Local-realist Bell-test experiment with momentum entanglement
Within our search for a local-realist description of quantum mechanics, two recent posts have discussed the local-realist rules of motion for particle pairs that are entangled in momentum and how position probability distributions are built upon.
We want now to describe a scenario for which these rules allow to retrieve typical quantum correlations between the two particles, which ultimately lead to violations of Bell's inequalities as predicted for QM by Bell's theorem.
The scenario consists of a two-slit interferometer, as depicted in the figure. This setting is equivalent to the double-source preparation discussed in the aforementioned posts. The two 'sources' are equally probable and the phase difference at each station is ε(I) = α, ε(II) = β. The detectors are placed at positions x± = ±t/(4D), where δ = 2D is the distance between the two slits, a parameter of the stations.
In the case of a single station active, with particles emitted one by one (non-entangled case), the situation is equivalent to a double-slit scenario, which has been discussed in this post, and gives the following result for the probability of hitting either of the two detectors:
=\frac{1\pm&space;\sin(\pi\alpha))}{2t} "\rho(x_{\pm};t)=\frac{1\pm \sin(\pi\alpha))}{2t}")
in agreement with Malus' law.
When the two stations are active and entangled particles are emitted in pairs, we just follow the reasoning of this post and apply equation (98) displayed therein. The joint probability density to obtain clicks at the corresponding detectors is
=\frac{1+\cos(\pi(\beta-\alpha)))}{4t^2}=\rho(x_-,x_-;t) "\rho(x_+,x_+;t)=\frac{1+\cos(\pi(\beta-\alpha)))}{4t^2}=\rho(x_-,x_-;t)")
while for opposite detectors, a factor π appears in the cosine argument of the aforementioned equation (98), leading to
=\frac{1-\cos(\pi(\beta-\alpha)))}{4t^2}=\rho(x_+,x_-;t) "\rho(x_-,x_+;t)=\frac{1-\cos(\pi(\beta-\alpha)))}{4t^2}=\rho(x_+,x_-;t)")
Of course, these are joint probabilities conditioned to the fact that both particles reach either of the two detectors. Particles that do not reach a detector after t iterations are not accounted for. Thus, the sum of the four probabilities gives 1/t2, that is, a constant less than one.
The correlations obtained are again in perfect agreement with QM, where they are calculated using complex wavefunctions. Some numerical simulations of the local-realistic model leading to the joint pdf above have been published in a 2017 arXiv paper. Those figures are reproduced here below.
The first two panels show the joint frequencies of arrivals at detectors placed at nodes 土25 (D = 1) after t = 100 iterations, for two different choices of β and varying α. Clearly, the patterns tend to the values computed above and they would approach the theoretical values even more for larger values of t. However, larger times would also require larger numbers of emissions to record a significant number of detection coincidences, as this number scales with the factor Np/t2, making the simulation time unpractically long.
The third and fourth panels show two correlation factors that are often used to discuss Bell's theorem, namely, the correlation factor C (the expected value of the product O(I)O(II) where O(R) = ± 1 is an integer associated to the arrival to detector x±) and the Bell-CHSH factor S (calculated for four pairs of angles that are equispaced between 0 and 3/4π). The results show that both these factors tend to the QM values C = cosθ and S = 3cosθ - cos(3θ), where θ is defined as the angle difference π(β-α).
In particular, the figure clearly shows that, in agreement with the theoretical values, the computed value of S exceeds the Bell limit value of 2 for some values of θ (peaking for "Bell test angles" ±π/4). In other terms, despite being a local and realistic model, the proposed model is able to violate the Bell-CHSH inequalities. Why is that possible will be discussed in future posts.
We want now to describe a scenario for which these rules allow to retrieve typical quantum correlations between the two particles, which ultimately lead to violations of Bell's inequalities as predicted for QM by Bell's theorem.
The scenario consists of a two-slit interferometer, as depicted in the figure. This setting is equivalent to the double-source preparation discussed in the aforementioned posts. The two 'sources' are equally probable and the phase difference at each station is ε(I) = α, ε(II) = β. The detectors are placed at positions x± = ±t/(4D), where δ = 2D is the distance between the two slits, a parameter of the stations.
In the case of a single station active, with particles emitted one by one (non-entangled case), the situation is equivalent to a double-slit scenario, which has been discussed in this post, and gives the following result for the probability of hitting either of the two detectors:
in agreement with Malus' law.
When the two stations are active and entangled particles are emitted in pairs, we just follow the reasoning of this post and apply equation (98) displayed therein. The joint probability density to obtain clicks at the corresponding detectors is
while for opposite detectors, a factor π appears in the cosine argument of the aforementioned equation (98), leading to
Of course, these are joint probabilities conditioned to the fact that both particles reach either of the two detectors. Particles that do not reach a detector after t iterations are not accounted for. Thus, the sum of the four probabilities gives 1/t2, that is, a constant less than one.
The correlations obtained are again in perfect agreement with QM, where they are calculated using complex wavefunctions. Some numerical simulations of the local-realistic model leading to the joint pdf above have been published in a 2017 arXiv paper. Those figures are reproduced here below.
The first two panels show the joint frequencies of arrivals at detectors placed at nodes 土25 (D = 1) after t = 100 iterations, for two different choices of β and varying α. Clearly, the patterns tend to the values computed above and they would approach the theoretical values even more for larger values of t. However, larger times would also require larger numbers of emissions to record a significant number of detection coincidences, as this number scales with the factor Np/t2, making the simulation time unpractically long.
The third and fourth panels show two correlation factors that are often used to discuss Bell's theorem, namely, the correlation factor C (the expected value of the product O(I)O(II) where O(R) = ± 1 is an integer associated to the arrival to detector x±) and the Bell-CHSH factor S (calculated for four pairs of angles that are equispaced between 0 and 3/4π). The results show that both these factors tend to the QM values C = cosθ and S = 3cosθ - cos(3θ), where θ is defined as the angle difference π(β-α).
In particular, the figure clearly shows that, in agreement with the theoretical values, the computed value of S exceeds the Bell limit value of 2 for some values of θ (peaking for "Bell test angles" ±π/4). In other terms, despite being a local and realistic model, the proposed model is able to violate the Bell-CHSH inequalities. Why is that possible will be discussed in future posts.
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