A proof of Bell’s theorem via counterfactuals, which clarifies the nature of Quantum Randomness and its relation to Quantum Probability.
We use methods from causal inference to derive Bell’s theorem, which was originally inspired by the super-deterministic theory proposed by De Broglie and Bohm [1,2,3]. In the paradigm of Pilot Wave theory, developed by De Broglie and Bohm, Quantum Probabilities determine the Algorithmic Probability with which an event occurs in a frequentist and deterministic manner.
A counter-intuitive Quantum mystery concerns pairs of spin \(\frac{1}{2}\) particles prepared in an entangled state with a very curious property. When the spins of both particles are measured along a common spatial axis, the measurement of one particle’s spin perfectly predicts the spin of the other even if separated by many light years. To be precise, if the first particle’s spin is up, the second particle’s spin is down which we shall call the Principle of Complementarity. For three decades, this mystery was known as the EPR paradox(1935) after Einstein, Podolsky and Rosen who maintained the possibility of a local hidden variables theory [4].
However, it was not until John Bell proved Bell’s theorem in 1964 that this mystery which some considered impossible to solve was finally settled. The conclusion of Bell’s analysis was that any hidden variable theory of Quantum phenomena must be explicitly non-local.
In the following analysis, we shall replicate a well-known proof of Bell’s theorem using methods from causal inference [5].
Let’s suppose we have two particles, and can use devices to measure the spin of each, along any axis of our choice.
Let \(X_1\) and \(X_2\) be two interventions each taking values in \(\{0,1,2\}\) where \(X_1\) records the angle at which particle 1 is measured, and \(X_2\) measures the angle at which particle 2 is measured.
We shall note that 0,1,2 correspond to three particular angles.
Let \(Y_1(x_1,x_2)\) be the binary spin(\(+1\) or \(-1\)) of particle 1 and \(Y_2(x_1,x_2)\) be the spin of particle 2, where particle 1 is measured at angle \(x_1\) and particle 2 is measured at \(x_2\). In the language of the Neyman model, \(Y_i(x_1,x_2)\) is the counterfactual response of particle \(i\) under the joint intervention \((x_1,x_2)\).
Let \(M(x_1,x_2) = 1\{Y_1(x_1,x_2) = Y_2(x_1,x_2)\}\) be an indicator function so that \(M(x_1,x_2)=1\) if the spin directions and \(M(x_1,x_2)=0\) otherwise.
According to Quantum experiments, if the spins of both particles are measured along a common spatial axis the measurement of one particle’s spin perfectly predicts the spin of the other. Hence, the Principle of Complementarity predicts:
\[\begin{equation} M(i,i) = 1\{Y_1(x_1=i,x_2=i) = Y_2(x_1=i,x_2=i)\} = 0 \tag{1} \end{equation}\]
In general, we have the relation:
\[\begin{equation} \forall i,j \in \{0,1,2\}, \mathbb{E}[M(x_1 = i, x_2 = j)] = \sin^2(\Delta_{ij}/2) \tag{2} \end{equation}\]
where \(\Delta_{ij}\) is the angle between angles \(i\) and \(j\).
In what follows, we take:
\[\begin{equation} \forall x_1, x_2 \in \{0,1,2\}, \mathbb{E}[M(x_1, x_2)] \tag{3} \end{equation}\]
as known, based on experimental data.
Meanwhile, it is worth noting that \(\sin(\Delta_{ii})=0\) for \(i \in \{0,1,2\}\) with probability 1.
Local hidden variables is equivalent to the hypothesis that the spin measured of one particle doesn’t depend on the angle of the second particle. In principle, this means that for all \(x_1, x_2 \in \{0,1,2\}\):
\[\begin{equation} \forall i,j \in \{1,2\}, Y_i(x_i,x_j) = Y_i(x_i) \tag{4} \end{equation}\]
In experiments, the time separation of the measurements was sufficiently close and the spatial separation of the particles sufficiently great that even a signal travelling at close to the speed of light wouldn’t be able to influence the outcome of the experiment. Hence, refuting the hypothesis of local hidden variables implies that any hidden variable theory must be explicitly non-local.
This hypothesis asserts both locality and reality:
Locality: the angle \(x_2\) at which particle 2 is measured has no effect on the spin \(Y_1\) of particle 1.
Reality: the spin \(Y_i(x)\) of a particle measured along axis \(x\) is assumed to exist for every \(x\), even though for each \(i\) only one of the \(Y_i(x)\) is observed.
All other \(Y_i(x)\) are missing data in the language of statisticians, or hidden variables in the language of physicists. Furthermore, using the language of counterfactual theory the hypothesis of local reality is the hypothesis of no interference between measurements.
For the sake of clarity, we shall decompose Bell’s theorem into two parts before considering its correspondence with Quantum experiments.
The local hidden variables hypothesis, \(\forall i,j \in \{1,2\}, Y_i(x_i,x_j) =
Y_i(x_i)\)
is false.
It’s worth noting that in order to experimentally verify that the local hidden variables hypothesis is false, it is sufficient to verify the following conditions:
\[\begin{equation} M(1,2)=1 \tag{5} \end{equation}\]
\[\begin{equation} M(0,0)= M(0,2) = M(1,0)=0 \tag{6} \end{equation}\]
are not simultaneously satisfiable.
If we consider two mutually exclusive scenarios:
\[\begin{equation} Y_1(1)=Y_2(2)=1 \tag{7} \end{equation}\]
\[\begin{equation} Y_1(1)=Y_2(2)=-1 \tag{8} \end{equation}\]
by analysing case (7), we find:
\[\begin{equation} M(0,2)=0 \land Y_2(2)=1 \implies Y_1(0)=-1 \tag{9} \end{equation}\]
However,
\[\begin{equation} M(1,0)=0 \land Y_1(1)=1 \implies Y_2(0)=-1 \tag{10} \end{equation}\]
where (10) may be reconciled with condition (4) i.e. \(M(0,0)=0\) only if \(Y_1(0)=1\). Thus, we have a contradiction.
On the other hand, if we consider case (8) we find:
\[\begin{equation} M(0,2)=0 \land Y_2(2)=-1 \implies Y_1(0)=1 \tag{11} \end{equation}\]
However,
\[\begin{equation} M(1,0)=0 \land Y_1(1)=-1 \implies Y_2(0)=1 \tag{12} \end{equation}\]
where (12) may be reconciled with condition (5) only if \(Y_1(0)=-1\) which yields another contradiction.
If \(\mathbb{E}[M(1,2)]-\mathbb{E}[M(0,2)]-\mathbb{E}[M(1,0)]-\mathbb{E}[M(0,0)] > 0\) then the hypothesis of local hidden variables is false.
We shall proceed by demonstrating that if there are Quantum Measurements such that:
\[\begin{equation} \mathbb{E}[M(1,2)]-\mathbb{E}[M(0,2)]-\mathbb{E}[M(1,0)]-\mathbb{E}[M(0,0)] > 0 \tag{13} \end{equation}\]
then there must be a Quantum System with \(M(1,2)=1\) and \(M(0,2)=M(1,0)=M(0,0)=0\).
We shall proceed by contradiction. Let’s suppose there is no Quantum System satisfying (5) and (6) simultaneously. Then, for all Quanta:
\[\begin{equation} M(1,2)-M(0,2)-M(1,0)-M(0,0) \leq 0 \end{equation}\]
which implies that:
\[\begin{equation} \mathbb{E}[M(1,2)]-\mathbb{E}[M(0,2)]-\mathbb{E}[M(1,0)]-\mathbb{E}[M(0,0)] \leq 0 \tag{14} \end{equation}\]
This combination of theorems is generally referred to as Bell’s theorem. What may be verified from the Quantum Theory of two-particle systems is that:
\[\begin{equation} \mathbb{E}[M(x_1=i,x_2=j)] = \sin^2(\Delta_{ij}/2) \tag{15} \end{equation}\]
Hence, we may deduce that the expression:
\[\begin{equation} \xi = \mathbb{E}[M(1,2)]-\mathbb{E}[M(0,2)]-\mathbb{E}[M(1,0)]-\mathbb{E}[M(0,0)] \tag{16} \end{equation}\]
equals:
\[\begin{equation} \xi = \sin^2(\Delta_{12}/2) - \sin^2(\Delta_{02}/2) - \sin^2(\Delta_{10}/2) \tag{17} \end{equation}\]
since \(\sin^2(\Delta_{ii}/2)= \sin^2(0)=0\).
From this equality, we may infer that the local hidden variables assumption is rejected if:
\[\begin{equation} \xi > 0 \implies \sin^2(\Delta_{12}/2) > \sin^2(\Delta_{02}/2) + \sin^2(\Delta_{10}/2) \tag{18} \end{equation}\]
where the angles \(0,1,2\) may be easily chosen to satisfy the inequality in experimental settings. In fine, we may conclude that the local hidden variables hypothesis is false.
Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables I.Physical Review. 1952.
Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables II.Physical Review. 1952.
de Broglie, L. “La mécanique ondulatoire et la structure atomique de la matière et du rayonnement”. Journal de Physique et le Radium. 1927.
Einstein, A; B Podolsky; N Rosen. “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?” 1935.
James M. Robins, Tyler J. VanderWeele, Richard D. Gill. A proof of Bell’s inequality in quantum mechanics using causal interactions. Arxiv. 2012.
For attribution, please cite this work as
Rocke (2022, April 9). Kepler Lounge: Bell's theorem via counterfactuals. Retrieved from keplerlounge.com
BibTeX citation
@misc{rocke2022bell's,
author = {Rocke, Aidan},
title = {Kepler Lounge: Bell's theorem via counterfactuals},
url = {keplerlounge.com},
year = {2022}
}