Chapter 15 Chapter 10 Inequality-2

Even if powerful quantum computers do come out, the prospects for electronic security are not bleak. As the saying goes, God closed the door here, but opened a window elsewhere.Quantum theory not only provides us with incomparably powerful computational cracking capabilities, but also allows us to see another possibility: an encryption method that can never be cracked.Here's another hot topic: quantum cryptography.If space permits, we will briefly describe the situation in this regard at the end of the history.The reason why this kind of encryption can be realized is that the magical quantum can break through the shackles arranged by Einstein's God-that fateful and mysterious inequality.And that's what we're going to discuss right now.

However, at the end of this section, we return to the multiverse explanation.How do we explain the magical computing power of quantum computers?The only possibility, Deutsch claims, is that it takes advantage of multiple universes, places calculations in multiple parallel universes at the same time, and finally aggregates that result.Take Xiao's algorithm as an example, we have already mentioned that when it decomposes a 250-digit number, it performs 10^500 calculations at the same time.De Yiqi indignantly asked those who did not believe in MWI to explain this fact: If the calculation is not carried out in 10^500 universes at the same time, where does it have the resources to perform such amazing calculations?In particular, he pointed out that the entire universe contains only about 10^80 particles.But while it's possible (though still weird-sounding) to put computations in multiple parallel universes, MWI isn't the only explanation.Basically, quantum computers rely on the fundamental equations of quantum theory, not an explanation.Its model is constructed mathematically, and has nothing to do with how you explain it.You can think of it as every computer in 10^500 universes is doing calculations, but it can also be explained in accordance with Copenhagen, imagine that there are 10^500 computers in this universe before observation (output results) Superimposed computers working simultaneously!As for how this is achieved, we have no right to discuss it, just as we don't know how electrons pass through the double slit at the same time, or how cats are dead and alive at the same time.This sounds unbelievable, but in the eyes of many people, it is no less weird than the sudden split of 10^500 universes in an instant.As Coveney said in "The Arrow of Time", even if such a computer is built, it may not prove that many worlds are necessarily superior to other explanations.The point is, we haven't got the evidence that we can really judge. Maybe we should still see if there are other roads, and which more exotic directions they all lead to.

Four We can finally retreat from the road of many worlds, and reflect on the meaning of quantum theory.The "Monster of Consciousness" sign we left behind is still vivid in my mind, and our situation here in the multiverse is not much better. Maybe we can use DeWitt's exact words to erect a "schizophrenia" sign Come to alert the world.Whereas in Copenhagen, our constant concern is how to collapse the wave function, in the multiverse, the question becomes what is "I" in the universe.If we are constantly being projected into countless worlds all the time, which one is the real "I"?Or, the concept of "I" should simply be defined as a collection of all "I" in the n forks in the universe from now on?If this is the case, then "quantum immortality" does not sound so absurd: in this set, "I" is always alive on a certain branch.If you disagree and think that "I" is nothing more than a being at a certain time, splitting into countless new and different "Is" with each quantum measurement, then is our spirit nothing but a momentary concept, which has no continuity at all?Living in a universe that is splitting all the time, there are infinite new "I" clones being created all the time, God knows why we still feel that time is smooth and continuous, God knows why our "self-awareness" is continuous Not split.

Whether it is Copenhagen or the multiverse, they are actually trying to explain such a wonderful property in the quantum world: superposition.As we have repeatedly revealed to you in the history, before the observation, the weird quantum elves are always in an uncertain state, and must be described as the superposition of all possibilities.The electron is both here and there, and before the actual observation, it does not have a uniquely definite position as we tacitly assumed in the previous classical world.When a photon moves from point A to point B, it does not have a definite trajectory by default in classical mechanics.Instead, its trajectory is a blur, the sum of all possible trajectories!And not just all possible space trajectories, in fact, it is the sum of paths in all spaces and all times!In other words, the photon from A to B is a superposition of all possible routes in the past, present and future.On this basis, Feynman established his "path integral" (path

integral) method to calculate the probability amplitude of a quantum system in four-dimensional space.We have seen Heisenberg's matrix and Schrödinger's wave earlier in the story, and Feynman's path integral is the third means to describe the quantum system.But it can also be proved that it is completely equivalent to the first two, but it is just a different form of mathematical expression.With the Feynman diagram, this method is simple, practical, and very clever.Applying it to the atomic system, we will be surprised to find that on most of the paths, the actions cancel each other out, leaving only a few possible "orbits", which is consistent with the observation!

We must admit that quantum theory is successful in reality, and it can perfectly explain and explain the observed phenomena.But to admit the superposition, whether it is a Copenhagen-style superposition or a multi-universe superposition, this always has a huge conflict with our common sense of the real world.We still can't help but miss the golden classical era. At that time, the "real world" still retained the noble and objective lineage. It was simple and clear, in line with common sense. An electron always has a definite position and momentum, independent of our will or It will not be divided inexplicably by the observation behavior, but will be meticulously operated according to the strict law of causality under the rule of a beautiful universe rule.Oh, such a scene is warm and heart-warming, it is simply the peach blossom garden in the dream of physicists, can we really not reproduce such an ideal, and return to that memorable era?

Wait a minute, there is a road here, with a big billboard: Back to the classics.It even pulls out Einstein as its spokesman: the road leads to Einstein's dream.God, isn't Einstein's dream the world of classical objectivity, simplicity and clarity, where everything is ruled by strict causality?There is neither a dice-throwing God nor a plethora of copies of the universe, what a heart-pounding scene.What are we hesitating about, let's go and have a look! Reversing time and space, we first have to go back to 1927, to the Fifth Solvay Conference in Brussels, and then recall the great debate that determined the rise of quantum theory.We have described some scenes of this famous meeting in the eighth chapter of the history. We still remember that the French nobleman de Broglie talked about his "guided wave" theory at the meeting, but it was rejected. Lee's doubts.At the 5th Solvay Conference, Bohr's principle of complementarity had just been introduced, and particles and waves were still playing happily. De Broglie's "guided wave" was just an attempt to solve this contradiction .We all remember de Broglie's discovery that whenever a particle travels, it is accompanied by a wave, which profoundly revealed the conundrum of wave-particle duality.But de Broglie did not believe in Bohr's principle of complementarity, the explanation that electrons are particles and waves at the same time.De Broglie imagined that the electron is always a real particle, but it is indeed affected by the wave that always accompanies it. This wave is like a navigation dog for the blind, detecting the surrounding road conditions for it and guiding it. How to move is why we call it "guided wave".In de Broglie's theory there is no place for Born's statistical interpretation, it is entirely deterministic and realist.The randomness on the surface of quantum effects is completely caused by some variables that we don't know. In other words, quantum theory is an incomplete theory. It does not take into account some invisible variables, so it appears unpredictable.If those additional variables are taken into account, the whole system is deterministic and predictable, consistent with strict causality.Such a theory is called "hidden variable theory" (Hidden variable theory).

Variable Theory). De Broglie's theory was not born at the right time, just at the moment when the great principle of complementarity was introduced, coupled with its own immaturity, it was criticized by many, and it was Von Noy in 1932 who finally sentenced it to death Man.We may recall that von Neumann laid the rigorous mathematical foundation for quantum theory in that year, and he proved some peculiar properties of quantum systems such as "infinite regression".However, in addition to these, he also proved one thing in passing, that is: it is impossible for any hidden variable theory to give a definite prediction of the measurement behavior.In other words, hidden variable theory's attempt to drive randomness out of quantum theory is impossible, and any hidden variable theory—no matter what it is—is doomed to fail.

Von Neumann's gorgeous genius captivated everyone, and no one doubted that he was one of the greatest mathematicians of the 20th century.The helpless efforts of hidden variable theory seemed doomed, and Einstein's belief in strict causality seemed doomed.De Broglie accepted this reality, and in his heart he was not as stubborn and combative as Bohr, but with an aristocratic demeanor abandoned his views.Throughout the 1930s and 1940s, the Copenhagen interpretation dominated the world, and the spirit of quantum uncertainty was deeply rooted in the blood of physics. Numerous electrons and photons incarnated as wave functions mysteriously diffused in the universe, and the stars set off like the moon. Come out of the magic of that great wise man - Niels Bohr.

Gell-Mann, the winner of the Nobel Prize in Physics in 1969, later said jokingly: "Bohr brainwashed a whole generation of physicists into believing that things have finally been solved." John Bell said angrily: "De Broglie put forward his theory in 1927. At that time, in a way that I now think is humiliating, it was laughed off by the physics community because his arguments were not refuted. , was simply trampled upon." Who would have thought that even a genius like von Neumann would sometimes capsize in the gutter.His proof is not valid!Von Neumann's proof that implicit function theory cannot give a unique and definite solution to observations is based on five assumptions. Among these five assumptions, the first four are not problematic, and the key lies in the fifth one. There.We all know that in quantum mechanics, we cannot obtain a definite result when we observe a definite system. It outputs according to randomness, and the result may be different every time.But we can calculate its expected (average) value according to the formula.If we observe X for a certain state vector Φ, then we can write its collapsed expectation value as <X, Φ>.As we have repeatedly emphasized, quantum theory is linear, it can be superposed.If we observe X and Y twice, their expected values ​​are also linear, that is, there should be a relationship:

<XY, Φ> = <X, Φ> + <Y, Φ> But in the implicit function theory, we think that the system light is not completely described by the state vector Φ, and it also has an invisible hidden function, or a hidden state vector H.After taking H into account, the result of each observation is no longer random, but uniquely determined.Now, von Neumann assumes that for a definite system, even after including the implicit function H, they can be superposed.That is: <XY, Φ, H> = <X, Φ, H> + <Y, Φ, H> There is greatly problem here.For the previous formula, we are talking about the average case.In other words, if there is an implicit function H, then when we only consider Φ, it actually includes all possible distributions of H, and what we get is the average value of H.But when specific H is taken into account, we are not talking about the average case!On the contrary, after considering H, according to the spirit of implicit function theory, there is no expectation value, but a unique definite result every time.The point is that just because averages add up, it doesn't mean that individual cases add up! Let's make an analogy like this: Suppose we throw a dice, and the dice can throw 1-6 points, then we get 3.5 points on average every time we throw a dice.This is an average number that can be superimposed linearly, that is to say, if we throw two dice at the same time, the average points obtained can be regarded as the sum of the average numbers obtained by throwing one dice twice, that is, 3.5+3.5 =7 points.To be more popular, assuming that ABC three people throw the dice at the same time, A throws two dice at a time, and B and C both throw one dice at a time, then in the long-term average situation, the average points obtained by A are equal to the sum of B and C. But von Neumann's hypothesis turned sour.He is in fact assuming that any time we throw two dice at the same time, it must be equal to the sum of the points of two people who each throw a dice!That is to say, as long as three people throw the dice at the same time, no matter which time, the points obtained by A must be equal to B plus C.This is very unlikely, when A rolls 12 points, B and C are likely to only roll 1 point each.While it's true that A is equal to B plus C on average, that doesn't mean it has to be every turn! Von Neumann's proof rested on such a shaky foundation that, of course, it eventually collapsed.The man who ended him was David Bohm, one of the most famous quantum mechanics experts of our time.Born in Pennsylvania, Bohm studied under both Einstein and Oppenheimer (in fact, he was Oppenheimer's last graduate student at Berkeley), and was deeply moved by Einstein's ideals Bohm made him decide to pursue a theory that returns to the strict law of causality and restores the original order of the universe. In 1952, Bohm revived de Broglie's guided wave and successfully created a complete implicit function system.Physicists all over the world were speechless with astonishment: Hasn't von Neumann completely ruled out this possibility?Now someone actually came up with a counterexample! Oddly, it didn't take much mathematical skill or insight to spot von Neumann's error, yet it managed to go unnoticed for 20 years. David Mermin teases that he really doesn't know if any experts or students have actually studied it since it was published.Bell said unceremoniously in the interview: "You can quote me like this: von Neumann's proof is not only wrong, but also stupid!" It seems that we still need to be very careful on the way forward. *********** Gossip after dinner: the fifth postulate Von Neumann fell on his fifth postulate, which seems to be a cycle of heaven and earth. 2000 years ago, the great Euclid also stumbled on his fifth postulate one time. No matter how you describe the greatness of "Elements of Geometry", it will not be too exaggerated. The axiomatic thought and deductive system it established directly gave birth to modern science and provided it with the most powerful force. "Elements of Geometry" builds all the propositional reasoning of geometry on the 5 axioms and 5 postulates given at the beginning, and builds an unattainable building with these most basic bricks and stones. People can accept the 5 axioms given by Euclidean and the first 4 postulates (which he called postulates for geometry).But for the fifth postulate, people feel that there is some dissatisfaction.The original form of this assumption is relatively lengthy, and people often change it into an equivalent expression: "Through a specific point outside the known straight line, it is possible and only possible to draw a straight line parallel to the known straight line".For a long time, people have not doubted the correctness of this postulate, but feel that it seems too complicated, maybe it should not be regarded as an axiom, but can be deduced from other axioms.But 2000 years have passed, and no mathematician has done this (many times someone claims that he has proved it, but their proofs are all wrong)! Euclid himself was obviously disturbed by this postulate. Compared with the other 4 postulates, the fifth postulate is so complicated (the other 4 postulates are: 1, a straight line can be drawn between any two points. 2, a line can be extended Segments make a straight line. 3. The center and radius determine a circle. 4. All right angles are equal).In "Elements", he carefully avoids using this postulate as much as possible, and only uses it when there is no other way, such as when it is necessary to prove that "the sum of the interior angles of any triangle is 180 degrees". The long-term failure makes people wonder, is the fifth postulate unprovable?If we use the method of proof by contradiction and assume that it is not established, then if we derive a contradiction, we can naturally prove the correctness of the fifth postulate itself in turn.But what if the assumption that the fifth postulate is not true results in no contradiction? The Russian mathematician N. Lobatchevsky did exactly that.He assumed that the fifth postulate was not valid, that is, through a point outside the straight line, more than one straight line can be made parallel to the known straight line, and deduced based on this.As a result, he got a series of weird results, but they are a self-contained system, they have no contradictions, and are logically self-consistent!A geometry other than Euclidean - non-Euclidean geometry was born! Starting from other assumptions than the fifth postulate, we can get some theorems that are slightly different from Euclid's original version.For example, "the sum of the interior angles of a triangle is equal to 180 degrees" is deduced from the fifth postulate. If more than one parallel line can be drawn after a point, then the sum of the interior angles of a triangle is less than 180 degrees.On the contrary, if the parallel line of the known straight line cannot be made after a point, the result is that the sum of the interior angles of the triangle is greater than 180 degrees.For the latter it is easy to imagine a sphere, where any lines that appear to be parallel must eventually meet.For example, at the Earth's equator all longitude lines appear to be parallel to each other, but they all eventually meet at the poles.If you draw a triangle on the surface of the earth, the sum of its interior angles will exceed 180 degrees. Of course, you have to draw it large enough to measure it.It is said that Gauss once used three peaks as the three vertices of a triangle to measure the sum of their interior angles, but he did not seem to find anything, but if he made such a measurement among galaxies, the result would be obvious: the mass of the galaxies caused The apparent curvature of space. Lobachevsky assumed that after a point, more than one straight line can be made parallel to the known straight line. Another mathematician, Riemann, assumed that such parallel lines could not be made, and created Riemannian non-Euclidean geometry.He extended the situation to n-dimensional, and completely laid the foundation of non-Euclidean geometry.More importantly, his system was applied to physics, and eventually gave birth to the most outstanding scientific giant of the 20th century-general relativity. Fives Bohm's hidden variable theory is an enhanced version of de Broglie's guided wave, but he replaced the so-called "guided wave" with the concept of "quantum potential".In his description, an electron or photon is always a real particle, whether we observe it or not, it has a definite position and momentum.However, in addition to some usual properties, such as electromagnetic potential, an electron also has a so-called "quantum potential".This is actually something similar to a wave, which develops according to the Schrödinger equation and spreads around the electrons.However, the effect of the quantum potential has nothing to do with its strength, but only its shape, which allows it to extend to the end of the universe without decay. In Bohm's theory, we have to think of the electron as something that is essentially a classical particle but that emanates from it a potential field that permeates the universe, making it Be aware of your surroundings at all times.When an electron travels toward a double slit, its quantum potential senses the presence of the double slit before it reaches it, directing it to behave in a standard interference pattern.If we try to close a slit, the ubiquitous quantum potential senses this change and guides the electron to change its behavior.In particular, if you try to measure the exact position of an electron, your measuring instrument will first interact with its quantum potential, which will cause the electron itself to undergo subtle changes that are unpredictable because the What's more is some "hidden variables", you can't detect them directly. The mathematical techniques used by Bohm are very superb, and his system has indeed basically achieved everything that traditional quantum mechanics can do!But, to our discomfort, such a theory of hidden variables always seems redundant.Quantum mechanics has been going all the way since the beginning of the century, and masters of physics have created a golden basic mathematical form for it.It's so beautiful and simple, and so useful in practice, that we see no reason to force unwieldy and ugly additional assumptions on it, unless absolutely necessary.Bohm's theory of implicit functions is complicated and unconvincing. He assumes that an electron has a definite trajectory, but stipulates that because of the disturbance relationship of hidden variables, we will never observe such a trajectory!This undoubtedly violates the principle of Occam's razor: existence is absolutely unobservable, what is the difference between this and non-existence?Do we have to give up the beauty, clarity and simplicity of physical principles for the reality of this world?Even Einstein himself would object to this. He has a deeper yearning and nostalgia for the beauty of science than anyone else.In fact, neither Einstein, nor even de Broglie expressed positive support for Bohm's theory during his lifetime. What is even more unforgivable is that after Bohm restored the reality and determinism of the world at all costs, he gave up another equally important thing: Locality.Localization means that within a certain period of time, all causal relationships must be maintained within a specific area, and cannot act and spread instantaneously beyond time and space.To put it simply, it means that there can be no causality of action at a distance, and any information must be sent at the upper limit of the speed of light, which is the spirit of the theory of relativity!But in Bohm, his quantum potential can instantly extend its tentacles to the end of the universe. Once something happens in a certain place, its information will be conveyed to every electron's ear immediately.If Bohm's theory is true, faster-than-light communication will be everywhere in the universe, and Einstein will not tolerate all this! However, Bohm did break through the ice caused by von Neumann's mistakes, at least opening up a difficult path for hidden variables from the thorns.In any case, hidden variable theory is possible in principle after all, then, do we still have at least a glimmer of hope that we can develop a perfect hidden variable theory, so that we can have a definite, real , and a warm world with localization?Isn't such a world Einstein's ultimate dream? On July 28, 1928, more than a year has passed since the composition of the most wonderful chapter of quantum theory - the uncertainty principle.On this day, John Stewart Bell (John Stewart Bell) was born in Belfast, the capital of Northern Ireland.Little Bell showed extraordinary intelligence when he was a child. At the age of 11, he made up his mind to his mother to become a scientist. At the age of 16, because he was not yet old enough to enter university, Bell first worked as an intern in the laboratory of Queen's University Belfast for a year. However, his talent has deeply infected the professors and staff there.A year later, he naturally entered Queen's University to study physics. Although he majored in experimental physics, he also showed extraordinary interest in theoretical physics.In particular, the fledgling quantum theory, its profound philosophical connotation made Bell quite obsessed. When Bell was in college, the construction of the main part of the quantum theory building had already been settled. The basic theoretical framework had been built by Heisenberg and Schrödinger, and Bohr had made the most philosophical interpretation for it. The most exciting years in the history of physics in the 20th century have passed, and it is of course a pity not to be able to participate in them, but perhaps it is because of this that people can calm down a little and not be too excited for the great cause. Involuntarily fell under the almost irresistible personal magic of Niels Bohr.Bell was surprised to find that he did not agree with the orthodox interpretation of quantum theory given by his teachers and textbooks.Heisenberg's Uncertainty Principle - it sounds so subjective it's really unflattering.What Bell wanted was a definite, objective theory of physics, and he described himself as a devoted follower of Einstein. After graduation, Bell first entered the British Atomic Energy Research Institute (AERE) and later transferred to the European Particle Center (CERN).His main work is in the field of accelerators and particle physics, but he still maintains a strong interest in quantum physics, following its development closely in his spare time. Bell was quite excited when Bohm's theory came out in 1952.He was fascinated by the idea of ​​hidden variable theory, believing that it restored realism and determinism, and undoubtedly took the first step towards that ultimate dream.This ultimate dream, which we have been talking about, is to bring the world back to the track of objectivity, independence, elegance and strict adherence to causality.Bell felt that the theory of hidden variables was exactly what Einstein required, and it could complete the completion of quantum mechanics.However, this may be Bell's wishful thinking, because it is extremely ironic that even Einstein himself did not agree with Bohm! In any case, Bell is going to take a closer look at whether there is a practical refutation of de Broglie and Bohm's ideas, that is, whether it is true that, as they claim, we can discard all quantum phenomena. determinism, and use some kind of realism to describe it. In 1963, Bell met Professor York in Geneva, the two had an in-depth discussion on this, and Bell gradually formed his ideas.If our universe is really what Einstein dreamed, what should it be like?To explore this point, we must recall a thought experiment mentioned by Einstein when he debated with Bohr in the past—the EPR paradox. If you have forgotten what EPR is, you can review 8-4 of our history first.What we describe is actually Bohm's simplified version of EPR, but they are essentially the same.Now let us redo the EPR experiment: a parent particle splits into two small particles A and B flying in opposite directions, they have opposite spin directions in theory, but before observation, according to the quantum theory, Their spins are in an indeterminate superposition state, and Einstein insisted that the states of A and B are deterministic from the moment of separation. We use a vector to represent the spin direction, and now A and B stand at the two ends of the distant sky and wait for the arrival of A and B respectively (for example, A is in the direction of Sagittarius, and B is in the direction of Gemini).At a certain critical moment arranged according to the standard time of the universe (for example, at 9 o'clock on August 12, 767 in the cosmic calendar, how does it sound like the legend of the silver hero, haha), the two of A and B's self Rotate to make measurements in the same direction.Then, as we have already discussed, for the overall conservation to be maintained, the two spins must be opposite, in either direction.If A measures A's spin in a certain direction as positive (+), then B's measurement of B's ​​spin in this direction must be negative (-) at the same time! In other words, A and B—no matter how far apart they are—always seem to have agreed that when A is +, B must be -, and their cooperation rate is 100%!In statistics, in a slightly more formal term, the correlation (correlation) of (A+, B-) is 100%, which is 1.We need to be familiar with the concept of correlation, which is a variable that expresses the degree of cooperation. If A and B cooperate every time, for example, when A is +, B is always -, then the correlation will reach the maximum value of 1. Conversely, if B does not cooperate with A every time, and whenever A is + or B must be +, then the correlation rate of (A+, B-) reaches the minimum value of -1.Of course, from another perspective at this time, the correlation of (A+, B+) is 1.If B does not cooperate with A or intends to confront, and its value has nothing to do with A, it seems completely random, then B is not related to A, and the correlation is 0. In EPR, no matter whether the state of two particles is uncertain before observation, the final result is certain: either (A+, B-) or (A-, B+) in the same direction, the correlation it's 1.But this is in the same direction, suppose it is in a different direction?Suppose A measures the spin of A along the x-axis, and B measures B along the y-axis, what will be the correlation rate of the results?A faint sixth sense tells us that the moment of fate is coming. In fact, we live in a 3-dimensional space and can observe in 3 directions. We assume these 3 directions as x, y, and z.They don't necessarily need to be perpendicular to each other, they can be taken arbitrarily.The spin of each particle is nothing more than positive and negative in a specific direction, so there are 8 possibilities in total in 3 directions (think of each direction as a line, then the combined result is nothing more than 8 hexagrams) . xyz - - - - - - - - - - - - There are 8 possibilities for A, so what about A and B in general?Obviously there are also 8 possibilities, because once we observe A, B will be determined.If A is (+, +, -), then because of conservation, B must be (-, -, +).Now let us assume that the quantum theory is wrong, and the observation results of A and B are doomed early when they are separated, and we cannot predict it, but it is just because we do not know how many hidden variables therein are.But it doesn't matter, we assume that this hidden variable is H, which can take values ​​from 1 to 8, corresponding to a possibility of observation.Let us assume that, corresponding to each possibility, the probability of its occurrence is N1, N2... until N8.We now have a summary table of possible observations: Ax Ay Az Bx By Bz probability of occurrence - - - N1 - - - N2 - - - N3 - - - N4 - - - N5 - - - N6 - - - N7 - - - N8 Each line above represents a possible result. For example, the first line means that A observes that the spins of A in the three directions of x, y, and z are all +, while B observes that B is in the three directions The spins of are correspondingly -, and the possibility of this result is N1.Because the observation result is 8 must be one of them, so N1+N2+...+N8=1, everyone can understand this, right? Now let's do a correlation exercise, using a little elementary school math.For the time being, we only look at the x direction. In this direction, what is the correlation of (Ax+, Bx-)?We need to do this: when a record meets one of two conditions: when A is + and B is - in the x direction, or A is not + and B is not - at the same time, if so, it meets our Requirement marks the cooperative attitude towards (Ax+, Bx-), so we add the corresponding probability.On the contrary, if A is + on x and B is also + at the same time, or A is - and B is also -, this is a kind of destruction and resistance to the (Ax+, Bx-) combination, and we must subtract the corresponding probability . It can be seen from the above table that the first four possibilities are that Ax is + and Bx is - at the same time, and the latter four are all that Ax is not + and Bx is not -, so the 8 lines all meet our conditions and are all positive. No.Our result is N1+N2+...+N8=1!Therefore, the correlation of (Ax+, Bx-) is 1, which is not surprising, our table was compiled based on this premise.If we want to calculate the correlation of (Ax+, Bx+), then none of the 8 lines meet the conditions, all of which are negative signs, and our result is -N1-N2-...-N8=-1. Next we have to go a little further, A is + in the x direction, and B is + in the y direction, what is the correlation between these two observations?现在是两个不同的方向，不过计算原则是一样的：要是一个记录符合Ax为＋以及By为＋，或者Ax不为＋以及By也不为＋时，我们就加上相应的概率，反之就减去。让我们仔细地考察上表，最后得到的结果应该是这样的，用Pxy来表示： Pxy＝－N1－N2＋N3＋N4＋N5＋N6－N7－N8 嗯，蛮容易的嘛，我们再来算算Pxz，也就是Ax为＋同时Bz为＋的相关： Pxz＝－N1＋N2－N3＋N4＋N5－N6＋N7－N8 再来，这次是Pzy，也就是Az为＋且By为＋： Pzy＝－N1＋N2＋N3－N4－N5＋N6＋N7－N8 好了，差不多了，现在我们把玩一下我们的计算结果，把Pxz减去Pzy再取绝对值： Pxz－Pzy ＝ －2N3＋2N4＋2N5－2N6 ＝2 N3＋N4－N5－N6 这里需要各位努力一下，超越小学数学的水平，回忆一下初中的知识。关于绝对值，我们有关系式x－y ≤ x ＋ y ，所以套用到上面的式子里，我们有： Pxz－Pzy ＝2 N3＋N4－N5－N6 ≤2（ N3＋N4 ＋ N5＋N6 ） 因为所有的概率都不为负数，所以2（ N3＋N4 ＋ N5＋N6 ）＝2（N3＋N4＋N5＋N6）。最后，我们还记得N1＋N2＋... N8＝1，所以我们可以从上式中凑一个1出来： 2（N3＋N4＋N5＋N6）＝1＋（－N1－N2＋N3＋N4＋N5＋N6－N7－N8） 看看我们前面的计算，后面括号里的一大串不正是Pxy吗？所以我们得到最终的结果： Pxz－Pzy ≤1＋Pxy 恭喜你，你已经证明了这个宇宙中最为神秘和深刻的定理之一。现在放在你眼前的，就是名垂千古的“贝尔不等式”。它被人称为“科学中最深刻的发现”，它即将对我们这个宇宙的终极命运作出最后的判决。 （我们的证明当然是简化了的，隐变量不一定是离散的，而可以定义为区间λ上的一个连续函数。即使如此，只要稍懂一点积分知识也不难推出贝尔不等式来，各位有兴趣的可以动手一试。）