Algebra is a programming language for geometry.
During my personal musings about the idea of quantum entanglement I have come to the conclusion that it's simply one huge mystification. But it's a mystification laid on very prosaic grounds. Such prosaic that it's too obvious for most people to be spotted, because they tend to expect something more involved.
And the main culprit is (as usual) the very idea of quantum probability, which (as I've said it many times) is not a probability at all!
It was Richard Feynman who smelled out that something is really wrong with that idea of quantum probability, and he described his doubts in his "Lectures on Physics". But, unfortunately, I haven't seen him taking this line of thought any further, closer to the real solution. On the contrary: in other places he seemed to fall back into that vicious circle of point particles and probabilistic quantum theory again :-/
Nevertheless, his doubts were right. The so called "quantum probability" is not a probability at all. True mathematical probabilities add. Directly. But in quantum physics you cannot add the probability wave functions just like that, because quantum probabilities are the wave functions' amplitudes moduli squared (that is, the amplitude of the wave function multiplied with its complex conjugate), so you need to add those amplitudes of wave functions instead. And this is not how probability works in mathematics!
So how can one talk about probability of finding some point particle at some place, if it doesn't even work as a probability from the mathematical point of view?! :-P Since it's the amplitudes what adds, which is exactly the same as in classical wave physics, then why don't we just admit that those are not probabilities, but simply classical wave phenomena? :-P
Many things seem to show that this is really the case. So we only need to stop digging in these waves for some ephemerous point particles which are not there, and never been. How fortunate that physicists haven't tried to apply the probability theory to water waves on the lake's surface, because then they would probably wonder to this day, trying to figure out the probability of falling of some imaginary water droplets into some place, instead of studying the motion of the water surface as a whole.
But let's get back to the quantum entanglement...
Since physicists use the probability theory for quantum physics, they also use so called "quantum state vectors". Such a vector is simply an arrow which points into some direction in space. Next, they associate each of the independent directions in space (dimensions) with one of the degrees of freedom of the quantum system. The more degrees of freedom the system has, the more dimensions you need to express its state by the state vector.
Now, when the scientists are not sure in which of those independent states the quantum system could be, they set its state vector pointing somewhere in between those directions, according to the "probability" of which of these states it is closer, and call it a "superposition of states" (that is, positing one on another). They say that it means that the system is partly in each of those states, at the same time.
This works quite well, and they can analyze it as probability. But tings start to break when each of those independent states is associated with different "particle" of the multi-particle system. Then, when the state vector turns out to point somewhere in between of those states, they call such a strange superposition a "quantum entanglement", because they think that both of these particles, which are usually separate from each other (which, for me, is already a bullshit, because there are no isolated systems, and everything interacts with everything else continuously), suddenly became "entangled" and dependent on each other, and they seem to "share" their state vectors, so that the changes of one of those "particles" seem to influence the other one "instantly".
And this is a huge breach in the other main theory of physics, that is, Special Relativity, which forbids such non-local "spooky actions at a distance" (as Einstein used to call it). But, since they have experiments to support their claims, then they've said: "To the hell with Einstein!" :-P No one could see that maybe it's not the experiments which contradict the Relativity, but just their interpretations of those experiments are wrong. And that there could be different interpretations which agree with Einstein.
Erwin Schrödinger once proposed a thought experiment, which is now famous and highly over-used in mass media, but no one really undestand it properly, as Schrödinger himself described it. He was very unhappy with what his fellow physicists have done with quantum theory, and with the whole idea of mixing probability theory with it, so he invented this experiment to shake his colleagues and let them see that something is wrong with their beliefs.
He knew that they're easy to be mistaken in the microscopic quantum domain of single atoms, because they cannot see what really happens there, and that they tend to use it as an excuse (following Neils Bohr) to throw in stinky ideas, so he brought the problem into our human-scale domain we are familiar with.
He proposed to put a living cat into an opaque box, together with a flask with a poison in it, and a mechanism which will open that flask and kill the cat when it detects a radiation from a decay of some radioactive sample closed in the box with the cat. The radioactive decay is considered a random event, because we can't tell for sure when it will decay. We can only say it with some probability. So, after the box is closed, you cannot be sure whether the cat is dead or alive until you open the box and see for yourself.
By proposing this experiment, he "entangled" the quantum state of those tiny atoms in a radioactive sample, ruled by pure probability (as his colleagues thought), with a human-scale cat, which is not so random at all (as for the cat, at least). And in this way he forced his colleagues to apply their mistaken probabilistic thoughts to the not-so-probabilistic cat, which were supposed to lead to a paradox and tell them they're wrong.
Unfortunately, they misunderstood the experiment too. They use it to this day as an example of their being right. And that's why Schrödinger, when he saw what they have done with his experiment, he said:
I don't like it, and I'm sorry I ever had anything to do with it.
The real point of this thought experiment is this: we know from our everyday experience that cats are never dead and alive at the same time (no matter their 9 lives). They're either dead or alive. So they couldn't be in such a mixed state in the "quantum" experiment either, no matter how strange theoretical trickery you use. And this experiments is not a proof for probabilistic quantum theory, but a proof that the probability applied to quantum physics is a total mistake. It works similar (and, by that, easy to confuse), but it's not identical. I'll try to explain it to you now with more human-scale examples.
So let us now consider the following situation, not so "quantum" at allSuppose I have a plain (non-quantum ) coin. Let's say that I associate the occurence of "heads" with the horizontal axis, and the occurence of "tails" with the vertical axis. These two axes are independent, just as how flipping "heads" or "tails" in my next coin toss are independent. The probability of flipping "heads" is the same as for flipping "tails", that is, 50:50.
Now I toss my coin, let it make some random turns in the air, and I catch it as it falls and smash on my hand with my other hand to cover it. For now, I don't see what side has been flipped, so, for me, its state is undefined. It's still fuzzied over those 50:50. So what happens with my description of this experiment?
Because I don't know what side has been flipped, my state vector for the coin is in a superposition of possible states: it would be at 45 degrees to both axes, half-way between horizontal and vertical axes which represent "heds" and "tails"!
Now, does it mean that my coin lays on my hand both sides up at the same time? Of course not! It's always one of its sides up, never both, because we know from our experience it's physically impossible. The fact that I don't know which one side is up shouldn't influence the reality of the coin. But the theoretical description of this experiment says that. So who's right and who's wrong? Is it Nature, or physicists which use such a lousy description for this phenomenon?
But maybe the problem is not with the model of this phenomenon, but with our interpretation of it? Maybe this doesn't tell us that the coin is in both these states at the same time, but simply that physics cannot answer this question this or that until we support it with more data? It's enough to take off the hand and uncover the coin to figure out that the coin is, say, "heads up". And it's been in that state from some time ago! But only now, when we uncovered it, our model of this phenomenon "collapses" itself into one of those states (of course to the one we know now it's true). Now our state vector lays perfectly along the horizontal axis, meaning "heads". What happened with the vertical component? It zeroed out. After the fact, the probability is always 100% in one state and 0% in others. But in the case of our coin this doesn't amaze us, because we don't consider it "quantum". (Have you noticed too that this word is currently used in a meaning similar to "magical"? :-P ). It's only in the "quantum" realm of atoms where we expect wonders, because we cannot peek into it directly with our naked eyes.
But if, from some strange reasons, we consider the sides of the coin as separate and isolated from each other, this will surely amaze us! Because now tossing "heads" caused the "tails" to loose all its probability, from no apparent reason! :-P In real life we're not amazed by that, because we know and
But when we face the "atomic particles" we stay in awe when we see how atoms influence each other at a distance, without any apparent physical contact. Because this old man Einstein (supposedly) said they cannot. But it's exactly the same situation!: Some fragments of our reality, which we consider separate, really aren't. They're parts of the same vibrating membrane which we call "space" or – using more modern words – "quantum vacuum" :-P (Earlier it was called "Aether", but would a rose smell less sweet if we call it differently? ). And the whole paradox comes from the fact that we forcifully seek for some point particles hanging in there, separate from each other, instead of seeing the continuous vibrating space which forms them. But it's the same as observing the bumps on a bubbling soup and considering them separate from the rest of the soup.
If the coin didn't convince you, we can also proceed with another experiment. Suppose we have two cards: one is red and the other is blue. We tell our friend to put these cards into opaque envelopes and seal them, so that we couldn't tell which card is inside which envelope. Now, we give one of these envelopes to a fellow astronaut to open it exactly after 10 minutes, write down its color, and put it back to the envelope and seal it again. We put him into a rocket and shoot him out as far as we can from out mother Earth. He flies through space, and after 10 minutes he opens the envelope and he sees that the card is, let's say, red. So he writes down his readout and puts the card back into the envelope, and turns back to Earth.
At the same time, here on Earth, we are doing exactly the same: after exactly 10 minutes we open our envelope, write down the color of the card we've found inside, put it back to the envelope and seal it, and then we wait for the astronaut to come back.
When the astronaut has returned, some independent arbiter is appointed to compare our readouts. And it turns out (suprise, surprise!) that the colors of our cards were exactly opposite: our fellow astronaut discovered the red card, and we here on Earth have discovered the blue one. And it all happened exactly at the same moment of time!
But there's more: The astronaut, after opening his own envelope and finding the red card there, immediately know that here on Earth we have discovered the blue card! :-o Is it an instant superluminal communication? Quantum entanglement? Spooky action at a distance? No, it's merely a card trick :-P
I think anyone will agree with me that there's no miraculous faster-than-light communication involved in this experiment, because there was no communication at all! The cards has been "entangled" (that is, correlated, oppositely in this case) with each other already on Earth, and that was the place where the information has been uniquely encoded into the system for ever after. Although the cards are separate entities in themselves and independent from each other, from this very moment their states (colors) are not independent anymore (together with all subsequent events of reading out their colors), because now their probabilities are conditional: They cannot be both red or both blue, nor anything in between. They can only be oppositely correlated: if one is red, the other is blue for sure, and vice versa.
And we don't need any superluminal or instant communication for that, because the information about the color of the card is encoded in it, and travels through space along with the card itself, with slower-than-light speed. But notice (and this is the most important thing in this article!) that, along with this card, there also travels the information about the color of the other card! So that wherever the card travels, the information about both cards comes along with it!
Are you seeing it now? :->
If we were to describe this phenomenon with a quantum model (that is, by using the state vector), then we have two options: either we describe each card separately, or we describe them both together with one state vector.
If we describe them separately, then we have two separate state vectors, and each could be pointed either horizontally (which means "this card is red") or vertically (which means "this card is blue"). This model would be adequate when we have two separate cards and their colors are independent of each other. That is, when each card can be picked independently, and it might happen that both of them are red, or both of them are blue. Or the first one blue and the second red, or the other way around. Four possible combinations allowed.
But when we state a condition that the cards have to be different, then we have only two possibilities: the first one blue and the other one red, or the other way around. So their probabilities are not independent anymore. They're not picked completely at random, so the probability is biased.
In our experiment, after shuffling the cards and putting them into sealed envelopes, we don't know which one contains which card, so their state vectors have to be (both) at 45 degrees angle, half-way between "red" and "blue" directions. But only when we consider them as separate! And this picture will reflect our knowledge about the system, not the system itself: The probability of the first card being red is 50% – the same as for it being blue.
How is this calculated from our model? Here's how: We have our state vector (which is a unit vector) at 45 degrees. Its "shadows" projected on each of the axes have length , which is about 0.707…. This is the "probability amplitude" of our vector for each of the colors. When we square it, we'll get , that is, our 50% probability for each of the colors :->
See? This is not probability theory, but simple trigonometry! This is a good example of how quantum physicists can obfuscate even the simplest things on Earth :-P But don't let them mislead you.
At the moment when the astronaut opens his envelope and see the color of his card as red, his state vector have to be laid fully along the "red" axix, because now we're 100% sure that it's red and not blue. Physicists call this phenomenon the "wavefunction collapse", because they thinkt that the wave function, which up to this moment were in a superposition of states, suddenly "collapsed" to only one state. Instantly (here we go again). And, accordingly, the other vector associated with the other card should "collapse" to its vertical axis (the "blue" one).
If we were to look at these events as separate (which is wrong, since after they've been entangled on Earth they're no longer independent), then indeed it might seem freaky to us. But if we instead describe (correctly) the state of both these cards with just one vector, then all will become clear. But we would need more dimensions for that: four. Because now we have four degrees of freedom (two for each card). (This multi-dimensional approach could be another factor making it difficult for people to see the truth.)
We will have one vector, which is somehow directed in four-dimensional space. But the directions in that space are not independent any more (when entangled): when such a vector rotates more into the state "The astronaut's card is red", then it automatically turns farther from the state "The card on Earth is red", and closer to the state "The card on Earth is blue". It's exactly the same as with the coin: when its vector were nearer to "heads", it became farther from "tails". The superposition of states and quantum entanglement are exactly the same phenomenon!
Also the "collapse" of the "probability" shouldn't amaze us, because the same goes in classical probability theory: For example, chances of winning the lottery are about 1 to 13 983 816 (in Polish "Lotto" lotery, where you have to spot 6 from 49). But this is before the lottery! Because when at the time of the lottery your numbers has been picked, then the probability of this event have just "collapsed" to 1 (100%): heck, it have just happened, so it's a sure thing! :-)
So why in case of coins, or cards, or any other objects in human scale we're not surprised with these results, and in case of atoms or polarization of light we say in awe? Do we tend to feed ourselves with such awe-inspiring fairy tales full of quantum miracles? :-P
There's one more explanation of this (not-so mysterious) quantum entanglement, which involves classical theory of information.
When someone sends us, say, 40 bits (signals, which could be either "on" or "off", "0" or "1", "true" or "false" etc.), then how many bits of information we've got? Is it 40? More? Less? Or maybe 0? :-) Well… it depends :-P Because, in theory of information, an information is only what you don't know yet. If, for exampe, all those bits encode a message which states "2+2=4", then none of these bits doesn't carry any information for us! :-P But if there's encoded a message, for example, "x=137", and we didn't know the value of `x` up to this moment, then all of those bits carry information.
Well, unless we know already that every message contains something in a form "x=...", where the only thing that changes is the number after `=`. Then, first 16 bits encoding the fragment "x=" doesn't carry information for us. Only the bits which follow, which encode, for example, "459". They could send us that message less the beginning 16 bits and we would still know exactly the same.
The number of bits needed to encode some information is called "entropy". The more bits is needed (or, the more complex the information is), the more is its entropy.
So let us now think about how many bits we need to encode the information about our cards.
If it comes to a single card, which could be either red or blue, obviously we need just one bit: we can encode "This card is red" as "0", and "This card is blue" as "1".
We can think that to describe two such cards we would require two such bits: one for each card. And often it will suffice. But not in the case when the cards are "entangled" :-) Because we can use "0" to encode "The card on Earth is red, and the card in space is blue", and "1" to encode "The card on Earth is blue, and the card in space is red".Because if the cards are always correlated oppositely (eg. when one is blue, the other is red, and vice versa), then this time it's enough if we have just one bit!
The correlation makes the entropy to reduce, because it elliminates possibilities. Can we blame this to the quantum miracles too?Or rather we realize that not everything we get is full of information? (I want you to think about it next time you'll be wathing news on TV )
And that not every information needs to be communicated at a distance. When you shuffle the cards on Earth, you're encoding the information in both of them. Each card stores information about its own color, but also about the color of the other card. So wherever the first card travels, the information about the other card travels along with it. By spreading them out, yo don't change the information stored in that system – they're still encoded both with the same single bit. You are just spreading this bit out as a rubber band. You can have two cards, but the information is one. So it doesn't have to be communicated at a distance faster than light, because it's already there, ready to be read out.
I hope you clearly see now how quantum physicists are mistaken, and how they plague other scientists by this mistake by confusing these simple things over the limit of comprehension. But now, when you know the trick, would it still be black magic for you? I hope you won't be overwhelmed now by the revelations like this one: "Uh oh: entangled particles break second law of thermodynamics" (thaks for dr James Neighbors for sending me this link). And that you will know that information is a relative thing, and it not always have to be communicated at a distance (especially "instantly").
But the real deal here is to spot out the real villian here, that is, the idea of quantum probability of finding a point particle somewhere in space, because it is the root of all evil throughout all quantum mechanics, and we will meet him many times yet. So don't let him scare you! :-)
You may also want to take a look at my other article, "Radical introduction to quantum computing", to find out how this idea of entanglement applies to quantum computers and how it could help us understand better how such computers work.
I recommend only the books I have read, and which I've found good and life-changing.