The polarization behaviour of entangled photons is widely discussed. A general overview is given by Laloe . The paper of Einstein, Podolsky and Rosen (EPR)  had initiated a discussion whether the theory of quantum mechanics (QM) is incomplete. The EPR argument when applied to entangled photons says as follows (see also Aspect et al. ): suppose we have a pair of entangled photons in singlet state. Both photons are propagating in opposite directions towards polarization measuring devices A and B consisting of polarizing beam splitters and detectors. A coincidence measuring device encounters coinciding events at the two detectors. When after measurement of the polarization of one photon the polarization of the other photon can be predicted in advance with certainty, there must exist an element of physical reality corresponding to this physical quantity according to EPR.
John Bell  has stated that it is impossible to explain the predictions of quantum mechanics with a model based upon real local variables. QM were a nonlocal theory, many authors claim , and there would be action at a distance in order to have a far distant photon forced to a particular polarization upon measurement of the peer photon. This action at a distance would be faster than light as was experimentally proved by some authors including Weihs . However, it was stated, that no information transport is possible through the quantum channel of entangled photons . Some authors have established nonlocal models in order to explain the predictions of QM, see Leggett . Branciard and others  have experimentally falsified Leggett’s model in agreement with QM. Englert  has stated “It is true that joint probabilities that result from quantum processes can have stronger correlations than those available by non quantum simulations;” and finally concludes quantum theory is a local theory. This assertion connotes that the measured value is defined before measurement. Otherwise, if the measured value would be created with the measurement, a correlation between the outcomes of measurements at different sides would demand nonlocal effects. The measurement problem is thus closely connected with the EPR paradox. This paper will help to understand the experimental results. Such an understanding is intended to rely on local effects only. An argument is given in Section 2. Following it there are no nonlocal actions involved with polarization measurements of photons in singlet state. In Section 3 a model is presented which describes the behaviour of polarized photons using a statistical parameter. Whether this is in contradiction to Bell’s theorem is also discussed in Section 3. The model will give an idea how entanglement works.
2 Photons in singlet state do not exhibit action at a distance
Suppose we have a pair of entangled photons in singlet state. Both photons are propagating in opposite directions towards polarization measuring devices consisting of polarizer P1 with detectors D1-1 and D1-2 at side 1 and polarizer P2 with detectors D2-1 and D2-2 at side 2. A coincidence measuring device encounters simultaneous events at the two detectors D1-1 and D2-1 or at D1-2 and D2-2, respectively. The polarizing angles can be adjusted at each side.
The Hilbert space for the combined system is H12 = H1⊗H2. Normalized base vectors are |H1〉 and |V1〉 for system 1 at side 1 and |H2〉 and |V2〉 for system 2 at side 2. |H1〉 and |H2〉 correspond to the x-axis and |V1〉 and |V2〉 correspond to the y-axis. Photon 1, as a convention, is measured before photon 2.
Suppose now the polarizer P1 is set to 0∘/90∘. Before photon 1 entered the polarizer P1 the pair was in the singlet state |Ψ12〉. In the following we only look at those photon 1 which are detected in detector D1-1 behind the 0∘ exit of P1. They are experimentally identified by the time stamp of the detector click of D1-1. After detecting photon 1 at time t2 in the detector D1-1 we know photon 1 was in state |H1〉 between t1 and t2. This is due to preparation by P1.
Upon hitting the detector D1-1 photon 1 has ceased to exist and the singlet state no longer describes the state of the photon pair. Photon 2 must thus be in another state. We can calculate this state from the evaluation of the experimental findings. With the polarizer P2 set to an arbitrary angle δ the probability P for photon 2 to pass the polarizer at δ after photon 1 had passed P1 at 0∘ is according to the experimental results  P = sin2(δ).
According to Born’s rule this is possible only if photon 2 was in state |V2〉 before measurement at P2 so that (1)
As there is no physical action upon photon 2 after it had left the source it is in state |V2〉 just after t0. There is no action at a distance from photon 1 upon photon 2. Nonlocality is therefore not a necessary consequence of entanglement. The abovementioned does also apply to the detection of photon 1 at any polarization angle α and the polarizer setting β at side 2 with δ = β − α because the experimental results are rotationally invariant. Photon 2 is then in state –sin(α)|H2〉 + cos(α)|V2〉 just after t0. With δ = 90∘ follows P = 1 and photon 2 is found with certainty behind the β exit of P2. How it can be understood that the setting of the polarizer at side 1 defines the polarization of photon 2 is subject of the model discussed in the following section.
3 Model describing the statistical behaviour of entangled photons
EPR  have stated quantum physics were incomplete as it does not account for the physical reality given by the fact that measured values can be predicted in advance to a measurement as we have seen in the previous section. See Fine  for a detailed discussion of the EPR paradox. Below a model is presented which describes the behaviour of polarized photons using a statistical parameter.
That such a model may be in contradiction to Bell’s theorem which forbids the existence of hidden parameters will be discussed also. The model assumptions show which conditions have to be met in order to reproduce the experimental results and to fulfil the requirements of quantum physics. The model should and does explain the following characteristics:
The measured value of a polarization measurement is determined before the measurement,
The QM predictions of polarization measurements with single photons according to Born’s rule,
The mechanism of entanglement by a parameter,
The locality of the polarization measurements with entangled photons,
Rotational invariance of the polarization measurements with entangled photons,
The QM predictions of polarization measurements with entangled photons,
The model violates Bell’s inequality,
Measurement results of subsequent measurements are not predictable.
Six model assumptions M1-M6 are made:
introduces the physical entity propensity state, called p-state, determining the polarization behaviour.
controls the statistical distribution of p-states.
accounts for the fact that the polarization of entangled photons is unknown but is defined after a measurement at one side as shown in Section 2.
establishes how the p-state depends on the statistical parameter λ.
accounts for the entanglement of the singlet state.
accounts for the fact that photons don’t have a memory of previous states after a measurement.
The six assumptions are described in detail in the following:
Model assumption M1
A propensity state called p-state determines which polarizer output a photon will take. A photon in p-state α would pass a polarizer set to α with certainty.
Purpose: introducing the physical entity propensity state, called p-state, determining the polarization behaviour.
Model assumption M2
A statistical parameter is introduced, the value of which determines in which of two orthogonal p-states a photon is. λ has the value range -1<λ<+1 and a normalized probability distribution ρ(λ) = ½ with dλρ(λ) =1.
Purpose: controlling the statistical distribution of p-states.
Model assumption M3
Photons from an ensemble in the same p-state have the polarization (state) given by the p-state. This assumption refers to entangled photons only where the polarization of the incoming photons is unknown. With not entangled prepared photons the polarization is determined by the preparation.
Purpose: Accounting for the fact that the polarization of entangled photons is unknown but is defined after a measurement at one side as shown in Section 2.
It is assumed that a photon can be simultaneously in different p-states depending on the value of a parameter λ and a chosen direction relative to the polarization of the photon. That means some of the photons with polarization α are also in p-state β and thus pass a polarizer set to β with certainty. As p-state and polarization are different physical entities ambiguous polarization states are excluded.
Entangled photons are generated by a common source on the side 1 with the polarization ϕ1 = 0∘ and on the side 2 with the polarization ϕ2 = 90∘ or on the side 1 with the polarization ϕ1 = 90∘ and on the side 2 with the polarization ϕ2 = 0∘. We now look at side 1 with the polarizer setting α/α+π/2: Incoming photons have the polarization ϕ1 = 0∘ or ϕ1 = 90∘. Outgoing photons leave the polarizer either through the output with the setting α or through the output with the setting α+π/2.
Let δ be the angle between the polarizer setting α and the polarization of the incoming photon. Then we get δ = α-ϕ1 and a polarization function A(δ, λ) can be defined, which indicates the p-state of the photon before a subsequent measurement. A(δ, λ) can have the values +1 and -1. The representation of the photon p-states occurs in the Bloch circle. From the projection onto the double angle 2δ, a rule can be constructed which determines whether the arriving photon of polarization ϕ1 is in p-state δ+ϕ1 corresponding to A(δ, λ) = +1 or in p-state δ+ 90∘+ϕ1 corresponding to A(δ, λ) = -1. See Figure 2. Being in p-state α = δ + ϕ1 means the photon would pass the polarizer output α with certainty.
Model assumption M4
For incoming photon 1 with polarization ϕ1 = 0∘ and polarizer P1 setting α: (2)
meaning photon 1 is in p-state α and (3)
meaning photon 1 is in p-state α + π/2 with δ = α – 0 = α.
For incoming photon 1 with polarization ϕ1 = 90∘ and polarizer P1 setting α + π/2 (4)
meaning photon 1 is in p-state α+π/2 and (5)
meaning photon 1 is in p-state α, with δ = α+π/2 − π/2 = α.
Purpose: establishing how the p-state depends on the statistical parameter λ.
This explains property #1: The measured value of a polarization measurement is determined before the measurement.
Geometrical calculations yield 1 + cos(2δ) = 2 cos2(δ) and 1 − cos(2δ) = 2sin2(δ). Using equation (2) the horizontally polarized photon 1 is found behind the output α of the polarizer P1 with the probability (6)
With δ = α we obtain the same Pδ for a photon in state |H〉 to be found in state cos(α)|H〉 + sin(α)|V〉 according to QM from Born’s rule.
This explains property #2: The QM predictions of polarization measurements with single photons according to Born’s rule.
With polarizers P1 and P2 orientated perpendicularly to each other measurement results at both sides are of opposite sign. Having P1 set to α means P2 is set to β = α+π/2. If the incoming photon on side 1 comes with polarization ϕ1 = 0∘ the partner photon on side 2 comes with polarization ϕ2 = 90∘. Let B(δ, λ) be the polarization function at side 2 and in this case δ = β–ϕ2 we get δ = α+π/2–π/2 = α. Here δ is again the angle between the polarizer setting and the polarization of the incoming photon.
Model assumption M5
In order to meet the experimental results with entangled photons, it is defined: (7)
Purpose: accounting for the entanglement of the singlet state.
This explains property #3: The mechanism of entanglement by a parameter and property #4: The locality of the polarization measurements with entangled photons.
The rules for the distribution of the incoming photons onto the two output directions of the polarizer are also valid for the partner photon on side 2. With equation (7) it follows:
For incoming photon 1 with polarization ϕ1 = 0∘ having incoming photon 2 with polarization ϕ2 = 90∘ and polarizer P2 setting β = α + π/2 we obtain from equation (4) (8)
meaning photon 2 is in p-state α+π/2 and from equation (5) (9)
meaning photon 2 is in p-state α, where δ = β – π/2 = α.
For incoming photon 1 with polarization ϕ1 = 90∘ having incoming photon 2 with polarization ϕ2 = 0∘ and polarizer P2 setting β = α we obtain from equation (2) (10)
meaning photon 2 is in p-state α and from equation (3) (11)
meaning photon 2 is in p-state α + π/2, where δ = β−0∘ = α.
With equations (2) and (5) it follows A(δ, λ) = 1 for each value of δ and for –1 < λ < 1 although from different incoming photons. From equations (8) and (11) we get B(δ, λ) = –1 for –1 < λ < 1, again from different incoming photons. Photons 1 with A(δ, λ) = +1 are part of an ensemble, selected by polarizer P1, with the polarization α and the partner photons 2 with B(δ, λ) = –1 are part of an ensemble with the polarization α+π/2 coupled with photon 1 by equation (7). This follows from model assumption M3.
With photon 1 in state ϕ1= α and photon 2 in state ϕ2 = α +π/2 we can rotate the coordinate system by an angle α and get and β′ = β – α.
This explains Property #5: Rotational invariance of the polarization measurements with entangled photons.
For incoming photon 1 with polarization having incoming photon 2 with polarization and polarizer P2 setting β′ we obtain from equation (8)
meaning photon 2 is in p-state β′ and from equation (9) (13)
meaning photon 2 is in p-state β′ –π/2, with δ = β′ – π/2.
For incoming photon 1 with polarization having incoming photon 2 with polarization and polarizer P2 setting β′–π/2 we obtain from equation (10)
meaning photon 2 is in p-state β′–π/2 and from equation (11) (15)
meaning photon 2 is in p-state β′ with δ = β′ – π/2 – 0∘.
The expectation value of a common measurement with polarizers P1 and P2 is (16)
where α and β are the polarizer settings at side 1 and side 2 respectively and
where δ(α) = α – α = 0 and
where δ(β)= β – α – π/2.
The probability Pα,β to measure a photon 2 polarization at β after photon 1 was measured at α can be obtained from (18)
This explains property #6: The QM predictions of polarization measurements with entangled photons.
As the expectation value E(α,β) from equation (17) does exactly match the predictions of quantum physics it also violates Bells  inequality. This explains property #7: The model violates Bells inequality as quantum physics does.
What is the reason for this? With δ = β – α – π/2 the value of B(β,λ) in equation (16) depends on the setting angle α of the polarizer at side 1 whereas in Bells model the measured value at side 2 does not depend on the setting of polarizer at side 1. This difference is the reason that the model presented here violates Bells inequality whereas Bells model does not. Bell thought, “the result B for particle 2 should not depend on the setting … for particle 1”. But we have seen with model assumptions M1-M6 that a local model is possible without Bells restriction cited above. Bell had cited Einstein  “But one supposition we should in my opinion, absolutely hold fast: the real factual situation of the system S2 is independent of what is done with the system S1, which is spatially separated from the former.” This supposition is fulfilled with the model above. With model assumption M3 the polarization of the incoming photon at side 2 depends on the setting of the polarizer at side 1 although there is no nonlocal action. As only those photons 2 are considered which match with a measured photon 1 it is not unexpected to see the result at side 2 depending on the polarizer setting at side 1. The rule that determines which polarizer exit a photon will take is the same for both sides. Dependencies between the photons on either side originate from the shared parameter λ and not from a nonlocal influence of photon 1 upon photon 2.
Model assumption M6
We assume that λ is indeterminate and uniformly distributed after a measurement.
Purpose: accounting for the fact that photons don’t have a memory of previous states after a measurement.
After a preparation the polarizer output of the next measurement is determined by the parameter λ, but it can not be predicted which polarizer output the photon will pass after a further subsequent measurement. How this indeterminacy is realized can not be said at the moment. However, this is a local effect which applies to single photons as well as to entangled photons. The ensemble of photons covers the full range –1 < λ < +1 after passing a polarizer and a photon has the polarization α after passing a polarizer with setting α. That explains property #8: Measurement results of subsequent measurements are not predictable.
From the model above we get an idea how entanglement works. From the six model assumptions M1-M6 only M3 and M5 refer to entanglement. The other four assumptions apply to single photons as well. Particularly M3 is significant. It states that an ensemble of photons with a particular polarization state can arise from two different input photons. This is the real reason for the rotational invariance of the polarization behaviour of entangled photons.
4 Results, Discussion and Conclusion
Photons in singlet state do not exhibit action at a distance. Nonlocality is therefore not a consequence of entanglement. Experimental results on entangled photons can be explained without assuming non-local effects. This means measured values are not generated upon the measurement, they already exist beforehand. If the measured value exists before the measurement, the measurement is a selection of existing states. This indicates an ensemble.
It was argued by Einstein, Podolsky and Rosen  quantum physics were incomplete as it does not predict the exact measurement result for each photon. A hidden parameter model was presented, able to complete the formalism of quantum physics for the case of polarized photons. It is not ruled out by Bells theorem. The reason for this was discussed. Relations between measurement results of the photons on either side do not come from nonlocal effects but rather originate from the shared parameter λ which connects them from the moment they are created by the common source. The model also made use of a propensity state which determines a subsequent polarization measurement. According to the model the quantum mechanical state again describes an ensemble. The individual parts differ by the value of a parameter.
However, the model presented above is an enhancement but no replacement of the formalism of quantum physics. Enhancement means that the model correctly describes certain phenomena in detail where QM does not give answers. These phenomena are the polarization measurements at linearly polarized photons including entangled photons where QM only describes probabilities. The model is valid if it is free of contradictions. Generally, the formalism of QM is not in question.
I would like to thank Prof. Dr. Harald Weinfurter and Prof. Dr. Robert B. Griffith for fruitful discussions on the subject.
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Published Online: 2017-12-29