On Bell’s Joint Probability Distribution Assumption and Proposed
Experiments of Quantum Measurement
Hai-Long Zhao
27th branch post office 15th P.O.BOX 11#, Lanzhou, 732750, China
Abstract: In the derivation of Bell’s inequalities, there is an additional assumption of joint
probability distribution besides the assumptions of locality and realism. It has been shown that
if this assumption does not hold, then Bell’s inequalities fail. Our further analysis shows that
Bell’s joint probability distribution holds only for two independent events but not for the joint
measurements of EPR pairs. We point out that polarization entangled photon pair is actually a
pair of circularly polarized photons or a circularly polarized photon and a linearly polarized
photon whose hidden variables are correlated. This hypothesis can explain the experimental
results of EPR pairs. Several experiments are proposed to test the relevant problems in
quantum measurement. The first use delayed measurement on one photon of EPR pair to
demonstrate directly whether measurement on the other could have any non-local influence
on it. The other two are used to reveal the constituents of polarization Bell states. The last one
verifies the deduction of determinism that two particles with the same hidden variable will
behave the same under the same measuring condition.
PACS: 03.65.Ta; 03.65.Ud; 42.50.Xa
1. Introduction
Quantum theory gives only probabilistic predictions for individual events based on the
probabilistic interpretation of wave function, which leads to the suspicion of the completeness
of quantum mechanics and the puzzle of the non-locality of the measurement of EPR pairs [1].
Indeed, if hidden variable theory is not introduced into quantum measurement, we can hardly
understand the distant correlation of EPR pairs, e.g. quantum teleportation and quantum
swapping [2,3]. Bell pointed out that any theory that was based on the joint assumptions of
locality and realism conflicted with the quantum mechanical expectation [4]. Since then,
various local and non-local hidden variable models against Bell’s inequalities have been
proposed (see, for example, [5-12]), of which the most attractive one is the time-related and
setting-dependent model suggested by Hess and Philipp [11,12], but was criticized by Gill et
al and Myrvold for being non-local [13,14]. As a matter of fact, there is an additional
assumption of joint probability distribution besides the assumptions of locality and realism
[15-17]. Bell supposed that joint probability distribution is only a function of hidden variable
and is irrelevant to measuring condition. However, the validity of this assumption is dubious.
As pointed out by de la Peňa et al and Nagasawa that if this assumption does not hold, then
Bell’s inequalities fail [16,17]. On the other hand, it has been shown that even if non-locality
is taken into account, Bell’s inequalities may also be violated [18,19]. So we focus on Bell’s
joint probability distribution assumption and discuss its validity. We point out that this
assumption is equivalent to the measurement of two independent events and is inapplicable to
the joint measurement of EPR pairs. In the meanwhile, we suggest polarization uncertainty as
the hidden variable for polarization degree of freedom.
In terms of quantum entanglement, the spin (polarization) of a pair of EPR particles is
1
indefinite and dependent on each other. By analysis of existing experiments [20-31], we show
that polarization Bell states (maximally entangled states) can be composed of two circularly
polarized photons or a circularly polarized photon and a linearly polarized photon (given their
hidden variables are correlated). The experimental results of EPR pairs are explained based on
this assumption. If hidden variable does exist, then the quantum state of one of the EPR pair
will not change when measurement is made on the other, and the outcomes of a pair of
particles with the same hidden variable will be the same under the same circumstance. We
propose three types of experiments to test above hypotheses. The experiments are easy to
realize for the experimental setups are very simple.
2. Discussion on Bell’s joint probability distribution assumption and
suggested hidden variable
In local hidden variable theory Bell’s inequalities play an important role. Bell regarded
that his correlation function was founded on the vital assumption of Einstein that the result of
B does not depend on the setting of measuring device a , nor A on b [4].
P(a, b) = ∫ A(a, λ ) B(b, λ ) ρ (λ ) dλ ,
(1)
where A(a, λ ) = ±1 , B(b, λ ) = ±1 , ρ (λ ) satisfies normalized condition
∫ ρ (λ )dλ = 1 . Bell
regarded joint probability distribution as only a function of λ . de la Peňa et al suggested that
joint probability distribution may depend on measuring condition [16]. Nagasawa later made
a detailed analysis based on strict mathematical definition [17]. We now discuss this problem
in a simple way. Eq. (1) includes four joint probabilities, which are P+ + ( A = 1, B = 1) ,
P+ − ( A = 1, B = −1) ,
P− + ( A = −1, B = 1) ,
P− − ( A = −1, B = −1) , respectively. Then we have
P (a, b) = P+ + − P+ − − P− + + P− − . Since P (a, b) actually denotes joint probabilities, ρ must be
the joint probability density function with respect to the measurement outcomes A and B ,
i.e. ρ = ρ ( A = ±1, B = ±1) . As the results of A and B depend on the settings of measuring
devices and hidden variables of the pair, we have ρ = ρ (a, b, λ ) . If it does not vary with
measuring condition, then it becomes the case that Bell considered. For a pair of EPR
particles it’s easy to understand that they share a same hidden variable. But there is no prior
reason that joint probability distribution is irrelevant to measuring condition. Although it
satisfies the normalized condition, its expression may vary with the settings of measuring
devices. Two possible curves are plotted in Fig.1 representing the different measuring
conditions a , b and a ′ , b ′ , respectively.
ρ ( a , b, λ )
ρ (a′, b′, λ )
0
1
λ
Fig. 1. Possible joint probability distributions under different measuring conditions.
One might think that ρ = ρ (a, b, λ ) conflicts with the locality. In fact, locality
assumption has already been included in the expressions of A = A(a, λ ) and B = B (b, λ ) .
2
Since joint measurement outcomes are related to a , b and λ , it’s natural that joint
probability distribution is a function of a , b and λ , which is unconcerned with
non-locality.
Now we discuss joint probability in another way. As A = A(a, λ ) , B = B (b, λ ) , we have
P (a) = ∫ A(a, λ ) ρ (a, λ )dλ , P(b) = ∫ A(b, λ ) ρ (b, λ )dλ , i.e. the probability spaces of the two
events are different. In order to calculate the joint probability we must carry it out in the same
probability space. For convenience we calculate the joint probability in the probability space
of A . Then Eq. (1) is modified as
P(a, b) = ∫ A(a, λ ) B (b, λ | A) ρ (a, λ )dλ = ∫ A( a, λ ) B(a, b, λ ) ρ (a, λ )dλ ,
(2)
where B(a, b, λ ) does not mean that the setting of measuring device a could have any
non-local influence on the result of B . It denotes the result of B conditioned to the settings
of measuring device that is related to a and b . To be specific consider the case that a pair
EPR photons is incident upon a pair of polarizers. B(a, b, λ ) represents the measurement
outcome of B under the condition that the orientation of polarizer is set to the direction
a ± b and the result of A is already known. It can be seen that Eq. (2) is just the joint
probability expressed with conditional probability, which applies to the calculation of joint
probability of two dependent events. If the two events are independent, then Eq. (2) is turned
into (1). Similarly, we have
P(a, c) = ∫ A(a, λ ) B(a, c, λ ) ρ (a, λ )dλ ,
(3)
P (b, c) = ∫ A(b, λ ) B (b, c, λ ) ρ (b, λ ) dλ .
(4)
Substituting ρ (b, λ ) in Eq. (4) with ρ (a, λ ) , we have
P (b, c) = ∫ A(a, b, λ ) B (a, b, c, λ ) ρ (a, λ )dλ .
(5)
With the Eqs. (2), (3) and (5), Bell’s inequalities cannot be obtained. We do not discuss the
detailed derivation process.
Most of the researchers think that the measurements of EPR pairs are stochastically
independent events, and they attribute this independence to locality [13-15]. Then Eq. (1) is
valid. Such assumption is unreasonable. Although the experimental results of A and B do
not depend on the setting of measuring device on the other side, this does not necessarily
imply that the two events are independent. In fact, dependent events that have the feature of
locality exist widely in macroscopic and microscopic worlds. For example, suppose there are
ten holes, of which a little cat can pass through six and a big cat can pass through four. Then
the probability that both cats can pass through a hole is not 0.6×0.4=0.24 but 0.4. This is
because if the big cat can pass through, then the little cat can pass through with certainty. For
a pair of particles in singlet state considered by Bell, perfect anticorrelation exists in the case
of a = b , i.e. A(a, λ ) = − B (a, λ ) . So we cannot conclude independence from locality, just as
we cannot think that the joint probability distribution is unrelated to measuring condition.
From above analysis we see that the problem with Bell’s inequalities is not the
assumptions of locality and realism but the assumption of joint probability distribution (or the
independence of the measurement outcomes of EPR pairs), which is the fundamental reason
that Bell’s inequalities conflict with the prediction of quantum mechanics. For a pair of EPR
particles whose hidden variables are correlated, their measurement outcomes are correlated. In
3
the derivation of Bell’s inequalities, only the perfect anticorrelation is considered when the
settings of measuring devices on the two sides are the same. In other cases this correlation is
not considered. So Eq. (1) actually represents the correlation of two independent events. In
order to indicate the intrinsic correlation of two dependent events, one way is to suppose that
joint probability density function varies with measuring condition, the other is the expression
with conditional probability. In both cases we cannot derive Bell’s inequalities.
In the following we discuss the problem of quantum measurement based on the
assumption that local hidden variable exists. We first explore the physical meaning of hidden
variable. Due to wave-particle duality and the fact that the fluctuation of a quantum state is in
three-dimensional space, the internal fluctuation states of two particles may be different even
if their external motion states are the same. We take the spin (polarization) of a particle as an
example. In classical theory angular momentum is a vector, whose magnitude and the
projections in three directions are all well-defined. In quantum mechanics, the magnitude of
angular momentum is well-defined, and we can only determine its projection l z in one
direction. The angular position φ and the other two projections l x and l y are all indefinite.
φ and l z satisfy the uncertainty relation ∆φ∆l z ≥ h / 2 . Both ∆φ and ∆l z indicate the
fluctuation state of the spin (polarization) of a particle in the projective direction. So the two
parameters may be used as the hidden variables.
In classical mechanics and quantum field theory, we have principle of least action. We
now try to introduce this principle into quantum measurement. We define ∆φ∆l z as the
action for spin (polarization) of a particle. When a photon is incident upon a polarizer, it has
two choices. Consequently, there are two possible collapsed polarization directions. We
suppose photon always chooses the direction with a less action. For a linearly polarized
photon, its polarization direction may be regarded as the direction with the least action,
namely in this direction we have ∆φ∆l z = h / 2 . Circularly polarized light may be thought of
being composed of two orthogonal polarized components, so circularly polarized photon has
two directions with a least action. Similarly, we define the product of the uncertainties of
position and momentum as the action for the motion of center of mass of a photon. As spin
(polarization) is a relativistic quantum effect, it’s likely that the corresponding hidden
variables are irrelevant to time. We will test this hypothesis in the experiment below.
In general cases, when measurement is made on a particle, its quantum state will collapse
into another one, and the collapsing process is nonlinear and irreversible. A small change of
external circumstance or internal fluctuation state may lead to a different result, i.e. the
measurement outcome is sensitive to external circumstance and internal fluctuation state of a
particle. So the collapse of quantum state is chaotic in essence. From this point of view, the
evolutions of microcosm and macrocosm, and even the universe are chaotic in essence.
3. Interpretation of EPR-type experiments
The experiment used to test Bell’s inequalities with polarization state of photon pairs is
shown in Fig. 2. A pair of EPR photons is incident from opposite directions upon a pair of
polarization analyzers a and b with different orientations. We denote the transmitted
channel and reflected channel with “+” and “–”, respectively. The results for | φ + 〉 state in
quantum mechanics are [25]
4
Coinc counter
Coinc counter
D3
Coinc counter
D4
D2
D1
S
a
b
Coinc counter
Fig. 2. Experimental test of Bell’s inequalities.
P+ (a) = P− (a) = 1 / 2 ,
P+ (b) = P− (b) = 1 / 2 ,
(6)
(7)
1
cos 2 (a − b) ,
2
1
P+ − (a, b) = P− + (a, b) = sin 2 (a − b) .
2
P+ + (a, b) = P− − ( a, b) =
(8)
(9)
In terms of quantum entanglement the polarization of a pair of EPR photons is indefinite.
If hidden variable exists, the polarization of each photon should be well-defined. Consider the
experiment of photon pairs emitted by the J = 0 → J = 1 → J = 0 cascade atomic calcium
[20,21]. According to classical theory, the two photons are circularly polarized. For the
experiment of J = 1 → J = 1 → J = 0 cascade atomic mercury [22], one photon is linearly
polarized and the other circularly polarized. In the down-conversation of nonlinear crystal
[23-31], the wave functions of two orthogonally polarized photons overlap at crystal or beam
splitter. If a π / 2 phase difference exists between the two photons, they will form two
circularly polarized photons. The combination of half-wave plate and quarter-wave plate can
transform a Bell state into other Bell states [25]. From these facts, we may think that Bell
state can be composed of a pair of circularly polarized photons or a circularly polarized
photon and a linearly polarized photon (two linearly polarized photons cannot form a Bell
state, which we will discuss below). For the twin photons generated in cascade radiation or
down-conversation, we may think that their hidden variables are correlated. Additionally,
when the wave functions of two photons with nearly equal wavelengths overlap at crystal or
beam splitter, the hidden variables of the two photons will become correlated owing to the
exchange of their spin (polarization) angular momentums (exchange effect will be weak for
large wavelength difference [31]).
We first consider Bell state composed of a pair of circularly polarized photons. For a
circularly polarized photon, the probabilities of being transmitted and reflected are both 1/2
no matter how we orientate the polarizer. Thus for single probabilities we get the results of
Eqs. (6) and (7). We make use of projective geometry to calculate the joint probabilities of a
pair of dependent events. We suppose P+ + (a, b) to be the square of the projection of the
the
eigenstate
in
direction
onto
direction
,
i.e.
modulus
of
P+ + (a, b) = (
2
1
cos(a − b)) 2 = cos 2 ( a − b) . As the moduli of the eigenstates in directions a
2
2
5
a
b
and b are equal, we get the same joint probability by projecting b onto a . The other three
joint probabilities can be obtained with the same method. They all agree with the expectations
of quantum mechanics. If we use conditional probability and Malus’ law, we will get the same
results. But note that one must not think that measurement on one side can influence the result
on the other side. For example, do not think that measurement on the left side will lead to the
collapse of the quantum state of the photon on the right side and turns it into linearly
polarized photon. Then on the basis of Malus’ law Eqs. (8) and (9) are obtained. We should
understand in this way. If the photon on the left side can pass through a polarizer with the
orientation of a , then the photon on the right side can certainly pass through a polarizer with
the same orientation. In the case that the photon on the left side can pass through a polarizer
with the orientation of a , the probability that the photon on the right side can pass through a
polarizer with the orientation of b is cos 2 (a − b) . Note that only for a pair of circularly
polarized photons with correlated hidden variables can we use this projective method. For a
pair of uncorrelated photons, we have P+ + (a, b) = P+ (a) P+ (b) = 1 / 4 .
As for the Bell state composed of a circularly polarized photon and a linearly polarized
photon, we suppose the former is incident upon polarizer a and the latter upon polarizer b .
We first project a onto b . Since P+ (a) = 1 / 2 and the angle between the orientations of the
1
2
two polarizers is a − b , we have P+ + (a, b) = cos 2 (a − b) . We then project b onto a . Let
the angle between the polarization direction of photon and the orientation of polarizer b is
x . Then according to Malus’ law, we have P+ (b) = cos 2 (b − x) . Suppose the polarization
directions of all the photons distribute uniformly in space, the joint probability is
P+ + (a, b) =
1
2π
2π
∫0
cos 2 (b − x) cos 2 (a − b)dx =
1
cos 2 (a − b) .
2
(10)
If the polarization directions of photons distribute only in two orthogonal directions, we have
P+ + (a, b) =
1
1
1
cos 2 x cos 2 (a − b) + sin 2 x cos 2 (a − b) = cos 2 (a − b) ,
2
2
2
(11)
which also agrees with the expectation of quantum mechanics. But the projective relation of
two linearly polarized photons is much complicated and we cannot obtain the above result,
which we will discuss in the following.
4. Proposed experiments of quantum measurement
4.1. Experimental test of the locality of the measurements of Bell states
In terms of quantum entanglement, measurement on one particle of the EPR pair will lead
to the collapse of the quantum state of the other. While according to hidden variable theory,
this is not the case. For example, suppose | ϕ + 〉 state is composed of two circularly polarized
photons. When we measure one photon with polarizer and turn it into linearly polarized
photon, the other will instantaneously collapsed into linear polarization according to quantum
entanglement. In terms of hidden variable theory, the other will remain circular polarization.
Does this violate the conservation of angular momentum? If we only consider the system
composed of a pair of photons, the angular momentum of the system is certainly not
6
conserved. In the measuring process, a third component—the measuring device is involved. If
the measuring device is included, the momentum and angular momentum of the system are
still conserved.
In order to discriminate between the two hypotheses, we must seek a material which can
exhibit different effects when circularly and linearly polarized photons pass through it
respectively. Here we make use of roto-optic effect (or Faraday effect) to distinguish between
circularly and linearly polarized photons. This is because a linearly polarized photon can be
regarded as the combination of left-handed and right-handed circularly polarized components.
When it passes through a roto-material, the velocities of the two components are different
according to Fresnel’s roto-optic theory. Then there exists a phase shift between the two
components. The polarization plane of the photon will rotate and the quantum state will
change. As a circularly polarized photon passes through the roto-material, only a neglectable
overall phase is added to the quantum state. The experimental setup is shown in Fig. 3, where
I and II are a pair of polarizers with the same orientation, and Ro is a roto-material which
rotates the polarization plane of linearly polarized photon by π / 2 . Circularized polarized
| φ + 〉 state photon pairs are generated from source of SPDC. Let the distance between source
S and Ro be larger than that between S and polarizer I (L2>L1). Then the leftwards-traveling
photon will first be analyzed. Co is an optical path length compensator which is used to
guarantee the simultaneous detection of a pair of photons within the coincidence window of
counters D1 and D2. If roto-material is a Faraday rotator, then the compensator can be used
with another same one that is power-off.
L1
Co
D1
S
L2
Ro
II
I
D2
Coinc counter
Fig. 3. Experimental test of the locality of the collapse of polarization Bell state.
We now see the expectations of the two theories. According to quantum entanglement,
when the leftwards-traveling photon passes through polarizer I, the polarization direction of
the rightwards-traveling photon will instantaneously collapse to the orientation of polarizer I.
Its polarization plane is then rotated by π / 2 when it passes through Ro. Thus it will be
reflected by polarizer II. If the leftwards-traveling photon is reflected by polarizer I, the
coincidence rate is zero whatever the rightwards-traveling photon is transmitted or reflected.
So the expectation of coincidence rate is zero in terms of quantum entanglement. According to
hidden variable theory, measurement on one photon does not influence the other. On the other
hand, roto-material does not change the quantum state of circularly polarized photon. So the
coincidence counting rate will remain unchanged and is always 1/2. If | ϕ + 〉 state is
composed of a circularly and a linearly polarized photons, and suppose that they are emitted
uniformly into both sides, the coincidence rate will be 1/4. If hidden variable varies with time,
as suggested by Hess and Philipp [11,12], the coincidence rate will vary with the position of
the polarizer on the right side. Thus we can distinguish whether a pair of | φ + 〉 state photons
can influence each other. Similar experiments can be made for the other three Bell states.
7
4.2. Experimental test of the constituents of Bell states
We have supposed in the above that Bell states can be composed of circularly or linearly
polarized photon pairs. To verify this assumption, we use a pair of linearly polarized photons
generated from type-I (II) non-collinear SPDC source. Since the two photons are generated
from a same photon, the hidden variables of the two daughter photons are correlated. So
polarization Bell state is easy to obtain by converting one (or two) photon into circularly
polarized state, which can be realized by inserting one (or two) quarter-wave plate (QWP)
into one (or two) optical path. For example, | ϕ + 〉 state can be generated by inserting two
quarter-wave plates into the optical paths of type I non-collinear SPDC source, as in Fig. 4. If
type-II collinear SPDC source is used, a polarizing beam splitter (PBS) may be adopted to
separate the two orthogonal polarized photons. Then |ψ 〉 state can be obtained with one (or
two) QWP preceding the polarizer.
IF D1
QWP Pol
Pump laser
Coinc
counter
NC
Type-I
QWP Pol IF
D2
Fig. 4. Generation of | φ + 〉 state from Type-I down-conversation.
Although Bennett et al have suggested quantum nonlocality without entanglement and
Pryde et al have demonstrated this phenomenon by a specific parameter estimation
experiment [32,33], the above experiment will explicitly show that entangled state is not
mysterious, it is only quantum state with correlated hidden variables.
The following experiment uses the exchange effect of a pair of orthogonally polarized
photons at beam splitter to obtain |ψ 〉 state. The experimental setup is shown in Fig. 5. A
beam of linearly polarized laser enters Mach-Zehnder interferometer (MZI), and a half-wave
plate (HWP) is inserted into one arm of MZI to rotate the polarization plane by π / 2 . BS is
50/50 beam splitter. A pair of linearly polarized photons exchanges angular momentum at the
output port. If the relative phase of the two photons in the two arms is correctly chosen, the
output states exiting from the two output ports are circularly polarized states. Additionally, the
hidden variables of the two photons will be correlated due to the exchange of angular
momentum. In order to obtain | ψ ± 〉 state, a glass plate may be inserted into the other arm or
we can scan one of the mirrors of MZI to change the relative phase of the two photons in the
two arms. The advantages of the experiment are that Bell state can be obtained without SPDC
and interference filters while the intensity of photon pairs can be arbitrary bright. If we
replace the first BS with a PBS, the HWP can be removed, and the polarization of pump laser
should be set to ± 45 o .
8
D2
Pol
M2
D1
BS
Pol
Laser
BS
M1
HWP
Fig. 5. |ψ 〉 state obtained by the exchange effect of a pair of photons in beam splitter.
4.3. Experimental test of determinism in quantum measurement
If quantum measurement is deterministic, then experimental result is determined by
measuring condition and hidden variable, and there are no random disturbances during the
measuring process. We may further infer that the collapsed quantum states of a pair of
particles with a same quantum state must be the same under the same measuring condition.
We now verify this assumption by experiment in Fig. 6. We add another pair of polarizers in
Fig. 2. The orientations of polarizers I and I′ are the same, and this also applies to the
polarizers II and II′. The source generated circularly polarized | φ + 〉 state photon pairs.
According to Eq. (7), as the photon pairs are detected by the first pair of polarizers, we have
P+ + = 1 / 2 . Half of the photon pairs will pass through the first pair of polarizers and reach the
second pair of polarizers. When they are analyzed by the second pair of polarizers, their
behaviors will still be the same, i.e. if one is transmitted, the other will also be transmitted.
Thus for the second pair of polarizers, we have P+ + = cos 2 θ , P−− = sin 2 θ , P+ − = P−+ = 0 ,
where θ is the angle between the orientations of the two pairs of polarizers. According to
quantum mechanics, the two photons are no longer entangled after the first measurement.
Since their subsequent experimental results are mutually independent, or if the result of
quantum measurement is random, the probabilities of P+ − and P− + will not be zero. Thus
there exists conceptual difficulty for quantum mechanics to explain the total correlation of a
pair of particles without entanglement, which can be readily understood in deterministic
hidden variable theory.
Coinc counter
Coinc counter
D3
Coinc counter
D4
D2
D1
S
II
I
I′
II′
Coinc counter
Fig. 6. Two successive polarization measurements on EPR photon pairs.
9
As the collapsed quantum states of a pair of photons after the first measurement are the
same, they can be restored into | φ + 〉 state by inserting two quarter-wave plates into the
optical paths between the two pairs of polarizers. If only one quarter-wave plate is inserted,
| φ − 〉 state will be obtained.
We now see the coincidence counting result when the orientations of the second pair of
polarizers are different. Suppose the orientation of the first pair of polarizers is in the x axis,
and the orientations of the second pair of polarizers in the directions of a and b ,
respectively. Let a , b and x lie in one plane. a and b are the directions perpendicular
to a and b , respectively, as shown in Fig. 7.
a
b
a
b
θ
x
Fig. 7. Orientations of the two pairs of polarizers.
As the collapsed states of a pair of photons are the same after the first measurement, their
subsequent outcomes are still correlated and we can use projective method as stated above but
with a bit of difference. In the case of a pair of circularly polarized photons, the single
probabilities P+ (a) and P+ (b) are equal, we can get the same joint probability P+ + ( a, b)
whether by projecting a onto b or by projecting b onto a . In the case of a pair of
linearly polarized photons, the single probabilities that the two photons pass through the
second pair of polarizers respectively are not equal. Then different projective sequences will
lead to different results. If we project a onto b , we get P+ + (a, b) = cos 2 a cos 2 θ , where
θ = a − b . If we project b onto a , we obtain P+ + (a, b) = cos 2 b cos 2 θ . As joint probability
cannot be greater than single probabilities, and the latter may not satisfy this requirement, we
choose P+ + (a, b) = cos 2 a cos 2 θ for the moment.
We now consider the expression of P+ − (a, b) . According to the rule of projecting from
one channel with a smaller probability onto the other with a larger probability, we obtain
P+ − ( a, b) = cos 2 a sin 2 θ
in the case of
cos 2 a ≤ sin 2 b
and
P+ − (a, b) = sin 2 b sin 2 θ
for
cos 2 a ≥ sin 2 b . As the requirement of P+ + (a, b) + P+ − (a, b) = P+ (a) = cos 2 a must be satisfied,
and considering the smooth joining of the expression of probability, we take
cos 2 a cos 2 θ
P+ + (a, b) =  2
cos a − sin 2 b sin 2 θ
cos 2 a ≤ sin 2 b
cos 2 a ≥ sin 2 b
.
(12)
It can be verified that in addition to satisfying the projective relation in the instances of
θ = 0 and θ = π / 2 , Eq. (12) also meets the expectations of P+ + ( a, b) = cos 2 a for b = 0 and
P+ + (a, b) = 0 for a = π / 2 . So it is a reasonable probability formula. With the expression of
P+ + (a, b) we can calculate the other three joint probabilities P+ − (a, b) , P− + (a, b) and
P− − (a, b) using the relations of P+ + (a, b) + P+ − ( a, b) = cos 2 a , P+ + (a, b) + P− + ( a, b) = cos 2 b
10
and P+ − (a, b) + P−− (a, b) = sin 2 b .
As the single probabilities that the two photons pass through the second pair of polarizers
respectively are not equal, a and b are in the asymmetric situations. So there may exist
another projective relation. In the case of Fig. 7, when b rotates between 0 and a , the joint
probability P+ + (a, b) may remain unchanged and is always equal to cos 2 a , i.e. joint
probability takes the smaller one of the two single probabilities. This implies that for two
dependent events under certain conditions (for example, a and b both lie in the same
quadrants), if one event with a smaller probability occurs, then another event with a larger
probability will occur with certainty, just like the instance of cats passing through holes we
have cited in the above. The four joint probability expressions can be written as
 P+ + ( a, b) = cos 2 a

 P+ − (a, b) = 0
.

2
2
 P− + (a, b) = cos b − cos a
 P (a, b) = sin 2 b
 −−
(13)
It can be seen that in the instance of θ = 0 we get the same joint probability as Eq. (12),
i.e. P+ + (a, b) = P+ (a) = P+ (b) = cos 2 a . In other cases, we cannot determine whether Eq. (12) or
(13) is correct, which can only be verified by experiment. But for a deterministic
measurement theory, the requirement that joint probability equals the single probabilities must
be satisfied in the case of θ = 0 .
If we suppose the polarization directions (the x axis in Fig. 7) of photon pairs distribute
uniformly in space, and then average over polarization direction to get joint probability, we
find that whether using Eq. (12) or (13) the result will not agree with the expectation of
quantum mechanics. We do not present the detailed calculation process. So a pair of linearly
polarized photons cannot form a Bell state.
5. Conclusion
Compared with the assumptions of locality and realism, the reliability of Bell’s joint
probability distribution assumption is much weak. There is no prior reason to support this
assumption. Our further analysis shows that this assumption holds only for two independent
events but not for dependent events, such as the joint measurement of a pair of EPR particles.
This is the reason that Bell’s inequalities conflicts with quantum mechanics. The results of
Bell-type experiments can be explained with the projective relation of the quantum states
composed of circularly or linearly polarized photons.
So far there is no experiment suggested to distinguish between the locality and
non-locality assumptions. Our first experiment is aimed for this purpose, which uses
roto-optic effect to distinguish between circularly and linearly polarized photons, and delayed
measurement is made on one photon of EPR pair to test whether measurement on the other
could influence it. The second and third experiments are used to verify the hypothesis that two
circularly polarized photons or a circularly polarized and a linearly polarized photons can
make up of a polarization Bell state. The last experiment tests determinism, which shows that
measurement outcome is determined by the intrinsic property of the particle and measuring
condition, and random disturbances do not exist in quantum measurement. Whatever the
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experimental results, the above experiments will deepen our understanding of quantum
measurement and the foundation of quantum mechanics.
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