Quantum Confusion for Dummies
In discussions of a recent topic, I noticed several messages from people who think that “physicists have agreed” on the existence of a superposition. That this is just a convenient mathematical / physical model, which has no real experiments to prove the presence of quanta in a superposition. That quanta, in fact, 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 well-known Bell theorem (Bell inequality), which was later improved by other scientists and repeatedly tested experimentally. For those who wish to familiarize themselves directly with his theorem, I advise you to skip this article, and immediately proceed to reading the books, links to which are given below, and in the comments. Understanding its fundamentals does not require deep knowledge of physics and mathematics. For those who even find a Wikipedia article to be difficult to understand, I’ll give a rather simplified analogy.
For simplicity, say, a quantum has some 3 characteristics: A, B, and C, which can take the values 1 or 0. Take two entangled quanta, such that:
1) If, when measuring at the first quantum of one of the characteristics, we get 1, then at the other quantum, the same characteristic when measuring will be equal to 0.
2) If we choose a characteristic for comparison in a random way, then in half of the cases we get the same values, and in 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 out statistically, programmatically, who both wants and can, let him do his own research: Put such an experiment: Create N predetermined 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 measure random parameters, the opposite values will appear in more than 50% of cases . That contradicts clause 2.
Namely, in our experiment, the probability of finding opposite values will lie in the range [5/9; 2/3] (0.555; 0.667). In the best model, we can not achieve different results in less than 55.5% of cases. While in reality you can make a similar experiment with quanta, in which it will remain equal to 1/2.
This is explained very simply: In the presence of "pre-determined quanta" we always have an "advantage" of its values in one direction. There either two units, and one zero, or two zeros and one, or in general all 3 values are equal to either one or zero.
It was this thought experiment that showed me that in the world of quanta there is no place for deterministic parameters. He forced me to study the topic in more detail and to find in it a lot of unusual, interesting and exciting.
PS Very well this experiment was described in the book of Richard Feynman (I hope the community will tell you which one, I'm a little confused)
PPS But no, this is Brian Green. The fabric of the cosmos. The space, time and texture of reality. ” That's just this moment . Maybe so someone will become clearer.
Upd1
Explanation from the mathematical side:
For example, 1 quantum has such characteristics (1,1,0), and confused with it (0,0,1). We randomly select and measure the characteristic of the first quantum and randomly choose 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 the combinations with 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 an advantage of 5: 4 in favor of different pairs.
For entangled 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, which means that a tangled 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 tangled threes. Of these, 5 are different values, 4 are the same. (p = 5/9)
Total, overall probability of pairs with different values: 5/9 * 6/8 + 1 * 2/8 = 2/3> 1/2
Upd2
I want to express a special thank you to the user Shkaff , for pointing out errors in the original version of the article, and for useful links in his comments . The article had to be slightly modified, but I tried to keep the original idea.
For simplicity, say, a quantum has some 3 characteristics: A, B, and C, which can take the values 1 or 0. Take two entangled quanta, such that:
1) If, when measuring at the first quantum of one of the characteristics, we get 1, then at the other quantum, the same characteristic when measuring will be equal to 0.
2) If we choose a characteristic for comparison in a random way, then in half of the cases we get the same values, and in 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 out statistically, programmatically, who both wants and can, let him do his own research: Put such an experiment: Create N predetermined 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 measure random parameters, the opposite values will appear in more than 50% of cases . That contradicts clause 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; using System.Threading.Tasks; namespace ConsoleApplication14 { class Program { static void Main(string[] args) { int confirm = 0, notconfirm = 0; Random rnd = new Random(); List<Tuple<bool, bool, bool>> p = new List<Tuple<bool, bool, bool>>(); for (int i = 0; i < 500000; i++) { var t = new Tuple<bool, bool, bool>(rnd.Next(2) == 1 ? true : false, rnd.Next(2) == 1 ? true : false, rnd.Next(2)==1?true:false); p.Add(t); } 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)); Console.ReadKey(); } } } Namely, in our experiment, the probability of finding opposite values will lie in the range [5/9; 2/3] (0.555; 0.667). In the best model, we can not achieve different results in less than 55.5% of cases. While in reality you can make a similar experiment with quanta, in which it will remain equal to 1/2.
This is explained very simply: In the presence of "pre-determined quanta" we always have an "advantage" of its values in one direction. There either two units, and one zero, or two zeros and one, or in general all 3 values are equal to either one or zero.
It was this thought experiment that showed me that in the world of quanta there is no place for deterministic parameters. He forced me to study the topic in more detail and to find in it a lot of unusual, interesting and exciting.
PS Very well this experiment was described in the book of Richard Feynman (I hope the community will tell you which one, I'm a little confused)
PPS But no, this is Brian Green. The fabric of the cosmos. The space, time and texture of reality. ” That's just this moment . Maybe so someone will become clearer.
Upd1
Explanation from the mathematical side:
For example, 1 quantum has such characteristics (1,1,0), and confused with it (0,0,1). We randomly select and measure the characteristic of the first quantum and randomly choose 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 the combinations with 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 an advantage of 5: 4 in favor of different pairs.
For entangled 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, which means that a tangled 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 tangled threes. Of these, 5 are different values, 4 are the same. (p = 5/9)
Total, overall probability of pairs with different values: 5/9 * 6/8 + 1 * 2/8 = 2/3> 1/2
Upd2
I want to express a special thank you to the user Shkaff , for pointing out errors in the original version of the article, and for useful links in his comments . The article had to be slightly modified, but I tried to keep the original idea.
Source: https://habr.com/ru/post/181692/
All Articles