31:T40d,function render(experiments) { const margin = { top: 10, right: 10, bottom: 30, left: 25 }; const height = width * .5; const svg = d3.select(DOM.svg(width, height)); const sampleSize = 1000; const x = d3 .scaleLinear() .domain([0, 360]) .range([margin.left, width - margin.right]); const y = d3 .scaleLinear() .domain([1, 0]) .range([margin.top, height + margin.top - margin.bottom]); const xAxis = g => g .attr('transform', `translate(${0},${y(0)})`) .call(d3.axisBottom(x).ticks(18)); const yAxis = g => g.attr('transform', `translate(${x(0)},${0})`).call(d3.axisLeft(y)); svg.append('g').call(xAxis); svg.append('g').call(yAxis); experiments.forEach(e => { e.data.forEach(d => { svg .append('circle') .attr('cx', x(d.angle)) .attr('cy', y(d.collisions)) .attr('r', 1) .attr('opacity', .75) .attr('fill', e.fill) .attr('stroke-width', 0); }); }); return svg.node(); }32:[] 2f:{"title":"Yet Another Naïve Attempt To Grokk Bell's Theorem","cells":[{"id":0,"value":"md`# Yet Another Naïve Attempt To Grokk Bell's Theorem\n\nThis page is a mental experiment I have made over a weekend. I am not pretending to be an expert in quantum physics, and my interpretation is most probably not precise.\n\nNevertheless, it may help to build an intuition about these things.\n\nLet's imagine a world where 2-dimensional particles exist. Each particle is a vector that points to a random direction from 0 up to 2π radians (i.e., from 0˚ up to 360˚):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":43,"value":"function Particle() {\n return {\n angle: randomAngle()\n };\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":655,"value":"new Particle()","pinned":true,"hidden":false,"mode":"ojs"},{"id":661,"value":"md`Each particle has only one property - angle, but according to this world's rules, the inhabitants can't measure a particle's angle directly - they can only use two-dimensional detectors:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":146,"value":"function Detector(angle) {\n return {\n angle: angle\n };\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":672,"value":"md`A detector is nothing more than another vector that an inhabitant can rotate to any angle. We provide the inhabitants a detect function that accepts a detector and a particle and returns either +1 or -1.\n\nHere is an example of a very classical detect function. It will return one (+1) if the angle between the particle and the detector is less than 90˚. In other words if the cosine of this angle is greater than zero (0). Otherwise, the function will return minus one (-1):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":535,"value":"function veryClassicalDetect(detector, particle) {\n return Math.cos(detector.angle - particle.angle) > 0 ? 1 : -1;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":683,"value":"test('veryClassicalDetect returns +1 if the angle is 0', () => {\n // Arrange\n const particle = new Particle();\n const detector = new Detector(particle.angle);\n const expected = 1;\n // Act\n const actual = veryClassicalDetect(detector, particle);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":691,"value":"test('veryClassicalDetect returns -1 if the angle is 45˚', () => {\n // Arrange\n const particle = new Particle();\n particle.angle = 0;\n const detector = new Detector(Math.PI / 4);\n const expected = 1;\n // Act\n const actual = veryClassicalDetect(detector, particle);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":699,"value":"test('veryClassicalDetect returns -1 if the angle is 135˚', () => {\n // Arrange\n const particle = new Particle();\n particle.angle = 0;\n const detector = new Detector((3 * Math.PI) / 4);\n const expected = -1;\n // Act\n const actual = veryClassicalDetect(detector, particle);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":709,"value":"test('veryClassicalDetect returns -1 if the angle is 180˚', () => {\n // Arrange\n const particle = new Particle();\n particle.angle = 0;\n const detector = new Detector(Math.PI);\n const expected = -1;\n // Act\n const actual = veryClassicalDetect(detector, particle);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":610,"value":"sampleSize = 1000","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"angles = Array.from(Array(3600).keys()).map(k => k / 10)","pinned":false,"hidden":false,"mode":"ojs"},{"id":733,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":713,"value":"md`Now, let's experiment: for each angle from 0˚ to 360˚, let's take two detectors that have this angle between each other. Then, let's prepare two particles, which angles are random except that each particle's angle is always opposite to another. Let's do this experiment, say, 1000 times per each angle from 0˚ to 360˚, and chart how many times our detectors will return the same results:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":472,"value":"veryClassicalExperiment = {\n const results = [];\n for (let i = 0; i < angles.length; i++) {\n let collisions = 0;\n const alice = new Detector(0);\n const bob = new Detector((angles[i] * Math.PI) / 180);\n for (let j = 0; j < sampleSize; j++) {\n const particleA = new Particle();\n const particleB = new Particle();\n particleB.angle = inverseAngle(particleA.angle);\n if (\n veryClassicalDetect(alice, particleA) ===\n veryClassicalDetect(bob, particleB)\n ) {\n collisions++;\n }\n }\n results.push({ angle: angles[i], collisions: collisions / sampleSize });\n }\n return results;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":625,"value":"render([{ data: veryClassicalExperiment, fill: 'blue' }])","pinned":false,"hidden":false,"mode":"ojs"},{"id":718,"value":"md`We shall not be surprised by what we see. Indeed, if one particle is 0˚, then another (the opposite) must be 180˚. If our detectors are less than 90˚ between each other, they always show the opposite results. If the detectors are 180˚ to each other, they both show the same results. The rest of the cases falls between these two.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":721,"value":"md`A significantly different situation would be if we did the same experiment with a semi-quantum detect function. It will not deterministically return +1 or -1, but instead, it will return either +1 or -1 with a probability of cosine of the angle between the two detectors:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":629,"value":"function semiClassicalDetect(detector, particle) {\n return probabilisticOne(Math.cos(detector.angle - particle.angle));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":634,"value":"semiClassicalExperiment = {\n const results = [];\n for (let i = 0; i < angles.length; i++) {\n let collisions = 0;\n const alice = new Detector(0);\n const bob = new Detector((angles[i] * Math.PI) / 180);\n for (let j = 0; j < sampleSize; j++) {\n const particleA = new Particle();\n const particleB = new Particle();\n particleB.angle = inverseAngle(particleA.angle);\n if (\n semiClassicalDetect(alice, particleA) ===\n semiClassicalDetect(bob, particleB)\n ) {\n collisions++;\n }\n }\n results.push({ angle: angles[i], collisions: collisions / sampleSize });\n }\n return results;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":636,"value":"render([\n // { data: veryClassicalExperiment, fill: 'blue' },\n { data: semiClassicalExperiment, fill: 'red' }\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":727,"value":"md`Now, the picture is very different. But here, we still have two deterministic particles that are not connected when they are detected. What John Bell proves in his theorem is that a pair of entangled particles is detected like these particles are connected even if they are far away from each other. To emulate this, we introduce a quantum detect function. It does not only return -1 or +1 with a probability of cosine between the particle and the detector, but it changes the particle's angle to the detector's angle if the result is +1 or to the opposite angle if the result is -1.\n\nWe will emulate the entanglement by measuring the same particle and inverting the results of the second detector:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":314,"value":"function quantumDetect(detector, particle) {\n const result = probabilisticOne(Math.cos(detector.angle - particle.angle));\n\n if (result > 0) {\n particle.angle = detector.Angle;\n } else {\n particle.angle = inverseAngle(detector.angle);\n }\n\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":439,"value":"quantumExperiment = {\n const results = [];\n for (let i = 0; i < angles.length; i++) {\n let collisions = 0;\n const alice = new Detector(0);\n const bob = new Detector((angles[i] * Math.PI) / 180);\n for (let j = 0; j < sampleSize; j++) {\n const particle = new Particle();\n if (probabilisticOne(.5) > 0) {\n if (quantumDetect(alice, particle) === -quantumDetect(bob, particle)) {\n collisions++;\n }\n } else {\n if (-quantumDetect(bob, particle) === quantumDetect(alice, particle)) {\n collisions++;\n }\n }\n }\n results.push({ angle: angles[i], collisions: collisions / sampleSize });\n }\n return results;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":643,"value":"render([\n // { data: veryClassicalExperiment, fill: 'blue' },\n // { data: semiClassicalExperiment, fill: 'red' },\n { data: quantumExperiment, fill: 'green' }\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":738,"value":"md`Let's plot all three detect functions on the same chart to see the difference:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":650,"value":"render([\n { data: veryClassicalExperiment, fill: 'blue' },\n { data: semiClassicalExperiment, fill: 'red' },\n { data: quantumExperiment, fill: 'green' }\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":743,"value":"md`Please, fork this notebook and fix it if you see that my explanation is not precise or is entirely wrong.\n\nThank you!`","pinned":false,"hidden":false,"mode":"ojs"},{"id":40,"value":"html`

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"md`## Helper functions`","pinned":false,"hidden":false,"mode":"ojs"},{"id":581,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"md`### Trigonometry`","pinned":false,"hidden":false,"mode":"ojs"},{"id":112,"value":"function square(n) {\n return Math.pow(n, 2);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":117,"value":"function magnitude(v) {\n return Math.sqrt(square(v.x) + square(v.y) + square(v.z));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"function angleBetweenVectors(v0, v1) {\n const dotProduct = v0.x * v1.x + v0.y * v1.y + v0.z * v1.z;\n return Math.acos((dotProduct / magnitude(v0)) * magnitude(v1));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":90,"value":"test('Angle between [1,0,0] and [1,0,0] should be 0.', () => {\n // Arrange\n const v0 = { x: 1, y: 0, z: 0 };\n const v1 = { x: 1, y: 0, z: 0 };\n const expected = 0;\n // Act\n const actual = angleBetweenVectors(v0, v1);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":72,"value":"test('Angle between [1,0,0] and [0,1,0] should be π/2', () => {\n // Arrange\n const v0 = { x: 1, y: 0, z: 0 };\n const v1 = { x: 0, y: 1, z: 0 };\n const expected = Math.PI / 2;\n // Act\n const actual = angleBetweenVectors(v0, v1);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":95,"value":"test('Angle between [1,0,0] and [-1,0,0] should be π', () => {\n // Arrange\n const v0 = { x: 1, y: 0, z: 0 };\n const v1 = { x: -1, y: 0, z: 0 };\n const expected = Math.PI;\n // Act\n const actual = angleBetweenVectors(v0, v1);\n // Assert\n expect(actual).to.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":98,"value":"test('Angle between [1,1,0] and [1,0,0] should be π/4', () => {\n // Arrange\n const v0 = { x: 1, y: 1, z: 0 };\n const v1 = { x: 1, y: 0, z: 0 };\n const expected = Math.PI / 4;\n // Act\n const actual = angleBetweenVectors(v0, v1);\n // Assert\n expect(actual).to.closeTo(expected, .0001);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":122,"value":"function x(inclination, azimuth) {\n return Math.sin(inclination) * Math.cos(azimuth);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":126,"value":"function y(inclination, azimuth) {\n return Math.sin(inclination) * Math.sin(azimuth);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":130,"value":"function z(inclination, azimuth) {\n return Math.cos(inclination);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":444,"value":"function inverseAngle(angle) {\n return angle > Math.PI ? angle - Math.PI : angle + Math.PI;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":447,"value":"test('Inverse to 0 is 180', () => {\n // Arrange\n const angle = 0;\n const expected = Math.PI;\n // Act\n const actual = inverseAngle(angle);\n // Assert\n expect(actual).to.be.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":467,"value":"test('Inverse to 270 is 90', () => {\n // Arrange\n const angle = (Math.PI / 2) * 3;\n const expected = Math.PI / 2;\n // Act\n const actual = inverseAngle(angle);\n // Assert\n expect(actual).to.be.equal(expected);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":620,"value":"function degreesToRadians(degrees) {\n return (degrees * Math.PI) / 180;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":225,"value":"md`### Testing`","pinned":false,"hidden":false,"mode":"ojs"},{"id":6,"value":"expect = (await require('https://cdnjs.cloudflare.com/ajax/libs/chai/4.2.0/chai.min.js'))\n .expect","pinned":false,"hidden":false,"mode":"ojs"},{"id":10,"value":"test = (title, fn) => {\n fn();\n return html`✓ : ${title ||\n 'Test passing '}`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":139,"value":"md`### Probability`","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"function random(min, max) {\n return Math.random() * (max - min) + min;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":22,"value":"function randomAngle() {\n return random(0, Math.PI * 2);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"test('random(0, Math.PI * 2) should return (pseudo)random values in a range [0, 2π)', () => {\n // arrange\n const fn = () => random(0, Math.PI * 2);\n const randomAngles = Array.from({ length: 1000 }, fn);\n // assert\n expect(d3.min(randomAngles)).to.be.at.least(0);\n expect(d3.max(randomAngles)).to.be.below(2 * Math.PI);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":60,"value":"test('random(0, Math.PI) should return (pseudo)random values in a range [0, π)', () => {\n // arrange\n const fn = () => random(0, Math.PI);\n const randomAngles = Array.from({ length: 1000 }, fn);\n // assert\n expect(d3.min(randomAngles)).to.be.at.least(0);\n expect(d3.max(randomAngles)).to.be.below(Math.PI);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":33,"value":"{\n const data = Array.from({ length: 1000 }, () => random(0, Math.PI * 2));\n const height = 25;\n\n const svg = d3.select(DOM.svg(width, height));\n\n const x = d3\n .scaleLinear()\n .domain([0, Math.PI * 2])\n .range([0, width]);\n\n svg\n .append('g')\n .selectAll('line')\n .data(data)\n .enter()\n .append('line')\n .attr('x1', d => x(d))\n .attr('x2', d => x(d))\n .attr('y1', d => 0)\n .attr('y2', d => height)\n .attr('stroke', 'black')\n .attr('opacity', .5)\n .attr('stroke-width', 1);\n\n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":65,"value":"{\n const data = Array.from({ length: 1000 }, () => random(0, Math.PI));\n const height = 25;\n\n const svg = d3.select(DOM.svg(width, height));\n\n const x = d3\n .scaleLinear()\n .domain([0, Math.PI])\n .range([0, width]);\n\n svg\n .append('g')\n .selectAll('line')\n .data(data)\n .enter()\n .append('line')\n .attr('x1', d => x(d))\n .attr('x2', d => x(d))\n .attr('y1', d => 0)\n .attr('y2', d => height)\n .attr('stroke', 'black')\n .attr('opacity', .5)\n .attr('stroke-width', 1);\n\n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":195,"value":"// [-1, +1]\nfunction probabilisticOne(probability) {\n const round = 1000;\n const randomValueInRange =\n (Math.round(Math.random() * round) / round) * 2 - 1;\n if (randomValueInRange >= probability) {\n return -1;\n } else {\n return +1;\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"test('probabilisticOnes(1) averages to +1 -2.5%', () => {\n // arrange\n const probability = +1;\n // act\n const actual = d3.mean(\n Array.from({ length: 1000 }, () => probabilisticOne(probability))\n );\n // assert\n expect(actual).to.be.closeTo(probability, 0.025);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":202,"value":"test('probabilisticOnes(.45) averages to +0.45 ±5%', () => {\n // arrange\n const probability = .45;\n // act\n const actual = d3.mean(\n Array.from({ length: 1000 }, () => probabilisticOne(probability))\n );\n // assert\n expect(actual).to.be.closeTo(probability, 0.05);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":207,"value":"test('probabilisticOnes(0) averages to 0 ±5%', () => {\n // arrange\n const probability = 0;\n // act\n const actual = d3.mean(\n Array.from({ length: 1000 }, () => probabilisticOne(probability))\n );\n // assert\n expect(actual).to.be.closeTo(probability, 0.05);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":210,"value":"test('probabilisticOnes(-.33) averages to -0.33 ±5%', () => {\n // arrange\n const probability = -.33;\n // act\n const actual = d3.mean(\n Array.from({ length: 1000 }, () => probabilisticOne(probability))\n );\n // assert\n expect(actual).to.be.closeTo(probability, 0.05);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"test('probabilisticOnes(-1) averages to -1 +2.5%', () => {\n // arrange\n const probability = -1;\n // act\n const actual = d3.mean(\n Array.from({ length: 1000 }, () => probabilisticOne(probability))\n );\n // assert\n expect(actual).to.be.closeTo(probability, 0.025);\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":143,"value":"md`### Last, but not least`","pinned":false,"hidden":false,"mode":"ojs"},{"id":25,"value":"d3 = require(\"https://d3js.org/d3.v5.min.js\")","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}