CHSH: there lies a vector of length √2
In my last entry I showed the correlation between EPR pairs composed of 2D spins is identical to that found in the CHSH form of Bell’s Inequalities. A 2D spin has two orthogonal axes of quantization rather than one and each axis carries equal correlation of √2. However only one axis can be measured at a time due to the Heisenberg Principle. Since Bell’s Inequalities are derived with classical arguments only, it does not take into account the quantum property of non-commutation. Here is a summary.
The probabilities for a single 2D spin to produce a + or – click are from one spin,
and EPR coincidences are products of these with filter settings at angle a for Alice and b for Bob (click equation to enlarge),
Half the correlation between an EPR pair comes from one axis (even quadrants) while the other half is associated with the second axis (odd quadrants). Since both cannot be diagonal simultaneously, the undetected axis of quantization has its coherence stored as quantum coherences in the off-diagonal elements. The representation that diagonalizes one axis means the other is off-diagonal and undetected. Consequently each EPR pair is capable of two coincidences rather than one, but only one can be detected in one experiment, while the other axis gives counterfactual coincidences.
This approach generates the EPR violation of the Bell’s inequalities by exactly 2√2.
An isolated spin is depicted by Figure 1a which can be directly compared to the filter settings that maximize the CHSH form of Bell’s Inequalities, Figure 1b (click to enlarge)
The CHSH form of Bell’s Inequalities expresses a limit that bounds classical correlations, E(a,b), by 2,
The correlation between a pair of spins in a singlet state is obtained from quantum mechanics, This had been used in the CHSH equation above. In the last entry I showed that the above probabilities are related to the CHSH equation by
These are maximized, and violate the above inequalities, when |a.(b+c)| = √2 and |d.(b-c)| = √2 and this requires a.d=0 and b.c = 0.
From the structure of the 2D spin, the vectors b and c are orthogonal and lie along the two axes of quantization in the body frame of a spin. The vectors a and d are two field directions. The CHSH equation is maximized giving 2√2 when a and d are orthogonal. This is depicted in Figure 1b. Finally because b and c are orthogonal, their sum and difference are vectors that bisect the quadrants formed from both b and c. These vectors lie in different quadrants and have length √2. The filter settings in Figure 1b maximize both the CHSH equation and the probabilities for coincidences for 2D spins.
EPR pairs have the same Local Hidden Variables, which means that both partners have their axes aligned i.e. (b±c) must be collinear for both Alice and Bob’s spins, and this is determined by their common origin.
The coefficients a+- and a-+ can be found elsewhere and are not repeated here.
In this entry I want to study the problem from the opposite direction. That is rather than showing the consistency of the 2D spin with Bell’s Inequalities, we show the CHSH equation predicts the hidden spin. Starting with the CHSH form of Bell’s Inequalities, a vector of length √2 is found that maximizes the CHSH equation: that is the 2D spin is hidden inside the CHSH equation.
Actually this has been done. In 2007 while attending a meeting on the Foundations of Quantum Mechanics in Växjo, Sweden I met Karl Gustafson who had studied various forms of Bell’s Inequalities. He used non-commutative trigonometry to analyze the CHSH equation and found, indeed, that to maximize it, a vector of length √2 must be present. You can find more details in his paper.
NONCOMMUTATIVE TRIGNOMETRY AND QUANTUM MECHANICS
Gustafson, K. Eds. N. M. Chuong et al.(pp. 341-360) copyright 2007 World Scientific Publishing Co. Advances in Deterministic and Stochastic Analysis, World Scientific Pub Co Inc, 2007, 341
Start with the form found above for Bell’s Inequalities,
While the first vector is normalized (hence the hat),
With these vectors the CHSH equation becomes,
In order to find the conditions that maximize this equation choose cos2θa,b+c+cos2θa,b-c =1+1=2 giving u2 a maximum length of √2 (see Eq.(*)). This requires the vector a to be parallel to b+c, and d to be parallel to b-c, compare with Figure 1b.
Finally the maximum occurs when the two vectors û1 and u2 are collinear. Under these conditions we have
Therefore indeed Gustafson found vectors of length √2 that maximize Bell’s inequalities for quantum systems. These vectors are also collinear to the hidden states of an isolated spin as shown in Figure 1a. Spin displays both horizontal and vertical components of angular momentum and this structure is consistent with the CHSH equation.
Whether spin actually has one or two axes of quantization cannot be verified experimentally. Only subjective arguments can decide between the two. Previously it was shown that the 2D spin has maximum coincidence correlation under the same conditions that maximize the CHSH equation. This leads to a purely quantum yet locally realistic reconciliation of the EPR paradox. Additionally a vector of length √2 is needed to maximize the CHSH equation, as shown here. The 2D spin has these properties while 1D point particle spin does not. These are compelling arguments to accept that, observed or not, spin has two orthogonal axes of spin quantization.
Since we have a choice, then it seems to me that accepting 2 orthogonal axes of quantization is less weird than quantum weirdness itself.