_{1}

^{*}

Based on the violation of Bell inequalities, it has been believed that the derivation of Bell correlations requires a quantum description that depends on entanglement. However, the present paper computes Bell correlations among polarization analyzer output intensities from two spatially separated sets of superposed random wave pairs. To obtain proper Bell correlations, the general statistics must be modified to represent single event pair selection. The correlations between analyzer output components are then in one-to-one correspondence with those computed from the entanglement formalism.

The Bell theorem and experimental violation of Bell’s inequalities are believed to make the description of Bell correlations impossible without the use of entanglement. If valid, this view would rule out the possibility of local descriptions of these correlations. However, it is shown in [

Below, it is shown that properly normalized covariances resulting from a statistical model of local “twin beams” from a down converter (see

tum source boundary conditions, result in modification of a general wave stochastic model to produce results in agreement with the entanglement description.

In Section 2 below, the correlation of polarization analyzer intensity outputs is first derived from the superpositions of orthogonally polarized light wave output from a type II down-converter as used in Bell experiments (see

The formalism uses statistical assumptions common in the description of chaotic optical waves [

The overall model may be thought of as a generalization of wave particle duality. In two-slit interference, discrete entities assumed to be emanating from a source are manifested as discrete counts at a detector but exhibit the intervening effects of wave interference. In the model below, pairs of counts at initial analyzer angles equal to zero are inserted into, and determine, the statistical parameters of a wave system that determines the count correlations at non-zero analyzer angles.

Based on the geometry of parametric down converter sources for Bell correlation experiments (see

and

In these equations, horizontal and vertical polarization components are indicated by subscripts

The action of polarization analyzers placed in each of Beams 1 and 2 is now described. These transmit and reflect components of

From these, and

Thus, two polarized output amplitudes from each input beam occur in directions determined by the rotation angle of the analyzers.

The intensities of the analyzer output components

where, for example,

yields analyzer 1 outputs

and

where

and

To proceed, further conditions on the parameters in Equations (2.6)-(2.9) must be specified. First, imposing source characteristics used in Bell experiments [

This is implied by the fact that photons are produced in pairs of opposite polarization in Beams 1 and 2. By applying Equations (2.10) to Equations (2.6)-(2.9) it follows that

One concludes that for lossless analyzers, the sum of powers in the output polarizations equals the sum of powers in the input polarizations, which by Equation (2.10) are equal for the two beams.

A phase relation between the input beams must be imposed to make the analysis consistent with requirements of the nonlinear optics process [

where the

The condition

When quantities specified in Equations (2.10) and (2.14) are inserted into Equations (2.6)-(2.9), one obtains

Note, if

with one set rotated 90˚ with respect to the other. The output wave intensities of the analyzers are interchanged due to Equations (2.10) and (2.14).

From Equations (2.15a)-(2.15d), and setting

and

To compute the correlation of

1) The complex amplitude variables introduced in Equations (2.1) and (2.2) are circular complex Gaussian random variables [

The properties that will be used below follow from assumption 1.

2) From 1, it follows that the wave intensities

buted. They are assumed to be statistically independent, but they have the same mean

3) The phases are statistically independent of the amplitudes and are uniformly distributed from

Conditions 1 - 3 are discussed in [

From points 2 and 3 above, two results are obtained that are used repeatedly in the following. First, if

Second, a uniform distribution of

The latter condition plus property 2 applied to Equation (2.15c) implies that

Thus, for Equations (2.15a)-(2.15d),

An important result of this section is now obtained by computing the correlation of

Using Equations (2.17b) and the independence of the

while

Inserting the averages of Equations (2.19a) and (2.19b) into Equation (2.18) yields

From the definition of the correlation of any two variables, the correlation of

The square root variances of

In spite of Equation (2.21), correlations among individual pairs of

form that allows interpretation in terms of discrete quantum counts. However, the use of normalized covariances rather than the intensites of Equations (2.15) does result in such a form. It will be shown below that the covariances result from modification of the statistics to represent selection of single event pairs in a detector time-constant (as implemented in experiments), rather than arbitrary numbers of event pairs as implied by the statistics thus far.

The appropriate modifications are obtained by subtracting their common mean value

tensity components of

The right-hand side of Equation (3.1) equals that of Equation (2.21), and it appears that nothing has been changed. However, Equation (3.1) may be written as a sum of terms resulting from the contributions of products

such as

Similarly, the contributions of products

are given by

However, while Equations (3.2a) and (3.2b) are equal to the corresponding quantum results, it is important to realize that at this point, little discreteness has been introduced into the model.

To achieve a quantum interpretation of Equations (3.1)-(3.2), the continuous variables used are defined in terms of equivalent photon count statistics [

1) Wave intensities are defined as count densities in time.

2)

cate count pairs as seen by analyzers at

implies that

3) Experimental parameters and observational protocols are adjusted so that single count pairs are selected as assumed in the entanglement formalism.

To understand the reason for the quantization analysis to be given in Section 3.1, first compute individual averages among the variables of Equations (2.15),

Using Equations (2.17a) and (2.17b), this becomes

In a similar manner, the other averages may be computed. The relevant averages are

While Equations (3.4a)-(3.4c) suggest Equations (3.2a) and (3.2b) (and quantum mechanical results), they have an additional constant offset, and an amplitude

First, consider Equation (3.4c). To measure correlations at the source, analyzer angles are set to

Writing wave intensities as count averages yields

The 0’s and 1’s in the last two terms are subscripted to indicate to which variable they belong. They represent selected count pair products for which each product equals zero. Note that the values in the right-most double sum occur with low probability, and are deleted under the protocols used in experiments. After averaging over the exponentially distributed variables, the average over the double sum is

Thus, the constant on the right side of Equation (3.4c) corresponds to events that are excluded if single event

pairs are selected from the range of possibilities. Clearly, the situation is the same for

ther, Equation (3.4c) holds for all equal analyzer angles so that the boundary condition at

From Equations (2.15a)-(2.15d), when

As above, Equation (3.4b) represents observation under the condition of single pair selection if the additive constant

The same reasoning may be extended to Equation (3.4a). Observe from Equations (2.15) that

Under this condition, Equation (3.4a) reduces to Equation (3.4c):

The end result is that Equations (3.4a)-(3.4c) represent observations under single count pair selection, if

Similar considerations apply to the interpretation of the multiplicative factor

where

The probability product for duplicate photons equals the probability for one photon squared, i.e.

same. Given that the probability of two counts on a side is made small compared to one count,

pendent processes with equal parameters, the second component of Equation (3.4a) yields

Given Equation (2.17a),

It is now necessary to apply the normalization used in the definition of correlation (the denominator of Equation (3.2a) to Equation (3.10), again taking pair selection into account. For beam 1, since

From count selection at

after ensemble averaging as in Equation (2.19a). Then

with anequalresult for^{ }

From Equations (3.10) and (3.11c), one has_{ }

with an equal result for

used in experiments, Equation (3.4a) is modified to Equation (3.2a). From the above analysis, the same procedure modifies Equation (3.4b) to Equation (3.2b).

It has been shown above that the interference of twin pairs of waves yields polarization analyzer intensity outputs whose appropriate cross correlations may be used to compute Bell correlations. The wave statistical parameters used are set equal to the basic observational count parameters found in SPDC. These results depend on interpreting wave intensity variables as counts per unit time and on the experimental restriction of correlations to represent single event pairs. When the general wave optical statistics are adjusted to represent single source-pair selection at analyzer angles equal to zero, the formalism provides the same results as the entanglement formalism at non-zero angles. This is a principle result of this paper.

It has been believed that the violation of the Bell inequalities from Bell’s theorem implies that only the formalism of quantum entanglement can describe the observed correlations of Bell experiments. Although it produces the experimentally observed correlations, the entanglement formalism has the troubling implication of non-lo- cality, or what some call “non-reality”. The purpose of the present investigation has been to examine an alternative formulation of the problem that does not have these troubling properties. It is based on the superposition of duplicate pairs of orthogonally polarized waves that, due to quantum source boundary conditions, produce “twin” count pairs and hence “twin” wave intensity pairs. Non-locality does not appear to be a consequence of the formalism. Results of measurements on the spatially separated beams are correlated due to the duplicate wave intensities and counts.

No hidden variables have been used in the calculation. The wave optics formalism and assumed statistics for describing correlations among twin beam polarization components depend on empirical results. These lead to the following assumptions: 1) intensities are defined as counts/time; 2) counts are not divided at polarization analyzers; and 3) photons are created in duplicate at the source, with the two members of each source pair having opposite polarization. Condition 3) provides initial conditions for the physical processes that follow in each down-converter beam. The fact that photons occur in duplicate does not in itself imply nonlocality, since initially, in the down-converter crystal where the photons are created, the output waves are superimposed.

The formalism is silent on the question of the connection of photons to waves other than through the usual intensity definition of count rate. Thus, one obtains a statistical description in both the semi-classical wave optics picture presented, and the quantum entanglement picture.

The possibility of an alternative to the entanglement description has been thought to be ruled out by violation of Bell’s inequalities. However, the Bell theorem is fundamentally flawed, as shown in [

Some of the analyses given in the present paper were described in a preliminary (unpublished) form in [

The author is grateful to Armen Gulian, and Joe G. Foreman of Chapman University, Burtonsville, MD. for stimulating discussions, and especially to Joe G. Foreman for critiques of the presentation.