How was Bell's theorem proved by Bell's theorem for the "ghostly action at a distance" that plagued Einstein?

We take it for granted that something that happens somewhere in the world will not immediately affect things far away. This theory is called locality by physicists, and it has long been regarded as a basic assumption about the laws of physics.

But the proposal of quantum mechanics seems to overturn this hypothesis. In 1935, Einstein and his two colleagues co-wrote a paper-"Is Quantum Mechanics a Complete Description of Physical Reality?" "(Also known as EPR paradox). The central idea is: According to quantum mechanics, for a pair of particles that have a certain relationship before departure, but completely lose contact after departure, the measurement of one particle can instantly affect the properties of another particle at any long distance. Even if there is no connection between the two. The speed of the influence of one particle on another particle can exceed the speed of light. Einstein called it "spooky super-distance action", thinking that this is simply impossible, so as to prove that quantum mechanics is incomplete. of.

Is there anything missing in quantum mechanics? This issue has caused physicists to quarrel for decades.Northern Irish physicist John Stewart Bell is also quite worried about this controversy.

Finally, in 1964, Bell proposed Bell's theorem and Bell's inequality , turning the debate about the completeness of quantum mechanics into a question that can be verified by experiments. In the following years, quantum mechanics has withstood experimental verification time and time again.

Bell's theorem subverts one of our deepest intuitions about physics, prompting physicists to explore how quantum mechanics can accomplish tasks unimaginable in the classical world. Krister Shalm, a quantum physicist at the National Institute of Standards and Technology of the United States, said, "The quantum revolution that is happening now, and all these quantum technologies, are 100% thanks to Bell's theorem."

Let’s talk about Bell. How does the theorem help researchers prove that the "ghostly action at a distance" does exist.

Quantum entanglement

The "ghost distance" that plagued Einstein is a quantum phenomenon called "entanglement". In this phenomenon, two particles that are different entities lose their independence. As we all know, in quantum theory , the position, polarization and other properties of a particle are uncertain before it is observed. But if we observe these entangled particles, we will find that their observations are strongly correlated, even if they are far apart and are observed almost simultaneously. In other words, the unpredictable observations of one particle seem to immediately affect the observations of the other, no matter how far apart they are.This violates the principle of locality.

To understand quantum entanglement more accurately, we take the spin characteristics of electrons or other particles as an example. The spin particle behaves a bit like a small magnet. When an electron passes through the magnetic field generated by a pair of north-south magnetic poles , it will deflect to a certain degree towards a certain pole of the magnetic field. This shows that electron spin is a quantity that can only take one of two values: "up" means deflection to the North Pole, and "down" means deflection to the South Pole.

Imagine one, let an electron pass through a magnetic field, with the south pole above and the north pole below. Observing its deflection, we can get the electron's spin up or down along the vertical axis. Now, rotate the axis between the two magnetic poles so that it is no longer perpendicular, and then measure the deflection along the new axis. The electrons will always deflect to one of the poles with the same amplitude. In other words, no matter which axis you measure along, you will get a binary spin value-either up or down.

In fact, it is impossible for us to construct an observation device to measure the spin of a particle along multiple axes at the same time. Quantum theory asserts that this characteristic of the spin observer is actually the characteristic of the spin itself: if an electron has a certain spin along a certain axis, then its spin along any other axis is undefined.

local latent variable

With the understanding of spin , we can design a thought experiment to prove Bell's theorem. Here is a specific example of an entangled state: there is a pair of electrons with a total spin of 0, that is, their spin results are opposite no matter which axis is measured along which. The unique feature of this entangled state is that although the total spin is a constant value in all directions, the individual spin of each electron is uncertain.

Suppose these entangled electrons are separated and transported to remote laboratories, and scientists in these laboratories can rotate the magnets of their respective observers arbitrarily when performing spin measurements. When two teams measure along the same axis, they will get the opposite result 100%. But is this evidence to overturn the principle of locality? The answer is not necessarily.

Einstein proposed that each pair of electrons may have a set of related "hidden variables" that specify the spin of particles along all axes at the same time. These hidden variables do not exist in quantum descriptions that include entangled states, but quantum mechanics may not be complete.

The latent variable theory can explain why coaxial measurement always produces the opposite result without violating locality: the measurement of one electron will not affect the other electron, on the contrary, this measurement only reveals a hidden variable Pre-existing value.

Bell proved that you can overturn the theory of local latent variables and locality by measuring the spins of entangled particles along different axes.

First, suppose that a laboratory happens to rotate its observer 180 degrees relative to another laboratory’s observer. This is equivalent to flipping its south pole and north pole. Therefore, the "up" result of one electron will never be accompanied by the "down" result of another electron. Scientists can also choose to rotate a different angle, such as 60 degrees. Depending on the relative orientation of the two lab magnets, the probability of producing opposite results may be between 0% and 100%

Without specifying any specific direction, assume that two teams have three possible measurement axes Agreed, we can label it as A, B, and C. For each pair of electrons, each laboratory measures the spin of one of the electrons along one of the three measurement axes (selected at random).

Now we assume that the hidden variable theory is valid,Quantum mechanics does not hold. In this case, each electron will have its own spin value in three directions. This leads to eight possible values ​​of latent variables, which are expressed as follows:

For example, the spin value of serial number 5 means: It is "up", and the measurement result along the B-axis and C-axis will be "down"; the measurement result of the second electron is the opposite.

For any pair of electrons in 1, 8, the measurement results of the spin value of the two laboratories are always opposite, no matter which axis the researcher chooses to measure along. The other six sets of spin values ​​produced opposite results in 33% of the different axis measurements (again taking the fifth group as an example, when one laboratory measures along the B axis and the other laboratory measures along the C axis , The two laboratories will get opposite results; this represents one-third of the possible options.)

Therefore, at least 33% of the time, when measuring along different axes, the two laboratories Will get the opposite result. In other words, the probability of them getting the same result does not exceed 67%. This number is the upper limit allowed by the theory of local latent variables, and it is also the core inequality of Bell's theorem.

exceeds the upper limit

With this experimental design, we are interested in how likely the two laboratories will get the same result when measuring electron spin along different axes. The equations of quantum theory provide the formula for this probability, which is a function of the angle between the measurement axes.

According to this formula, when the distance between the three coordinate axes is as far as possible, that is, the three axes are at an angle of 120 degrees (similar to the Mercedes-Benz logo), the two laboratories will get the same result in 75% of the cases .This exceeds the 67% upper limit of Bell's inequality.

This is the essence of Bell's theorem: if the locality is true, that is, the observation of one particle will not immediately affect the observation result of another distant particle, then, in a specific experimental setting, the correlation of the results cannot exceed 67%. But if entangled particles can affect each other even if they are far apart (as described by quantum mechanics), the results of certain measurements will show stronger correlation.

Since the 1970s, physicists have conducted more and more precise experimental tests on Bell's theorem. Each one confirms the completeness of quantum mechanics. In the past 5 years, various loopholes have also been closed. In this regard, the researchers concluded that locality—that has long been considered a basic assumption about the laws of physics—is not a true characteristic of our world.

link to the original text:

https://www.quantamagazine.org/how-bells-theorem-proved-spooky-action-at-a-distance-is-real-20210720 _p