2.1 Model overview
In polarization measurements, photons can choose one of two perpendicular exits of the polarizer. A model with hidden variable must describe which of these two possible exits a photon will take. Four model assumptions are introduced, which are outlined and then described in italics:
MA1 introduces the statistical parameter 𝜆 which controls the polarizer exit that a photon will take. This model assumption is the same as MA1 in [4].
MA2 describes the polarization of a selection of photons from an entangled pair. This is a new model assumption.
MA3 describes the coupling of photons from an entangled pair. This is a new model assumption.
MA4 states that photons carry the complete set of the hidden variable after a measurement. This model assumption is the same as MA4 in [4].
2.2 Model assumptions
Model assumption MA1
The statistical parameter 𝜆, uniformly distributed between 0 and 1, controls which of the two polarizer exits the photon will take. Given the polarizer setting α and the photon polarization ϕ we define δ = α  ϕ as the difference between the polarizer setting and the polarization of the photon. The function A(δ,λ) indicates which polarizer exit the photon will take.
A(δ,λ) can have values +1 and − 1. For 0 ≤ δ < π/2, we define
A(δ,λ) = + 1 for 0 ≤ 𝜆 ≤ cos 2 (𝛿), (1)
meaning the photon takes polarizer exit α and
A(δ,λ) = 1 for cos 2 (𝛿) < 𝜆 ≤ 1, (2)
meaning the photon takes polarizer exit α + π/2. MA1 is valid for single photons as well as for each wing of entangled photons.
The case π/2 ≤ δ < π is covered referring to the other exit of the polarizer. Then Eq. (2) applies and the range of values of 𝜆 for positive results is cos2(𝛿) < 𝜆 ≤ 1. The case δ < 0 is covered by reversing the polarizer direction by 180°. Thus, π ≤ δ < π/2 is equivalent to 0 ≤ δ < π/2 and
π/2 ≤ δ < 0 is equivalent to π/2 ≤ δ < π.
Thus A(δ,λ) = + 1
for 0 ≤ δ < π/2 and 0 ≤ 𝜆 ≤ cos2(𝛿), (3.1)
for π/2 ≤ δ < 0 and cos2(𝛿) < 𝜆 ≤ 1, (3.2)
for π/2 ≤ δ < π and cos2(𝛿) < 𝜆 ≤ 1, (3.3)
for π ≤ δ < π/2, and 0≤ 𝜆 ≤ cos2(𝛿) and (3.4)
A(δ,λ) = 1 otherwise. (3.5)
Model assumption MA2
If the fractions of horizontally and vertically polarized photons from an entangled state that contribute to a photon stream selected by a polarizer are cos 2 (α) and sin 2 (α) respectively, then they obtain a common polarization of α or  α, because of the indistinguishability of the photons.
The fractions of horizontally and vertically polarized photons that leave a polarizer exit α are cos2(α) and sin2(α) respectively. This makes up for the common polarization. The selection comprises all photons that take the same polarizer exit. Photons with polarization α and α + π/2 come in equal shares, due to symmetry reasons. MA2 accounts for the fact that the polarization of photons from the entangled state is undefined because of their indistinguishability, but is changed and redefined by entanglement. Thus, the photons of a selection cannot be distinguished by their polarization. This argument has already been made in [4] but only for photon pairs with common hidden variables. MA2 is a contextual assumption, because the polarization of a selection coincides with the setting of a polarizer. However, this is a local realistic assumption, because it assigns a real value to the physical quantity polarization. MA2 leaves open whether the polarization of a selection is positive or negative. To distinguish this we use the initial conditions taking into account the conservation of angular momentum. This leads to
Model assumption MA3
Each Bell state is a mixture of indistinguishable constituent photon pairs in equal shares whose components have the same polarization 0° or 90° for Φ + and Φ and an offset of π/2 for Ψ + and Ψ. The constituent photon pairs make up the initial state.
The coupling of a selection on wing A with polarization α and the corresponding selection of the partner photons on wing B with polarization β is a relation between the signs of the polarizations on both sides and is given
for Ψ + and Φ + as sign(α) A = sign(β) B , and (4)
for Ψ and Φ as sign(α) A =  sign(β)B,(5)
where all angles are in the interval [π/2, +π/2].
From angles outside this interval we subtract π because α and απ denote the same polarization.
With this definition we obtain
sign(α) =  sign(α + π/2) = sign(α π/2). (6)
Model assumption MA4
Photons having left a polarizer exit α have polarization α with λ evenly distributed in the range 0 ≤ λ ≤ 1.
MA4 emphasizes that photons carry the full set of hidden variables after leaving the polarizer.
2.3 Predicting measurement results for single photons
Using equations (3.1 or 3.4), a photon with polarization ϕ is found behind the exit α of a polarizer with probability
Pδ = \({\int }_{0}^{{\text{c}\text{o}\text{s}}^{2}\left(\right)}d\) = cos2(δ), (7)
where δ = α ϕ with 0 ≤ δ < π/2 or π ≤ δ < π/2.
Using equations (3.2 or 3.3) for π/2 ≤ δ < 0 or π/2 ≤ δ < π we refer to the other exit of the polarizer and have, with
ϑ* = δ  π/2
Pδ = \({\int }_{{\text{c}\text{o}\text{s}}^{2}\left(\right)}^{1}d\) = 1cos2(ϑ*) = cos2(ϑ), as well. (8)
With δ = α ϕ we obtain the same Pδ for a photon in state cos(ϕ)*H > + sin(ϕ)*V > by projection onto
cos(α)*<H + sin(α)*<V according to QM (i.e., Born’s rule).
2.4 Conclusions from the model assumptions
MA2 has the consequence that the selection by a polarizer in position α on one side corresponds to a selection with polarization α + π/2 or απ/2 on the other side.(for Ψ + or Ψ) This can be seen from the following consideration: According to equations (7,8) a polarizer PA set to α selects a fraction of cos2(α) of horizontally polarized photons 1 and a fraction of sin2(α) of vertically polarized photons 1. This means that partner photons 2 are also selected, but with perpendicular polarization, resulting in a selected fraction of cos2(α) = sin2(α + π/2) of vertically polarized photons 2 and a selected fraction of sin2(α) = cos2(α + π/2) of horizontally polarized photons 2. Due to MA2 the polarization of the selected photons 2 is α + π/2 or α π/2.
From equations (4) and (6) we obtain for Ψ + the polarization α π/2 of the partner photon 2 with the same sign as that of the polarization α. For Ψ we obtain the polarization α + π/2 of partner photon 2 with an opposite sign of the polarization α in accordance with Eqs. (5) and (6).
For Φ + and Φ we find that the selection by a polarizer in position α on one side corresponds to a selection with polarization α or α on the other side. Again a polarizer PA set to α selects a fraction of cos2(α) of horizontally polarized photons 1 and a fraction of sin2(α) of vertically polarized photons 1. This means that partner photons 2 are also selected, but in this case with the same polarization, resulting in a selected fraction of cos2(α) of horizontally polarized photons 2 and a selected fraction of sin2(α) of vertically polarized photons 2. Due to MA2 the polarization of the selected photons 2 is α or α.
According to Eq. (4) we obtain the polarization of the partner photons 2 of α for Φ + as sign(α)A = sign(α)B and for Φ the polarization of partner photon 2 is α as sign(α)A =  sign(α)B in accordance with Eq. (5). The results for all four Bell states are presented in Table 1.
Table 1
polarization of partner photons 2 at wing B for different Bell states for a selection of photons 1 with a polarizer set to α at wing A.
Bell state

A

B

Ψ−

α

α + π/2

Φ+

α

α

Ψ+

α

α π/2

Φ−

α

α

The Bell states Ψ and Φ + are known to be rotationally invariant. The same applies to the states Ψ + and Φ as well if the coordinate system on wing B is changed from left to righthanded. In this case, the polarization values for Ψ + and Φ in column B in Table 1 change sign, so that the difference between A and B is constant and therefore independent of α. Model assumption MA3 reproduces the conservation of spin angular momentum. This is shown in the following section.
2.5 Conclusions from conservation of spin angular momentum
Conservation of spin angular momentum requires that the total spin of a Bell state is zero. Let R > and L > denote the state of the right and left polarized photons, respectively. These are related to the spin direction. The connection to the linear polarization is given by
R > = 1/√2 *(H > + iV>) and
L > = 1/√2 *(H>  iV>) with (9)
H > = 1/√2 *(R> + L>) and
V> = i/√2 *(R>  L>). (10)
This gives for the four Bell states with the suffixes A and B denoting the wings of the entangled states:
Φ+ = 1/√2 *(HA>HB> + VA>VB>)
= 1/√2 *(RA>LB> + LA>RB>), (11)
Ψ− = 1/√2 *(HA>VB>  VA>HB>)
= i/√2 *(RA>LB>  LA>RB>), (12)
Φ− = 1/√2 *(HA>HB>  VA>VB>)
= 1/√2 *(RA>RB> + LA>LB>), (13)
Ψ+ = 1/√2 *(HA>VB> + VA>HB>)
= i/√2 *(RA>RB>  LA>LB>). (14)
For Φ + and Ψ the total spin of the photon pairs vanishes because left and right polarization cancel. This also applies to Φ and Ψ + if the coordinate system on wing B is rotated by 180°, i.e. the photons exit the source in the opposite direction.
Φ+ and Ψ− are rotationally symmetrical. So it also applies
Φ+ = 1/√2 *(H‘A>H‘B> + V‘A>V‘B>) (15)
for each angle α of a rotation of the coordinate system, with
H’> = cos(α) *H > + sin (α) *V > and
V’> = sin(α) *H > + cos (α) *V>. (16)
Projection onto < H‘A yields
<H‘A Φ+> = H‘B > = cos(α) *H B > + sin (α) *V B >. (17)
So we see that a projection or selection of Φ + by a polarizer PA in position α means the state or polarization of the partner photons in direction α. The projection for Ψ yields
<H‘A Ψ− > = V‘B> = sin (α) *H B > + cos(α) *V B >. (18)
This state is orthogonal to α. A projection or selection of Ψ−by a polarizer PA in position α results in the direction α + π/2. for the state or polarization of the partner photons.
Φ− and Ψ+ are also rotationally symmetrical if the coordinate system on wing B is rotated by 180°, i.e. the photons exit the source in the opposite direction. This means with the corresponding horizontal coordinate
−HB > = HB> that (19)
Φ− = 1/√2 *(HA>−HB >  VA>VB>). (20)
Because of the rotational symmetry also applies
Φ− = 1/√2 *(H‘A>−H’B >  V‘A>V‘B>) (21)
for each angle α of a rotation of the coordinate system, with
H’> and V’> given by Eq. (16) and
−H’B > = cos(α) *−HB > + sin (α) *V B > .
Projection onto < H‘A yields with Eq. (19)
<H‘A Φ−> = −H’B > = (cos(α) *−HB > + sin (α) *V B>)
=cos(α) *H B > + sin (α) *V B >. (22)
So we see that a projection or selection of Φ by a polarizer PA in position α means the state or polarization of the partner photons in direction α in the original coordinate system.
For Ψ+ we get with Eq. (19)
Ψ+ = 1/√2 *(H’A>V’B>  V’A>−H’B >), and the projection gives with Eq. (16)
<H‘A Ψ+ > = V‘B> = sin (α) *−HB > + cos(α) *VB >
= sin (α) *H B > + cos(α) *V B >. (23)
This state is orthogonal to α. A projection or selection of Ψ + by a polarizer PA in position α results in the direction α π/2 for the state or polarization of the partner photons in the original coordinate system.
Altogether it follows that of the two possibilities given by MA2, only the one given by MA3 is consistent with conservation of angular momentum. For the relationship between the position of the selective polarizer and the polarization of the partner photons, the conservation of the spin angular momentum means the same sign of α on both sides for Φ+ and Ψ− and the opposite sign for Φ− and Ψ+ as shown in Table 1.
2.6 Calculating expectation values for photons in singlet state
We have seen above that all selected photons 1 from the singlet state which take PA exit α have polarization α while their partner photons 2 have polarization α + π/2. Seen from the other side we can conclude that if photon 2 leaves polarizer PB at β we have matching events if those photons 2 with polarization β would leave PB with an assumed setting of α + π/2. Note, that λ is evenly distributed in the value range
0 ≤ λ ≤ 1 for the photons 2 with polarization β. This can be seen examining the initial states and applying equations (3.1)  (3.4) to horizontally polarized photons and vertically polarized photons. For example, the horizontally polarized photons with 0 ≤ 𝜆 ≤ cos2(𝛿) and the vertically polarized photons with
cos2(𝛿) < 𝜆 ≤ 1 contribute to a selection with polarization β for 0 ≤ δ < π/2.
Thus, the probability that photons 2 with polarization β would pass PB at α + π/2 can be obtained by equations (7) and (8), using δ = α + π/2  β thus yielding
Pδ = cos2(δ) = cos2 (α + π/2  β ) = sin2(αβ), (24)
where δ is the angle between the PB polarizer setting β and
the polarization α + π/2 of photons 2 selected by PA.
The expectation value for a joint measurement with photon 1 detected behind detector PA at 𝛼 and partner photon 2 detected behind detector PB at 𝛽 is as obtained from
([4], Eq. (13))
E(𝛼,𝛽) = cos(2(αβ)), (25)
in accordance with QM. As the expectation value E(α,β) in Eq. (25) exactly matches the predictions of quantum physics, it also violates Bell's inequality.
2.7 Applying the model to entanglement swapping
Entanglement swapping uses a protocol in which two wings of different systems, each in singlet state, are entangled by a Bell state measurement of the two remaining wings[1, 2, 8].
Let AB and CD be the two initial systems in singlet state. Then we define the outer pair AD and the inner pair BC. With a Bell state measurement between B and C we want to entangle A and D. However, this coupling is random in the case of entanglement swapping. Therefore four resulting Bell states are possible. How are these results for the inner pair BC related to the state of the outer pair AD? This is determined by applying Table 1 to the pairs of channels. AB and CD are always in state Ψ. BC is obtained by the Bell state measurement.
Thus, we obtained the results of Table 2. Compared with Table 1 we see that the Bell state of the outer pair AD is equal to the measured Bell state of the inner pair BC according to QM [8]. Note that the polarizations α + π and α are equal
Table 2
polarization of the photons of wings B,C and A,D for different Bell states obtained between B and C by applying Table 1 with an assumed selection of photons by a polarizer set to α at wing A.
Bell
state ssstatestate BC

A

B

C

D

Ψ−

α

α + π/2

α (+ π)

α + π/2

Φ+

α

α + π/2

α + π/2

α (+ π)

Ψ+

α

α + π/2

α  π

α  π/2

Φ−

α

α + π/2

α  π/2

α

Multilevel entanglements can also be explained locally by multiple applications of the relationships from Table 1.
2.8 Applying the model to teleportation
Teleportation uses a protocol in which an unknown state β is transferred to another wing B of a singlet state by Bell state measurement between the unknown β and wing A of the singlet state [9]. Using MA3 and Table 1 we obtain the polarizations at wing A and B. AB are always in the Ψ state. The polarization of the pair βA is obtained by measuring the Bell state.
Thus, we obtained the results shown in Table 3. The results at wing B can be converted to the state β by simple rotation or mirroring. This result is in accordance with quantum mechanical calculations [9]. Note that the polarizations β + π and β are equal.
Table 3
polarization of the photons of wings A and B for different Bell states obtained between the unknown β and wing A.
Bell state βA

A

B

Ψ−

β + π/2

β (+ π)

Φ+

β

β + π/2

Ψ+

βπ/2

β

Φ−

β

β + π/2
