Entanglement Made Simple
quantamagazine.org
quantamagazine.org
Say there are particles a and b that can be in states 1 or 2
entangled: a1 * b2 - a2 * b1, vs not entangled: (a1+a2)*(b1+b2)
So, when you apply a measurement operator on the state of particle a, you know something about particle b in the entangled case.
Schrodinger said: "the best possible knowledge of the whole doesn't necessarily include the best possible knowledge of the parts." An entangled state should be treated as a whole, and not a sum of parts, kind of like how a book is treated as a continuous story, not a sum of letters.
Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being ‘best possibly known,’ i.e., of possessing, each of them, a representative of its own.
The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself.
Attention has recently been called to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary.
It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it.
" Schrödinger, 1935 (o)
where i disagree with schrödinger is thinking there are two systems stead one that simply contains the entanglement.. if that's what he's going to call it..
"
When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own.
I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.
By the interaction the two representatives [the quantum states] have become entangled.
" Schrödinger, 1935 (o)
You do now know something about particle b, but your explanation suggests that information existed all along.
This is why I object to classical analogies using envelopes and cards and pennies that end with "Yeah QM is strange". It's really a fundamental physical/mathematical effect, and it's much less intuitive if you make a mapping to objects in a classical world. I often feel conceptual maps to information worlds (like books and the internet) are more intuitive.
It also leaves you unequipped to reason about quantum computing, by the way.
Also:
> 1. A property that is not measured need not exist.
> 2. Measurement is an active process that alters the system being measured.
These are both misleading enough that they can probably just be called 'wrong' as well. We shouldn't assign mysterious properties to "measurement" as though it's some magical thing that just makes quantum mechanics happen. Just call it what it is: entangling the state of your brain with the results of an experiment. That conveniently also provides for a good jumping off point into Many Worlds / Relative-State Formulation of QM.
Measurement is at its most mysterious in the MW interpretation, because measurements are probabilistic and probabilities are hard to get out of a purely deterministic theory like MW.
This is a particular viewpoint on the role of the Schroedinger equation in theory, incompatible with the Born interpretation of |\psi|^2 which is the orthodox way to use it. It is a different thought scheme that has no support in experiment. No experiment uses Schr. equation to consider how experimenter's brain gets entangled with their measurement apparatus; the ones that use it at all use it to calculate measurable values and probabilities of measurement results.
And, are you saying that your brain isn't entangled with an experimental result after observing it (at the latest)? If so, what is special about brains, that shields them from it?
Regarding the entanglement of brain with experiment, that is totally unfounded extrapolation of applicability of many-particle Schroedinger's equation. It is useful for atoms and molecules, but it is practically intractable for systems of few atoms. For macroscopic objects, like arm indicator of an ammeter, the Born rule assumes that results of measurements are definite, so probabilities can be assigned to them. In the picture where the indicator or brain is just a part of the system that gets entangled, no definite results of measurements are obtained and it makes no sense to talk about their probability in the way all successful applications do.
However, my objection is to the very idea of using quantum-theoretic formalism to describe macroscopic things. Because: even if we imagine we somehow obtained a solution to Schroedinger's equation for this system, we have no need for it. The purpose of \psi function is to get probabilities of system configuration or results of measurement of some quantity. This is useful for comparing the theory of natural phenomena with the reality.
Much less so for describing the system of experiment+experimenter. Would you want to apply the quantum formalism to obtain probabilities of the end states of the joint system, when you can just ask the experimenter on what was the result of measurement and thus obtain definite answer with more value to physics?
> Regarding the entanglement of brain with experiment, that is totally unfounded extrapolation of applicability of many-particle Schroedinger's equation. It is useful for atoms and molecules, but it is practically intractable for systems of few atoms.
It seems like you're saying that a different physics governs reality once the number of particles is large enough. Is your sole objection instead that rigorously describing the state of a brain is computationally intractable? If so, what bearing does that have on this:
> We shouldn't assign mysterious properties to "measurement" as though it's some magical thing that just makes quantum mechanics happen. Just call it what it is: entangling the state of your brain with the results of an experiment.
To answer your question, I don't need formally specify the experimenter in rigorous detail, in order to reason about what is going to happen when I ask the experimenter the result. It's also weird that you mention that I can "just ask the experimenter" when that doesn't really help me make any predictions. (Unless I try to predict what the experimenter will say, at which point we're back to entangled brains again.) What were you getting at with that?
It's all so bizarre that I don't see how anyone can look that deeply into it and not come away thinking our entire existence is one big pointless hoax.
There is a (subtly, and not in a terribly important way) flawed conclusion regarding personal identity toward the end, and parts of it are a bit long-winded (I guess esp if you don't already enjoy reading Yudkowsky as I do), but it is solid.
tl;dr: QM's reputation is more a function of it being poorly taught (cf this article), not really the theory itself - ultimately it makes the world less confusing, as any sound scientific theory should.
- trying to describe quantum mechanics without mathematics (since we have no direct intuition/experience, many analogies do not hold),
- inventing some profound philosophy to describe facts.
For example, my entry on that: https://johncarlosbaez.wordpress.com/2015/03/13/quantum-supe... (not entanglement, but - an electron being in two places at once).
[Full disclaimer: I am an ex theoretical physicist. I don't claim I am sane, though.]
You sound sane :)
Quantum mechanics understanding will probably be a scientific revolution, something comparable to understanding the earth is round or the sun does not circle earth.
There are a few scientist that are trying to break the old model of thinking.
http://www.theepochtimes.com/n3/1714898-breaking-through-con...
The weirdness of QM comes from the following, slightly more complicated, situation.
You have two cards, a red one and a blue one. You also have two envelopes. Without looking at anything, you put one card in each envelope. You label them, and send one each to Alice and Bob.
Now, we know for sure that when Alice opens her envelope and looks at the card, we immediately know which card Bob has. There is no information being exchanged, so Alice knows what card Bob has immediately when she opens her envelope.
The weirdness of QM comes from looking at the intermediate states. i.e. the states before Alice opens her envelope. There are (waving wands) ways to tell what the intermediate states could be, by varying certain (QM) magical properties of the experiment.
The weirdness of QM is this. Every single idea you have about the intermediate state is wrong.
What do I mean by that?
You could say there are "hidden variables". i.e. one envelope "really has" a red card in it, and the other envelope "really has" a blue card in it. This interpretation is wrong.
Every experiment shows that the envelopes are in a "superposition" of states. That is, the envelopes don't contain either a red card or a blue card. They carry... something... that is maybe red and maybe blue.
That seems strange, but OK. What happens when Alice opens her envelope?
When Alice opens her envelope, the card which is "maybe blue, maybe red" suddenly becomes either 100% blue, or 100% red.
The weird part is the next step.
When Alice opens her envelope, Bob's card immediately becomes the opposite color. This happens no matter how far away Bob is from Alice.
So is information traveling faster than the speed of light? No. We knew Alice's card was red or blue, so Bob's card must have been blue or red. Lo and behold, Bob's card turns out to be one of those colors.
And yes, Bob's card really is in an indeterminate state prior to Alice opening her envelope. And once Alice opens her envelope, Bob's card really is 100% either red or blue.
The confusion comes from what is really happening behind the scenes. i.e. What is the real explanation?
The current QM research shows that the answer is "Uh... right... we're not actually sure."
Somehow the two cards are connected (QM people say "entangled"), because when you look at one, the other immediately "decides" (?) what color it is. But there's no information being passed between the cards. There's nothing exchanged between the cards. Yet somehow Bob's card "knows" that Alice has looked at her card.
This is profoundly strange and confusing. No one has any idea what's going on. Physicists have many theories, but it's difficult to either prove or disprove most of them. That takes time, money, expertise, etc.
So we're left with "QM is strange". As Richard Feynman said, "QM is not only stranger than you can imagine, it's stranger than you can possibly imagine."
You've done a lot better in your correction of it, but in doing so you've had to basically render the analogy worthless. There is no analogy for what's happening. It's simpler and clearer to just state the facts.
Don't think Feynman said that. It's kind of the antithesis of 'the pleasure of finding things out'.
It seems to be a quote from an essay by JBS Haldane. 'The Universe is not only queerer than we suppose, but queerer than we can suppose.'
That's not necessarily true, there are perfectly valid deterministic interpretations of quantum mechanics.
I am not sure that this is the correct or best way to describe it. That looking at the cards does not allow exchanging any information between Alice and Bob does not necessarily imply that the process of determining the colors of the cards does not involve any information exchange. It seems almost necessary that an information exchange happens between the two cards, after all they have to acquire different colors and they have no definite color before looking at them. That this process can not be exploited by Alice and Bob to exchange any information is a wholly different issue.
That's how physicists describe it. So it is the best way.
The thing to remember is that information (in physics) means "particles". The screen gives your eyes information in the form of photons.
In the case of QM entanglement, nothing is exchanged. So no information is exchanged.
> It seems almost necessary that an information exchange happens between the two cards, after all they have to acquire different colors and they have no definite color before looking at them.
That's the issue with QM. You would like to think that something happens. But as I said, whatever really happens is different than what you think.
I am not sure that this is correct. I think the usual statement is that it does not allow Alice and Bob to exchange information, not that no information exchange is involved. When Alice looks at her card and sees that it is red something happens, i.e. red as a possible colors for Bob's card is eliminated. The wave function collapses if you want to use that picture. The state of Bob's cards changes instantly from maybe red or maybe blue to definitely blue. One can now argue whether that change should be called information or whether the term information should be reserved for things that can be exploited by Alice and Bob but that does not change the fact that a non-local instantaneous effect changed the state of Bob's card.
The thing to remember is that information (in physics) means "particles".
That is pretty vague and might or might not be true. Do you mean every information exchanges between two spatially separated parties necessarily requires the exchange of (real) particles?
That's the issue with QM. You would like to think that something happens.
Something happens, the wave function changes.
But even disregarding this, there seems to be a need to propagate something through space. The wave function of the entangled pair could be spread out in space and when Alice performs a measurement on her particle you have to notify the wave function at Bob's place that it should also change.
You could also try to imagine that the entire wave function for the two entangled particles is not spread out in space and located next to Alice which makes it easier to imagine how the entire wave function could change at once. But now how does Bob's particle know that the far away wave function changed in order to behave appropriately? We still have to propagate that notification over to Bob. And what if Bob decides to perform the measurement instead of Alice but the wave function is sitting around at Alice's place?
There is some inherent nonlocality in quantum mechanics and the effects of measurements have to propagate through space even when they can not be used to exchange classical information. One idea that I find really interesting is ER = EPR [1] by Susskind. The idea is vaguely that entanglement determines locality, i.e. two things are close together if they are entangled, especially two entangled and spatially separated particles are connected by a wormhole and therefore actually remain close together all the time. This would make spooky actions at a distance a lot less spooky because it would mean that you can never really separate entangled particles. It would probably also mean that the topology of space is actually extremely complex and only superficially looks like Euclidean three dimensional space.
That's the problem with QM. So far as we can tell, nothing propagates through space. Even if something did propagate through space, it does so faster than the speed of light. And we know that information travelling faster than the speed of light is impossible. Because if it was possible, we would have causality violations.
i.e. if faster than light travel was possible, the universe would necessarily look very different than it does.
So we have multiple experiments, all of which are true, and which apparently disagree with each other in certain esoteric ways. Since the universe is consistent, we must conclude that the failure is in our understanding, not in the universe.
The limitation of the number of threads to r^2 might be because they all have to puncture the surface to create the appearance of a volume in the first place. The theory of Loop Quantum Gravity can even generate a discrete spectrum of area values in response to various meshed (graph) 'edges' puncturing a topological surface. Geometry is quantized.
QM is obviously an incomplete theory, and there is no point repeating the philosophical arguments of the 20th century, ad nauseam. In the 21st century, physicists are beginning to put together new theories of quanta, space, time and gravity that will be a lot more coherent and satisfying, even though we will lose the cherished notion of locality.
The Dining Philosophers should stop arguing over the forks and start to discuss the spaghetti!
P.S. Douglas Adams was right, it's The Great Spaghetti Monster after all.
So if it we're possible for entangled particles to have one wave function, collapsing it makes the observed result the one that applies all the time, back to when it got entangled in a pair. I don't see this a communicating anything over distance. It's more like 'communicating' backward in time, but I don't really think that's communicating any more than opening the box communicates back in time to effect an outcome.
<sigh> So you're smarter than every physicist alive.
Well... no.
> One can now argue whether that change should be called information ...
No, you can't. You're using colloquial English to reason about physics. This is wrong. Physicists have a very specific definition for "information" (as I already said, and you ignored). And in this case, no information is exchanged.
> That is pretty vague and might or might not be true.
As the published nuclear physicist in the argument... yes, yes, it's true.
> Do you mean every information exchanges between two spatially separated parties necessarily requires the exchange of (real) particles?
Yes.
Well... no.
The usual statement is that entanglement can not be used to exchange information between spatially separated observers, right? I totally agree with that. But what about the wave function? If the wave function is spatially extended, then the change of the wave function has to propagate through space if one assumes that the change is caused by a local measurement. And the wave function contains the information that fully describes the state of the system but one can not learn the wave function in general because some operators are not commutative. Nonetheless a change of the wave function means a change of the information describing the system whether an observer can detect that change or not. That are of course two different things, the information describing the state of the system and the information an observer can learn about the state of the system, but I see no conflict here.
No, you can't. You're using colloquial English to reason about physics. This is wrong. Physicists have a very specific definition for "information" (as I already said, and you ignored). And in this case, no information is exchanged.
If I measure a spin without prior knowledge of its state, I obtain one bit of information, I guess that is the sense in which you want to understand information, right? If I repeat the experiment with identical preparation over and over again, then I can learn the wave function, maybe 30 % up, 70 % down. If Alice and Bob do this with their entangled cards, then Bob can decide between the case that Alice saw a red card in which case he will always see his card blue and the case that Alice did not look at her card in which case he will see his card as randomly red or blue. When Alice looks at her card she must of course perform a post selection to get rid of the pairs where she saw her card as blue.
I hope you notice that I am not trying to argue that you are wrong, I am just trying to understand what you say, especially why one would consider the result of a measurement as information but not consider the wave function as information.
Looking at the wave function changes nothing. It's all the same physics, and all the same concepts. You can't get different behavior from the physical system by "looking at the wave function" instead of looking at the particles.
> If the wave function is spatially extended, then the change of the wave function has to propagate through space
... via real particles. Which can't go faster than the speed of light.
> And the wave function contains the information that fully describes the state of the system but one can not learn the wave function in general because some operators are not commutative.
That... doesn't make any sense from a physics point of view.
You're trying to understand which is good. But you're mangling the concepts.
> If I measure a spin without prior knowledge of its state, I obtain one bit of information, I guess that is the sense in which you want to understand information, right? If I repeat the experiment with identical preparation over and over again, then I can learn the wave function, maybe 30 % up, 70 % down
No. You're learning the probability distribution of the wave function. i.e the statistical distribution of the wave function, taken over many measurements.
> If Alice and Bob do this with their entangled cards,
... they learn that 50% of the cards are blue, and 50% of the cards are red. Which they already knew.
> ... Bob can decide between the case that Alice saw a red card in which case he will always see his card blue and the case that Alice did not look at her card in which case he will see his card as randomly red or blue
I'm not even sure what that means. It's based on a misunderstanding of the underlying concepts, so the sentence doesn't really make sense to me.
> I am just trying to understand what you say, especially why one would consider the result of a measurement as information but not consider the wave function as information.
I never said that.
The wave function is a probability distribution.
Information is exchanged via real particles.
You can learn new information through measurements... but not when the underlying measurements are random. i.e. with entangled particles, all you measure is that (in this case) half of the cards are red, and half are blue. Which you already knew.
Since you already knew that half of the cards are red and half are blue, the measurements give you no new information.
The concepts are really quite simple, once you throw away your "common sense" understanding of what is going on.
Alice hands Bob a particle that is either spin up or a superposition of spin up and spin down. Bob can not distinguish those two cases with a measurement. But if Bob would know the wave function he could tell the difference and Bob could learn the wave function be repeating the experiment many times. So there is a difference between looking at the outcome of a measurement and looking at the wave function.
... via real particles. Which can't go faster than the speed of light.
If Alice measures her particle of an entangled pair the wave function of Bobs particle will change without any particle transporting any information from Alice to Bob. Bob is unable to detect that change with a local measurement but the wave function changed and no particle has been exchanged.
That... doesn't make any sense from a physics point of view. You're trying to understand which is good. But you're mangling the concepts.
What is wrong with that? The wave function fully describes the state of a system but you can not learn that state in general because you would have to perform several different measurements and those measurements do not commute in general and change the state to an eigenstate of the measured observable. Only in an experimental setup with repeated measurements on identically prepared systems can you learn the wave function of the system.
No. You're learning the probability distribution of the wave function. i.e the statistical distribution of the wave function, taken over many measurements.
No, I said identically prepared systems, so the wave functions are identical in every run of the experiment. I then obtain a distribution of the eigenstates of the operator by repeating the experiment and that determines the wave function up to the phase.
I'm not even sure what that means. It's based on a misunderstanding of the underlying concepts, so the sentence doesn't really make sense to me.
Alice prepares a set of entangled particle pairs. On two thirds of the sets she measures her particle and sorts them by outcome. Now Alice has three sets, unmeasured, measured up and measured down. Bob can determine which set is which by measuring his particles, he gets 50/50 up and down, 100 % down and 100 % up respectively.
I never said that.
You did or at least I understood it that way. You say that information is what an observer can learn about the state of a system by performing measurements but there is also the information describing the state of the system, the wave function, which is not necessarily accessible to an observer.
The wave function is a probability distribution.
It is not, it is a description of the state of the system. You can obtain a probability distribution of eigenstates by repeatedly performing measurements on identical wave functions and that distribution is determined by the magnitude of the amplitude of the wave function but the wave function itself is not a probability distribution.
You can learn new information through measurements... but not when the underlying measurements are random. i.e. with entangled particles, all you measure is that (in this case) half of the cards are red, and half are blue. Which you already knew.
Since you already knew that half of the cards are red and half are blue, the measurements give you no new information.
I did not dispute that, observers obtain information about the state of a system by performing measurements, but that is not the same as the information describing the state of the system. The later is the truth, the former is what an observer knows about the truth.
You're using colloquial English to reason about physics. This is wrong. Physicists have very specific definitions of the terms they use. Which are sort of similar to the common ones, but differ in key points. Those key points are what you're getting hung up on. And it's difficult to explain the differences without explaining all pf physics.
On top of that, you've disagreed with the common definitions used by physicists "I'm not sure that's correct". Well, it is. If you think that's wrong, you either think you're smarter than all physicists alive, or you're wrong. Pick one.
> ... But if Bob would know the wave function he could tell the difference ...
No. No. A thousand times no.
It just doesn't work like that.
> If Alice measures her particle of an entangled pair the wave function of Bobs particle will change without any particle transporting any information from Alice to Bob. Bob is unable to detect that change with a local measurement but the wave function changed and no particle has been exchanged.
That is a bunch of pseudo-physics words put together in a sentence.
>> You're trying to understand which is good. But you're mangling the concepts. > What is wrong with that?
Everything. If you're not using the correct terms in the correct way, you might as well be putting random words together in a sentence.
>> No. You're learning the probability distribution of the wave function. i.e the statistical distribution of the wave function, taken over many measurements. > No, I said identically prepared systems, so the wave functions are identical in every run of the experiment.
So... you know better than the nuclear physicist.
This should set off alarm bells that you either don't understand the topic, or you don't care to understand it.
> You did or at least I understood it that way. You say that information is what an observer can learn about the state of a system by performing measurements but there is also the information describing the state of the system, the wave function, which is not necessarily accessible to an observer.
<sigh> The observer can do multiple measurements to determine the wave function.
>> The wave function is a probability distribution. > It is not, it is a description of the state of the system.
Again, you're arguing with the nuclear physicist.
The probability distribution is a description of the system.
>> Since you already knew that half of the cards are red and half are blue, the measurements give you no new information. > I did not dispute that, observers obtain information about the state of a system by performing measurements,
Yes.
> but that is not the same as the information describing the state of the system. The later is the truth, the former is what an observer knows about the truth.
<sigh> I'm trying to educate you on physics, and you're arguing pseudo-philosophical metaphysics.
Please stop. Your understanding of the terms is largely wrong. As a result, your arguments are based on falsities, and therefore also wrong.
Please go read a popular book about QM before discussing this with anyone.
And stop arguing with the nuclear physicist. This is one situation where I can appeal to authority without it being a logical fallacy. I've explained repeatedly why you're wrong. You don't seem to understand.
Wave function and probability distribution. You repeatedly insisted on precise language, that is why I objected your statement that the wave function is a probability distribution. And I still do, nuclear physicist or not. A wave function is complex-valued, a probability distribution is usually defined by a real-valued probability density function or a real-valued cumulative distribution function. The state 1/sqrt(2)(|0> + |1>) has probability amplitudes of magnitude 1/sqrt(2) for both states, the probability distribution you obtain when you repeatedly measure particles in that state has probability 1/2 for both states, the square of the magnitude of the amplitude. The wave function certainly defines a probability distribution but it is not itself one.
<sigh> I'm trying to educate you on physics, and you're arguing pseudo-philosophical metaphysics. Please stop. Your understanding of the terms is largely wrong. As a result, your arguments are based on falsities, and therefore also wrong. Please go read a popular book about QM before discussing this with anyone. And stop arguing with the nuclear physicist. This is one situation where I can appeal to authority without it being a logical fallacy. I've explained repeatedly why you're wrong. You don't seem to understand.
I haven't read everything again but most of your responses, especially in your last comment, simply state that I am wrong without pointing out why or providing a supposedly correct point of view. It is of course not you job to teach me physics, but simply stating that I am wrong is only of limited help to overcome misunderstandings. Besides that we are discussing a topic that has not been definitely settled and where there is disagreement even among professional physicists, so maybe we should not be too surprised about a certain amount of disagreement.
I would very much appreciate if we could continue this discussion a bit once I had time for a full reply - and I agree, we have to back up quite a bit, the discussion got fragmented pretty quickly - but I won't blame you if you declare me a lost case.
It isn't. All known underlying models of QM are either wrong, or (as yet) untested.
My understanding of Bell's inequality is that experiments had two electrons traveling in two different directions, through two different measuring devices measured ended up being measure as spin-up or spin-down when measured at different angles. There were certain levels of correlation when the measurements were done at different angles, but those didn't fit when compared to other correlations (IE, you couldn't have 90 and 45 degrees match 85% of the time, 45 and 0 match 85% of the time, and 90 and 0 match only 20% of the time).
Worth noting that this doesn't necessarily mean, as some claim, that "the cards" aren't 100% red or blue at any given moment in time. There are certain valid interpretations that are deterministic (De Broglie-Bohm, for example). I believe Bell was in favor of deterministic interpretations.
>This “spooky action at a distance,” as Einstein called it, might seem to require transmission of information — in this case, information about what measurement was performed — at a rate faster than the speed of light.
>But does it? Until I know the result you obtained, I don’t know what to expect. I gain useful information when I learn the result you’ve measured, not at the moment you measure it. And any message revealing the result you measured must be transmitted in some concrete physical way, slower (presumably) than the speed of light.
The thing is when the results are obtained you can write them down or similar. Unless you are presuming the written results and other records change as you mail them to each other you are left with the faster than light type paradox. I'm with Einstein on that one in that something odd is happening.
My guess is that quantum mechanics is the fundamental nature of the universe and the appearance of space and time are a resulting epiphenomenon. As a result of that the particles appear far apart but are in some ways are still connected.
That's not paradoxical, that's just two objects with the same random shape and the same random color. Quantum entanglement is stranger than that, it can't be explained by hidden correlated information about both objects.
Here's an example of quantum entanglement. Imagine that Alice and Bob live far from each other and can't communicate. There are three possible yes/no questions you can ask them, but each person can only answer one question. These facts are known:
1) If you ask Alice and Bob the same question, they always give opposite answers.
2) If you ask Alice question 1 and ask Bob question 2, the probability that both will say yes is 5%.
3) If you ask Alice question 2 and ask Bob question 3, the probability that both will say yes is 5%.
4) If you ask Alice question 1 and ask Bob question 3, the probability that both will say yes is 20%.
Under classical assumptions, there's no way for Alice and Bob to prepare a strategy beforehand that would lead to such probabilities. The reason is that case 4 is completely covered by cases 2 and 3, but has higher probability than the two of them combined. Here's a proof:
(4)
=> A1Y and B3Y
=> A1Y and B3Y and (A2N or A2Y) // tautology
=> A1Y and B3Y and (B2Y or A2Y) // by (1)
=> (A1Y and B2Y and B3Y) or (A1Y and A2Y and B3Y) // logical operations
=> (A1Y and B2Y) or (A2Y and B3Y) // logical operations
=> (2) or (3)
Before you ask, it's provable that such tricks cannot be used to send information faster than light. Quantum "spooky action" is somehow more powerful than hidden correlations, but less powerful than outright communication. It's pretty subtle and most popular accounts don't get it right.
To be clear: it can't be explained by locally deterministic mechanisms, assuming a nondeterministic universe.
(A && B && C) || (A && D && C) == (A && B) || (D && C)
How do you get from this line:
(A1Y and B2Y and B3Y) or (A1Y and A2Y and B3Y)
(A1Y and B2Y) or (A2Y and B3Y)
Let's write it differently: Why are these terms equivalent?
(u && v && w) || (u && x && w)
(u && v) || (x && w)Now I also get your previous comment. I've read your => as arrows instead of "equal or greater".
Yes it can, as long as that information is non-local. That's where the EPR "paradox" comes in, because Einstein believed non-locality was impossible due to relativity.
For clarity, is this correct (you are simplifying something) or am I reading what you wrote incorrectly?
> Yet using your rules 2 and 3, you could manipulate the odds of Bob's response by asking Alice different questions.
Can you explain in more detail? Assume that the odds of Bob's response are 50/50 at the start.
The following statement is true, correct?
Even though the probability that Alice and Bob's chance of both answering yes varies, the individual chance that either of them answers yes without any other information does not change.
1) If you ask Alice and Bob about the same coin, their answers always agree.
2) If you ask Alice about coin 1 and Bob about coin 2, they are different about 5% of the time.
3) If you ask Alice about coin 2 and Bob about coin 3, they are different about 5% of the time.
4) If you ask Alice about coin 1 and Bob about coin 3, they are different about 20% of the time!
Why is it shocking you ask? Well, if coin 1 is different from coin 3, then at least one of them must be different from coin 2, which you didn't ask about, but could have. So the probability of the last situation can't be larger than the sum of the previous two. But here it is, almost twice larger. Quantum magic :-)
Alain Aspect's comment regarding "if you just want answers you simply compute" in context of "understanding" entanglement is spot on.
> Entanglement arises in situations where we have partial knowledge of the state of two systems.
No, there are no squares and circles, and no cakes. It isn't simpler an there's no insight to be gained by thinking about the cakes and squares and circles.
Much better learning about what is. For example, to get some intuition for the things you can't directly observe try considering what's going on in the "simple" act of water drop forming and detaching.
https://youtu.be/c4MUTij8f6I?t=120
It's waves all the way down.