Bell's theorem does not dismiss local-realism
In a previous post, we have discovered that quantum mechanics can described equivalently to standard approaches, and in particular Bell's inequalities can be violated in a Bell test with momentum-entangled pairs of particles, despite making use of local realist assumptions. However, it is a fact that Bell's theorem and its descendants are regularly used to dismiss any possibility that a local-realist quantum mechanical model could even exist.
How happens that our local-realist model captures BI violations and correctly reproduces QM statistics? How to solve this apparent paradox?
Despite its mathematical simplicity, interpretation of Bell's theorem has given rise to a vast literature, in particular concerning its assumptions and the conclusions that can be drawn. The usual assumptions used in deriving Bell inequalities are realism (properties of physical systems are elements of reality, outcomes of tests are determined by some hidden variables), factorability (these outcomes cannot be influenced faster than the speed of light), and measurement independence (the measurement setting choices are independent of the hidden variables and vice versa).
Since BI are experimentally violated, at least one of the Bell's assumptions above must be false. Rejection of one particular of these assumptions corresponds to one of the admissible interpretations or solutions of Bell’s theorem. The standard approach (indeterminism) is to reject realism, that is, the existence of any hidden variable (HV) completing quantum mechanics and thus the fact that the values of the outcomes even exist before their measurement. Another possible solution is to reject the factorability condition (FC). Since this assumption is often, probably incorrectly, equated to no-signaling and thus locality, such approach leads to non-local theories that have many advocates (e.g., Bohmian mechanics). The last possibility concerns the validity of measurement independence (MI). It is often believed that MI represents the freedom of the experimenter to choose the measurement setting at will and thus is also referred to as freewill hypothesis. The fact that MI is not satisfied have been often explained by some kind of (super)determinism or conspiracy.
However, other, less unpleasant reasons to renounce to these assumptions exist. We want to present a simple, "toy" model that illustrates how MI or FC can be violated without appealing to the existing solutions (nonlocality, superdeterminism, etc.).
This toy model is inspired by the actual treatment of this Bell test. Consider two binary outputs σi = ±1 (i = 1,2), that are deterministic functions of two "intermediate" HV, v = {v1,v2},
&space;-&space;\delta\left(v_i+\frac{1}{2}&space;\right&space;) "\sigma_i=\delta\left(v_i-\frac{1}{2} \right ) - \delta\left(v_i+\frac{1}{2} \right )")
where δ denotes here Dirac delta function. Thus, σi is 1 (-1) only when vi equals 1/2 (-1/2). The intermediate HV are functions of the parameters a = {a1,a2} and of a pair of "original hidden variables" λ = {λ1,λ2} via the deterministic transformations
)}{2\pi}=\lambda_{1}+a_1-\lambda_2 "v_1+\frac{\sin(2\pi(v_1-a_1+\lambda_2))}{2\pi}=\lambda_{1}+a_1-\lambda_2")
)}{2\pi}=-\lambda_{1}+a_2-\lambda_2 "v_2+\frac{\sin(2\pi(v_2-a_2+\lambda_2))}{2\pi}=-\lambda_{1}+a_2-\lambda_2")
=\delta\left(\frac{\sigma_1}{2}-\frac{1}{2\pi}\sin(2\pi(-a_1-\lambda_2))-\lambda_1-a_1+\lambda_2\right) "P(\sigma_1|a_1,a_2,\lambda)=\delta\left(\frac{\sigma_1}{2}-\frac{1}{2\pi}\sin(2\pi(-a_1-\lambda_2))-\lambda_1-a_1+\lambda_2\right)")
=\delta\left(\frac{\sigma_2}{2}-\frac{1}{2\pi}\sin(2\pi(-a_2-\lambda_2))+\lambda_1-a_2+\lambda_2\right) "P(\sigma_2|a_1,a_2,\lambda)=\delta\left(\frac{\sigma_2}{2}-\frac{1}{2\pi}\sin(2\pi(-a_2-\lambda_2))+\lambda_1-a_2+\lambda_2\right)")
We shall further assume that the original HV are uniformly distributed between -1 and 1. The joint probability density of the intermediate HV is evaluated from the Jacobian of the transformation and reads
=\frac{1+\cos(\pi(a_2-a_1)+\pi(v_1-v_2))}{4} "P(v_1,v_2|a_1,a_2)=\frac{1+\cos(\pi(a_2-a_1)+\pi(v_1-v_2))}{4}")
(1)
Consequently, the joint probability density of the outputs given the parameter setting is evaluated as
=\frac{1+\cos(\pi(a_2-a_1)+\pi\frac{\sigma_1-\sigma_2}{2})}{4} "P(\sigma_1,\sigma_2|a_1,a_2)=\frac{1+\cos(\pi(a_2-a_1)+\pi\frac{\sigma_1-\sigma_2}{2})}{4}")
(2)
Finally, after some algebraic manipulation, the expected value of the product of the two outputs is evaluated as
) "\langle \sigma_1\cdot \sigma_2 \rangle=\cos(\pi(a_2-a_1))")
We immediately recognize the QM statistics (correlation factor) obtained for entangled particles in this post. Therefore, it is clear that such a model violates the BI. To do so, one of the Bell's assumptions must be false. Certainly, that is not locality: clearly σ1depends only on v1 which in turn depends only on a1 but not on a2, and vice versa.
We could analyse which Bell's assumption fails by considering both the original and the intermediate HV. Here, we shall discuss only the second approach. With the aforementioned change of variables, we have that
=\delta\left(v_1-\frac{\sigma_1}{2}\right)\delta\left(v_2-\frac{\sigma_2}{2}\right) "P(\sigma_1,\sigma_2|a_1,a_2,v)=\delta\left(v_1-\frac{\sigma_1}{2}\right)\delta\left(v_2-\frac{\sigma_2}{2}\right)")
=\int&space;P(\sigma_1,\sigma_2|v_1,v_2,a_1,a_2)P(v_1,v_2|a_1,a_2)d\lambda "P(\sigma_1,\sigma_2|a_1,a_2)=\int P(\sigma_1,\sigma_2|v_1,v_2,a_1,a_2)P(v_1,v_2|a_1,a_2)d\lambda")
we need to use (1), which clearly does not fulfill MI. In fact, the distribution of the HV depends on the settings! (In the original HV framework, it would be FC that is not satisfied).
This demonstration of a toy HV model that violates BI while being based on local relations is very simple. However, it has the merit to illustrate how dangerous it is to draw aprioristic conclusions on the validity of alternative models (or interpretations) of quantum mechanics. In particular, with the local-realistic model that we defend in this blog, Bell's inequalities are violated in a Bell test with momentum-entangled pairs of particles and QM results are reproduced, despite the fact that local theories are generally thought to be ruled out by Bell's theorem. In fact, the proposed model is not of the form postulated by Bell’s assumptions, thus it is not forbidden by Bell’s theorem to violate Bell’s inequalities.
John Stewart Bell (1928-1990)
How happens that our local-realist model captures BI violations and correctly reproduces QM statistics? How to solve this apparent paradox?
Despite its mathematical simplicity, interpretation of Bell's theorem has given rise to a vast literature, in particular concerning its assumptions and the conclusions that can be drawn. The usual assumptions used in deriving Bell inequalities are realism (properties of physical systems are elements of reality, outcomes of tests are determined by some hidden variables), factorability (these outcomes cannot be influenced faster than the speed of light), and measurement independence (the measurement setting choices are independent of the hidden variables and vice versa).
Since BI are experimentally violated, at least one of the Bell's assumptions above must be false. Rejection of one particular of these assumptions corresponds to one of the admissible interpretations or solutions of Bell’s theorem. The standard approach (indeterminism) is to reject realism, that is, the existence of any hidden variable (HV) completing quantum mechanics and thus the fact that the values of the outcomes even exist before their measurement. Another possible solution is to reject the factorability condition (FC). Since this assumption is often, probably incorrectly, equated to no-signaling and thus locality, such approach leads to non-local theories that have many advocates (e.g., Bohmian mechanics). The last possibility concerns the validity of measurement independence (MI). It is often believed that MI represents the freedom of the experimenter to choose the measurement setting at will and thus is also referred to as freewill hypothesis. The fact that MI is not satisfied have been often explained by some kind of (super)determinism or conspiracy.
However, other, less unpleasant reasons to renounce to these assumptions exist. We want to present a simple, "toy" model that illustrates how MI or FC can be violated without appealing to the existing solutions (nonlocality, superdeterminism, etc.).
This toy model is inspired by the actual treatment of this Bell test. Consider two binary outputs σi = ±1 (i = 1,2), that are deterministic functions of two "intermediate" HV, v = {v1,v2},
where δ denotes here Dirac delta function. Thus, σi is 1 (-1) only when vi equals 1/2 (-1/2). The intermediate HV are functions of the parameters a = {a1,a2} and of a pair of "original hidden variables" λ = {λ1,λ2} via the deterministic transformations
It should be clear that the outputs depend on the original HV and the parameters such that their conditional probability density (denoted with P) read
Finally, after some algebraic manipulation, the expected value of the product of the two outputs is evaluated as
We immediately recognize the QM statistics (correlation factor) obtained for entangled particles in this post. Therefore, it is clear that such a model violates the BI. To do so, one of the Bell's assumptions must be false. Certainly, that is not locality: clearly σ1depends only on v1 which in turn depends only on a1 but not on a2, and vice versa.
We could analyse which Bell's assumption fails by considering both the original and the intermediate HV. Here, we shall discuss only the second approach. With the aforementioned change of variables, we have that
which fulfills the FC. However, in order to express (2) in the required Bell form (for which the Bell's theorem holds),
This demonstration of a toy HV model that violates BI while being based on local relations is very simple. However, it has the merit to illustrate how dangerous it is to draw aprioristic conclusions on the validity of alternative models (or interpretations) of quantum mechanics. In particular, with the local-realistic model that we defend in this blog, Bell's inequalities are violated in a Bell test with momentum-entangled pairs of particles and QM results are reproduced, despite the fact that local theories are generally thought to be ruled out by Bell's theorem. In fact, the proposed model is not of the form postulated by Bell’s assumptions, thus it is not forbidden by Bell’s theorem to violate Bell’s inequalities.
Comments
Post a Comment