to avoid pitfalls, F(x, C, U) where now F(.) is a vector-valued function of its we rst devise an algorithm that simulates the perfect arguments and C and U represent the known condition- laboratory experiment and then construct a probabilistic s/settings and unknown inuences, respectively. In gen- model of this algorithm. A graphical representation of the eral, there is no reason why U should be a constant from modeling philosophy that we adopt in this paper is shown trial to trial. In other words, F(.) is unknown. Therefore, in Figure 2. Note that our approach starts from the experi- data produced by laboratory EPRB experiments (or any ments and results in a theory, which we believe is the only other laboratory experiments) cannot, as a matter of prin- direction one should go. In contrast, Bell’s approach was ciple, be cast in the form of data generated by a CFD- the design of an experiment starting from his theoretical compliant model. On the other hand, performing com- point of view. puter experiments in a CFD-compliant manner is not dif- cult nor is it much work to change a CFD-compliant al- gorithm into one that does not meet all the requirements 4 Counterfactual deniteness of CFD simulation. In other words, computer experiments can be carried out in both CFD and non-CFD mode, provid- ing quantitative information about the dierences of these Counterfactual reasoning plays a signicant role in the lit- two modes of modeling. erature related to Bell’s work and is seen by many a con- In the realm of nite sets of two-valued data, a strict ditio sine qua non to derive Bell-type inequalities [18, 40, derivation of Bell-type inequalities [5, 6], such as the Bell- 56, 67]. However, as explained below, the actual EPRB ex- CHSH [5, 14] and Eberhard’s inequalities [27] require that periments do not permit any proof of CFD compliance. This these data are generated in a CFD-compliant manner [23, fact demonstrates an unexpected conceptual advantage of 40]. In other words, CFD is a prerequisite for deriving Bell- computer experiments. We can turn on and o CFD com- type inequalities. Therefore, to test whether or not a sim- pliancy at will in our algorithm and simulate the conse- ulation model produces data that do not satisfy such in- quences and thus distinguish the precise conditions that equalities, it is necessary to perform a CFD-compliant sim- may or may not lead to violations of Bell-type inequalities. ulation. Otherwise, there is no mathematical justication This is our reason to dedicate signicant sections of this for the hypothesis that these data should satisfy Bell-type paper to counterfactual deniteness (CFD) as dened be- inequalities in the rst place. Of course, we can always re- low, and include CFD-compliant models at the side of more vert to the non-CFD algorithm and check if e.g. averages faithful models of the actual EPRB experiments. exhibit the same features as the averages obtained from So called counterfactual “measurements” yield values the CFD-compliant algorithm (see section 12). that have been derived by means other than direct obser- In an earlier paper [23], we have adopted this strategy vation or actual measurement, such as by calculation on to demonstrate that in the case of EPRB experiments, the basis of a well-substantiated theory. If one knows an equation that permits deriving reliably, output values from 1. CFD-compliant simulations can reproduce the aver- a list of inputs to the system under investigation, then one ages and correlation of two particles in the singlet has “counterfactual deniteness” (CFD) in the knowledge state, of that system [79]. 2. CFD does not distinguish classical from quantum The word “counterfactual” is a misnomer [79] but is physics because our computer models do not contain well established. It is therefore helpful to have a clear- any quantum concepts, yet yield results that lead to cut operational denition of what is meant with CFD. In conclusions (e.g. entanglement) that are commonly essence, CFD means that the output state of a system, rep- regarded as signatures of quantum physics. resented by a vector of values y, can be calculated using an explicit formula, e.g. y = f(x) where f(.) is a known vector-valued function of its argument x. If x denotes a In this paper, we adopt the same strategy. We construct vector of values, the elements of this vector must be inde- a CFD-compliant simulation model of the laboratory ex- pendent variables for the mathematical model to be CFD- periments [34, 85], meaning that we simulate a perfected, compliant [23, 40]. idealized realization of these laboratory experiments. Of In laboratory EPRB experiments, every trial takes course, this does not mean that we omit essential features place under dierent conditions, dierent settings etc. of the laboratory experiments. These features have to be which may or may not aect the outcome of a single trial. included, otherwise the simulation model is not applica- 718 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

tion is a two-valued variable x = ±1 and a detector-related variable vmin ≤ v ≤ vmax. The correspondence between the data produced by the experimental realization of an observation station and those generated by the computational model is as follows. The variable x encodes the detector outcomes (either D+,i or D−,i in station i = 1, 2) that red. In the laboratory ex- periments [34, 85] there is only one detector in each sta- tion but in the computer experiment we can easily simu- late the complete EPRB experiment (see Figure 1), hence we consider both the “+” and “−” events. The variable v Figure 3: Block diagram of an observation station. The specic form represents the voltage signal produced by the electronics x x a ϕ r v v a ϕ r of the input-output relations = ( , , ) and = ( , ,b) is that amplies the transition-edge detector current (see the not important to follow the discussion on the conceptual level and is only essential to perform the simulations reported in this paper. description in section IV of the supplementary material to These relations are dened by Eqs. (19) and (20) in section 11. Ref. [34]). If necessary, we label dierent events by attaching the subscript i = 1, 2 of the observation station and/or the subscript k where k = 1, ... , N and N denotes the to- ble to these laboratory experiments. see section 7 for a gen- tal number of input events to a station. In full detail, for eral discussion of this aspect. the kth input at station i, the observation station i gen-

erates the output values xi = xi(ai , ϕi,k , ri,k) and vi = vi(ai , ϕi,k ,bri,k) according to the rules which will be spec- 5 Computational model of the ied in full detail in section 12. Occasionally, we write x a x a ϕ r v a v a ϕ r laboratory experiments [34, 85] i( i) = i( i , i,k , i,k) and i( i) = i( i , i,k ,bi,k) to simplify the notation while still emphasizing that the x’s and v’s only depend on variables that are local to the re- In this section, we introduce the essential elements of the spective station. simulation model of the laboratory experiments reported in Ref. [34, 85]. The details of the simulation algorithm are given in section 11. For concreteness, we adopt the termi- 5.2 Photon identication nology that is used in Ref. [34] when we connect the ele- ments of the simulation model to those of the laboratory In the following, we use the term detection event when- experiments. ever the negative voltage signal produced by the electron- As shown in Figure 1, in a typical EPRB experiment ics that amplies the transition-edge detector current is there are three dierent components. There is a source and smaller (we are dealing with negative voltages) than the there are two observation stations. The algorithm that sim- “trigger threshold” (terminology from Ref. [34] (supple- ulates the source is described in full detail in section 11. In mentary material)), and speak of the observation of a pho- this section, we focus on the observation stations which, ton whenever the same negative voltage signal is smaller because we are performing “perfected” experiments, are than the “photon identication threshold” (about 4/3 assumed to be identical. times the “trigger threshold”) (terminology from Ref. [34] (supplementary material)). 5.1 Observation station From the description of the laboratory EPRB experi- ments under scrutiny, it follows immediately that not ev- ery detection event is regarded as the observation of a In Figure 3, we show a graphical representation of the photon [34, 85]. Indeed, after all the voltage traces of an function of an observation station. Input to an observa- experimental run have been recorded, a part of the col- tion station is the setting a (representing the orientation lected trace is analyzed by software, the photon iden- of the polarizer), two numbers 0 ≤ r, r < 1 taken from a b tication thresholds are “calibrated” and assuming that list of uniform random numbers (see section 11 for further the relevant properties of the whole set of traces is sta- details) and an angle 0 ≤ ϕ < 2π (representing the initial tionary in real time, the remaining set of traces is ana- polarization of the photon). Output of the observation sta- lyzed [34]. In Ref. [34] there is no specication of the cost The photon identication loophole Ë 719 function that is being minimized by the calibration proce- is v ≤ V. Recall that we adopted the convention of the lab- dure whereas Ref. [85](supplementary material) explicitly oratory experiments [34] in which V takes negative values. states that “Because the experiment was calibrated to maxi- In practice, we have vmin ≤ vi ≤ vmax and vmin ≤ mize violation of the CH inequality...”. This seems to suggest V ≤ vmax with nite vmin and vmax, hence the condition for that the software is designed to adjust the photon identi- counting a detection event as photon may be written as cation thresholds such that the desired result, namely a vi − v V − v violation of a Bell-type inequality, is obtained. 0 ≤ min ≤ min , i = 1, 2. (2) vmax − vmin vmax − vmin In our simulation approach, we may assume that all units are identical. Therefore, unlike in Ref. [34], one and Dening a dimensionless “time” ti ≡ (vi − vmin)/(vmax − the same value of photon identication threshold, de- vmin) and a dimensionless “time window” W = (V − noted by V, can be used to identify photons. The eect vmin)/(vmax − vmin), Eq. (2) reads of the photon identication threshold is captured by the t W i function 0 ≤ i ≤ , = 1, 2, (3) which expresses the condition to observe a photon at sta- w(ai) = wi(ai , ϕi,k ,bri,k) = Θ(V − vi) , i = 1, 2, (1) tion i = 1, 2 in terms of locally dened time slots of size where Θ(x) is equal to one if x > 0 and is zero otherwise. W. From Eq. (3) we have −t2 ≤ t1 − t2 ≤ W − t2 and using Recall that, as in Ref. [34], V is negative. In the simulation, −W ≤ −t2 we nd we do not “calibrate” V but simply generate the data and |t t | W analyze the results as a function V. 1 − 2 ≤ , (4) The correspondence with the data collected in the lab- which is exactly the same criterion as the one used in oratory experiment is as follows: a detection event is rep- most EPRB experiments with photons [3, 15, 57, 89] (with resented by xi(ai) = +1 and wi(ai) = 0 and the observation t1, t2 and W representing physical times in that case) of a photon in station i = 1, 2 is represented by xi(ai) = +1 and in computer simulation models thereof [19, 20, 22– (because there is only one, not two, transition-edge de- 26, 78, 90]. tectors at each station) and wi(ai) = +1 (implemented in In summary: although physically very dierent, lo- software), both exactly as in the simulation model. Recall, cal voltage thresholds, local time windows or time- and this is new and important, that also in the simulations coincidence counting are mathematically equivalent and the photon identication is performed locally, i.e. without all serve the same purpose, namely to give an operational communication between the observation stations. meaning to the statement “a single photon (pair)” has been identied. 6 Equivalence of local time-window and time-coincidence processing 7 Loopholes in experimental tests of Bell’s theorem In this section we show that in spite of the conceptually very dierent setup, from an operational point of view, A useful model of an experiment needs to encompass all employing local photon identication thresholds is equiv- relevant parameters that aect the experimental outcomes alent to local time-window selection and also to time- and, of course, the most important elements of physical coincidence counting that is used in most EPRB experi- reality, namely the data itself. Specically, a physical the- ments with photons [3, 15, 57, 89]. ory that describes pair-correlations of space-like separated As explained above, in the laboratory experiments a systems, must account for and include all procedures that detection event is classied as being a photon if the (neg- determine the detection of the particles and the knowledge ative) voltage signal, denoted by v, produced by the elec- which pair of particles and data belongs together. There- tronics that amplies the transition-edge detector current fore, any model which purports to describe the laboratory (see the description in section IV of the supplementary ma- experiments [34, 85] that we consider in this paper must terial to Ref. [34]) is smaller than the photon identica- necessarily account for the photon identication thresh- tion threshold V. This rejection criterion is implemented old mechanism that is instrumental in the data-processing through Eq. (1) from which it follows directly that the cri- step of these experiments, see section 5.2. Likewise, the terion to observe a photon in these laboratory experiments earlier generation of EPRB experiments that employ time 720 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

Figure 4: Schematic layout of the computational equivalent of the laboratory EPRB experiments reported in Refs. [34, 85]. The input-output relation for i = 1, 2 is given by xi = xi(ai , ϕi , ri) and vi = vi(ai , ϕi ,bri) dened by Eqs. (19) and (20). Alternatively, the input-output rela- tion may be written as (x1, v1, x2, v2) = F(a1, a2, ϕ1, ϕ2, r1, r2,br1,br2) showing that for xed (a1, a2) the simulation model satises the denition of a CFD theory [40].

Figure 5: Computational model for the EPRB experiment satisfying the criterion of a CFD theory.

coincidence to identify pairs [3, 13, 89] can only be faith- The photon-identication loophole that we introduce fully be described by models that incorporate the time- in this paper accounts for coincidence window selection process that is an essen- tial component of this class of experiments [19, 20, 22– (i) the fact that laboratory experiments [34, 85] employ a 26, 78, 90]. threshold to decide whether or not a detection event Drawing a conclusion about a world view from mod- is considered to be a photon, els (such as those of Bell and his followers, see the Ap- (ii) the assumption that voltage signals produced by the pendix) that do not properly account for the photon iden- detection equipment may depend on the analyzer tication threshold mechanism which, in the laboratory setting (see Eq. (1)). Regarding this latter assumption, experiments [34, 85], is essential for identifying the pho- it is of interest to recall that since the early days of tons, requires a drastic departure from rational reasoning. the Bell-test experiments, it is well-known that appli- If we allow for such a departure, we might equally well cation of Bell-type models requires at least one extra wonder what it means for our world view when we con- assumption. We reproduce here the relevant passage struct and analyze a model of an airplane that excludes the from Ref. [13] (p.1890): “The approach used by CHSH engines and then observe that a real airplane can take o is to introduce an auxiliary assumption, that if a par- by itself. Any reasonable person would rightfully question ticle passes through a spin analyser, its probability of our ability to represent the airplane (or laboratory experi- detection is independent of the analyser’s orientation. ments) by such a model and regard the idea that we may Unfortunately, this assumption is not contained in the have to change our world because of the contradictions to hypotheses of locality, realism or determinism.” such a model as unfounded. In other words, the only logi- (iii) the requirement that a relevant model of an experi- cally correct conclusion that one can draw from the failure ment needs to encompass all elements that aect the of Bell-type models to describe the qualitative features of experimental outcomes. the experimental data is that these models are too simple. The photon identication loophole Ë 721

There is a large body of theoretical work that considers all putes the vector-valued function kinds of loopholes in experiments that must be closed be-     x x = x (a , ϕ , r ) fore a denite conclusion about the consequences of Bell 1 1 1 1 1 1  v   v = v (a , ϕ , r )  inequality tests for certain wold views can be drawn. A de-  1  =  1 1 1 1 b1   x   x = x (a , ϕ , r )  tailed, comprehensive discussion of a large collection of  2   2 2 2 2 2  v v = v (a , ϕ , r ) loopholes is given in Ref. [68]. Also in this respect, the 2 2 2 2 2 b2 digital computer – laboratory experiment metaphor oers = F(a1, a2, ϕ1, ϕ2, r1, r2,br1,br2), (5) unique possibilities because we can open and close loop- which clearly denes a CFD-compliant model. Neverthe- holes at will. As this and our earlier paper [23] demon- less, with this CFD-compliant model we cannot construct strate, computer simulation models of EPRB experiments the quadruple (x , x , x′ , x′ ) in a CFD-compliant manner. can easily be engineered to be free of e.g. detection, coin- 1 2 1 2 Indeed, by construction, there is no guarantee that the cidence, and memory loopholes [68] and, in addition, in- (ϕi k , ri k)’s that determine say the x ’s for the pair of set- clude features such as CFD compliance that close the con- , , 1 tings (a , a ) will be the same as the (ϕi k , ri k)’s that deter- textuality loophole [75–77]. 1 2 , , mine that values of the x′ ’s for the pair of settings (a′ , a ). Wrapping up: in this paper we construct a minimal 1 1 2 Of course, with a simulation on a digital computer being model of the perfected version of the laboratory experi- an ideal, fully controllable experiment, we could enforce ment [34]. With the exception of the photon-identication CFD-compliance by re-using the same (ϕi,k , ri,k ,bri,k)’s for loophole, this minimal model is free of the known loop- every pair of settings. This would make the simulation holes and reproduces the quantum results of the EPRB CFD-compliant. However, in this paper we do not so but thought experiment, from which violations follow auto- instead generate new values of the (ϕi,k , ri,k ,bri,k)’s for ev- matically. This approach oers the unique possibility to ery new instance of input. confront all kinds of reasonings and assumptions, such as The layout of a CFD-compliant computer model of the (ii) above, with actual facts. the EPRB experiment is depicted in Figure 5. It uses the same units as the model shown in Figure 4, the only dierence being that the input ϕi is now fed into 8 CFD compliance an observation station with setting ai and into another ′ one with setting ai, something which, for obvious rea- In section 5, the operation of the simulation model of an sons, is impossible to realize in laboratory experiments observation station has been dened such that for every with photons. As each of the four units operates accord- input event (a, ϕ, r,br), we know the values of all outputs ing to the rules given by Eq. (19) and (20), we have ′ ′ ′ ′ variables x = x(a, ϕ, r) and v = v(a, ϕ,br). Therefore, (x1, x1, x2, x2) = X(a1, a1, a2, a2, ϕ1, ϕ2, r1, r2) and ′ ′ ′ ′ the input-output relation of this unit, represented by the (v1, v1, v2, v2) = T(a1, a1, a2, a2, ϕ1, ϕ2,br1,br2). As the diagram of Figure 3, satises the requirement of a CFD- arguments of the functions X and T are independent and compliant model. may take any value out of their respective domain, the The computational equivalent of the EPRB experi- whole system represented by Figure 5 satises, by con- ments [34, 85] is shown in Figure 4. Each time the source struction, the criterion of a CFD theory. S is activated, it sends one entity carrying the data ϕ1 to Note that the actual EPRB experiments produce only station 1 and another entity carrying the data ϕ2 to station pairs of data. The three pairs of data considered by Bell in- 2. The procedure for generating the ϕ’s, r’s and br’s is spec- volve, therefore, six local measurements and the four pairs ied in section 12. of CHSH involve eight local measurements. Our CFD com- Upon arrival of the entities, observation stations i = pliant model considers only quadruple (= four local) mea- 1, 2 execute their internal algorithm (completely specied surements to simulate the actual eight possible measure- by Eqs. (19) and (20)) and produces output in the form of ment outcomes of a CHSH type experiment. the pair (xi , vi). The scheme represented by Figure 4 com-

9 Bell-type inequalities

It is evident from the formulation of his model that Bell and all his followers, including Wigner, do not deal with the issue of identifying particles and take for granted that 722 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess the measured pairs correspond to the correlated particles. that in order for all equalities to be satised simultane- ′ ′ ′ ′ The common prejudice that additional variables cannot ously we must have ex1 = x1, ex1 = x1, ex2 = x2, ex2 = x2. possibly defeat Bell-type inequalities is based on the as- In other words, imposing the constraints es = −2, +2 and sumption that all sent out correlated pairs, or a represen- eb1 = −1, 3, ... , eb4 = −1, 3 on data obtained in a non-CFD tative sample of them, are measured. This reasoning does setting forces this data to form quadruples, i.e. to be CFD not account for the photon identication or pair-modeling compliant. It then follows immediately that CFD is nec- loophole: the necessary particle or pair identication may essary and sucient for the equalities Eqs. (6) – (10) to necessarily select in a way that is not representative for hold. all possible measurements of all possible pairs emanat- Attaching the subscript k (k = 1, ... , N) to label ing from the source. In this section, we adopt Bell’s view- the events, the algorithm generates the set of quadruples { ′ ′ | } point by ignoring the v-variables and demonstrate that (x1,k , x1,k , x2,k , x2,k) k = 1, ... , N . Introducing the CFD-compliance and the existence of Bell-type inequali- Bell-CHSH function ties are mathematically equivalent. N 1 X bS s Figure 5 shows the CFD compliant arrangement of the = N k , (11) computer experiment. The two stations on the left of the k=1 source S receive the same data ϕ1 from the source. The set- it follows immediately from |sk| = 2 (see Eq. (10)) that ′ tings a1 and a1 are xed for the duration of the N repeti- |bS| ≤ 2 for all N ≥ 1, that is we obtain the Bell-CHSH in- tions of the experiment. The same holds for the two sta- equality constraining four correlations of pairs of actual tions on the right of the source, with subscript 1 replaced data. Put dierently, if the output consists of quadruples of by 2. Clearly, the algorithm represented by Figure 5 gen- two-valued data generated by the setup shown in Figure 5 ′ ′ erates quadruples of output data (x1, x1, x2, x2) in a CFD- and we ignore the v-variables then the Bell-CHSH inequal- compliant manner. ity |bS| ≤ 2 is always satised, independent of the number ′ ′ For any such quadruple (x1, x1, x2, x2) in which the of events N ≥ 1 considered. x’s only take values +1 and -1, it is straightforward to verify Similarly, from the fact that for example b1,k = −1, 3 that the following equalities hold: we obtain the Leggett-Garg inequality [41, 69] for three cor-

( relations of pairs of actual data and by combining b1,k = ′ ′ −1 b1 = x1x + x1x2 + x x2 = (6) −1, 3 with the equalities obtained by substituting x1 → 1 1 +3 −x we obtain the Bell inequality involving three correla- ( 1 ′ ′ ′ ′ −1 tions of pairs of actual data [5]. In other words, Bell-type b2 = x1x + x1x + x x = (7) 1 2 1 2 +3 inequalities follow directly from the fact that quadruples ( of data satisfy rather trivial arithmetic identities such as ′ ′ −1 b3 = x1x2 + x1x + x2x = (8) Eq. (10). 2 2 +3 It then also follows immediately that CFD is a nec- ( essary and sucient x x b x′ x x′ x′ x x′ −1 condition for the data ( 1,k , 2,k), 4 = 1 2 + 1 2 + 2 2 = (9) ′ ′ ′ +3 (ex1,k , x2,k), (x1,k , ex2,k), and (ex1,k , ex2,k) with k = 1, ... , N ( to satisfy simultaneously for all N ≥ 1, all Bell-type in- ′ ′ ′ ′ −2 s = x1x2 − x1x + x x2 + x x = . (10) equalities involving three and four dierent correlations 2 1 1 2 +2 of pairs. We emphasize that this conclusion follows from Other equivalent sets of equalities can be obtained by re- elementary arithmetic only. Concepts such as “locality” or placing e.g. x1 by −x1 etc. Note that e.g. Eq. (6) follows from any other physical argument are irrelevant for establishing ′ Eq. (10) if we set x2 = x1. this result. In a non-CFD setting, the data is collected as four pairs Similar reasoning yields Eberhard’s inequality which ′ which we may denote as (x1, x2), (ex1, x2), (x1, ex2), and diers from the Bell-CHSH inequality in the sense that it ′ ′ (ex1, ex2) where the tilde is used to indicate that the value of can account for reduced detector eciencies [27]. For con- ′ e.g. x1 obtained with setting (a1, a2) may be dierent from venience of comparison with the original work, we tem- the one obtained with setting (a1, a2). Instead of Eq. (10), porarily adopt Eberhard’s parlance and notation. Central ′ ′ we now consider the expression es = x1x2 − ex1x2 + x1ex2 + to Eberhard’s derivation is the so-called fate of a photon. ′ ′ ex1ex2 = −4, −2, 0, +2, +4 and similar ones for eb1, ... , eb4, This fate can be either detected in the ordinary beam (la- each of them taking values −3, −1, +1, +3. If we now im- beled o), or detected in the extraordinary beam (labeled pose that es = −2, +2 and eb1 ... , eb4 = −1, 3, simple enu- e), or undetected (labeled u). For counting purposes, we meration of all possible values of the x’s and the ex’s shows represent the fate of a photon by the symbol f , taking the The photon identication loophole Ë 723 values +1, 0, and −1 corresponding to o, u and e, respec- encounter data that violate a Bell-type inequality. The log- tively. Introducing the variables no = f (f + 1)/2, ne = ically correct conclusion that one can draw from such a 2 f (f − 1)/2, and nu = (1 − f ), it is clear that one of them violation is that these data have not been generated in a takes the value 1 with the other two taking the value 0. CFD-compliant manner. For a given pair of settings, say (α1, β2), the number of pairs with both photons suering fate (o) is then given by noo(α1, β1) = no(α1)no(β1) = f1,1(f1,1 + 1)f2,1(f2,1 + 1)/4 10 An inequality accounting for where f1,1 = f1,i(αi) and f2,i = f2,i(βi) for i = 1, 2. There are similar expressions for neo(α1, β2), nuo(α1, β2), etc. Fol- photon identication lowing Eberhard, we consider the expression [27] In this section, we address the modications to the in- j = noe(α1, β2) + nou(α1, β2) + neo(α2, β1) equality |bS| ≤ 2 that ensue when we take into account the n α β n α β n α β + uo( 2, 1) + oo( 2, 2) − oo( 1, 1). (12) fact that laboratory experiments employ the photon iden- It is straightforward to enumerate all possible 81 values of tication threshold to decide whether or not a detection the 4 dierent f -variables that appear in Eq. (12). This enu- event corresponds to the observation of a photon. meration proves that j ≥ 0, independent of the values of The average detection event counts and detection the settings. Attaching the subscript k (k = 1, ... , N) to la- event pair correlation are given by bel the events as we did to derive the Bell-CHSH inequality 1 X Ebi(ai) = xi(ai) , i = 1, 2 and introducing the Eberhard function N 1 X N Eb a a x a x a X ( 1, 2) = N 1( 1) 2( 2), (16) JEberhard = jk , (13) k=1 respectively, and we have similar expressions for the other it follows immediately that J ≥ 0 for all N ≥ 1. choices of settings. In Eq. (16) and in the equations that Eberhard P PN In the laboratory experiments [34, 85] there is only one follow, it is understood that means k=1, i.e. the sum detector per observation station. Hence it makes sense to over all input events, represented by values of the ϕ’s. As regard also say, the e photons, as undetected. In terms of shown in section 9, if the x’s that enter Eq. (16) have been the “fate” variables f introduced above this amounts to let- obtained by a CFD-compliant procedure, the correlations ting f taking the values +1 and 0 corresponding to o and Eb(a1, a2), ... satisfy Bell-type inequalities. u, respectively. Instead of Eq. (12), we now consider the ex- In contrast to Eq. (16), the average photon counts and pression photon pair correlation for the settings (a1, a2) are given by j n α β n α β n α β n α β CH = ou( 1, 2) + uo( 2, 1) + oo( 2, 2) − oo( 1, 1). P w(ai)xi(ai) (14) Ei(ai) = , i = 1, 2 P w a Enumerating all possible 16 values of the 4 dierent f- ( i) P w(a )w(a )x (a )x (a ) variables that appear in Eq. (13) proves that jCH ≥ 0, in- E(a , a ) = 1 2 1 1 2 2 , (17) 1 2 P w a w a dependent of the values of the settings. Attaching the sub- ( 1) ( 2) script k (k = 1, ... , N) to label the events as before and where, as explained in section 5.2, the w’s in Eq. (17) ac- introducing the CH function count for the eect of the photon identication thresholds N X and take values 0 or 1. Clearly, Eq. (17) is very dierent JCH = jCH,k , (15) from Eq. (16) unless all the w’s that appear in Eq. (17) are k=1 equal to 1, in which case the photon identication thresh- it follows immediately that the CH inequality [13] JCH ≥ 0 old mechanism is superuous and unlike as in the labora- holds for all N ≥ 1. tory experiment [34], the number of photon and detection In short: if for all N, the x’s (f’s) are generated accord- events is the same. ing to a CFD-compliant procedure the Bell-CHSH (the Eber- In the analysis of the experimental data, the pho- hard and CH) inequality is (are) satised. In essence, this ton identication threshold is chosen such that many of result is embodied in the work of George Boole [8], see also the w’s are zero [34]. Hence from the discussion in sec- Ref. [21, 83]. Moreover, as CFD implies that all Bell-type in- tion 9, it follows immediately that with some w’s zero, it equalities hold for all N ≥ 1, there is no room for specu- is impossible to prove that the Bell-CHSH function S = ′ ′ ′ ′ lating without violating at least one of the rules of Aris- S(a1, a2, a1, a2) ≡ E(a1, a2) − E(a1, a2) + E(a1, a2) + ′ ′ totelian logic that something “spooky” is going on if we E(a1, a2) satises the inequality |S| ≤ 2. 724 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

However, it directly follows from the proof given in our computer simulation, the detectors used in these labora- earlier paper [23] that if the x’s and w’s have been gener- tory experiments are not perfect. As already mentioned, ated by a CFD-compliant procedure, the Bell-CHSH func- Eberhard’s inequality can account for reduced detector ef- tion S can never violate the inequality ciencies and this feature can be put to good use through

minimizing the value of JEberhard with respect to the corre- |S| E a a E a a′ E a′ a E a′ a′ δ = ( 1, 2) − ( 1, 2) + ( 1, 2) + ( 1, 2) ≤ 4−2 . lation [27]. This is what is done in the laboratory experi- (18) ments [34, 85]. However, in our simulation model, the de- δ The term 2 in Eq. (18) is a measure for the number of tectors are perfect. Hence the minimum value of JEberhard paired events that have been rejected relative to the num- will be obtained by choosing maximally correlated, or- ′ ber of emitted pairs. In detail, 0 ≤ δ ≡ N /Nmax ≤ 1 where thogonal polarizations [27]. Recall that our aim here is to ′ N denotes the number of input events for which the neg- simulate the most ideal, perfect experiment that accounts ative voltage signal of all the photons is smaller than the for all the essential features of the laboratory experiments, photon identication threshold V and Nmax is the maxi- not to simulate a real laboratory experiment including mum number of contributing pairs per setting. If all paired trams passing by [34] etc. events would be regarded as photon pairs then δ = 1 and Upon receiving the input ϕ an observation station (see then, and only then we recover the Bell-CHSH inequality Figure 3) executes the following two steps. First it retrieves |S| x w ≤ 2. If the ’s and ’s have not been generated by a two uniform random numbers r and br from a list of such CFD-compliant procedure, there is only the trivial bound numbers (or, more conveniently, generates these numbers |S| ≤ 4. on the y) and then The inequality Eq. (18) is a rigorous mathematical fact that holds if, for all N ≥ 1, the x’s and v’s are generated in 1. computes a CFD-compliant manner and none of the denominators in c a ϕ s a ϕ Eq. (17) is identically zero (in which case no photon pairs = cos[2( − )] , = sin[2( − )], (19) have been detected). Conversely, if we nd a set of x’s and 2. sets v’s that yields a value of |S| that exceeds 4 − 2δ, we can only conclude that these data have not been obtained from d x = sign(1 + c − 2 * r) , v = br|s| (Vmax − Vmin) − Vmax, a CFD-compliant procedure. Any other conclusion would (20) not be logically justied. In analogy with the derivation of Eq. (18), one may de- rive an Eberhard-type or CH-type inequality that accounts where d, is an adjustable parameter to be discussed later for the w’s but as such inequalities do not add anything and 0 ≤ Vmax, and 0 ≤ Vmin ≤ Vmax set the range of the to the discussion that follows, we do not discuss them any voltage signal. From Eq. (20) it follows that −Vmax ≤ v, V ≤ further. −Vmin, as in the laboratory experiment [34]. Our choice for the specic functional forms of x = x(a, ϕ, r) and v = v(a, ϕ,br) is inspired by previous work in 11 Discrete-event simulation which it was shown that a similar model, which employs time-coincidence to identify pairs, exactly reproduces the algorithm single particle averages and two-particle correlations of the singlet state if the number of events becomes very In this section, we specify the algorithm and the simula- large [26, 72]. tion procedure in full detail. The algorithm that mimics the Equations (19) and (20) form the core of the simulation operation of the particle source is very simple. For each algorithm which has the following key features: event k = 1, ... , N, a uniform random generator is used – For any xed value of ϕ and uniformly distributed to generate a oating-point number 0 ≤ ϕ1,k ≤ 2π. This random numbers r, the unit generates a sequence of ′ number is input to the stations with setting (a1, a1) and randomly distributed x’s such that the average of the another number ϕ2,k = ϕ1,k + π/2 is input to the sta- x’s agrees (within statistical uctuations) with Malus’ ′ tions with setting (a2, a2). Because of ϕ2,k = ϕ1,k + π/2, law, i.e. the normalized frequencies to observe x = +1 the kth event simulates the emission of a photon pair with and x = −1 are given by cos2(a − ϕ) and sin2(a − ϕ), maximally correlated, orthogonal polarizations. In this re- respectively. spect, we deviate from what is done in the laboratory ex- – The presence of an output variable v which serves to periments [34, 85] in the following sense. Unlike in the mimic the detector traces recorded in the laboratory The photon identication loophole Ë 725

experiments. Note that the explicit expression of v = 1 (a) v(a, ϕ,br) shows a dependence on the local setting of

the station. Such a dependence cannot be ruled out a 0.5 posteriori but nds a post-factum justication in the ) fact that the simulations reproduce the results for two 2 ,a particles in a singlet state, see section 13. 1 0 E(a – The use of the random numbers 0 ≤ r,br ≤ 1 mimics the uncertainties about the outcomes, as observed in -0.5 experiments. Thereby, it is implicitly understood that

for every instance of new input, new values of the uni- -1 form random numbers r and br have been generated. 0 30 60 90 120 150 180 θ – By construction, the algorithm is a metaphor for (degrees) a a 1 Einstein-local experiments: changing 1 ( 2) does not (b) aect the present, past or future values of x2 (x1) or v2 (v1). In plain words, the output of one particular unit 0.5 depends on the input to that particular unit only. ) 2 ,a

1 0 For the settings of the observation stations, we take a =

1 E(a ′ ′ θ + π/8, a1 = a1 + π/4, a2 = π/8, a2 = 3π/8 and let θ vary from 0 to π. For this choice of settings, quantum the- -0.5 ory for a system in the singlet state predicts E(a1, a2) = E a′ a′ θ E a a′ E a′ a θ ( 1, 2) = cos 2 , (√1, 2) = ( 1, 2) = sin 2 and -1 ′ ′ 0 30 60 90 120 150 180 S(a1, a2, a , a ) = −2 2 cos(2θ + π/4), the latter reach- 1 2 √ θ (degrees) ing its maximum 2 2 at θ = 3π/8. When we operate the computer model in non-CFD mode, random numbers are Figure 6: The correlation E(a1, a2) (○), the single-particle averages ′ E a a 4 E a a 5 θ a a used to make a choice between the settings ai and ai, for 1( 1, 2) ( ) and 2( 1, 2) ( ) as a function of = 1 − 2. i = 1, 2, exactly as in the experiments [34, 85]. (a) CFD-compliant model; (b) non-CFD model. Solid line: quantum theoretical prediction E(a , a ) = − cos 2θ. Dashed line: quantum The simulation procedure is quite simple. We choose 1 2 theoretical prediction E (a , a ) = E (a , a ) = 0. The photon V 1 1 2 2 1 2 a xed photon identication threshold , generate input identication threshold is V = −0.995. pairs k = 1, ... , N, collect corresponding outputs in terms of x’s and w’s, and compute the single- and two-particle averages according to Eq. (18), the Bell-CHSH function ′ ′ ′ ′ ger threshold” (terminology from Ref. [34] (supplementary S(a1, a2, a1, a2) = E(a1, a2) − E(a1, a2) + E(a1, a2) + ′ ′ material)) are important to the laboratory implementation E(a1, a2), and the Eberhard function J given by Eq. (13) . but are superuous, meaning that they do not aect the Because the computer experiment is “perfect”, it dif- results of our computer experiments in any way. Indeed, fers from the laboratory experiment in the sense that all in our perfect experiments, there is no ambiguity in deter- pairs are created “on demand” and all emitted pairs create mining when a particle arrives at the observation station. one detection event in each station (there are no “false” de- Nevertheless, to counter possible (pointless) critique that tection events) but exactly as in the laboratory experiment, we have not incorporated into our simulation model the the local photon identication threshold at each observa- two thresholds that are essential to the laboratory imple- tion station serves to decide whether a photon has arrived mentation, we have chosen V = 1/2 in order to leave or not. min room for introducing these thresholds. We limit the discussion to the case d = 4 because we know from earlier work [20, 72, 90], which uses time- 12 Computer simulation results coincidence selection, that for d = 4 the computer model reproduces the quantum theoretical result of the correla- This section reports the results of simulations with N = tion of two particles in the singlet state, Malus’ law for 105 events for the CFD-compliant and N = 105 events the single-particle averages etc. if N → ∞ followed by per setting for the non-CFD model, with Vmin = 1/2 and V → −Vmax. Vmax = 1. Note that the “time-tag threshold” and a “trig- 726 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

4 90000 (a) +2*sqrt(2.) (a) 80000 3 +2 -2 70000 2 -2*sqrt(2.) S(x) 60000 1 50000 ) 2 40000 ,a

1 0 30000 Eberhard S(a -1 J 20000 -2 10000 0 -3 -10000 -4 -20000 0 30 60 90 120 150 180 0 30 60 90 120 150 180 θ (degrees) θ (degrees)

4 1 (b) +2*sqrt(2.) (b) 3 +2 -2 0.8 2 -2*sqrt(2.) S(x) 1 ) 0.6 2 ,a δ 1 0

S(a -1 0.4

-2 0.2 -3

-4 0 0 30 60 90 120 150 180 0 30 60 90 120 150 180 θ (degrees) θ (degrees)

Figure 7: The Bell-CHSH function S(θ + π/8, θ + 3π/8, π/8, 3π/8) ob- Figure 8: (a) The Eberhard function JEberhard given by Eq. (13) as a tained from the data shown in Figure 6 as a function of θ = a1 − a2. function of θ, for the CFD-compliant (open symbols) and non-CFD (a) CFD-compliant model; (b) non-CFD model. Solid line: quan- model (solid symbols), with (circles) and without (squares) using S θ π θ π π π δ tum√ theoretical prediction ( + /8, + 3 /8, /8, 3 /8) = the photon identication threshold (see text). (b) The value of −2 2 cos(2θ + π/4). The photon identication threshold is entering the inequalities Eq. (18) as a function of θ. The photon V = −0.995. identication threshold V = −0.995.

It is not dicult to see that single-particle averages In Figure 7 we show the data of the Bell-CHSH func- E1(a1, a2), E2(a1, a2) etc. are expected to be zero, up to tion S(θ + π/8, θ + 3π/8, π/8, 3π/8) as a function of θ. uctuations. The reason is that ϕ → ϕ + π/2 changes the Clearly the simulation results are in excellent agreement x v S θ π θ sign of the ’s but has no eect on the values of the ’s (see with the quantum theoretical√ prediction ( + /8, + Eq. (20)). Therefore, if the ϕ′s uniformly cover [0, 2π[, the 3π/8, π/8, 3π/8) = −2 2 cos(2θ + π/4). In both Figs. 6 number of times that x = +1 and x = −1 appear is about and Figure 7, there are deviations from the quantum the- the same. All our simulation results are in concert with this oretical prediction which are not due to statistical uctua- prediction. tions. These deviations can be reduced systematically and In Figure 6, we present the simulation data of the eventually vanish by letting V → −Vmax (V →> −Vmax), a correlation E(a1, a2) (○), the single-particle averages fact that can be proven rigorously for the probabilistic ver- E1(a1, a2) (4) and E2(a1, a2) (5) as a function of θ = sion of the simulation model [20, 72, 90]. a1 − a2, as obtained from a CFD-compliant (Figure 6(a)) We note in passing that the observation that the fre- and non-CFD (Figure 6(b)) simulation. All the simulation quency distribution of many events agrees with the prob- data are in excellent agreement with the quantum theo- ability distribution of a singlet state is a post-factum char- retical description of a two-particle system in the singlet acterization of the repeated preparation and measurement state which predicts E1(a1, a2) = E2(a1, a2) = 0 and process only, not a demonstration that at the end of the E(a1, a2) = − cos 2θ. Within statistical uctuations, it is preparation stage, each pair of particles actually is in an dicult to distinguish between CFD-compliant and non- entangled state. The latter describes the statistics, not a CFD simulation data, in concert with our earlier work [23]. property of a particular pair of particles [4]. The photon identication loophole Ë 727

Results of the Eberhard function Eq. (13) as a function 1 (a) of θ are given in Figure 8(a). The correspondence between the symbols used in Eberhard’s and this paper are as fol- 0.5 ′ lows: o ⇔ +1, e ⇔ −1, α1 ⇔ a1, α2 ⇔ a1, β1 ⇔ a2,

′ ) and β2 ⇔ a2. As expected from the requirements to de- 2 ,a rive Eq. (13) (see section 9), the CFD-compliant simula- 1 0 tion without the photon identication threshold satises E(a

JEberhard ≥ 0 for all θ whereas processing the data as in the -0.5 laboratory experiment, i.e. by employing a photon identi- J cation threshold, yields Eberhard < 0 for a non-zero interval -1 of θ’s. As is clear from Figure 8(a), the results of Eberhard 0 30 60 90 120 150 180 θ function Eq. (13) do not change signicantly if we use re- (degrees) 4 place the CFD-compliant simulation model by its non-CFD (a) +2*sqrt(2.) version. The reason for this apparent violation is that the 3 +2 -2 data obtained through the application of the photon iden- 2 -2*sqrt(2.) S(x) tication mechanism do not satisfy the mathematical re- 1 ) 2

quirements for deriving Eq. (13). ,a

1 0 For completeness, in Figure 8(b) we present results for S(a -1 the function δ which determines the upperbound to the Bell-CHSH function in the case that the photon identi- -2 cation threshold is being used to discard detection events -3 (see Eq. (18)). From Figure 8(b), it follows that δ < 0.8. -4 0 30 60 90 120 150 180 Hence Eq. (18) predicts an upperbound that is not smaller θ (degrees) than√ 3.4, large enough to include the maximum value of 2 2 ≈ 2.83 predicted by the quantum theory of the po- Figure 9: Demonstration that the simulation model cannot repro- larizations of two photons (or, equivalently, two spin-1/2 duce the correlation of the quantum system if the voltage signals v1 v a a particles). and 2 are independent of the settings 1 and 2, respectively. (a) The correlation E(a , a ) (○), the single-particle averages Finally, it is instructive to compare the number of de- 1 2 E1(a1, a2) (4) and E2(a1, a2) (5) as a function of θ = a1 − a2 tection events that the photon identication threshold re- as obtained from a CFD-compliant simulation. (b) The Bell-CHSH jects as being a photon. In the laboratory experiment [34], function S(θ + π/8, θ + 3π/8, π/8, 3π/8) obtained from the data the number of trials is about 3.5 × 109 and the total num- shown in Figure 9(a) as a function of θ. The photon identication V ber of so-called “relevant counts”, i.e. the number of times threshold = −0.995, Vmin = 0.95, Vmin = 1, and d = 0. Solid that at least one photon was identied by means of the lines: quantum theoretical predictions. photon identication thresholds (by software), is about 1.8 × 105. Thus, in this experiment the overall number of events considered to be relevant for the physics, relative 13 Post-factum justication of the to the number of detection events is about 0.005%. For comparison, in the simulations, a photon identication simulation model threshold V = −0.995 identies about 23% of the detec- tion events as photons, several orders of magnitude larger We have already drawn attention to the fact that the ex- than in the laboratory experiment. Clearly, the quality of plicit expression of the voltage v = v(a, ϕ,br) shows a de- the data collected in the laboratory experiments are not on pendence on the local setting of the station through the d par with the quality of the data produced by the computer factor | sin[2(a − ϕ)]| (see Eqs. (19) and (20)) and men- experiments but obviously, the latter is much easier to re- tioned that such a dependence cannot be ruled out a pos- alize and use than the former. teriori. In this section, we examine the consequences of re- moving this dependence. In Figure 9, we show results for d = 0, in which case ′ the random variations of the voltage signals v1, v1, v2, and ′ ′ ′ v2 do not depend on a1, a1, a2, and a2, respectively. In- stead of E(a1, a2) ≈ − cos 2θ for d = 4, the simulation for d = 0 yields E(a1, a2) ≈ −(1/2) cos 2θ and, as Fig- 728 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess ure 9(b) shows, |S(θ + π/8, θ + 3π/8, π/8, 3π/8)| ≤ 2. 14 Conclusion Therefore, the only way to have simulation models of the laboratory experiment [34] reproduce the quantum theo- The general message of this paper is that a model that retical prediction of the polarizations of two photons (or, purports to describe the data produced by an experiment equivalently, two spin-1/2 particles) is to assume that v , 1 should account for all the data that are relevant for the v′ , v , and v′ depend on a , a′ , a , and a′ . Of course, 1 2 2 1 1 2 2 analysis of the experimental results. In the case at hand, there is no good argument why, in a particular experiment, the situation is as follows: this dependence should be of the form Eq. (20). We re- peat that we have chosen the form Eq. (20) because our (i) experimental data [34] are interpreted in terms of a main goal is to reproduce by a CFD-compliant, manifestly Bell-type model that uses only half of the variables non-quantum model, the quantum theoretical predictions (the x’s), of the polarizations of two photons (or, equivalently, two (ii) in the actual experiments [34] the other half of the |S θ spin-1/2 particles), which as a by-product, yields ( + variables (the v’s) is essential for the identication of π θ π π π | /8, + 3 /8, /8, 3 /8) > 2. the photons but are ignored in Bell-type models, Disregarding the original motivation to perform the (iii) the failure of Bell-type models to describe the exper- Bell-test experiments, the experimental setup shown in imental data is taken as a proof that “local realism” Figure 1 can be regarded as a tool to characterize the re- (local in Einstein’s sense) is incompatible with quan- sponse of the observation stations to the incoming sig- tum theory and is therefore is declared dead. nals. In the case at hand, what is under scrutiny is the re- sponse of the observation station, i.e. of its optical com- ponents, the transition-edge detector and the electronics We believe that it requires an exotic form of logic to recon- that amplies its current, under the condition that the in- ciliate the last statement (iii) with the second one (ii). cident light is extremely feeble. Viewed from this perspec- To head o possible misunderstandings, the authors tive, our simulations support the hypothesis that the labo- of this paper do not necessarily subscribe to all or any ratory experiments [34, 85] convincingly demonstrate that forms of what is called local realism, CFD theories, or ... the statistics of the observed photons, as dened by the We are of the opinion that the arguments based on Bell’s photon detection threshold, depends on the settings (and theorem in conjunction with Bell-type experiments suer hence on the polarizations assigned to the photons) of the from what we earlier called the photon identication loop- observation stations. hole. One simply cannot blame a model that only accounts It is of interest to mention here that since the early for part of the data for not describing all of them. Regard- days of the Bell-test experiments, it is well-known that ap- ing the previous sentence, Albert Einstein’s quote “make plication of a Bell-type model requires at least one extra it as simple as possible, but not simpler” is more pertinent assumption. We reproduce here a passage from Ref. [13] than ever. (p.1890): “The approach used by CHSH is to introduce an The challenge for the Bell-experiments community is, auxiliary assumption, that if a particle passes through a therefore, to construct an EPRB-type experiment with a spin analyser, its probability of detection is independent of photon (pair) identication that cannot, from the perspec- the analyser’s orientation. Unfortunately, this assumption is tive of the simple Bell models, be turned into a “loophole”. not contained in the hypotheses of locality, realism or de- Our general proofs of the derivation of Bell-type inequali- terminism.” It is stunning that although there is at least ties for actual data (see section 9), indicate that this chal- one auxiliary assumption involved in testing e.g. the CHSH lenge cannot be met. inequality with Bell-test data, the possibility that this as- sumption is not valid is, to the best of our knowledge, ig- Acknowledgement: We like to thank D. Willsch, F. Jin, and nored in the experimental studies. As a matter of fact, as M. Nocon for useful comments and discussions. we have argued above, all Bell-test experiments with pho- tons performed up to this day can be regarded as direct ex- perimental proof that this auxiliary assumption is invalid. In view of the intricate atomic-scale processes that are in- A Probabilistic models volved when light passes through a material, this conclu- sion seems very reasonable but is, of course, way less spec- Traditionally, mathematical models of the EPRB experi- tacular than the conclusion that Bell-type experiments can ments are formulated in terms of probabilistic models [1, be used to rule out certain world views. 2, 5, 9, 10, 14, 16, 17, 21, 29–33, 35, 39, 42–55, 58, 60– The photon identication loophole Ë 729

67, 70, 71, 73–77, 80, 83, 84, 86], often without explicitly the mathematical subtleties that arise when dealing with mentioning Kolmogorov’s axiomatic framework of proba- innite sets [37, 59]. bility theory [37, 59]. However, there is a considerable, con- The next step is to assign a real number, a proba- ceptual gap between a laboratory EPRB experiment and a bility measure, between zero and one that expresses probabilistic model thereof. The (over)simplications re- the likelihood that an element of the set Ω oc- quired to come up with a tractable, proper probabilistic curs [37, 59]. We denote this (conditional) probability ′ ′ ′ ′ model of a laboratory EPRB experiment are key to the un- measure by P(x1, x1, x2, x2|a1, a1, a2, a2, Z), the part ′ ′ derstanding of the phenomena involved. |a1, a1, a2, a2, Z) indicating that the settings and all When we use the “digital computer – laboratory exper- other conditions, denoted collectively by Z, do not iment” metaphor, both the simplication and replacement change during the imaginary probabilistic experi- are made during the formulation of the computer model. A ment. By denition, the probability measure satises P ′ ′ ′ ′ computer simulation algorithm entails a complete speci- x x′ x x′ ∈Ω P(x1, x , x2, x |a1, a , a2, a , Z) = ( 1 , 1 , 2 , 2) 1 2 1 2 cation of how the data are generated. In this respect, all 1 [37, 59]. “physically relevant” processes are well-dened (by con- Note that a probability measure is a purely mental con- struction) and known explicitly in full detail. There are no struct [59]. If it were not, we could interchange the exper- uncertainties or unknown inuences. Note that the reverse iment/computer simulation, the results of which directly operation, i.e. to construct an algorithm for a digital com- connect to our senses, with the imaginary world of math- puter out of a probabilistic model is, as a matter of princi- ematical models and prove theorems, not only about the ple, impossible. At most, a probabilistic model can serve mathematical description, but also about our sensory ex- as a guide to construct an algorithm, see also [66]. periences, a tantalizing possibility. One such example that In the particular case where the observed phenomena exploits intricate features of set theory is given in Ref. [81], take the form of data generated by simulations of EPRB in which it is explicitly stated that there does not exist an experiments on a digital computer, the transition from algorithm to actually calculate the relevant functions. In the observed phenomena to suitable mathematical mod- other words, this example cannot be realized on a physical els does not suer from the many “uncertain” factors that device such as a digital computer, not even approximately. may or may not play an essential role in the laboratory ex- Moreover, unlike the simulation algorithm executing on periments and is, therefore, a fairly simple transition. In a digital computer, the probabilistic description does not this section, we start from a computer simulation model contain a specication of the process that actually pro- and construct the probabilistic model thereof. We start by duces an event: we have to call up Tyche to do this for us. In showing that the simplest of these models are identical to other words, a probabilistic model is incomplete in that it those proposed and analyzed by Bell [6] and then move on only describes the outcomes of the simulation procedure, to the construction of a probabilistic model for the com- not the procedure itself. However, this incompleteness is puter model of the recent Bell-type experiments [34, 85] partially compensated for by the fact that the calculation that we use in our simulation work presented in section 12. of averages and correlations no longer involves the num- ber of events N. We have for instance

E a a′ a a′ A.1 Bell-type models e12( 1, 1, 2, 2) = (21) X ′ ′ ′ ′ x1x2P(x1, x1, x2, x2|a1, a1, a2, a2, Z), Consider the CFD simulation model in which we explicitly Ω ignore the v-variables. For a given input to the observation and we have similar expressions for the other two-particle stations, the outcome is one of the so-called elementary averages and also for the single-particle averages. Here P events [37, 59], in this case one of the 16 dierent quadru- and in the following, we use the shorthand notation Ω = x x′ x x′ P ples ( 1, 1, 2, 2). In the language of probability theory, x x′ x x′ ∈Ω. We have written Ee instead of Eb to empha- ( 1 , 1 , 2 , 2) the set of these 16 dierent quadruples is called the sample size that the former have been calculated within a proba- space Ω [37, 59], the set of elementary events, from which bilistic model whereas the latter involve calculations with we construct the so-called σ-eld F of subsets of Ω, con- actual data. From Eq. (21), we have taining the impossible (null) event and all the (compound) X   S a a a′ a′ x x x′ x x x′ x′ x′ events in whose occurences we may be interested [37, 59]. e( 1, 2, 1, 2) = 1 2 − 1 2 + 1 2 + 1 2 Ω In modeling the computer experiments, we only need to P x x′ x x′ |a a′ a a′ Z consider nite sets, hence we do not have to worry about × ( 1, 1, 2, 2 1, 1, 2, 2, ), (22) 730 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess and because the elementary events are quadruples, it which expresses the probability measure ′ ′ ′ follows directly from Eq. (10) that |eS(a1, a2, a1, a2)| ≤ P (x1, x2|a1, a2, Z) in terms of the single-variable proba- 2. Thus, in the probabilistic realm, not in the world of bility measures P(x1|a1, ϕ1, Z) and P(x2|a2, ϕ2, Z) and the observed two-valued data, the existence of the Bell- the measure µ(ϕ1, ϕ2) of the variables ϕ1 and ϕ2. CHSH inequality follows from the existence of a proba- The factorized form Eq. (26) is the landmark of the bility measure for the elementary events of quadruples so-called “local hidden-variable models” [5, 13]. Although ′ ′ (x1, x1, x2, x2). Moreover, it can be shown that with some “local” is often used to express the notion that physical in- additional requirements on its marginals, the existence of uences do not travel faster than the speed of light it is, in such a probability measure is necessary and sucient for the context of a probabilistic model (computer model), an Bell-type inequalities to hold [11, 30, 82, 87, 88]. expression of statistical (arithmetic) independence only. ′ It is clear from Figure 5 that xi and xi for i = 1, 2 Bell’s theorem uses the factorized form Eq. (26) to state ′ only depend on the corresponding ai and ai, respec- that quantum mechanics is incompatible with local real- tively. However, the probability measure for quadruples, ism, the world view in which physical properties of objects ′ ′ ′ ′ P(x1, x1, x2, x2|a1, a1, a2, a2, Z), does not express this exist independently of measurement and where physical basic property of the computer model, nor does it explic- inuences cannot travel faster than the speed of light [34]. itly express the dependence on the ϕ’s. In one respect, Eq. (26) is grossly deceiving, namely it A simple way to incorporate all these features of the does not reect the elementary fact that the parent prob- simulation model in a probabilistic description is to dene ability measure Eq. (23) from which Eq. (26) follows, con- a new joint probability measure for quadruples by cerns quadruples, not pairs. Without the knowledge that Eq. (26) is in fact a marginal distribution of the probability P′ x x′ x x′ |a a′ a a′ Z ( 1, 1, 2, 2 1, 1, 2, 2, ) = measure Eq. (23) for quadruples, one is inclined to think, Z ′ ′ as Bell did and his followers still seem to do, that there are P(x1|a1, ϕ1, Z)P(x1|a1, ϕ1, Z)P(x2|a2, ϕ2, Z) “physical” assumptions involved in justifying the factor- ′ ′ × P(x2|a2, ϕ2, Z)µ(ϕ1, ϕ2)dϕ1dϕ2, (23) ized form Eq. (26). However, this is not the case because the Bell-type inequalities hold if and only if there exists where P(x |a , ϕ , Z) etc. are the “local” probabilities to 1 1 1 a joint probability measure for the quadruples [30]. This observe x etc., the integration is over the whole domain 1 mathematical statement is void of any physical meaning. of ϕ and ϕ and µ(ϕ , ϕ ) is a non-negative, normal- 1 2 1 2 In summary: in this subsection we have shown that a ized density. With the new probability measure Eq. (23), probabilistic model of the CFD computer simulation model Eq. (21) simplies considerably. For instance, for the de- that does not account for the photon identication mech- tection events we have Ee (a , a′ , a , a′ ) = Ee(a , a ) 12 1 1 2 2 1 2 anism of the EPRB laboratory experiment, automatically where leads to the models introduced by Bell [5]. Within this Z X framework, the existence of Bell-type inequalities and the Ee(a, b) = xyP(x|a, ϕ1, Z)P(y|b, ϕ2, Z) x,y=±1 corresponding joint probability measures are mathemati- cally equivalent [30]. The latter statement, which relates to × µ(ϕ1, ϕ2)dϕ1dϕ2. (24) imaginary data only, corresponds to the statement that in Instead of Eq. (22), we now have the realm of actual two-valued data, the existence of Bell- type inequalities and CFD-compliant generation of all the |S a a a′ a′ | |E a a E a a′ e( 1, 2, 1, 2) = e( 1, 2) − e( 1, 2) quadruples are mathematically equivalent, see section 9. ′ ′ ′ + Ee(a1, a2) + Ee(a1, a2)| ≤ 2, (25) which is the Bell-CHSH inequality in probabilistic form [5, A.2 Incorporating the photon identication 14]. threshold From Eq. (23), it follows directly that

′ Referring to Eq. (5), the extension of the construction out- P (x1, x2|a1, a2, Z) X X lined in section A.1 to incorporate the local photon identi- = P′(x , x′ , x , x′ |a , a′ , a , a′ , Z) 1 1 2 2 1 1 2 2 cation mechanism is of purely technical nature. Instead x′ x′ 1=±1 2=±1 Z = P(x1|a1, ϕ1, Z)P(x2|a2, ϕ2, Z)µ(ϕ1, ϕ2)dϕ1dϕ2, (26) The photon identication loophole Ë 731 of Eq. (23), we now introduce have Z ′′ ′ ′ ′ ′ ′ ′ X P (x1, v1, x1, v1, x2, v2, x2, v2|a1, a1, a2, a2, Z) A(a1, a2) = x1x2Θ(W − t1)Θ(W − t2)Θ(t1)Θ(t2) Z x1 ,x2=±1 ′ ′ = P(v1|a1, ϕ1, Z)P(x1|a1, ϕ1, Z)P(v1|a1, ϕ1, Z) × P(t1|a1, ϕ1, Z)P(x1|a1, ϕ1, Z)P(t2|a2, ϕ2, Z) ′ ′ × P(x1|a1, ϕ1, Z)P(v2|a2, ϕ2, Z)P(x2|a2, ϕ2, Z) × P(x2|a2, ϕ2, Z)µ(ϕ1, ϕ2) dϕ1 dϕ2 dt1 dt2, P v′ |a′ ϕ Z P x′ |a′ ϕ Z and × ( 2 2, 2, ) ( 2 2, 2, ) Z × µ(ϕ1, ϕ2) dϕ1 dϕ2 dv1 dv2, (27) B(a1, a2)= Θ(W − t1)Θ(W − t2)Θ(t1)Θ(t2)P(t1|a1, ϕ1, Z)

P t |a ϕ Z µ ϕ ϕ dϕ dϕ dt dt where P(v1|a1, ϕ1, Z) etc. are the “local” probability den- × ( 2 2, 2, ) ( 1, 2) 1 2 1 2, sities to pick v1 etc., and all other symbols have the same (29) meaning as in Eq. (23). for the time-coincidence selection we have It is now straightforward to write down the proba- Z bilistic expressions that incorporate in exactly the same X A(a1, a2) = x1x2Θ(W − |t1 − t2|)P(t1|a1, ϕ1, Z) manner as in the analysis of the laboratory experiment x1 ,x2=±1 data [34], the eect of the photon identication threshold. × P(x1|a1, ϕ1, Z)P(t2|a2, ϕ2, Z)P(x2|a2, ϕ2, Z) For instance, we have for the photon counts × µ(ϕ1, ϕ2) dϕ1 dϕ2 dt1 dt2, A(a , a ) Ee(a , a ) = 1 2 , and 1 2 B(a , a ) Z 1 2 B(a , a ) = Θ(W − |t − t |)P(t |a , ϕ , Z) where 1 2 1 2 1 1 1 X Z A(a1, a2) = x1x2Θ(V − v1)Θ(V − v2) × P(t2|a2, ϕ2, Z)µ(ϕ1, ϕ2) dϕ1 dϕ2 dt1 dt2. x1 ,x2=±1 (30)

× P(v1|a1, ϕ1, Z)P(x1|a1, ϕ1, Z)P(v2|a2, ϕ2, Z) From earlier work based on representation P x |a ϕ Z µ ϕ ϕ dϕ dϕ dv dv × ( 2 2, 2, ) ( 1, 2) 1 2 1 2, Eq. (30) [20, 26, 72] and from the simulation results and Z presented in section 12, it follows directly that the B(a1, a2) = Θ(V − v1)Θ(V − v2)P(v1|a1, ϕ1, Z) probabilistic model dened by Eq. (27) is capable of repro- ducing the predictions of quantum theory for the single- × P(v2|a2, ϕ2, Z)µ(ϕ1, ϕ2) dϕ1 dϕ2 dv1 dv2, and two-particles averages of two photon polarizations (28) in the singlet state. This then should stop spreading the misconception that Bell has proven that quantum theory and, as before, we have similar expressions for the other is incompatible with all “local hidden-variable models”. expectations in Eq. (21) and for the single-particle aver- Of course, “local hidden-variable models” that do not ages. include the, for the laboratory experiment essential, The expressions for the single- and two-particles av- mechanism to identify single or pairs of photons are inca- erages that derive from the probabilistic model Eq. (27) pable to describe the salient features of the experimental all have the form that is characteristic of a genuine “local data but as explained in section 7, that is hardly more hidden-variable model”, as exemplied by Eq. (28). Only than a platitude. “local” detection and photon identication are involved. In summary, in this appendix we have shown how to The values of the variables local to one observation station construct probabilistic descriptions of computer simula- do not depend on variables that are local to another obser- tion models of EPRB experiments that rely on photon iden- vation station. The only form of “communication” between tication thresholds to decide whether or not a photon the stations is through the “hidden” variables ϕ1 and ϕ2. has been detected [34]. The resulting probabilistic mod- It directly follows from the general discussion of sec- els conform to the requirements of genuine “local hidden- tion 6 that A(a1, a2) and B(a1, a2) can be expressed in variable models” and if they account for a local mecha- terms of both the local time-window and time-coincidence nism to identify photons, are also capable of producing re- selection. In detail, for the local time-window selection we sults that are in full agreement with quantum theory. 732 Ë Hans De Raedt, Kristel Michielsen, and Karl Hess

References [24] De Raedt H., Michielsen K., Jin F., Einstein-Podolsky-Rosen- Bohm laboratory experiments: Data analysis and simulation. AIP Conf. Proc., 2012, 1424, 55–66. [1] Accardi L., Some loopholes to save quantum nonlocality, AIP [25] De Raedt K., De Raedt H., Michielsen K., A computer program Conf. Proc., 2005, 750, 1–20. to simulate Einstein-Podolsky-Rosen-Bohm experiments with [2] Adenier G., Khrennikov A.Y., Is the fair sampling assumption photons, Comp. Phys. Comm., 2007, 176, 642–651. supported by EPR experiments. J. Phys. B: At. Mol. Opt. Phys., [26] De Raedt K., Keimpema K., De Raedt H., Michielsen K., Miyashita 2007, 40, 131–141. S., A local realist model for correlations of the singlet state, Eur. [3] Aspect A., Dalibard J., Roger G., Experimental test of Bell’s in- Phys. J. B, 2006, 53, 139–142. equalities using time-varying analyzers, Phys. Rev. Lett., 1982, [27] Eberhard P.H., Background level and counter eciencies re- 49, 1804–1807. quired for a loophole-free Einstein-Podolsky-Rosen experiment, [4] Ballentine L.E., Quantum Mechanics: A Modern Development. Phys. Rev. A, 1993, 47, R747–R750. World Scientigic, Singapore, 2003 [28] Einstein A., Podolsky A., Rosen N., Can quantum-mechanical [5] Bell J.S., Speakable and Unspeakable in Quantum Mechanics. description of physical reality be considered complete?, Phys. Cambridge University Press, Cambridge, 1993 Rev., 1935, 47, 777–780. [6] Bell J.S., On the Foundations of Quantum Mechanics, World Sci- [29] Fine A., On the completeness of quantum theory, Synthese, entic, Singapore, New Jersey, London, Hong Kong, 2001 1974, 29, 257–289. [7] Bohm D., Quantum Theory, Prentice-Hall, New York, 1951 [30] Fine A., Hidden variables, joint probability, and Bell inequali- [8] Boole G., On the theory of probabilities, Phil. Trans. R. Soc. ties, Phys. Rev. Lett., 1982, 48, 291–295. Lond., 1862, 152, 225–252. [31] Fine A., Joint distributions, quantum correlations, and commut- [9] Brans C., Bell’s theorem does not eliminate fully causal hidden ing observables, J. Math. Phys., 1982, 23, 1306–1310. variables, Int. J. Theor. Phys., 1987, 27, 219–226. [32] Fine A., Some local models for correlation experiments, Syn- [10] Brody T., The Philosphy Behind Physics, Springer, Berlin, 1993 these, 1982, 50, 279–294. [11] Brody T.A., The Suppes-Zanotti theorem and the Bell inequali- [33] Fine A., The Shaky Game: Einstein Realism and the Quantum ties, Revista Mexicana de Física, 1989, 35, 170–187. Theory, University of Chicago Press, Chicago, 1996 [12] Christensen B., McCusker K., Altepeter J., Calkins B., Lim C., [34] Giustina M., Versteegh M.A.M., Wengerowsky S., Handsteiner Gisin N., Kwiat P., Detection-loophole-free test of quantum non- J., Hochrainer A., Phelan K., Steinlechner F., Kofler J., Larsson locality, and applications, Phys. Rev. Lett., 2013, 111, 130406 J.A., Abellán C., Amaya W., Pruneri V., Mitchell M.W., Beyer J., [13] Clauser J.F., Horn M.A., Experimental consequences of objective Gerrits T., Lita A.E., Shalm L.K., Nam S.W., Scheidl T., Ursin R., local theories, Phys. Rev. D, 1974, 10, 526–535. Wittmann B., Zeilinger A. Signicant-loophole-free test of Bell’s [14] Clauser J.F., Horn M.A., Shimony A., Holt R.A. Proposed exper- theorem with entangled photons, Phys. Rev. Lett., 2015. 115, iment to test local hidden-variable theories, Phys. Rev. Lett., 250401 1969, 23, 880–884. [35] Graft D.A. The Bell inequality cannot be validly applied to the [15] Clauser J.F., Shimony A., Bell’s theorem: Experimental tests and Einstein-Podolsky-Rosen-Bohm (EPRB) experiments, Phys. Es- implications, Rep. Prog. Phys., 1978. 41(12), 1881–1927. says, 2009, 22, 534–542. [16] De Baere W., Mann A., Revzen M. Locality and Bell’s theorem, [36] Graft D.A., Analysis of the Christensen et al. test of local realism, Found. Phys., 1999. 29, 67–77. J. Adv. Phys., 2015, 4, 284–300. [17] de la Peña L., Cetto A.M., Brody T.A., On hidden-variable theo- [37] Grimmet G.R., Stirzaker D.R., Probability and Random Pro- ries and Bell’s inequality, Lett. Nuovo Cim., 1972, 5, 177–181. cesses, Clarendon Press, Oxford, 1995 [18] de Muynck V., W. W.D., Martens H. Interpretations of quan- [38] Hensen B., Bernien H., Dreau A.E., Reiserer A., Kalb N., Blok tum mechanics, joint measurement of incompatible observ- M.S., Ruitenberg J., Vermeulen R.F.L., Schouten R.N., Abellan C., ables and counterfactual deniteness, Found. Phys., 1994, 24, AmayaW., Pruneri V., Mitchell M.W.,Markham M., Twitchen D.J., 1589–1664. Elkouss D., Wehner S., Taminiau T.H., Hanson R., Loophole-free [19] De Raedt H., De Raedt K., Michielsen K., Keimpema K., Miyashita Bell inequality violation using electron spins separated by 1.3 S., Event-based computer simulation model of Aspect-type ex- kilometres, Nature, 2015, 15759 periments strictly satisfying Einstein’s locality conditions, J. [39] Hess K., Einstein Was Right! Pan Stanford Publishing, Singa- Phys. Soc. Jpn., 2007, 76, 104005 pore, 2015 [20] De Raedt H., De Raedt K., Michielsen K., Keimpema K., Miyashita [40] Hess K., De Raedt H., Michielsen K., Counterfactual deniteness S. Event-by-event simulation of quantum phenomena: Applica and Bell’s inequality, J. Mod. Phys., 2016, 7, 1651–1660. tion to Einstein-Podolosky-Rosen-Bohm experiments, J. Com- [41] Hess K., De Raedt H., Michielsen K., From Boole to Leggett-Garg: put. Theor. Nanosci., 2007, 4, 957–991. Epistemology of Bell-type inequalities, Adv. Math. Phys., 2016, [21] De Raedt H., Hess K., Michielsen K. Extended, Boole-Bell in- 2016, 4623040 equalities applicable to quantum theory, J. Comput. Theor. [42] Hess K., Michielsen K., De Raedt H., Possible experience: from Nanosci., 2011, 8, 1011–1039. Boole to Bell, Europhys. Lett., 2009, 87, 60007 [22] De Raedt H., Jin F., Michielsen K., Data analysis of Einstein- [43] Hess K., Philipp W., A possible loophole in the theorem of Bell. Podolsky-Rosen-Bohm laboratory experiments, Proc. SPIE, Proc. Natl. Acad. Sci. USA, 2001, 98, 14224–14277. 2013, 8832, 88321N1–11 [44] Hess K., Philipp W., Bell’s theorem and the problem of decid- [23] De Raedt H., Michielsen K., Hess K., The digital computer as a ability between the views of Einstein and Bohr, Proc. Natl. Acad. metaphor for the perfect laboratory experiment: Loophole-free Sci. USA, 2001, 98, 14228–14233. Bell experiments, Comp. Phys. Comm., 2016, 209, 42–47. The photon identication loophole Ë 733

[45] Hess K., Philipp W., Bell’s theorem: Critique of proofs with and 1985, 9, 857–860. without inequalities, AIP Conf. Proc., 2005, 750, 150–157. [70] Loubenets E., “Local Realism”, Bell’s Theorem and Quan- [46] Jaynes E.T., Clearing up mysteries - The original goal. In: J. tum “Locally Realistic” Inequalities, Found. Phys., 2005, 35, Skilling, ed., Maximum Entropy and Bayesian Methods, Kluwer 2051–2072. Academic Publishers, Dordrecht, 1989, 36, 1–27 [71] Matzkin A., Is Bell’s theorem relevant to quantum mechanics? [47] Khrennikov A., EPR-Bohm experiment and interference of prob- On locality and non-commuting observables, AIP Conf. Proc., abilities. Found. Phys. Lett., 2004, 17, 691–700. 2009, 1101, 339–348. [48] Khrennikov A., Bell’s inequality: Nonlocalty, “death of reality”, [72] Michielsen K., De Raedt H., Event-based simulation of quantum or incompatibility of random variables? AIP Conf. Proc., 2007, physics experiments, Int. J. Mod. Phys. C, 2014, 25, 01430003 962, 121–131. [73] Morgan P., Bell inequalities for random elds, J. Phys. A, 2006, [49] Khrennikov A., Nonlocality as well as rejection of realism are 39, 7441–7445. only sucient (but non-necessary!) conditions for violation of [74] de Muynck W.M., The Bell inequalities and their irrelevance to Bell’s inequality. Inf. Sciences, 2009, 179, 492–504. the problem of locality in quantum mechanics, Phys. Lett. A, [50] Khrennikov A., After Bell, Fortschritte der Physik, 2017, 65, 1986, 114, 65–67. 1600044 [75] Nieuwenhuizen T., Kupczynski M., The contextuality loophole is [51] Khrennikov A.Y., Interpretations of Probability, VSP Int. Sc. Pub- fatal for the derivation of Bell inequalities: Reply to a comment lishers, Utrecht, 1999 by I. Schmelzer, Found. Phys., 2017, 47, 316–319. [52] Khrennikov A.Y., A mathematicians viewpoint to Bell’s theorem: [76] Nieuwenhuizen T.M., Where Bell went wrong, AIP Conf. Proc., in memory of Walter Philipp, AIP Conf. Proc., 2007, 889, 7–17. 2009, 1101, 127–133. [53] Khrennikov A.Y., Contextual Approach to Quantum Formalism. [77] Nieuwenhuizen T.M., Is the contextuality loophole fatal for Springer, Berlin, 2009 the derivation of Bell inequalities?, Found. Phys., 2011, 41, [54] Khrennikov A.Y., Violation of Bell’s inequality and non- Kol- 580–591. mogorovness, AIP Conf. Proc., 2009, 1001, 86 [78] Pascazio S. Time and Bell-type inequalities, Phys. Lett. A, 1986, [55] Khrennikov A.Y., On the role of probabilistic models in quan- 118, 47–53. tum physics: Bell’s inequality and probabilistic incompatibility, [79] Pearl J., Causality: models, reasoning, and inference, Cam- J. Comput. Theor. Nanosci., 2011, 8, 1006–1010. bridge University Press, Cambridge, 2000 [56] Khrennikov A.Y., Volovich Y., Discrete time dynamical models [80] Pearle P.M., Hidden-variable example based upon data rejec- and their quantum-like context-dependent properties, J. Mod. tion, Phys. Rev. D, 1970, 2, 1418–1425. Opt., 2004, 51, 1113–1114. [81] Pitowsky I., Resolution of the Einstein-Podolsky-Rosen and Bell [57] Kocher C.A., Commins E.D., Polarization correlation of pho- Paradoxes, Phys. Rev. Lett., 1982, 48, 1299–1302. tons emitted in an atomic cascade. Phys. Rev. Lett., 1967, 18, [82] Pitowsky I., Correlation polytopes: Their geometry and 575–577. complex- ity. Math. Program., 1991, 50, 395–414. [58] Kofler J., Giustina M., Larsson J.Å., Mitchell M., Requirements [83] Pitowsky I., George Boole’s ‘Conditions of Possible Experience’ for a loophole-free photonic bell test using imperfect setting and the Quantum Puzzle, Brit. J. Phil. Sci., 1994, 45, 95–125. gen erators, Phys. Rev. A, 2016, 93, 032115 [84] Santos E., Bell’s theorem and the experiments: Increasing em- [59] Kolmogorov A., Foundations of the Theory of Probability, pirical support to local realism? Stud. Hist. Phil. Mod. Phys., Chelsea Publishing Co., New York, 1956 2005, 36, 544–565. [60] Kracklauer A.F., Bell’s inequalities and EPR-B experiments: Are [85] Shalm L.K., Meyer-Scott E., Christensen B.G., Bierhorst P., they disjoint?, AIP Conf. Proc., 2005, 750, 219–227. Wayne M.A., Stevens M.J., Gerrits T., Glancy S., Hamel D.R., All- [61] Kupczynski M., On some tests of completeness of quantum me- man M.S., Coakley K.J., Dyer S.D., Hodge C., Lita A.E., Verma chanics. Phys. Lett. A, 1986, 116, 417–419. V.B., Lambrocco C., Tortorici E., Migdall A.L., Zhang Y., Kumor [62] Kupczynski M. Bertrand’s paradox and Bell’s inequalities. Phys. D., Farr W.H., Marsili F., Shaw M.D., Stern J.A., Abellán C.,Amaya Lett. A, 1987, 121, 205–207. W., Pruneri V., Jennewein T., Mitchell M.W., Kwiat P.G., Bienfang [63] Kupczynski M., Entanglement and Bell inequalities, J. Russ. Las. J.C., Mirin R.P., Knill E., Nam S.W., Strong loophole-free test of Res., 2005, 26, 514–523. local realism, Phys. Rev. Lett., 2015, 115, 250402 [64] Kupczynski M., Entanglement and quantum nonlocal- [86] Sica L., Bell’s inequalities I: An explanation for their experimen- ity demystied, AIP Conf. Proc., 2012, 1508(1), 253–264. tal violation, Opt. Comm., 1999, 170, 55–60. 10.1063/1.3567465 [87] Suppes P., Zanotti M., When are probabilistic explanations pos- [65] Kupczynski M., Causality and local determinism versus quan- sible?, Synthese, 1981, 48, 191–199. tum nonlocality, J. Phys.: Conference Series, 2014, 504(1), [88] Vorob’ev N.N., Consistent families of measures and their exten- 012015 sions, Theor. Probab. Appl., 1962, 7, 147–162. [66] Kupczynski M., Entanglement and quantum nonlocality demysti [89] Weihs G., Jennewein T., Simon C., Weinfurther H., Zeilinger A., ed. Found. Phys., 2015, 45, 735–753. Violation of Bell’s inequality under strict Einstein locality condi- [67] Kupczynski M., EPR paradox, quantum nonlocality and physi- tions, Phys. Rev. Lett., 1998, 81, 5039–5043. cal reality, Journal of Physics: Conference Series, 2016, 701(1), [90] Zhao S., De Raedt H., Michielsen K, Event-by-event simulation 012021 model of Einstein-Podolsky-Rosen-Bohm experiments, Found. [68] Larsson J.Å., Loopholes in Bell inequality tests of local realism, Phys., 2008, 38, 322–347. J. Phys. A: Math. Theor., 2014, 47, 424003 [69] Leggett A.J., Garg A., Quantum Mechanics versus Macroscopic Realism: Is the Flux There when Nobody Looks. Phys. Rev. Lett.,