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What does a violation of the Bell's inequality prove?

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In this paper I show that the Einstein-Podolsky-Rosen-Bohm Gedankenexperiment and so-called entanglement of photons have a simple explanation within the framework of classical electrodynamics if we take into account the discrete (atomic) structure of the detectors and a specific nature of the light-atom interaction. In this case we do not find such a paradox as "spooky action at a distance". I show that CHSH criterion in EPRB Gedankenexperiment with classical light waves can exceed not only a maximum value S_HV=2 which is predicted by the local hidden-variable theories but also the maximum value S_QM=2.828 predicted by quantum mechanics and in this case there is no desire to construct a local hidden-variable theory.
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1
What does a violation of the Bell’s inequality prove?
arXiv:1701.03700 [physics.gen-ph]
Sergey A. Rashkovskiy
Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow,
119526, Russia
Tomsk State University, 36 Lenina Avenue, Tomsk, 634050, Russia
E-mail: rash@ipmnet.ru, Tel. +7 906 0318854
January 11, 2017
ABSTRACT
In this paper I show that the Einstein-Podolsky-Rosen-Bohm Gedankenexperiment and so-called
entanglement of photons have a simple explanation within the framework of classical
electrodynamics if we take into account the discrete (atomic) structure of the detectors and a
specific nature of the lightatom interaction. In this case we do not find such a paradox as
spooky action at a distance. I show that CHSH criterion in EPRB Gedankenexperiment with
classical light waves can exceed not only a maximum value
2
HV
S
which is predicted by the
local hidden-variable theories but also the maximum value
22
QM
S
predicted by quantum
mechanics and in this case there is no desire to construct a local hidden-variable theory.
PACS number(s): 03.65.Ud, 03.65.Ta, 03.65.Sq, 03.50.De
Keywords: EPRB Gedankenexperiment; EPRB paradox; entanglement; semiclassical theory;
local hidden-variable theory.
I. INTRODUCTION
One of the most mysterious and intriguing predictions of quantum mechanics is the entanglement
phenomenon, which manifests in a strong correlation of the behavior of quantum objects, even
when they are separated by a large distance. According to quantum mechanics, the state of each
such an object cannot be described independently instead, a quantum state must be described
for the system as a whole. The entangled state cannot be factorized into a product of two states
associated to each object. According to this, it is considered 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 A. Einstein, B. Podolsky, and N. Rosen
[1] and was developed further by D. Bohm [2] who described what came to be known as the
EPRB Gedankenexperiment and EPRB paradox.
The first quantitative criterion which describes such a paradox was proposed by J. Bell (Bell’s
inequality) [3]. The Bell’s inequality, derived on the basis of the local hidden-variable theories,
2
contradicts in some cases the predictions of quantum mechanics. It is considered that an
experiment in which the violation of the Bell’s inequality occurs cannot be explained based on
the local realism view. Bell’s inequality gave the tool for experimental verification of the
counterintuitive predictions of quantum mechanics. Later, J. F. Clauser, M.A. Horne, A.
Shimony and R.A. Holt (CHSH) proposed a new criterion and an experiment to test the local
hidden-variable theories [4].
In these experiments, the two photons
1
and
2
, emitted in the entangled state, are analyzed by
linear polarizers in orientations a and b [5-7] (Fig. 1). Each polarizer is followed by two
detectors, giving results + or , corresponding to a linear polarization found parallel or
perpendicular to a and b.
Measuring the clicks of the detectors one can calculate the probabilities of events, both singles
and their coincidences.
Quantum mechanics predicts for singles probabilities
21)()()()( bbaa PPPP
(1)
where
)(a
P
and
)(b
P
are the probabilities of getting the results ± for the photons
1
and
respectively.
These results are in agreement with that each individual polarization 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 polarizer a and correspondingly the
clicks of the detectors
b
and
b
for polarizer b. Accordingly to [3,4] the entanglement of the
photons manifests in the probabilities
)( ba,P
of joint detections of
1
and
2
in the channels
+ or − of polarizers a and b. For entangled particles, quantum mechanics predicts:
2
2
sin
2
1
)()(
cos
2
1
)()(
baba
baba
,P,P
,P,P
(2)
where
is the angle between orientations of the polarizers a and b.
FIG. 1. Einstein-Podolsky-Rosen-Bohm Gedankenexperiment with photons [4-7].
In order to describe quantitatively the correlations between random events, one can introduce the
correlation coefficient [4]
S
+
_
+
_
b
a
2
1
3
)()()()(),( bababababa ,P,P,P,PE
(3)
Using (2), quantum mechanics predicts
2cos),( ba
QM
E
(4)
Carrying out the experiments for four different orientations
aa
,
and
bb
,
of the polarizers a
and b, one can calculate the parameter [4]
),(),(),(),( babababa
EEEES
(5)
The local hidden-variable theories predict [4]
22 HV
S
(6)
It is well known that the greatest conflict between quantum mechanical predictions and CHSH
inequalities (6) that follows from the local hidden-variable theories [4] is expected for the set
orientations
),(),(),( bababa
=22.5o and
),( ba
=67.5o. In this case, quantum mechanics
predicts
22
QM
S
(7)
CHSH inequality (6) was testable in numerous relevant experiments, starting with the pioneering
works [5-7], all of which have shown agreement with quantum mechanics rather than the
principle of local realism. Violation of Bell’s inequalities (6) was fixed for a wide range of the
distances and timings of the measurements [8-16].
These results have shown, in particular, that the EPRB experiments with entangled photons
cannot be described within the local hidden-variable theories and in general that it is impossible
to construct the local hidden-variable theories which are capable to describe 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 [5-16] is that these experiments can be physically explained based on
the photonic (corpuscular) representations only if one assumes that the interaction between the
two entangled particles and between the particles and measuring devices are propagated at a
velocity substantially exceeding the speed of light, which contradicts the relativity theory.
Einstein characterized 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 experiments [17,18] the results of which initially also have caused surprise. Later, the
simple explanation of the Hanbury Brown and Twiss effect was found within the framework of
4
semiclassical theory without quantization 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 classical electromagnetic (light) wave for time
t
(Fermi’s golden rule)
tbItw
(8)
where
w
is the probability of excitation of atoms per unit time;
2
~EI
is the intensity of the
classical light wave at the location of the atom; and
b
is a constant which does not depend on the
intensity of 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 Hanbury Brown and Twiss effect [19,20]. The Hanbury
Brown and Twiss correlation appears as a result of correlation of intensities of light waves
arriving at the two detectors due to splitting the incident light wave.
As shown in [21-23], the experiments with so-called “single photons” (but actually with the
weak classical light waves), namely the double-slit experiments and the Wiener experiments
with standing light waves, can completely be described using the expression (8) within the
framework of semiclassical theory without quantization of radiation.
Based on the results [19-23], one can conclude that the so-called “wave–particle duality” of light
has a simple explanation within the framework of classical electrodynamics if we take into
account the discrete (atomic) structure of detector, while the concept of a “photon” becomes
superfluous.
In this paper I show that the Einstein-Podolsky-Rosen-Bohm Gedankenexperiment and so-called
entanglement of “photons” have also a simple explanation within the framework of classical
electrodynamics if we take into account the expression (8) and atomic structure of the detectors.
II. 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 two opposite
directions; i.e.
EEE 21
(9)
for light waves
1
and
2
(Fig. 2).
5
The emitted waves
1
and
2
arrive at two spatially separated two-channel polarizers a and b,
each of which splits the incident light beam into two mutually orthogonal linearly polarized
beams that arrive at corresponding detectors. For each polarizer, we introduce its own coordinate
system
),( yx
, where the
x
axis is parallel to the axis of the polarizer, while the axis
y
is
perpendicular to it. Further, the beam with polarization parallel to the axis of the polarizer is
denoted by the index “+, while the beam with polarization perpendicular to the axis of the
polarizer is denoted by the index . Corresponding detectors will be denoted as
a
and
.
FIG. 2. Einstein-Podolsky-Rosen-Bohm Gedankenexperiment with classical light waves.
Each polarizer can rotate around the axis of the incident beam. Due to the isotropy of the system,
only a relative angle of rotation of the polarizers
has the meaning. Therefore, we choose a
coordinate system associated with polarizer a as the reference system. Furthermore, we assume
that polarizers are ideal, i.e. we neglect the energy losses of light wave at passing the optical
system. Then only the component
x
E
of the incident light wave
1
will arrive at the detector
a
while only the component
y
E
of this wave will arrive at the detector
a
. The intensities of light
waves arriving at the detectors
a
are as follows
2
)( x
EI
a
;
2
)( y
EI
a
(10)
Let polarizer
b
is turned with respect to the polarizer
a
at an angle
(Fig. 2).
We denote the own coordinate system of the polarizer
b
as
),( yx
, whose axes are parallel to
the corresponding main axis of the polarizer
b
. Then only the component
x
E
of the incident
light wave
2
will arrive at the detector
b
while only the component
x
E
of this wave will
arrive at the detector
b
.
Taking into account (9), one can write
x
y
x'
y'
a
b
a+
a
b+
b
S
Ex
Ey
Ex
Ey
2
1
6
sincos yxx EEE
,
cossin yxy EEE
(11)
The intensities of light waves arriving at the detectors
b
are, respectively,
2
)( x
EI
b
;
2
)( y
EI
b
(12)
Under the action of the incident light wave, the excitation of the atoms of a 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 registrar.
It is believed that after the triggering, the detector (an atom) returns again into the initial
(ground) state and 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 that of radioactive
decay. In particular, the probability of excitation of an atom during a time Δt is [21-23]
)exp(1)( twtP
(13)
Taking into account (8), one obtains
)exp(1)( tbItP
(14)
where
I
is the intensity of light wave arriving at a corresponding detector.
Assuming that a source of radiation is Gaussian, and that the components of the electric field
vector
),( yx EEE
of the light wave are statistically independent, one obtains the probability
density for the components of the electric field vector
0
22
02
exp
21
),( I
EE
I
EEp yx
yx
(15)
where
22
0yx EEI
(16)
...
denotes averaging.
Obviously, considering (9),
0
22 IEE yx
(17)
Equations (14)-(17) allow calculating the EPRB Gedankenexperiment in detail. Indeed,
calculating the single events of triggering the detectors
a
and
b
using the expressions (14)
and (15), one can determine their statistics both for single events and for their coincidences and
compare it with the predictions of quantum mechanics (1), (2).
7
III. MONTE CARLO SIMULATION OF EPRB GEDANKENEXPERIMENT
Let us consider the discrete time intervals (time windows)
Ni ,...,2,1
which have a duration
t
, 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 (14), in which we use the intensities (10) and
(12) of the light waves arriving at corresponding detector within a given time window. The
intensity of the light waves
1
and
2
emitted by a source for different time windows are
considered to be random and are described by the probability density function (15).
Let us introduce the nondimensional exposure time (nondimensional duration of time window)
tbI 0
(18)
In this case, the probability of excitation of the atom during a time window is
 
)(exp1)( 0
IItP
(19)
Further, we take the value
0
I
as a characteristic intensity of light. In this case we use the
parameter
21
0
I
as a scale for the field
E
. Taking into account the expressions (15) and (19), in
further calculations we will take
1
0I
, while the value
0
I
itself will be included into
nondimensional duration
of time window.
Then the expressions (15) and (19) can be written in nondimensional form:
 
IP exp1)(
(20)
2
exp
2
1
),( 22 yx
yx EE
EEp
(21)
where
2
EI
.
Thus in the problem under consideration, there is a single nondimensional parameter
varying
which we can change the experimental conditions”.
The calculation proceeds quite trivial using the Monte Carlo method: at each time window
i
, the
components of the electric field vector
E
of the light wave are generated using the probability
density (21). Using the components
x
E
and
y
E
, 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 (20). At the same time the random numbers
]1,0[R
are generated for each detector
using a random number generator. If the condition
PR
is satisfied for some detector, it is
8
considered to be excited and this event is recorded in corresponding time window. Thus, we
record all events of 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 [5-16], the duration of the time window
was significantly longer then 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 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
Ni ,...,2,1
were calculated, the statistical analysis 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 analyze the results of Monte Carlo simulations based on photonic
(corpuscular) representations. For this purpose, we will assume that each triggering the detector
is the result of hit on it a particle - a photon. At the same time, we will remember that in reality,
the results of Monte Carlo simulation were obtained within the framework of 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 analyze the results of experiments from the standpoint of the photonic
representations, 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 polarizer simultaneously triggered during one time
window, for indivisible photon, such an event can be explained either by a background or by
interferences in 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 the detectors behind the both polarizers will be
recorded. In other words, if a photon was detected behind the polarizer a, then the second photon
must also be detected behind the polarizer b. If a second event did not happen, then the result can
be explained either by an insufficient sensitivity of a detector, or by a malfunction of a detector,
or by the fact that a one photon of the entangled pair was lost on the way to a polarizer, which
is also perceived as an detection error and such time windows should not be taken into account
when calculating the photon coincidences.
9
Thus when counting the number of coincidences we have to reject not only the time windows in
which both detectors triggered behind any polarizer, but also those time windows at which one
detector triggered behind one of polarizers while there are no detectors triggered behind other
polarizer, because only such events are consistent with the “photonic representations” in this
experiment.
Thus, by statistical analysis of the results of semiclassical Monte Carlo simulations of EPRB
Gedankenexperiment with classical light waves, we calculate the number of the time windows in
which the pairs of corresponding events have occurred:
),(),,(),,(),,( babababa NNNN
. For example,
),( ba
N
is the number of time
windows in which the events were recorded simultaneously on the detectors
a
and
, while
the events on other detectors were not observed, etc. Then
),(),(),(),(
0babababa NNNNN
(22)
is the number of the time windows at which the events were recorded behind both polarizers but
only one detector has triggered behind each polarizer.
Then the probabilities of the corresponding pair events are determine by the expression
0
),(),( NNP baba
(23)
The probabilities (23) determined exactly in this way correspond to those calculated in quantum
mechanics.
Using the probabilities (23) for each relative orientation
of the polarizers 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
of time window processed statistically
based on the photonic representations are shown in Figs. 3-5. Ibid, the dependences (2) and (4)
predicted by quantum mechanics are shown.
We can see that at large
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
)0(
P
and
)0(
P
differ slightly from 0.5, which in a real
EPRB experiment, could be attributed due to a non-ideality of the optical equipment, as it was
done in [5-7].
At the same time, we see that the probabilities
),( ba
P
and the correlation coefficient
),( baE
are increasingly deviated from the predictions of quantum mechanics (2) and (4) with the
decrease of nondimensional width
of time window.
Thus, in the EPRB Gedankenexperiment with classical light waves, we observe exactly the effect
which is called entanglement of photon. We see that entanglement is observed at
1
only
10
after statistical processing the “experimental data” based on the photonic representations and is
related to nonlinear dependence (20) at
1
.
FIG. 3. (Colour online) Dependence of the probabilities of the pairs of events (left) and
correlation coefficient
),( baE
(right) on the angle between polarizers for
=20. Markers are the
results of semiclassical Monte Carlo simulations; lines are the predictions of quantum mechanics
(2) and (4).
FIG. 4. (Colour online) Dependence of the probabilities of the pairs of events (left) and
correlation coefficient
),( baE
(right) on the angle between polarizers for
=1. Markers are the
results of semiclassical Monte Carlo simulations; lines are the predictions of quantum mechanics
(2) and (4).
0
0.1
0.2
0.3
0.4
0.5
010 20 30 40 50 60 70 80 90
Probabilities
p++
p--
p+-
p-+
QM
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
010 20 30 40 50 60 70 80 90
E(a,b)
0
0.1
0.2
0.3
0.4
0.5
010 20 30 40 50 60 70 80 90
Probabilities
p++
p--
p+-
p-+
QM
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
010 20 30 40 50 60 70 80 90
E(a,b)
11
FIG. 5. (Colour online) Dependence of the probabilities of the pairs of events (left) and
correlation coefficient
),( baE
(right) on the angle between polarizers for
=0.1. Markers are the
results of semiclassical Monte Carlo simulations; lines are the predictions of quantum mechanics
(2) and (4).
IV. ANALYTICAL DESCRIPTION OF EPRB GEDANKENEXPERIMENT
Let us obtain the expressions for probabilities
),( ba
P
.
First of all, we note that in the experiment under consideration, the splitting the light beam at the
one polarizer is equivalent to the Hanbury Brown and Twiss experiment, with the only difference
being that here the polarizer selects two mutually perpendicular, and thus statistically
independent components of the vector
E
:
2
0
2222 IEEEE yxyx 
; at the same time, in the
Hanbury Brown and Twiss experiment, each light beam behind a splitter is a mixture of both
polarizations, and therefore both the beams behind a splitter are correlated: for example, for a
Gaussian beam
2121 2IIII
.
We first calculate
),( ba
N
in EPRB Gedankenexperiment with classical light waves.
Due to independence of the events on each detector, the probability that, at fixed
x
E
and
y
E
,
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
)](1)][(1)[()( baba PPPP
, where
 
)(exp1)( aa IP
;
 
)(exp1)( bb IP
(24)
are the probabilities of triggering the corresponding detectors behind the polarizers
a
and
b
.
Then, averaging over the all possible realizations of the parameters
x
E
and
y
E
, one obtains
)](1)][(1)[()(),( bababa PPPPNN
(25)
0
0.1
0.2
0.3
0.4
0.5
010 20 30 40 50 60 70 80 90
Probabilities
p++
p--
p+-
p-+
QM
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
010 20 30 40 50 60 70 80 90
E(a,b)
12
where
N
is the total number of the time windows; the averaging is carried out using the
probability density (21):
 
yxyx dEdEEEpPPPPNN ),()](1)][(1)[()(),( bababa
(26)
Taking (21) and (24) into account, after the simple calculations one obtains
)41( 1
861
2
sin441
1
),( 222
NN ba
(27)
Similarly
 
yxyx dEdEEEpPPPPNN ),()](1)][(1)[()(),( bababa
(28)
and after the simple calculations one obtains
)41( 1
861
2
cos441
1
),( 222
NN ba
(29)
It is easy to show that
),(),( baba NN
,
),(),( baba NN
(30)
Then for the conditional probabilities (23) one obtains the expression
 
),(),(2 ),(
),( baba ba
ba
NN N
P
(31)
Obviously, the normalization of probabilities takes place:
1),(),(),(),( babababa PPPP
(32)
Taking (30) into account one obtains
),(),( baba PP
,
),(),( baba PP
(33)
Using the probabilities (31) and expression (3), it is easy to calculate the correlation coefficient
),( baE
.
The results of calculations by the expressions (3), (27), (29)-(31) are shown in Figs. 6 and 7.
Figure 6 shows a comparison of the dependences of probabilities
),( ba
P
and
),( ba
P
on the
angle between polarizers for
=20 calculated by the analytical expressions (27), (29)-(31) and
obtained by the semiclassical Monte Carlo simulations and processed statistically based on the
photonic representations.
Figure 7 shows that the analytical dependences (3), (27), (29) - (31) at
1
are close to the
predictions of quantum mechanics (2) and (4) but do not match exactly with them. At the same
time at
1
, the theoretical dependences (3) (27) (29) - (31) are markedly different from the
predictions of quantum mechanics (2) and (4) for entangled photons.
13
Using expressions (3), (27), (29)-(31), one can calculate the parameter (5), the value of which
allows judging about possibility to describe 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), (27), (29)-
(31) and (5) for the set orientations
),(),(),( bababa
=22.5o and
),( ba
=67.5o.
FIG. 6. (Colour online) Comparison of the dependences of probabilities
),( ba
P
and
),( ba
P
on the angle between polarizers for
=20 calculated by the analytical expressions (27), (29)-(31)
(solid lines) and obtained by the semiclassical Monte Carlo simulations (markers); dashed lines
are the predictions of quantum mechanics (2).
It also shows the limit values predicted by the local hidden-variable theories
2
HV
S
and by
quantum mechanics
22
QM
S
. Calculations show that in the case under consideration, the
parameter S has a limit value
S
3.2794 which corresponds to
. Thus we see that
depending on the nondimensional width
of time window, the parameter S can vary from
S
1.4145 at
0
up to
S
3.2794 at
. In particular, in the semiclassical theory under
consideration, the limit value
2
HV
S
predicted by the local hidden-variable theories is easily
exceeded starting from
0.5, while at
3.7, the parameter S calculated based on the
semiclassical theory exceeds even the limit value
22
QM
S
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”. In this regard, no one will have
a desire to construct a hidden-variable theory which would describe the observed coincidences
using concept of some fictitious particles (photons).
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
010 20 30 40 50 60 70 80 90
Probabilities
p++
p+-
14
FIG. 7. (Colour online) Theoretical dependences of the probabilities
),( ba
P
of the pairs of
events (left) and correlation coefficient
),( baE
(right) on the angle between polarizers
for
different nondimensional width
of the time window. The dashed line corresponds to
prediction (4) of quantum mechanics.
FIG. 8. (Colour online) Theoretical values of the CHSH criterion (5) depending on the
nondimensional width
of the time window. Dashed lines show the critical values predicted by
the local hidden-variable theory (red line,
2
HV
S
) and quantum mechanics for entangled
photons (green line,
22
QM
S
), as well as the limit value (asymptote)
S
3.2794 which
corresponds to
(blue line).
V. HOW CAN ONE OBTAIN EXACTLY THE QUANTUM MECHANICAL
PREDICTIONS?
First of all note, that relations
5.0)2()0(
PP
(34)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
010 20 30 40 50 60 70 80 90
Probabilities
0.001
0.1
1.0
20
500
P++, P--
P+-, P-+
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
010 20 30 40 50 60 70 80 90
E(a,b)
0.001
0.1
1.0
20
500
0
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20
S
HV
QM
S
15
)0()0()2()2(
PPPP
(35)
follow from the theory under consideration, where
)0(
P
and
)2(
P
are the
probabilities
),( ba
P
at
0
and
2
, respectively, for a given nondimensional width
of the time window. Dependence of probability
)2(
P
on
is shown in Fig. 9.
FIG. 9. Dependence of the probability
)2(
P
on the nondimensional width
of the time
window.
Taking into account the probabilities (35), one can scale the probabilities
),( ba
P
, using the
expressions
)2(41 )2(),(
),(
PPP
Pba
ba
(36)
)2(41 )2(),(
),(
PPP
Pba
ba
(37)
)0(41 )0(),(
),(
PPP
Pba
ba
(38)
)0(41 )0(),(
),(
PPP
Pba
ba
(39)
Taking (33) and (35) into account we conclude that the condition (33) is conserved also for
scaled probabilities
),( ba
P
:
),(),(),,(),( babababa
PPPP
(40)
Taking (32) and (35) into account one obtains
1),(),(),(),(
babababa PPPP
(41)
This indicates that the parameters
),( ba
P
can also be considered as some probabilities.
0
0.05
0.1
0.15
0 5 10 15 20
P++(
=π/2)
16
As an example, the dependences of the scaled probabilities
),( ba
P
and
),( ba
P
on
are
shown in Fig. 10. Ibid, the markers show the predictions of quantum mechanics (2).
We see that at
1
, the scaled probabilities
),( ba
P
differ somewhat from the predictions of
quantum mechanics (2), however at
1
, the scaled probabilities
),( ba
P
practically
coincides with the quantum mechanical predictions (2). This means that the correlation
coefficient (3) and the limit value (7) of the parameter (5) calculated at
1
coincide with the
predictions of quantum mechanics.
FIG. 10. (Colour online) Dependencies of the scaled probabilities
),( ba
P
and
),( ba
P
on
for different nondimensional width
of the time window. Lines
=0.001 and
=0.1 practically
coincide, as well as the lines
=20 and
=500. Markers correspond to predictions of quantum
mechanics (2).
Let us analyze the scaled probabilities
),( ba
P
.
Taking into account a definition (23), one can write
0
0)2(),(
),( NPNN
P
ba
ba
(42)
0
0)2(),(
),( NPNN
P
ba
ba
(43)
0
0)0(),(
),( NPNN
P
ba
ba
(44)
0
0)0(),(
),( NPNN
P
ba
ba
(45)
where
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
010 20 30 40 50 60 70 80 90
Scaled probabilities P'++ and P'--
0.001
0.1
1.0
20
500
QM
17
)]2(41[
00
PNN
(46)
Let us introduce
)2(),(),( 0
PNNN baba
(47)
)2(),(),( 0
PNNN baba
(48)
)0(),(),( 0
PNNN baba
(49)
)0(),(),( 0
PNNN baba
(50)
Taking (22) and (35) into account, one obtains
),(),(),(),(
0babababa
NNNNN
(51)
Then for the scaled probabilities
),( ba
P
, one obtains the definition, similar to definition (23):
0
),(),( NNP
baba
(52)
Expressions (47)-(52) give us an algorithm for calculation of probabilities
),( ba
P
:, it is
necessary to leave only those time windows at which only one of the detectors behind each
polarizer triggered, but at the same time the detectors behind both polarizers triggered
simultaneously. As a result,
0
N
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 polarizers which are not connected with a “photons hit.
This background can be determined by considering the events (in the selected time windows
0
N
) on the detectors
a
and
b
at
2
, on the detectors
a
and
b
at
2
, on the
detectors
a
and
b
at
0
and on the detectors
a
and
b
at
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
(35), all these “background” events have the same probability, therefore, one can consider
)2(
P
only. Considering that the background does not depend on the angle of the mutual
pivot of the polarizers, 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
0
N
(for given
). Obviously, the number of such background time windows will be equal to
)2(
0
PN
. As a result, according to (47)-(50),
),( ba
N
good time windows remain,
for which a statistical processing (52) is performed. Thus the conditional probabilities
),( ba
P
defined in this manner for
1
exactly correspond to predictions of quantum mechanics for so-
called entangled photons" (Fig. 10).
Let us formalize this analysis for the case
1
.
18
Taking (24), (25), (28), (30) and (31) into account, in this case one obtains
)()(),( baba IIAP
(53)
where
A
is the parameter which is defined from the normalization condition (32).
Taking (10) and (12) into account, we write the expression (53) in the form
)()(),( 22 baba EEAP
(54)
where
)(a
E
are the components of the vector
E
of classical electromagnetic (light) wave at the
entrance of polarizer
a
, respectively, in parallel (+) or perpendicular (-) to the axis of polarizer.
Let us choose an arbitrary coordinate system
),( yx
in which the components of the vector
E
we
denote as
x
EE
1
and
y
EE
2
.
Then for components
)(a
E
and
)(b
E
one obtains the expressions
ii EE )()( aa
(55)
ii EE )()( bb
(56)
where
2,1i
; summation is carried out by repeated indices, while the parameters
)(a
i
and
)(b
i
are connected with the angles of pivot of the axes of polarizers
a
and
b
with respect to
the axis
x
of the chosen coordinate system similar to expression (11).
By virtue of isotropy of the system under consideration
2222 )()( yx EEEE ba
(57)
Taking into account (55) and (56), one obtains
kikikiki EEEEEE )()()(,)()()( 22 bbbaaa
(58)
For the normal distribution (21)
ikkiEE
(59)
As a result, one obtains
1)()( 22 ba EE
(60)
)()()(),()()( 22 bbbaaa iiii EE
(61)
1)()()()( bbaa iiii
(62)
According to (24), the probabilities of single events are determined by the expressions
)()( 2aa EBP
;
)()( 2bb EBP
(63)
where the parameter
B
is determined from the normalization conditions
1)()(;1)()( bbaa PPPP
(64)
which follow from the rule of selection of “appropriate” time windows.
19
Taking (60) and (63) into account, one obtains
2
1
B
, which is equivalent to the result (1) of
quantum mechanics.
Taking (61) into account, the expressions (63) can formally be written in form
)()(
2
1
)( aaa ii
P
;
)()(
2
1
)( bbb ii
P
(65)
Let us introduce the functions
 
)()(
2
1
)(,)()(
2
1
)( 2121 bbbaaa
ii
(66)
Then the expressions (65) take the form
22 )()(;)()( bbaa PP
(67)
It follows that the functions
)(a
and
)(b
are the wave functions of single events observed
at the detectors
a
and
b
.
Using (55) and (56), the expression (54) takes the form
mkjimkji EEEEAP )()()()(),( bbaaba
(68)
For the normal distribution (21) one obtains
jkimjmikkmijmkji EEEE
(69)
Then
)]()()()(2)()()()([),( bababbaaba kkiikkii
AP
(70)
Taking (62) into account one can write (70) in the form
 
])()(21[),( 2
baba ii
AP
(71)
If the angle between the axes of the polarizers a and b is equal to
2
, taking (21) and (54)
into account one obtains
AEEAEEAP yxyx 
)()()()()2( 2222 baba
(72)
Then taking (35)-(39) and (72) into account one obtains
 
2
)()(),( baba ii
AP
(73)
where the parameter
A
is determined from the normalization condition (41). Taking the
properties of the matrix
i
into account, one obtains
2
1
A
.
Let us introduce the functions
)()(
2
1
),( baba ii
(74)
20
or in expanded form
 
)()()()(
2
1
),( 2211 bababa
(75)
Taking into account (74), the expression (73) can be written in the form
2
),(),( baba
P
(76)
Thus, we have obtained (up to notation) the well-known result of quantum mechanics: the state
of the system which is in entangled state, is described by the wave function (75), which cannot
be factorized into a product of two states associated to each object, at the same time the
probability (76) of realization of any of the possible binary events for such a system is equal to
the square of the corresponding wave function (75).
VI. CONCLUDING REMARKS
Thus we see that the EPRB Gedankenexperiment and entanglement of photons have a simple
explanation within the framework of classical electrodynamics and classical optics without
involving such a concept as a “photon”.
Indeed, 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 hitting the particles (photons), but with an excitation
of the atoms of the detector by the classical electromagnetic wave according to the relation (8),
which is the result of solution of the Schrödinger equation. In this regard, it would be more
correct to talk not about the entanglement of photons, but about the entanglement of events for
different detectors, or, more precisely, about the correlation of events for different detectors. In
particular, if, as was proposed in [21,23], we will call by the photons, not some mythical light
particles, but the discrete events of triggering the detectors under the action of classical light
wave, we will not face with such paradoxes as the wave-particle duality and the spooky
action at a distance.
Thus, as in the case of Hanbury Brown and Twiss effect, the correlation of the events in the
EPRB Gedankenexperiment, which is called the entanglement of photons, is connected with the
correlation of intensities of classical light waves arriving at the different detectors. The
predictions of quantum mechanics for entangled photons are adapted to the experimental data
at the expense of an artificial rejection of the bad events which do not fit into the photonic
representations. We note that at processing of the results of real EPRB experiments [5-16], such
21
concepts as the detectors efficiency and the “background events” are actually used; this gives a
justification for rejection of the “wrong” time windows and events [24-26]. Therefore, the
selection of the “suitable” events for subsequent statistical processing considered in this paper, is
fully consistent with the existing practice of processing the results of real EPRB experiments.
Thus we can conclude that a violation of the Bell’s inequalities proves only that the intensities of
light waves arriving at different detectors are correlated in full compliance with classical
electrodynamics and classical optics.
In this connection, a question arises: can the classical “entangled” light waves be used for
quantum computing and quantum cryptography? Is it possible to use the classical correlated light
beams taking into account their specific character of interaction with the detectors to construct
the computational algorithms, similar to “quantum” algorithms?
ACKNOWLEDGMENTS
Funding was provided by the Tomsk State University competitiveness improvement program.
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... One of the most mysterious of quantum mechanics is the entanglement phenomenon, which shows strong correlation of the behavior of quantum objects, even when they are separated by a large distance [1,2]. According to quantum mechanics, a state of each such an object cannot be described independently, instead a quantum state must be described for the system as a whole [3]. By other words entangled state cannot be factorized into a product of two states associated to each object. ...
... As the pump beam interacts with these nonlinear crystals, single photon split into entangled "signal" and "idler" photons with wavelengths longer than the pump [4]. The entanglement phenomenon was considered for the first time by A. Einstein, B. Podolsky, and N. Rosen and was developed further by D. Bohm who described what came to be known as the EPRB Gedanken experiment and EPRB paradox [3]. ...
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