# Quantum entanglement for dummies

In discussions of a recent topicI noticed several messages from people who think that “physicists have agreed” on the existence of a superposition. That it’s just a convenient mathematical / physical model that doesn’t have any real experiments, proving the presence of quanta in superposition. With quanta, in fact, they are always in specific positions, and conducting an experiment only reveals these positions. For some time this was a controversy among physicists, until in 1964, John Stuart Bell formulated his famous Bell theorem (Bell inequalities), which was subsequently improved by other scientists and repeatedly tested experimentally. For those wishing to familiarize themselves directly with his theorem, I advise you to skip this article and go straight to reading the books, the links to which are given below, and in the comments. To understand its basics, a deep knowledge of physics and mathematics is not required. For those who evenWikipedia article seems difficult to understand, I will give a rather simplified analogy.

For simplicity, let's say, a quantum has some 3 characteristics: A, B and C, which can take values ​​1 or 0. Take two entangled quanta, such that:
1) If, when measuring the first quantum of one of the characteristics, we get 1, then for another quantum, the same characteristic during measurement will be 0.
2) If we choose a characteristic for comparison at random, then in half the cases we get the same values, and in the half - different. (!)

At first, it seems that it is very easy to fulfill these two conditions, by writing a simple program we can simulate this situation. BUT! Let's just check it statistically, programmatically, whoever wants and can, let him do his own research: Perform the following experiment: Create N predefined pairs of triples of values: (1,0,1) - (0,1,0); (1,1,0) - (0,0,1) ... etc., then build a model that will satisfy both of the above points.

It turns out that this is not only difficult to do, but in principle impossible. If we measure the same parameters with such initial data, we will get opposite values. What is clear and consistent with paragraph 1. But if, we will measure random parameters, then the opposite values ​​will appear in more than 50% of cases. Which contradicts paragraph 2.

A small piece of C # code, trying to write such a model
``````using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace ConsoleApplication14
{
class Program
{
static void Main(string[] args)
{
int confirm = 0, notconfirm = 0;
Random rnd = new Random();
List> p = new List>();
for (int i = 0; i < 500000; i++)
{
var t = new Tuple(rnd.Next(2) == 1 ? true : false, rnd.Next(2) == 1 ? true : false, rnd.Next(2)==1?true:false);
}
for (int i = 0; i < 500000; i++)
{
var t = p[i];
bool first, second;
switch (rnd.Next(3))
{
case 0: first = t.Item1;
break;
case 1: first = t.Item2;
break;
case 2: first = t.Item3;
break;
default:
first = false;
throw new Exception("first error");
}
switch (rnd.Next(3))
{
case 0: second = !t.Item1;
break;
case 1: second = !t.Item2;
break;
case 2: second = !t.Item3;
break;
default:
second = true;
throw new Exception("second error");
}
if (first != second)
confirm++;
else
notconfirm++;
}
Console.WriteLine((double)confirm / (double)(confirm+notconfirm));
}
}
}``````

Namely, in our experiment, the probability of detecting opposite values ​​will lie in the range [5/9; 2/3] (0.555; 0.667). In the best model, we will not be able to achieve different results in less than 55.5% of cases. While in reality you can put a similar experiment with quanta in which it will remain equal to 1/2.

The explanation is very simple: In the presence of "predefined quanta" we always have an "advantage" of its values ​​in one direction. There are either two units, and one zero, or two zeros and one, or in general all 3 values ​​are equal to either unity or zero.

It was this thought experiment that showed me that in the world of quanta there is no place for deterministic parameters. He made me study the topic in more detail and find a lot of unusual interesting and exciting in it.

PS Very well this experiment was described in a book by Richard Feynman (I hope the community will tell you which one, I'm a little confused)
PPS No, this is Brian Green “The fabric of space. Space, time and texture of reality. ” This is just this moment . Maybe so it will become clear to someone.

Upd1 The
explanation from the mathematical side:
For example, 1 quantum has such characteristics (1,1,0), and entangled with it (0,0,1). We randomly select and measure the characteristic of the first quantum and randomly select and measure the characteristic of the second quantum. With a large number of experiments, we will have the results of all possible combinations: A1A2, A1B2, A1C2, B1A2, B1B2, B1C2, C1A2, C1B2, C1C2 (9 pieces) with approximately the same probability of occurrence of each.
Now, if we write out all combinations from our pair of quanta, we get:
10,10,11,10,10,11,00,00,01. 5 pairs of different values. 4 pairs are the same. Thus, for such quanta we will have a 5: 4 advantage in favor of different pairs.
For tangled pairs (0,0,0) - (1,1,1) - we will always get different pairs.
We have 8 distribution options for three binary parameters: 000,001,010,100,011,101,110,111.
2/8 of them with three identical values, so an entangled pair will always be with opposite values ​​(p = 1).
6/8 of them with two identical and one opposite value. 9 different combinations with such intricate triples. Of these, 5 are different values, 4 are the same. (p = 5/9)
Total, the total probability of pairs with different values: 5/9 * 6/8 + 1 * 2/8 = 2/3> 1/2

Upd2
I want to express special thanks to the user Shkaff , for indicating errors in the original version of the article, and for useful links in his comment . The article had to be changed a little, but I tried to keep the original idea.