Thursday, 9 May 2013

A Local Realistic simulation of the EPR correlation.


Bryan C. Sanctuary

The Simulation Programs

The EPR paradox can be reconciled in a local realistic way by assuming that an isolated spin that makes up a quantum ensemble (the usual quantum state)  has two axes of quantization rather than one.  Using this model the EPR data is exactly simulated one EPR pair at a time. 

In this blog the Java programs that do the simulation are given. The program was written by Chantal Roth and was modified for my model by Mr. Michael Havas.

The research paper is submitted to Physical Review A. 
  • A Local Realistic Reconciliation of the EPR Paradox
  • A document with most of the derivations; 
  • A recorded seminar
These can be found on my blog along with some other information:

The following programs contain the source code and compiled versions of the program:

Source code

Linux installer (no source)

Mac installer (no source)

Windows installer (no source)

The program without an installer (no source)

The plot below is the simulation that gives exact agreement with the correlation observed experimentally and calculated from quantum mechanics.

Figure 1 A local realistic simulation of the EPR data calculated at every 22.5 degrees and the points connected. The smooth curve is -cosθab plotted as a function of θab.
The model treats a spin in complete isolation before it reaches a filter. The model assumes that an isolated spin has two components of angular momentum, σx and σz. Since it is impossible to measure them both simultaneously (Heisenberg), two simulations are needed. Each simulation gives half the correlation,

Figure 2  The result of simulating the EPR correlation for one axis of quantization. The quantum coherences are ignored and the correlation from one axis gives half the correlation predicted from quantum mechanics.

The Model

It is assumed that a single spin that makes up the quantum state is described by the state operator,                   


The body frame,

is related to the laboratory frame by Local Hidden Variables (LHV), θ, φ.  In addition, there are other LHV
and these correspond to the quadrants of the body frame (these quadrants are seen in Figure 3). In complete isolation, the states of this model have eigenstates of ±2,

Figure 3 Spin depicted in an isotropic environment showing the eight possible pure states with two per quadrant. The 2D spin can occupy only one state at any instant. The states bisecting the quadrants are the eigenstates  |√2,±,q,f>nx,nz.
And these are depicted in Figure 3 showing a “horizontal” and “vertical” component.


The details of the model can be found in my research paper.  There are a number of caveats which mainly follow from the Heisenberg Uncertainty Relations: 
  • It is impossible to experimentally prove the 2D model is correct because it is impossible to simultaneously measure two spin components, σx, σz , at the same time. 
  • Although the full quantum correlation 2√2 is obtained with my local realistic model, only half the correlation is obtained, √2  per axis, from each simulation.  Therefore the CHSH equation is not violated √2 < 2 whereas Bell’s original inequalities are violated √2<1.
  • The reason the CHSH equation is not violated is because in the classical derivation, Hidden Variable space is partitioned into two regions.  As the classical CHSH equation, it is assumed that the two regions commute and can be simultaneously sampled.  My local realistic model does not use any classical notions, and the two regions do not commute and therefore cannot be simultaneously measured.
  • Another way to state this is that one spin component is measured; the coincidences from the other components are counterfactual.
  • Quantum mechanics is a theory of measurement.  In this treatment, quantum theory is extended to treat one particle in complete isolation.  This leads to some changes in quantum theory that are not addressed in this work.  The model and its simulation are the main focus at this stage.
Since experiment cannot resolve the issue, subjective arguments alone must decide. If quantum mechanics is complete then we have to accept non-local indeterminism.  Non-locality is a concept that no one understands. Indeterminism goes against our basic intuition about Nature.


In contrast, the local realistic simulation using the 2D spin model:
  •    Agrees with both quantum theory and experiment
  •   Is consistent with Bell’s correlation and gives more insight into the  quantum correlations
  •    Predicts the filter settings that maximize the EPR correlation
  •   Agrees with the treatment of Gustafson which identifies a vector of length √2 in the CHSH equation.

The Experimental Data

Here I will only comment on one issue, (please see the paper for more details). Why experiments appear to show violation of the CHSH equation in one experiment?

Heisenberg again reconciles this because half the coincidences that are present cannot be measured when two complementary axes exist. The expression usually used to determine the correlation is,
When there are two axes of quantization, only one can be measured at one time because as one axis lines up with the filter, the other decohers and cannot be detected,

Figure 3. Depiction of how half the correlation is lost in the presence of a probe: (a) The two body fixed axes of spin quantization, z and x (heavy lines).  The two arrows depict two possible orientations of a vector probe field in the laboratory frame, one closer to the z axis and the other closer to the x axis (only one can exist at any instant): (b) The case when the probe is closer to the z axis. Then the 2D spin deterministically nutates until the z axis aligns with the probe while the x component precesses in the XY plane. (c) The same as case (b) but now the probe is closer to the x axis which now aligns in the X direction and the z component precesses in ZY plane. 
Here a coincidence is defined by a product.  But what happens when there are two axes of quantization?  Then there are two polarizations,
In the case of two axes, the number of coincidences is doubled because there are 2 coincidences for each EPR pair.  However only the Z or the X can be measured, so the correlation associated with the axis that can be measured is 
Where N is the total number of coincidences actually detected. This is half the number that actually exists. Another experiment, with the filter angles rotated to by 90 degrees would reveal the other half of the correlation in principle but due to rotational invariance, the results are indistinguishable.

This means the coincidences from the undetected axis are counterfactual, and it is impossible to experimentally distinguish between the usual description of spin, with a single axis of quantization, and the model here with two orthogonal axes of quantization.

Another way of expressing this is to consider a gedanken experiment where, in spite of non-commutation of the two Pauli spin components, both somehow can be measured simultaneously. In that case a single EPR pair would record a coincidence in the Z channels, and simultaneously another coincidence in the X channels. The total number of coincidences for N EPR pairs is again 2N. With this modification, the simulated correlation per axis is in agreement with the experimental data.

Consider also Figure 3.  As a 2D spin approaches a filter either the Z axis or the X axis aligns with the filter direction, but not both.  Therefore half the coincidences come from the X axis and the other half from the Z axis, in agreement with the above equation.   

Counterfactual events cannot be detected and therefore cannot be experimentally confirmed. This illustrates the limit of quantum measurement, making it impossible to experimentally confirm that two axes exist rather than one.

1 comment:

  1. Bell's original inequalities assume local realisn *and* perfect anti-correlation when equal measurement settings are used in each wing of the experiment. However your simulation only generates half the quantum-correlations so it must have r(A,A) = -1/2.