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Bell's Theorem in CHSH Form: An Interactive Derivation

Building

A proof against the Einstein Podolsky Rosen Paradox, including Bell's Theorem, plus subsequent work by CHSH and Tsirelson.

The ultimate goal of this project is to create an interactive webpage in the style of a Distill.pub article that rules out Local Realism in Quantum Mechanics. The article will use similar logical approaches to Bell’s Theorem and the works of John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH). Subsequently, we will go on to prove Tsirelson’s Bound.

Note: The website you're currently on is not the actual project. Once finished, the project will be linked at the top and bottom of the project proposal.

Learning Goals§

My aim with this project is to improve on various skills related to Math, Physics, and Computer Science:

  1. Formal mathematical writing skills
  2. Deepen understanding of different Quantum Mechanics concepts
  3. Simulation coding
  4. Derivation Skills
  5. Overall public writing skills

Introduction§

Naturally, the project must start with an introduction of what the subject, or problem in this case, actually is. The introduction will start by acknowledging the fact that it is difficult to choose “which interpretation of Quantum Mechanics is right”.

The project will then present the two clashing positions of this discussion:

  1. The statistical and probabilistic approach to Quantum Mechanics that is most often used today.
  2. Einstein’s, Podolsky’s, and Rosen’s (EPR) beliefs that Quantum particles must obey locality (measurement on one particle cannot affect another particle instantly at space-like distances), that the probabilistic model is incomplete, and that there must exist “Hidden Variables” whose properties exist prior and independent of measurement.

The Experiment:§

As part of the introduction, we must introduce the very experiment that will be used to rule out Local Realism. This experiment involves everyone’s favourite duo: Alice and Bob, along with a specialized electron emitter that will emit a singlet pair of electrons, which will then pass through two Stern-Gerlach magnets. Alice and Bob each get their own magnet, and they each get to choose two angles that the magnets will flip between at random during the experiment. The experiment is to be run NN times. When an electron is detected by the magnet, it will either return +1+1 (Spin Up) or −1-1 (Spin Down). Alice and Bob’s goal is to determine how correlated their results are over many tries. This correlation value is created through the product of their results. Meaning, when Alice and Bob get the same value (+1+1 and +1+1 OR −1-1 and −1-1), the product of their results will be +1+1 (positive correlation). When they get different results, for example, Alice getting +1+1 and Bob getting −1-1, their product will be -1 (Negative correlation). Ultimately, Alice and Bob will create the scalar SS, which we will define more rigorously in the proof. This scalar SS will be a trap that Local Realism cannot pass, but the Probabilistic Model can.

The Classical Trap§

This will be the first section of the proof, and it is where we introduce our variables and set up the trap for classical mechanics. Since this is just the plan, I will not write the whole derivation here; however, the purpose of this section will be to show, using a variable s1s1, which represents the outcome of one individual test, that the magnitude of the overall expectaiton value for correlation cannot exceed 2.

The Quantum Solution§

quantum take goes here

Tsirelson’s Bound§

boung goes here

Interactive Widget§

SAVE

Conclusion§

Absolutely wonderful conclusion goes here