EPRB Gedankenexperiment and Entanglement with Classical Light Waves

1 Introduction

One of the most mysterious and intriguing predictions of quantum mechanics is the entanglement phenomenon, which manifests in a strong correlation of the behaviour of quantum objects, even when they are separated by a large distance. According to quantum mechanics, the state of each such object cannot be described independently – instead, a quantum state must be described for the system as a whole. The entangled state cannot be factorised into a product of two states associated with each object. According to this, it follows that we cannot ascribe any well-defined state to each object.

The entanglement phenomenon is considered to be a basis for new hypothetical solutions, primarily in the field of information technologies.

This phenomenon was considered for the first time by Einstein et al. [1] and was developed further by Bohm [2] who described what came to be known as the Einstein-Podolsky-Rosen-Bohm (EPRB) Gedankenexperiment and the EPRB paradox.

The first quantitative criterion that describes such a paradox was proposed by Bell (Bell’s inequality) [3]. Bell’s inequality, which is derived on the basis of the local hidden-variable theories, contradicts in some cases the predictions of quantum mechanics. It is considered that an experiment in which the violation of Bell’s inequality occurs cannot be explained on the basis of the local realism view. Bell’s inequality gave the tool for experimental verification of the counterintuitive predictions of quantum mechanics. Later, Clauser, Horne, Shimony, and Holt (CHSH) proposed a new criterion and an experiment to test the local hidden-variable theories [4].

In these experiments, the two photons ν₁ and ν₂, which are emitted in the entangled state, are analysed by linear polarisers in orientations a and b [5], [6], [7] (Fig. 1). Each polariser is followed by two detectors, giving results + or −, corresponding to a linear polarisation found parallel or perpendicular to a and b.

Figure 1:

Einstein-Podolsky-Rosen-Bohm Gedankenexperiment with photons [4], [5], [6], [7].

By measuring the clicks of the detectors, one can calculate the probabilities of events, both single ones and their coincidences.

Quantum mechanics predicts for single probabilities

where P±(a) and P±(b) are the probabilities of getting the results ± for the photons ν₁ and ν₂, respectively.

These results are in agreement with the fact that each individual polarisation measurement gives a random result and with the point of view that the photon is indivisible and we cannot observe simultaneously the clicks of the detectors a₊ and a₋ for polariser a and correspondingly the clicks of the detectors b₊ and b₋ for polariser b. According to [3], [4], the entanglement of the photons manifests in the probabilities P_±±(a, b) of joint detections of ν₁ and ν₂ in the channels + or − of polarisers a and b. For entangled particles, quantum mechanics predicts

where α is the angle between the orientations of the polarisers a and b.

In order to describe quantitatively the correlations between random events, one can introduce the correlation coefficient [4]

Using (2), quantum mechanics predicts

By carrying out the experiments for four different orientations a, a′ and b, b′ of the polarisers a and b, one can calculate the parameter [4]

The local hidden-variable theories predict [4]

It is well known that the greatest conflict between quantum mechanical predictions and CHSH inequalities (6) that follow from the local hidden-variable theories [4] is expected for the set orientations (a, b)=(a′, b)=(a′, b′)=22.5° and (a, b′)=67.5°. In this case, quantum mechanics predicts

The CHSH inequality (6) was testable in numerous relevant experiments, starting with the pioneering works in [5], [6], [7], all of which have shown agreement with quantum mechanics rather than with the principle of local realism. Violation of Bell’s inequalities (6) was fixed for a wide range of distances and timings of measurements [8], [9], [10], [11], [12], [13], [14], [15], [16].

These results have shown, in particular, that the EPRB experiments with entangled photons cannot be described within the local hidden-variable theories and that, in general, it is impossible to construct the local hidden-variable theories that are capable of describing the quantum mechanical regularities. From this point of view, the entanglement is considered as a direct evidence of the existence of photons.

The paradox of the results in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] is that these experiments can be physically explained on the basis of the photonic (corpuscular) representations only if one assumes that the interaction between the two entangled particles and between the particles and the measuring devices are propagated at a velocity substantially exceeding the speed of light, which contradicts the relativity theory.

Einstein characterised it as “spooky action at a distance” and argued that the accepted formulation of quantum mechanics must therefore be incomplete.

We note that these conclusions are based on the photonic representations, i.e. on the fact that each click of the detector is associated with a hit of a particle – a photon.

Let us recall that the coincidence experiments were started with the pioneering Hanbury Brown and Twiss (HBT) experiments [17], [18], the results of which initially also have caused surprise. Later, the simple explanation of the HBT effect was found within the framework of the semiclassical theory without quantisation of radiation [19], [20]. The idea of this explanation is as follows.

Solution of the Schrödinger equation allows calculating the probability of excitation of an atom of the detector by the classical electromagnetic (light) wave for time Δt (Fermi’s golden rule)

where w is the probability of excitation of atoms per unit time, I~E² is the intensity of the classical light wave at the location of the atom, and b is a constant that does not depend on the intensity of the incident light. In this case, each click of the detector is considered to be the result of the excitation of one of the atoms under the action of light. Assuming that the components of the electric field vector of the light wave E are random variables and have a Gaussian distribution, one can calculate all regularities of the HBT effect [19], [20]. The HBT correlation appears as a result of the correlation of intensities of light waves arriving at the two detectors due to splitting the incident light wave.

As shown in [21], [22], [23], the experiments with single photons, namely the double-slit experiments and the Wiener experiments with standing light waves, can be reproduced with weak classical light waves if we take into account the discrete (atomic) structure of a detector and a specific nature of the light-atom interaction, which is described by the expression (8).

In this article we show that results similar to those of the EPRB Gedankenexperiment and entanglement of photons can also be obtained using weak classical light waves.

2 EPRB Gedankenexperiment with Classical Light Waves

Let us consider the EPRB Gedankenexperiment with classical light waves (Fig. 2). We assume that a source S emits two identical classical electromagnetic (light) waves in opposite directions; i.e.

Figure 2:

Einstein-Podolsky-Rosen-Bohm Gedankenexperiment with classical light waves.

for light waves ν₁ and ν₂ (Fig. 2).

The emitted waves ν₁ and ν₂ arrive at two spatially separated two-channel polarisers a and b, each of which splits the incident light beam into two mutually orthogonal linearly polarised beams that arrive at the corresponding detectors. For each polariser, we introduce its own coordinate system (x, y), where the x-axis is parallel to the axis of the polariser, while the y-axis is perpendicular to it. Further, the beam with polarisation parallel to the axis of the polariser is denoted by the index “+”, while the beam with polarisation perpendicular to the axis of the polariser is denoted by the index “−”. The corresponding detectorswill be denoted as a± and b±.

Each polariser can rotate around the axis of the incident beam. Because of the isotropy of the system, only the relative angle of rotation of the polarisers α has any meaning. Therefore, we choose a coordinate system associated with the polariser a as the reference system. Furthermore, we assume that the polarisers are ideal, i.e. we neglect the energy losses of the light wave on passing through the optical system. Then, only the component E_x of the incident light wave ν₁ will arrive at the detector a₊, while only the component E_y of this wave will arrive at the detector a₋. The intensities of light waves arriving at the detectors a± are as follows:

Let the polariser b be rotated with respect to the polariser a through an angle α (Fig. 2). We denote the own coordinate system of the polariser b as (x′, y′), whose axes are parallel to the corresponding main axes of the polariser b. Then, only the component E_x′ of the incident light wave ν₂ will arrive at the detector b₊, while only the component E_y′ of this wave will arrive at the detector b.

Taking into account (9), one can write

The intensities of light waves arriving at the detectors b± are, respectively,

Under the action of the incident light wave, excitation of the atoms of the detector can occur. We assume that the excitation of an atom of the detector inevitably causes an electron avalanche in the detector, which manifests in the form of a single event (click of detector), which is fixed by the recorder.

It is believed that after the triggering, the detector (an atom) returns again to the initial (ground) state and is ready for the next act of excitation.

The rate of atomic excitation w is described by the expression (8) and does not depend on the concentration of atoms. If the radiation intensity does not change within the time of exposure (within a time window), then the law of excitation of atoms will be similar to the law of radioactive decay. In particular, the probability of excitation of an atom during time Δt is [21], [22], [23]

P(t)=1−exp(−wΔt).

Taking into account (8), one obtains

where I is the intensity of light wave arriving at the corresponding detector.

Assuming that the source of radiation is Gaussian, and that the components of the electric field vector E=(E_x, E_y) of the light wave are statistically independent, one obtains the probability density for the components of the electric field vector as

where

and … denotes averaging.

Obviously, considering (9)

Equations (13)–(16) allow us to calculate the EPRB Gedankenexperiment in detail. Indeed, by calculating the single events of triggering the detectors a± and b± using the expressions (13) and (14), one can determine their statistics both for single events and for their coincidences and compare it with the predictions of quantum mechanics (1) and (2).

3 Monte Carlo Simulation of EPRB Gedankenexperiment

Let us consider the discrete time intervals (time windows) i=1, 2, …, N which have a duration Δt, during which the discrete events occurring at different detectors are recorded. The events on different detectors will be considered as simultaneous if they occurred within the same time window i. At the same time, the events occurring at different detectors are statistically independent, and are described by the probabilities (13), in which we use the intensities (10) and (12) of the light waves arriving at the corresponding detector within a given time window. The intensities of the light waves ν₁ and ν₂ emitted by a source for different time windows are considered to be random and are described by the probability density function (14).

Let us introduce the nondimensional exposure time (nondimensional duration of the time window)

τ=bI0Δt.

In this case, the probability of excitation of the atom during a time window is

Further, we take the value I₀ as the characteristic intensity of light. In this case, we use the parameter I01/2 as a scale for the field E. Taking into account the expressions (14) and (17), in further calculations we will take I₀=1, while the value I₀ itself will be included in the nondimensional duration τ of the time window.

Then, the expressions (14) and (17) can be written in the nondimensional form

where I=E².

Thus, in the problem under consideration, there is a single nondimensional parameter τ, varying which we can change the “experimental conditions”.

The calculation proceeds quite trivially using the Monte Carlo method [See Supplementary Material at (https://doi.org/10.1515/zna-2018-0049) which contains the model source code and the program for simulations.]: at each time window i, the components of the electric field vector E of the light wave are generated using the probability density (19). Using the components E_x and E_y, the intensities of the light waves (10) and (12) arriving at the corresponding detector are calculated taking into account the expression (11). Using these intensities, the probabilities of excitation of each detector are calculated using the expression (18). At the same time, the random numbers R∈[0, 1] are generated for each detector using a random number generator. If the condition R≤P is satisfied for some detector, it is considered to be excited, and this event is recorded in the corresponding time window. Thus, we record all the events triggering the detectors at different time windows. Note that in these calculations the assumption was made that no more than one discrete event can occur at one detector within one time window. In real experiments [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], the duration of the time window was significantly longer than the relaxation time of the detector. Therefore, generally speaking, there is a finite probability that the same detector will trigger several times during the same time window. Accounting for no more than one event of triggering the same detector within the same time window, in fact, means the rejection of such time windows in a real experiment.

After all time windows i=1, 2, …, N are calculated, the statistical analysis of both the single events and the coupled events (coincidences) for different pairs of detectors a± and b± is performed. This allows determining any statistical characteristics of such an experiment.

For us, it is of interest to analyse the results of Monte Carlo simulations based on photonic (corpuscular) representations. For this purpose, we will assume that each triggering of the detector is the result of a hit on it by a particle – a photon. At the same time, we recall that in reality, the results of Monte Carlo simulation were obtained within the framework of the semiclassical theory, in which light is considered as a classical electromagnetic wave, while the photonic model is just a fiction, the goal of which is a mechanistic (naive) “explanation”of discrete events of triggering the detectors.

As soon as we begin to analyse the results of experiments from the standpoint of the photonic representation, we immediately have to introduce a number of limitations related to our ideas about photons as indivisible particles.

First of all, if both detectors behind the same polariser simultaneously got triggered during one time window, for an indivisible photon such an event can be explained by a background, by interferences in the circuit, or by an error in the detector operation. In any case, this result leads to violation of the conditions (1), and therefore the time windows in which such events occurred should be rejected.

Further, assuming that the source S emits a pair of entangled photons, we can expect that within one time window, the simultaneous triggering of the detectors behind both polarisers will be recorded. In other words, if a photon was detected behind the polariser a, then the second photon must also be detected behind the polariser b. If a second event did not happen, then the result can be explained by an insufficient sensitivity of the detector, by malfunctioning of a detector, or by the fact that a one photon of the entangled pair was “lost” on the way to a polariser, which is also perceived as a detection error, and such time windows should not be taken into account when calculating the “photon coincidences.”

Thus, when counting the number of coincidences, we have to reject not only the time windows in which both detectors triggered behind any polariser but also those time windows at which one detector triggered behind one of polarisers while there were no detectors triggered behind the other polariser, because only such events are consistent with the photonic representations in this experiment.

Note that such a postselection of time windows is widely used in experiments testing the violation of Bell’s inequality and has an effect on the final results [24], [25], [26].

Thus, by statistical analysis of the results of semiclassical Monte Carlo simulations of the EPRB Gedankenexperiment with classical light waves, we calculate the number of the time windows in which pairs of the corresponding events have occurred: N₊₊(a, b), N(a, b), N₊₋(a, b), N₋₊(a, b). For example, N₊₊(a, b) is the number of time windows in which the events were recorded simultaneously on the detectors a₊ and b₊, while the events on other detectors were not observed, etc. Then

is the number of the time windows at which the events were recorded behind both polarisers but only one detector has triggered behind each polariser.

Then the probabilities of the corresponding pair events are determined by the expression

The probabilities (21) determined exactly in this way correspond to those calculated in quantum mechanics.

Using the probabilities (21) for each relative orientation α of the polarisers, one can calculate the correlation coefficient (3).

The results of Monte Carlo simulations of the EPRB Gedankenexperiment with classical light waves for different values of nondimensional width τ of time window processed statistically based on the photonic representations are shown in Figures 3–5. Also, the dependences (2) and (4) predicted by quantum mechanics are shown.

Figure 3:

Dependence of (a) the probabilities of the pairs of events and (b) the correlation coefficient E(a, b) on the angle between the polarisers for τ=20. Markers are the results of semiclassical Monte Carlo simulations; lines are the predictions of quantum mechanics (2) and (4).

Figure 4:

Dependence of the (a) probabilities of the pairs of events and (b) the correlation coefficient E(a, b) on the angle between the polarisers for τ=1. Markers are the results of semiclassical Monte Carlo simulations; lines are the predictions of quantum mechanics (2) and (4).

Figure 5:

Dependence of (a) the probabilities of the pairs of events and (b) the correlation coefficient E(a, b) on the angle between the polarisers for τ=0.1. Markers are the results of semiclassical Monte Carlo simulations; lines are the predictions of quantum mechanics (2) and (4).

We can see that at τ>>1, the results of semiclassical Monte Carlo simulations practically coincide with the predictions of quantum mechanics (2) and (4) for entangled photons. In particular, the probabilities P₊₊(α=0) and P₋₋(α=0) differ slightly from 0.5, which in a real EPRB experiment could be attributedto the non-ideality of the optical equipment, as was done in [5], [6], [7].

At the same time, we see that the probabilities P_±±(a, b) and the correlation coefficient E(a, b) increasingly deviate from the predictions of quantum mechanics (2) and (4) with the decrease of the nondimensional width τ of the time window.

Thus, in the EPRB Gedankenexperiment with classical light waves, we observe exactly the effect that is called the entanglement of photons. We see that entanglement is observed at τ>>1 only after statistical processing of the “experimental data” based on the photonic representations and is related to the nonlinear dependence (18) at τ>>1.

4 Analytical Description of EPRB Gedankenexperiment

Let us obtain the expressions for probabilities P_±±(a, b).

First of all, we note that in the experiment under consideration, the splitting the light beam at one polariser is equivalent to the HBT experiment, with the only difference being that here the polariser selects two mutually perpendicular, and thus statistically independent, components of the vector E: 〈Ex2Ey2〉=〈Ex2〉〈Ey2〉=I02; at the same time, in the HBT experiment, each light beam behind the splitter is a mixture of both polarisations, and therefore both the beams behind the splitter are correlated: for example, for a Gaussian beam I₁I₂=2I₁I₂.

We first calculate N₊₊(a, b) in the EPRB Gedankenexperiment with classical light waves.

Because of the independence of the events on each detector, the probability that at fixed E_x and E_y the clicks of both detectors a₊ and b₊ will occur simultaneously but at the same time the clicks of detectors a₋ and b₋ will not occur is equal to P₊(a)P₊(b)[1−P₋(a)][1−P₋(b)], where

are the probabilities of triggering the corresponding detectors behind the polarisers a and b.

Then, by averaging over the all possible realisations of the parameters E_x and E_y, one obtains

where N is the total number of the time windows, and the averaging is carried out using the probability density (19):

Taking (19) and (22) into account, after some simple calculations, one obtains (for details of these calculations, see Supplementary Material)

Similarly

and after some simple calculations one obtains

It is easy to show that

Then for the conditional probabilities (21), one obtains the expression

Obviously, the normalisation of probabilities takes place:

Taking (28) into account, one obtains

Using the probabilities (29) and expression (3), it is easy to calculate the correlation coefficient E(a, b).

The results of calculations by the expressions (3), (25), and (27)–(29) are shown in Figs. 6 and 7.

Figure 6:

Comparison of the dependences of probabilities P₊₊(a, b) and P₊₋(a, b) on the angle between the polarisers for τ=20 calculated by the analytical expressions (25), (27)–(29) (solid lines) and obtained by the semiclassical Monte Carlo simulations (markers); dashed lines are the predictions of quantum mechanics (2).

Figure 7:

Theoretical dependences of (a) the probabilities P_±±(a, b) of the pairs of events and (b) the correlation coefficient E(a, b) on the angle between the polarisers α for different nondimensional widths τ of the time window. The dashed line corresponds to prediction (4) of quantum mechanics.

Figure 6 shows a comparison of the dependences of the probabilities P₊₊(a, b) and P₊₋(a, b) on the angle between polarisers for τ=20 calculated by the analytical expressions (25) and (27)–(29) and obtained by the semiclassical Monte Carlo simulations and processed statistically on the basis of the photonic representations.

Figure 7 shows that the analytical dependences (3), (25), and (27)–(29) at τ>>1 are close to the predictions of quantum mechanics (2) and (4) but do not match exactly with them. At the same time, at τ<1, the theoretical dependences (3), (25), and (27)–(29) are markedly different from the predictions of quantum mechanics (2) and (4) for entangled photons.

Using expressions (3), (25), and (27)–(29), one can calculate the parameter (5), the value of which allows us judge the possibility of describing the results of quantum experiments using the local hidden-variable theories.

Figure 8 shows the dependence of the parameter S, calculated by the expressions (3), (25), (27)–(29), and (5) for the set orientations (a, b)=(a′, b)=(a′, b′)=22.5^o and (a, b′)=67.5^o.

Figure 8:

Theoretical values of the CHSH criterion (5) depending on the nondimensional width τ of the time window. Dashed lines show the critical values predicted by the local hidden-variable theory (red line, S_HV=2) and quantum mechanics for entangled photons (green line, SQM=22), as well as the limiting value (asymptote) S_∞=3.2794, which corresponds to τ=∞ (blue line).

It also shows the limiting values predicted by the local hidden-variable theories S_HV=2 and by quantum mechanics SQM=22. Calculations show that, in the case of weak classical light waves, the parameter S has a limiting value S_∞=3.2794, which corresponds to τ=∞. Thus we see that depending on the nondimensional width τ of the time window, the parameter S can vary from S≈1.4145 at τ→0 up to S_∞≈3.2794 at τ→∞. In particular, in the semiclassical theory under consideration, the limiting value S_HV=2 predicted by the local hidden-variable theories is easily exceeded starting from τ≈0.5, while at τ>3.7, the parameter S calculated on the basis of the semiclassical theory exceeds even the limiting value SQM=22 predicted by quantum mechanics.

Obviously, this fact does not cause much surprise, because the result was obtained within the framework of the classical wave theory of light without using any real particles, and therefore there is no need even to mention a “spooky action at a distance”. For the same reason, there is no need even to discuss a theory of hidden parameters that would describe the results of the described experiments with weak light waves.

5 How Can One Obtain Exactly the Quantum Mechanical Predictions for Classical Light Waves?

First of all, we note that the relations

follow from the theory under consideration, where P_±±(α=0) and P_±±(α=π/2) are the probabilities P_±±(a, b) at α=0 and α=π/2, respectively, for a given non-dimensional width τ of the time window. The dependence of the probability P₊₊(α=π/2) on τ is shown in Figure 9.

Figure 9:

Dependence of the probability P₊₊(α=π/2) on the nondimensional width τ of the time window.

Taking into account the probabilities (32), one can scale the probabilities P_±±(a, b) using the expressions

Taking (31) and (32) into account, we conclude that the condition (31) is conserved also for scaled probabilities P′±±(a, b):

P′++(a, b)=P′−−(a, b),  P′+−(a, b)=P′−+(a, b).

Taking (30) and (32) into account, one obtains

This indicates that the parameters P′±±(a, b) can also be considered as some probabilities.

As an example, the dependences of the scaled probabilities P′++(a, b) and P′−−(a, b) on α are shown in Figure 10. Also, the markers show the predictions of quantum mechanics (2).

Figure 10:

Dependence of the scaled probabilities P′++(a, b) and P′−−(a, b) on α for different nondimensional width τ of the time window. All lines with τ≤0.1 practically coincide, as well as lines with τ≥20. Markers correspond to predictions of quantum mechanics (2).

We see that at τ≥1, the scaled probabilities P′±±(a, b) differ somewhat from the predictions of quantum mechanics (2). However, at τ<<1, the scaled probabilities P′±±(a, b) practically coincide with the quantum mechanical predictions (2). This means that the correlation coefficient (3) and the limiting value (7) of the parameter (5) calculated at τ<<1 coincide with the predictions of quantum mechanics.

Let us analyse the scaled probabilities P′±±(a, b).

Taking into account the definition (21), one can write

P′++(a, b)=N++(a, b)−N0P++(α=π/2)N′0,P′−−(a, b)=N−−(a, b)−N0P−−(α=π/2)N′0,P′+−(a, b)=N+−(a, b)−N0P+−(α=0)N′0,P′−+(a, b)=N−+(a, b)−N0P−+(α=0)N′0,

where

N′0=N0[1−4P++(α=π/2)].

Let us introduce

Taking (20) and (32) into account, one obtains

Then, for the scaled probabilities P′±±(a, b), one obtains the definition similar to the definition (21):

Expressions (38)–(43) give us an algorithm for calculating the probabilities P′±±(a, b): it is necessary to save only those time windows at which only one of the detectors behind each polariser was triggered, but at the same time the detectors behind both polarisers were triggered simultaneously. As a result, N₀ windows, which are consistent with the photonic representations, will be selected. Further, we assume that there is some background – the random simultaneous triggering the detectors behind different polarisers that are not connected with a “photons hit”. This background can be determined by considering the events (in the selected time windows N₀) on the detectors a₊ and b₊ at α=π/2, on the detectors a and b at α=π/2, on the detectors a₊ and b at α=0, and on the detectors a and b₊ at α=0. Indeed, according to quantum mechanics (2), the probabilities of such events must be equal to zero, and if they are not equal to zero, it should be perceived as the background, which must be rejected. According to (32), all these “background” events have the same probability; therefore, one need consider P₊₊(α=π/2) only. Considering that the background does not depend on the angle of the mutual pivot of the polarisers, we should subtract the number of the time windows in which we expect that the events are connected to the background from all selected time windows N₀ (for a given τ). Obviously, the number of such “background” time windows will be equal to N₀P₊₊(α=π/2). As a result, according to (38)–(41), N′±±(a, b) “good” time windows remain, for which a statistical processing (43) is performed. Thus the conditional probabilities P′±±(a, b) defined in this manner for τ<<1 exactly correspond to predictions of quantum mechanics for the so-called entangled photons (Fig. 10).

Let us formalise this analysis for the case τ<<1.

Taking (22), (23), (26), (28), and (29) into account, in this case one obtains

where A is the parameter defined from the normalisation condition (30).

Taking (10) and (12) into account, we write the expression (44) in the form

where E±(a) are the components of the vector E of the classical electromagnetic (light) wave at the entrance of polariser a, respectively, in parallel (+) or perpendicular (−) to the axis of the polariser.

Let us choose an arbitrary coordinate system (x, y) in which the components of the vector E are denoted as E₁=E_x and E₂=E_y.

Then, for components E±(a) and E±(b), one obtains the expressions

where i=1, 2; the summation is carried out by repeated indices, while the parameters ψ±_i(a) and ψ±_i(b) are connected with the angles of pivot of the axes of polarisers a and b with respect to the axis x of the chosen coordinate system, similar to the expression (11).

By virtue of the isotropy of the system under consideration

〈E±2(a)〉=〈E±2(b)〉=〈Ex2〉=〈Ey2〉.

Taking into account (46) and (47), one obtains

〈E±2(a)〉=ψ±i(a)ψ±k(a)〈EiEk〉,  〈E±2(b)〉=ψ±i(b)ψ±k(b)〈EiEk〉.

For the normal distribution (19)

〈EiEk〉=δik.

As a result, one obtains

According to (22), the probabilities of single events are determined by the expressions

where the parameter B is determined from the normalisation conditions

P+(a)+P−(a)=1;  P+(b)+P−(b)=1,

which follow from the rule of selection of “appropriate” time windows.

Taking (48) and (51) into account, one obtains B=12, which is equivalent to the result (1) of quantum mechanics.

Taking (49) into account, the expressions (51) can formally be written in the form

Let us introduce the functions

Ψ±(a)=12[ψ±1(a)+iψ±2(a)],Ψ±(b)=12[ψ±1(b)+iψ±2(b)].

Then the expressions (52) take the form

P±(a)=|Ψ±(a)|2;  P±(b)=|Ψ±(b)|2.

It follows that the functions Ψ±(a) and Ψ±(b) are the wave functions of single events observed at the detectors a± and b±.

Using (46) and (47), the expression (45) takes the form

P±±(a, b)=Aψ±i(a)ψ±j(a)ψ±k(b)ψ±m(b)〈EiEjEkEm〉.

For the normal distribution (19), one obtains

〈EiEjEkEm〉=δijδkm+δikδjm+δimδjk.

Then

Taking (50) into account, one can write (53) in the form

P±±(a, b)=A[1+2(ψ±i(a)ψ±i(b))2].

If the angle between the axes of the polarisers a and b is equal to α=π/2, taking (19) and (45) into account, one obtains

Then, taking (32)–(36) and (54) into account, one obtains

where the parameter A′ is determined from the normalisation condition (37). Taking the properties of the matrix ψ±_i into account, one obtains A′=12.

Let us introduce the functions

or in expanded form

Taking into account (56), the expression (55) can be written in the form

Thus, we have obtained (up to notation) the well-known result of quantum mechanics: the state of the system that is in the entangled state is described by the wave function (57), which cannot be factorised into a product of two states associated to each object, and at the same time the probability (58) of realisation of any of the possible binary events for such a system is equal to the square of the corresponding wave function (57).

6 Conclusions

Thus we see that results similar to the EPRB Gedankenexperiment and entanglement of photons can be obtained using weak classical light waves if we take into account the discrete (atomic) structure of the detectors and the specific nature of the light-atom interaction.

Once again, we note that there are no particles (photons) in the model under consideration, while light is considered as a classical electromagnetic wave; in this case, the discrete events on the detectors (clicks of detectors) are associated not with the hitting particles (photons) but with the excitation of the atoms of the detector by the classical electromagnetic wave according to the relation (8), which is the result of the solution of the Schrödinger equation. In this regard, it would be more correct to talk not about the entanglement of photon but about the entanglement of events for different detectors, or, more precisely, about the correlation of events for different detectors. In this case, if we call by“photons” the discrete events triggering the detectors under the action of the classical light wave, we will not face with such paradoxes as the “wave-particle duality” and the “spooky action at a distance”.

Then, as in the case of HBT effect, the correlation of the events in the EPRB Gedankenexperiment, which can be interpreted asthe entanglement of photons, is connectedwith the correlation of intensities of classical light waves arriving at the different detectors. The predictions of quantum mechanics for such “entangled photons”are adapted to the experimental data at the expense of an artificial rejection of the “bad” events that do not fit into the photonic representations. We note that in processing the results of real EPRB experiments [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], such concepts as detectors efficiency and the “background events” are actually used; this gives a justification for rejection of the “wrong” time windows and events [27], [28], [29]. Therefore, the postselection of the “suitable” events for subsequent statistical processing considered in this article is fully consistent with the existing practice of processing the results of real EPRB experiments.