2f:T726,{ const probabilityPhotonPassesFilter = Math.cos(radians(demoFilter.filterAngle - demoFilter.photonSpin)) ** 2 * 100; const degreeBaseline = 85; const percentBaseline = degreeBaseline + 15; return html` `; }30:T5c5,{ demoCrystal; const hBaseline = 124; const spin = Math.round(Math.random() * 90); return htl.html` `; }31:Tf4a,{ const hBaseline = 124; const vBaseline = 300; const spin = Math.round(Math.random() * 90); const probabilityPhotonPassesFilter = Math.cos(radians(demoEntanglement.filterAngle - spin)) ** 2 * 100; return htl.html` `; }32:Td2c,{ const hBaseline = 124; const vBaseline = 300; const photon1Passes = Math.random() <= 0.5; const photon2Passes = !photon1Passes; return htl.html` `; }33:[] 2d:{"title":"A Physical Look at Quantum Entanglement","cells":[{"id":0,"value":"# A Physical Look at Quantum Entanglement","pinned":false,"hidden":false,"mode":"md"},{"id":6,"value":"Quantum entanglement is a surprisingly simple phenomenon, but at first glance it's not obvious what's so weird about it. Let's look at what happens in quantum entanglement at a physical level. We're not going to get into the experiments that proved entanglement happens, but instead focus on what's actually going on and what it means.","pinned":false,"hidden":false,"mode":"md"},{"id":11,"value":"## Tools at our Disposal\n\nWe have photons. They might have a spin. This spin (the direction of polarization) points in a direction: up, down, and everything in between. Let's say ${tex`0\\degree`} points straight up and ${tex`90\\degree`} points to the right.\n","pinned":false,"hidden":false,"mode":"md"},{"id":196,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":299,"value":"viewof demoPhoton = Inputs.form({\n photonSpin: Inputs.range([0, 90], {\n value: 35,\n step: 1,\n label: \"Photon spin (degrees)\"\n })\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":25,"value":"\nAlright, but how do we know what spin a particle has? We can't \"look\" at its spin, but we do have a polarized filter we can send it through. We can angle the filter however we want, and again, ${tex`0\\degree`} points straight up, and ${tex`90\\degree`} points to the right. There are only two outcomes: \n\n* The photon will pass through, in which case it will **have a spin along the axis of the filter**, or\n* The photon will get **blocked**.\n\nThere is no in-between. What's strange is that if the photon goes through, it \"acquires\" the angle of the filter regardless of what the spin was beforehand.","pinned":false,"hidden":false,"mode":"md"},{"id":306,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs"},{"id":362,"value":"viewof demoFilter = Inputs.form({\n photonSpin: Inputs.range([0, 90], {\n value: 0,\n step: 1,\n label: \"Photon Spin\"\n }),\n filterAngle: Inputs.range([0, 90], {\n value: 45,\n step: 1,\n label: \"Filter Angle\"\n }),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"If a photon's spin is ${tex`0\\degree`} and so is our filter, the photon will get sent through 100% of the time, because it was already pointing along the filter. If its spin is ${tex`90\\degree`}, we'd see it get blocked 100% of the time for the same reason.","pinned":false,"hidden":false,"mode":"md"},{"id":34,"value":"But if a photon's spin is ${tex`0\\degree`} and our filter is at ${tex`45\\degree`}, then 50% of the time the photon will go through, and 50% of the time the photon will get blocked. Because the photon's spin is halfway from the filter, it's a coin flip whether it goes through or not. If it goes through, it will have a ${tex`45\\degree`} spin on the other side.","pinned":false,"hidden":false,"mode":"md"},{"id":29,"value":"## The Setup\nNow lets take a photon with spin 0, and [send it through a crystal](https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion) that \"splits\" it into two halves. Because of the laws of conservation of energy and momentum, _they must spin at right angles to each other_, because their spin has to add up to 0. If one photon is vertically polarized, the second must be horizontally polarized. If one has a spin of ${tex`15\\degree`}, then the other must have a spin of ${tex`105\\degree`} (adding 90).","pinned":false,"hidden":false,"mode":"md"},{"id":494,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":619,"value":"viewof demoCrystal = Inputs.button(\"Send Another Photon Through the Crystal\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"## What We Would Usually Expect\n\nClassical mechanics thinks that when a photon goes through the crystal, the outgoing pair gets a random spin on the way out (still, always perpendicular to each other). Perhaps the pair would be ${tex`0\\degree`} and ${tex`90\\degree`}, or ${tex`45\\degree`} and ${tex`135\\degree`}. Each of those photons go off independently and hit a filter. If we got lucky and measured along the same axis as the photon's spin, we'd expect one photon to go through and the other to get blocked. But as we rotate the filter, we could see a mix. For example, suppose the photons had spins of ${tex`0\\degree`} and ${tex`90\\degree`}, but our filter pointed at ${tex`45\\degree`}. Then, each particle would have a 50% chance of going through. Sometimes one will go through and the other won't, but other times they'll *both* get blocked or get through, because they're independent particles that have their own spin.","pinned":false,"hidden":false,"mode":"md"},{"id":919,"value":"### Classical Model","pinned":false,"hidden":false,"mode":"md"},{"id":1001,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":675,"value":"viewof demoEntanglement = Inputs.form({\n filterAngle: Inputs.range([0, 90], {\n value: 45,\n step: 1,\n label: \"Filter Angle\"\n }),\n sendPhoton: Inputs.button(\"Send another photon through the crystal\")\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":58,"value":"## What Actually Happens\n### Quantum Model\n","pinned":false,"hidden":false,"mode":"md"},{"id":1017,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":926,"value":"**They always do the opposite of each other.** If one photon passes through the filter, it gets spin in the same direction as the filter. But even more strangely: _it behaves as if it always had that spin_. And if it always had that spin, then that means its partner always had a perpendicular spin, so it *always gets blocked.*\n\nWhy is this weird? Because it doesn't even matter how far apart the filters are: the moment one hits a filter, the other will invariably do the opposite. It's as if the photons don't get their spins after going through the crystal, instead *they get their spins after the first one hits the filter*. Before that, they exist in some sort of **entangled state** where they agree \"our spin is conserved at 0.\"\n","pinned":false,"hidden":false,"mode":"md"},{"id":157,"value":"## Unreal\nQuantum entanglement violates a Bell inequality known as the [CHSH inequality](https://en.wikipedia.org/wiki/CHSH_inequality). This inequality holds if both of the following are true:\n\n* **Locality** ${tex`-`} measurement of one photon cannot affect the measurement of another photon faster-than-light\n* **Realism** ${tex`-`} the spin of a photon exists independently of being measured. Essentially, it has a value even when nobody's looking.\n\nSince we have found that entanglement violates the CHSH inequality, we know that one (or both) of these assumptions does not hold. ","pinned":false,"hidden":false,"mode":"md"},{"id":187,"value":"# Appendix\n_Under the hood of this notebook._","pinned":false,"hidden":false,"mode":"md"},{"id":1041,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":189,"value":"Photon = ({\n render: ({ x = 0, y = 0, r = 1, spin = null }) => {\n return svg.fragment`\n \n \n \n ${\n spin != null &&\n svg.fragment``\n }\n \n \n `;\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":739,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":312,"value":"Filter = ({\n render: ({ x = 0, y = 0, size = 1, angle = 0 }) => {\n return svg.fragment`\n \n \n ${_.times(\n 5,\n (i) =>\n svg.fragment``\n )}\n \n `;\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":734,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":497,"value":"Crystal = ({\n render: ({ x = 0, y = 0, size = 1 }) => {\n return svg.fragment`\n \n\n\n `;\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":695,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":688,"value":"Arrow = ({\n render: ({ x = 0, y = 0, size = 1 }) => {\n return svg.fragment`\n\n\n\n `;\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":941,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":939,"value":"MissingPhoton = ({\n render: ({ x = 0, y = 0, size = 1 }) => {\n return svg.fragment``;\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1036,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1029,"value":"SuperpositionPhoton = ({\n render: ({ x = 0, y = 0, r = 1 }) => {\n return svg.fragment`\n\n`;\n }\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":237,"value":"Group = (transform, fragment) => svg.fragment`\n \n ${fragment}\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":386,"value":"radians = (angle) => angle * (Math.PI / 180)","pinned":false,"hidden":false,"mode":"ojs"},{"id":330,"value":"_ = require(\"lodash@4\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":191,"value":"html = htl.html","pinned":false,"hidden":false,"mode":"ojs"},{"id":231,"value":"svg = htl.svg","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}