That's not a bad analogy, but you have to be very careful here because no classical analogy can be a perfect fit for entanglement. The wave function is deeply and fundamentally different than our classical reality, and there is no way to reproduce its behavior classically. Among the fundamental differences is the fact that classical information can be copied but quantum states cannot be cloned. This is IMHO the single biggest disconnect between the wave function and classical reality because the nature of our (classical) existence is fundamentally intertwingled with copying (classical) information. It is happening right now even as you read this. Information is being copied out of my brain onto the internets and into your brain. At the same time, all our cells are busily copying the information in our DNA, and so on and so on.
A classical analogy for entanglement: suppose I have two balls in a bag. They are identical in every way, except one is red and the other is blue. I randomly grab one in each hand and show my hands closed. Now the states of the ball are entangled: as soon as you see the color of one ball, that "determines" the color of the other. (Not claiming that this is a perfect analogy, but I don't see where it diverges from how entangled quantum waves would behave.)
> Among the fundamental differences is the fact that classical information can be copied but quantum states cannot be cloned.
The no-cloning theorem says that there exists no universal quantum machine that can perfectly clone an arbitrary quantum state. However, that does not preclude a machine that can imperfectly clone any quantum state, or machines that can perfectly clone some but not all quantum states [1]. (Clearly the information transferred to my brain is not a perfect copy of your brain's state, and your DNA is not perfectly copied every time.)
>They are identical in every way, except one is red and the other is blue. I randomly grab one in each hand and show my hands closed. Now the states of the ball are entangled: as soon as you see the color of one ball, that "determines" the color of the other.
This gets used to explain entanglement but it really has absolutely nothing to do with it. This is nothing that the ancient Greeks wouldn't have known.
Not to pick on you specifically, but do people really think it took a major revolution in physics in order to understand that if there are two balls, one is blue and one is red, then if you see one of the balls is red, you can conclude the other ball is blue?
It's something that I think humans can solve at the age of 3.
The failure in your explanation is right when you state that "one of the balls is red and the other is blue". The entire point of entanglement is that such a statement is not possible, that's a strictly classical interpretation. Rather, both balls are in a superposition of being both red and blue simultaneously, and it is not possible in principle to assign a color to either one of them until the moment a measurement is made.
> This gets used to explain entanglement but it really has absolutely nothing to do with it. This is nothing that the ancient Greeks wouldn't have known.
To be fair, this usually crops up in entanglement discussions to deomonstrate how it can't be used for FTL communication and not to actually explain what entanglement is.
> Rather, both balls are in a superposition of being both red and blue simultaneously, and it is not possible in principle to assign a color to either one of them until the moment a measurement is made.
I don't disagree, and (clearly) I make a measurement when I show you the color of a ball. Before I show you a ball, I would also say that the colors of the balls are in a superposition.
> major revolution in physics in order to understand that if there are two balls, one is blue and one is red, then if you see one of the balls is red, you can conclude the other ball is blue?
Entanglement is really just this simple — entanglement itself is a statement about a wave function, classical or quantum. The major revolution in physics is that transformations of the wave functions do not behave as we would classically expect. Entangled particles are a tool that we can use to measure those transformations (and get surprising results).
Entanglement is not a property about wave functions and really has nothing to do with waves. It's a logical consequence of the uncertainty principle and was ironically deduced by Einstein, Rosen, and Podolsky (EPR Paradox) as a way to argue that quantum mechanics is an incomplete description of physical reality. Being that it's strictly a consequence of the uncertainty principle, it applies equally well to non-wave function formulations of quantum mechanics such as the matrix formulation which does not use a wave function.
Entanglement is precisely the principle that a physical system can exist such that no part of the system can be described without describing the rest of the system as a whole. Einstein argued that this made quantum mechanics incomplete, the idea that somehow two properties of a physical system separated potentially by light years could not be decomposed into two physical systems that behaved independently of one another violated basic notions of local realism.
The issue is that as soon as you stated that one ball is red you have made a statement about some property of the physical system that is independent of the rest of the system. That is fundamentally what entanglement states you can not do. All you can state is that there are two balls that are in a superposition of being red and blue and there is no way to describe one ball as red and the other as blue, they are both red and blue simultaneously.
That is what entanglement is and that is the new principle that was neither known to the ancient Greeks or something that a 3 year old could figure out. Not the idea that if there are two balls and one ball is red and the other is blue, then if you see the red ball you know that the other ball is blue. Nothing about that ever baffled any physicist.
> Entanglement is not a property about wave functions and really has nothing to do with waves. It's a logical consequence of the uncertainty principle...
I don't follow, and I can't find anything online that makes this claim. Could you explain more?
Maybe we disagree about the definition of entanglement. I'll take one from Griffith's Introduction to Quantum Mechanics. On page 422, Griffith writes [1]:
> An entangled state [is] a two-particle state that cannot be expressed as the product of two one-particle states....
(There is no mention of uncertainty in this section either.) Here I read "state" to mean "wave function" which implies that entanglement is a statement about a wave function, as I earlier claimed. "Cannot be expressed as a product" means not independent, just like the balls in my analogy (or electrons from neutral pion decay).
When I say "see the color of one ball," I am collapsing the wave function of the balls by making an observation (in the Copenhagen interpretation). This is analogous to measuring an electron's spin. If you replace "ball" with "electron," "bag" with "decay of a neutral pion", "red/blue" with "spin up/down," and "see the color of one ball" with "measure the spin of one electron," that's a completely valid statement in QM.
While I believe that entanglement is genuinely something new and interesting, your explanation of it simply feels like a semantic difference. There is no way in which the universe you describe would be different from a classical universe, at least up to the limits of your description. I'm simply "not allowed" to say that one of the balls is red and the other is blue, before I've looked? It's just, what, against the law to say that? There must be more to it than that.
There has to be some observation that would be different in a universe with entanglement than in a universe without entanglement, and you haven't described what that difference is. There must be one out there, though -- it's just not clear to me what it is. Does it have to do with the fact that the fastest I can spread the message "I just looked at ball A and it's red!" is the speed of light, and ball B could be very very far away? But I thought entanglement doesn't actually allow FTL communication?
Isn't this distinction exactly what the article is about? By saying ahead of time, "one ball is red, the other is blue", you're describing a hidden-variables theory of entanglement. It may be unknowable (before measurement) which color the ball in your left hand is, but it has a color.
But Bell's theorem provides a very measureable counterexample to this type of explanation of entanglement. Sure, in the article they talk about electron spins instead of ball colors, but the analogy is that there isn't a well defined "color of the ball" before it's measured.
Of course, the analogy breaks down a bit: electron spin can be measured in multiple axes with somewhat complicated interactions.
> By saying ahead of time, "one ball is red, the other is blue", you're describing a hidden-variables theory of entanglement.
No, consider the case of neutral pion decay, which emits one spin up electron and one spin down electron. We can clearly say ahead of time one electron will be spin up, and the other will be spin down. But there is no hidden variable that determines which.
If there were a hidden variable, then knowledge of that hidden variable would let you predict which electron is spin up (which ball was red). In the macroscopic world, the hidden variable might be the state of my brain when it chose which hand to grab which ball. But if you replaced me with a robot, and that robot used the measurement of a quantum event (such as an electron's spin) to determine which ball to choose, then there is no hidden variable.
> No, consider the case of neutral pion decay, which emits one spin up electron and one spin down electron.
No, it does emit two electrons with total spin zero which is not the same thing.
> We can clearly say ahead of time one electron will be spin up, and the other will be spin down.
Let’s imagine that one was really up and the other was down. But you decide to measure instead the spins along a perpendicular axis. You would expect to find no correlation between them.
However, what you actually see is that if you measure both spins along any (common) axis they will point in opposite directions.
It doesn’t make any sense to say that before any measurement one was up and the other down. The red and blue balls analogy is very misleading and has nothing to do with entanglement.
> The red and blue balls analogy is very misleading and has nothing to do with entanglement.
this is exactly why classical analogies should not be used to describe quantum entanglement - it gives the layperson a wrong impression. Those analogies makes it easy for the layperson to imagine the hidden variable hypothesis, which is proven to be wrong.
OPs explanation is that entanglement is when there is a red ball and a blue ball and when you know which ball is red, you determine that the other ball must be blue.
My explanation is that entanglement is when there is no red ball or blue ball, there are simply two balls and the color of both balls is both red and blue simultaneously. It's not simply that one ball is red, the other is blue, but we don't know which one is which until we measure them. It's that fundamentally there is no red ball and blue ball, there are just two balls whose colors are in a superposition of red and blue.
I will try to come up with an observable difference but it's hard to do so with colors because the typical examples used for entanglement involve properties that can cancel one another out, so that two entangled particles exhibiting a superposition of two properties will, after many trials, end up forming some kind of destructive or constructive interference that would not be possible if those two particles were in a definite state.
Bell's experiment itself is readily understandable to most laypeople - and comparing the outcome to what you'd expect with e.g. hidden variables is really the easiest way to see why the red/blue explanation misses the point IMO.
Ah, but that’s the tough part - there IS a measurable difference in behavior of the universe between these two examples! (albeit hard to experimentally prove exists, but it has been!)
They really are in a superposition, not just ‘not known’ until one is measured.
Just like light was proven to (truly, actually) be both a light and a wave through the double slit experiments. It doesn’t feel right, but it is - and that is where the progress is made, and why the pushback on some examples. It hides the actual truth behind a misleading, but easy to understand example, that teaches people the opposite of what is really going on.
It could also be that we simply don’t understand something about light phase, and that’s causing us to get confused about superpositions. After all, the experiments aren’t on single photons, they are on beams of photons.
Not sure if we're confusing threads here - double slit experiments have been run on single photons and the results are pretty conclusive. Even a single photon is a wave that interferes with itself.
I would expect similar here. Intuition is terrible at understanding what is going on at the atomic and smaller level, or anywhere relativistic anything is happening.
Not confusing threads; the double slit experiment is often given as evidence of superposition. My attempts at replicating the experiment myself have been foiled because it inevitably goes to phase calculations on lasers, which I don't have any idea how to do. I keep looking for a way to do this famous and supposedly simple experiment, but haven't found a way yet. In any case, when I go deep on what is there (as a layman) it inevitably seems to result in phase measurements as the smoking gun proving superpositions exist at all.
I think you're having a pedantic moment. Nobody claimed that the red/blue ball example was some big unsolved mystery. It's merely to give people a taste of entanglement in a way that your average person can understand.
Isn't it true that if you entangle two particles, separate them, then measure one it'll tell you something about the other particle? That's all the example is trying to communicate.
>Isn't it true that if you entangle two particles, separate them, then measure one it'll tell you something about the other particle?
Yes that's true, but that's also true of things that aren't entangled. I assure you if I went to Socrates, showed him a red ball and a blue ball, put them in a bag, and took out a ball at random that happened to be red, Socrates would have no problem realizing that the other ball must be blue. I am sure if I went to my 4 year old daughter, she'd figure it out as well because nothing about quantum mechanics or entanglement would be needed to understand this.
What entanglement tells us is that if two balls had their colors entangled, then both balls are both red and blue at the same time and it's simply not possible to reason about one ball being blue and one ball being red while they are entangled. They are in a superposition of both colors and remain so until a measurement is performed.
Once the measurement is performed, they are no longer entangled and only at that point can you call one ball red and the other blue.
I think what OP means is, spookiness comes out of the fact that one particle that can be separated by huge distance from another particle, and both particles being in superposition of states, observing one particle can affect the state of another.
It is not about state, that you do not know, but state that is not yet there.
When one particle's state decomposes from superposition of states to a single state, given the assumption of quantum theory, it also affects the state of particle that is physically separated from the particle. That is the spookiness.
If we assume that the quantum particles are always in superposition of states, the question is how can one particle's observation can affect another particle at distance.
If you take out, indeterminate state assumption, then it is indeed missing the point of 'spooky action at distance'.
Yes but I think what you both are missing is that this example is meant for laypeople. Nobody has ever claimed that this is literally entanglement and I can say from first-hand experience that it's useful to bridge the gap to actually understanding entanglement.
Well, I am layman regarding in general, particularly physics. I get what you are saying, but the analogy lose the point of what makes entanglement nonsensical and spooky for anybody, layman or not.
As I said before, if they had an analogy of balls which does not have a color and when you see one ball and it gets color and the other ball which was in contact with it become colored magically too, it would be fine. I am ranting and I am sure educators can come up with better analogy.
The point is, spookiness is important for understanding the significance of why this is big deal at all and some people think that should not get lost in translation.
> I make a measurement when I show you the color of a ball
You “make a measurement” well before that, when you say that you have a red ball and a blue ball.
The point of entanglement is that until you make a measurement they don’t have a color. You could measure something else than color and you would also find a correlation.
But if they have a defined color the entanglement is broken. Sure, one is red and the other is blue. But if you measure anything else (a non-commuting observable, that is) there will be no correlatiom.
And not knowing which one is what (already defined) color is not a superposition. It’s just a mixture.
so true. I have a similar beef about the popular explanation for uncertainty principle: "well you see the light hits the particle very hard so we know where it was but we don't know where its gone now". urgh.
As someone who doesn't know any better than the explanation you have a beef with, I would love if you could explain in layman's terms why it's wrong and what a more accurate understanding is. I always thought that analogy was exactly how it worked, but it seems I have been misled unawares.
It presupposes that position and momentum have definite states that are just uncertain to us, while in most interpretations of quantum mechanics (e.g. Copenhagen and many worlds) the particle exists as a wavefunction, lacking a specific position or momentum, but instead existing as a probability density function in this space.
The uncertainty principle here then relates to how much this probability density function 'peaks' in position space or momentum space. A higher peak in one space results in a wider spread in the other. This is because position and momentum are Fourier transforms of each other.
So is it true that there are multiple properties of a particle---such as location, position, and maybe its spin---that are all described as wave functions, and therefore they can all be entangled? Can anything that is described by a wave function be entangled?
Since position and momentum (assuming that's what you meant, since you said location and position which are synonyms) have this sort of dual relationship, I don't think it makes sense to talk about entanglement with respect to them - they intrinsically have to be related to each other, and the position state (i.e., function) a particle is in fully determines its momentum state.
But it is possible to imagine usually unrelated properties of a particle being entangled, e.g. a two-peaked position function, spin up if it's over here and spin down if it's over there. So that's possible. Usually when discussing entanglement, though, we're talking about 'distinct'* particles. Electron A's spin entangled with electron B's spin. Not that it has to be spin, of course. But that's a common case because of how naturally this sort of entanglement occurs, for example, in atoms where electrons have to form spin pairs.
* This is complicated by QFT where particles are not exactly distinct, but exist as excitations in a particle field. E.g. there aren't two electrons but the electron field is excited by two quanta. At least, that's my understanding; I never went to grad school for physics, so I'm limited to undergraduate knowledge and some extracurricular reading.
> Since position and momentum (assuming that's what you meant, since you said location and position which are synonyms)
Yep, sorry, artifact of the editing process.
> they intrinsically have to be related to each other, and the position state (i.e., function) a particle is in fully determines its momentum state.
Sure, but the momentum doesn't determine the position (due to the constant of integration) so you can have two particles with the same momentum functions and different locations, and that leads to my next question...
> Usually when discussing entanglement, though, we're talking about 'distinct'* particles.
That's what I actually meant to ask but didn't phrase clearly: since position and momentum are described by wave functions, can you entangle the positions of two particles? or entangle their momentum?
> Sure, but the momentum doesn't determine the position (due to the constant of integration) so you can have two particles with the same momentum functions and different locations, and that leads to my next question...
There's no constant of integration since the integral will be over all of space (or momentum space).
> That's what I actually meant to ask but didn't phrase clearly: since position and momentum are described by wave functions, can you entangle the positions of two particles? or entangle their momentum?
No, I'm sorry, I'm not going to pull out heaps of regurgitated quantum information to back this up but that's straight-up wrong.
The red ball and the blue ball exist as physical objects, it is us, the observers, who are unaware of whether they are red or blue at either position. There's no superposition here. They are red, or blue, assigned randomly. Not both, not none. These are facts - properties - about the balls that are real, that exist, but we simply don't have that information at that point. It is meaningless that there is no observer that can 'see through' our hands to know which is correct.
Sorry, this is just wrong. Bell‘s inequality and the very related Bell-Kochen-Specker theorem [1] state that local hidden variables (one ball is blue, one is red, but we just don’t know it) are not consistent with QM.
The problem with your classical analogy for entanglement is that it doesn't match the data. Or rather, it only matches the data for quantum properties that are similarly blue or red.
The non-classical properties of entanglement start appearing once you start measuring combinations of the redness and blueness of those balls.
Let's say that instead of looking at the balls, you pass them through some machine that will let a red ball pass through with some probability P that you control; if the ball is blue, the machine will let it pass with probability 1-P. Let's say further that you have three such machines. You set the first machine to P=1. You pass each ball falling from this machine through a second machine, which has P = 0. You will never see a ball pass through to the end - if it were red, it would pass the first machine, but not the second; if it were blue, it would not pass the first machine at all.
But, let's say you now put a third machine between the other two, and you set P = 0.5. With classical balls, nothing changes - a blue ball doesn't make it past the first machine, while a red ball goes through the first, may or may not pass the second, and never makes it through the third regardless.
However, a quantum ball actually has a chance to pass through the 3 machines if you set it up this way. In fact, that chance is pretty large - more than half of the balls will start passing once you add the middle filter machine.
Still, this is easy to explain if we assume that the middle machine actually paints the ball instead of just detecting its color. This is where the entanglement experiment comes in: if you pass the pair of balls through the three machines, with ball 1 passing through machines P=1 and P=0.5, and ball 2 passing through P=1, you will find that sometimes both balls make it through, even though both balls can't be red at the same time, and they can't communicate about passing through the P=0.5 machine (you can repeat the experiment with the balls being taken arbitrarily far away before passing through the filters).
This is a great thought experiment, thank you. I'm not totally clear how the machines could work without actually taking a measurement, though. It sounds like you're saying the 2nd machine (P = 0.5) takes measurements (and therefore "paints" the balls), but the other two don't?
I've heard of the apocryphal "half-silvered mirror", but I don't get why reflection isn't an observation/interaction there either.
> I don't get why reflection isn't an observation/interaction there either.
I know this comment is going to get lost in the noise, but that is a really excellent point, one of the best that has been raised here so far. This is a point that is often glossed over, but it is actually really important, and quite challenging to explain without getting deep into the weeds. The answer is that passing through a half-silvered mirror is an observation/interaction, but it is special because it can be practically reversed by using additional mirrors so that you can get back to a state where you can no longer tell what the outcome of the "measurement" was. All measurements are reversible in principle, but some are irreversible in practice because the number of things you'd have to reverse is just too large. And in particular, by the time a measurement has affected the state of any macroscopic system (like a ball) it is absolutely impossible to reverse in practice, though not in principle. This process of becoming irreversible-in-practice is called "decoherence".
I agree with this. But it's worth pointing out that lots of good physicists don't. It's a statement of the Everett/relative state/many worlds interpretation, which is simply too weird for many people to accept. That's why there are about a dozen other interpretations of quantum measurement theory, which are all weird in other ways that I can't accept.
The weird part is precisely this: "you can get back to a state where you can no longer tell what the outcome of the 'measurement' was." In other words, at lunchtime you believed that a horizontally polarised photon hit your nose at 10am in morning, and you were right. Now it's dinner time, you don't believe that, and you would be wrong if you did. If the Everett interpretation doesn't pose a massive challenge to your ideas about reality and human identity, you haven't understood it. There are physics professors who picture photons choosing which way to go at a beam splitter, then transmitting the news backwards in time, because that seems more plausible to them.
Of course, interpretations are not science. Everyone agrees how an experiment would go: any attempt to reverse the interaction of the photon with your nose and brain would fail, because thermodynamics. From a purely scientific viewpoint, it simply doesn't matter how many other yous are superposed in parallel universes, because their existence or lack of it has no consequences that (any of?) you can observe. But scientists are as fascinated by this as everyone else is.
> It's a statement of the Everett/relative state/many worlds interpretation
No, it isn't. I've said nothing about many-worlds, only reversibility. And on that point everyone agrees.
> Now it's dinner time, you don't believe that
You really need to read the link above. It goes into all that in great detail. But the TL;DR here is that if it's dinner time, you haven't actually reversed the measurement, notwithstanding your current mental state with respect to the photon.
I don't like claims to authority, but maybe you should read the papers I've published about Bell inequalities too? :-p
It sounds like you're saying that, in principle, physical processes are all reversible. (Although that is often thermodynamically impossible in practice.) You're also saying that it's impossible in principle for someone to learn the result of a measurement in the morning, then unlearn it when the measurement is reversed during the afternoon. I don't see how there could be a self-consistent interpretation of quantum measurement where both those things are true.
How am I supposed to do that? You haven't provided and references and your profile is empty.
> in principle, physical processes are all reversible
Correct. This is a straightforward mathematical property of the Schroedinger equation.
> it's impossible in principle for someone to learn the result of a measurement in the morning, then unlearn it when the measurement is reversed during the afternoon
That's right. But that's not because it's impossible to reverse the measurement. It's because when you reverse a measurement you don't just "unlearn" the result.
That's right. That's why when this experiment is actually done, the mirror is typically rigidly mounted to an optical bench, which is sitting on the surface of a planet. If the mirror were freely floating in zero G, the outcome would be different. It is a worthwhile exercise to calculate how small the mass of the mirror would have to be before you would actually notice a difference in the results.
> I'm not totally clear how the machines could work without actually taking a measurement, though.
Here you're hitting on the heart of the Measurement Problem. In QM as it is understood today, unlike classical mechanics, there are two fundamentally different kinds of interactions between objects: quantum interactions and measurement. Quantum interactions are linear changes to the wave function, while measurements perform a non-linear update to the wave function (it becomes one for the measured value and 0 everywhere else).
Unfortunately, we do not have any theory so far that explains what is the difference between a quantum interaction and a measurement. The experiment I described works with 'machines' that interact quantically with the 'balls', but does not reproduce if the machines measure the state of the balls.
I will note that in the Many Worlds Interpretation, the measurement problem is somewhat different - it states that the state of the universe is always described by a wave function, but that parts of the wave which are sufficiently separated can no longer perceive each other somehow, usually called branching. Precisely when, why or how this happens are just as unknown, though decoherence seems to play a role
Valid solutions to the Schrodinger equation give you the wave function amplitudes in multiple places; the particles in these places can interact with each other still, even if they are 'the same particle'.
However, the wave function at different places interacts with the environment and start to shift in phase, eventually becoming unable to interfere with itself - this is called decoherence, and is a valid explanation about why and how we can't observe wave-like behaviors at large scales or in hot systems.
On the other hand, we can only postulate, based on observations, that when a particle interacts with a measurement device, the measurement device will show a single value with a probability determined by the amplitude of the particle's wave function at that point. We can postulate that the wave function collapses, or we can postulate that the device branches out into different devices in different worlds (enough such devices&worlds to achieve the probability distribution through observer selection somehow), or many other ways of formulating the Born rule. But whichever way you put it, this rule must be added to your system to predict experimental results, it does not derive from the Schrodinger equation.
>Valid solutions to the Schrodinger equation give you the wave function amplitudes in multiple places; the particles in these places can interact with each other still, even if they are 'the same particle'.
I suppose it's destructive interference. It's qualitatively interesting, but its observation is complicated by orthogonal states: when you multiply orthogonal states you get zero. If you can thoroughly dismantle the state to observe it, you still can do it only on microscale, then you'll have a problem lifting it to macroscale evading destructive interference while orthogonal states are all over the place. Anyway, Schrodinger equation describes behavior of quantum states with mathematical precision and the math is quite conclusive that a linear equation behaves in a linear way. When you feel intuition doesn't get you much, you can resort to math, that's why math is seen as an indispensable part of science, because intuition isn't guaranteed to work, which is exactly your case.
>it does not derive from the Schrodinger equation
MWI derives it from the Schrodinger equation. Observation is experience of the observer and can be calculated. Unless you assume that the observer is supernatural and is thus unknowable.
> MWI derives it from the Schrodinger equation. Observation is experience of the observer and can be calculated. Unless you assume that the observer is supernatural and is thus unknowable.
This posits the notion of an observer that only observes one outcome, whereas the SE predicts that an observer will observe several different outcomes with different amplitudes. The MWI is postulating that we should only look at each outcome separately.
Furthermore, it is not possible to derive the actual probability value from the wave function amplitude without some additional postulate equivalent to the Born rule, for example that the number of observers that observe one outcome is proportional to the wave function amplitude of that outcome.
The result of calculation of the state of observer is linear evolution: the state of observer splits and entangles with the observed state and each part observes the respective outcome. Ironically Copenhagen gave the same result for Schrodinger's cat experiment: even before measurement it's known what states are in superposition and those states are "dead" and "alive", and it's still known without measurement too.
>that the number of observers that observe one outcome is proportional to the wave function amplitude of that outcome
If you mean the number, the norm of each state of observer that observes the respective outcome can be calculated. The statistics over the outcomes can be calculated too.
Bell's inequality (as you allude to) describes how transformations on quantum wave functions cannot behave classically. But classical wave functions can certainly be entangled as entanglement is a property of a wave function, not transformations on wave functions.
I'm not sure what you mean by classical wave functions - I've only seen the term 'wave function' used for quantum mechanics. Are you referring to classical wave equations? I'm not sure how the concept of entanglement is supposed to apply to classical waves though.
No, you can't, at least not one that behaves like the quantum wave function does. Classical probabilities are real numbers between zero and one. The wave function takes on complex values, which allows you to add two wave-function values with non-zero magnitude and get zero, i.e. produce destructive interference. Classical probabilities can't do that.
So basically a box with two colored balls could be described by a "classical wave functions" (a real-valued wave function), where the values of the two balls are entangled, and then yes, your experiment would exactly describe what happens.
But this sheds no light on quantum wave functions and quantum entanglement.
The balls in a bag experiment is exactly the kind that does use hidden variables that are local. No information has to be transmitted in either direction.
Bell showed that the correlation is even greater than you can get using that sort of thinking. Reality is more like this:
Bob and Alice each get a pair of bags, one black and one white. They open one of the bags in which they get either a red ball or a blue ball.
If they both choose the white bags, their balls are different colors. However if either or both of them choose the black bag, the colors of the balls are the same.
If you think about it, there's no way to put the balls in the bags to satisfy these conditions in all cases. This is a simplification of what's going on with Bell's theorem.
Thanks for the example. Could you point me to a resource where it explains why the reality is like that? If that’s an implication of a formula of quantum theory(which the article also mentioned briefly), I would like to learn about it and be able to derive this implication myself.
The analogy you mentioned is exactly the wrong one - it suggest that it’s just a matter of a hidden variable.
A proper (but less elegant) would be: you have two balls with the same color or a pattern.
You take one out. If you check the color first, you will find the other’s color the same, but the pattern sometimes different. If you check the pattern first, you will find the pattern the same, but the color sometimes different.
> it suggest that it’s just a matter of a hidden variable.
I disagree. Suppose that I create a machine that chooses which ball to place in each box. This machine makes the choice based on some measurement of a quantum particle (electron spin). Then the colors of the ball are entangled with the state of the quantum particle, which cannot be described by some local hidden variable.
Only if you can completely isolate the balls so their states don't decohere. That is not practically possible to achieve, particularly since in your scenario you reach into the bag and touch the balls. As soon as you interact with the balls in any way, you become entangled with them and the behavior of the system becomes classical.
No. The only way you can actually observe entanglement is in an isolated entangled system (this is the reason quantum computers are hard to build). It is true that at a philosophical level there is no difference, but from the point of view of physics, which is to say, what is observable, isolation is crucial. Non-isolated systems behave classically, notwithstanding that they are actually quantum systems.
Would you claim that when Einstein developed his theories of relativity, they were invalid (from the point of view of physics) because their consequences were not yet observable? For example, Einstein used thought experiments to develop special relativity in 1905, but since kinematic time dilation was only experimentally confirmed in 1971, his work was not a contribution to physics until then?
The difference is that the limits on observing relativistic effects in 1905 were technological, whereas in QM you cannot observe the effects of entanglement in a non-isolated system even in principle. This is a fundamental constraint imposed by the theory itself. You can't get around it even with arbitrarily advanced technology.
That too could be a valid analogy if "randomly grab one in each hand" isn't actually random, but only appears random. This would be analogous to superdeterminism.
A more faithful analogy: I have two boxes with a small hole and ball inside.
If I look through the holes, I will see that one ball is blue and the other is red.
If I touch them through the holes, I will feel that one ball is hot and the other is cold.
However, I cannot look and touch at the same time. And once I check the color or the temperature of one of the balls (so I know it for both) the link is broken.
If I look first, I’l see that one is red and the other is blue. But if now I touch them, each one will by hot or cold with 50/50 probability. Finding the temperature of one doesn’t tell me anything bout the temperature of the other. And when I touch a ball I don’t know its color anymore: if I look at it again it could be red or blue with 50/50 probability.
Reviewing Bell's theorem - described in this article - has resulted in experimental evidence that all classic analogies in the style of "some state was embedded in each particle at the moment of entanglement and the measurement just revealed something about what was in that single particle locally at that time" can not be true.
Bell's theorem describes the highest possible upper bound of correlations for spin measurements along different axis if it was as you say. But it turns out that in practice they are more correlated than what would be possible according to Bell's theorem, ergo, that analogy (which, in general, is plausible and reasonable) is not compatible with the physical reality we live in.
The classical analogy don't model superposition, that's what violates Bell's inequalities, but it illustrates the correlation aspect of entanglement well.
As others here have already noted, this analogy is wrong. But I think people here haven't given a convincing example of a system that behaves differently due to entanglement than it would if the behaviour were simply conditionally random (the ball example behaves identically if the balls are entangled or otherwise just classically random).
The issue in finding a good example is that the effect of entanglement is rather subtle and hard to interpret intuitively.
Here is an example that may make it more obvious:
There exists a game that can be played cooperatively between two players that share two random bits. It is possible to win this game only 75% of the time if the bits are not entangled. If the bits are entangled there is a strategy for winning the game about 85% of the time.
The details of the game and a good explanation can be found here:
https://www.scottaaronson.com/blog/?p=2464
Basically, there is a game that involves sharing two bits, if they are entangled, it can be won 85% of the time. If they are not entangled but otherwise random (like the red and blue ball example), it can be won only 75% of the time.
If you wanted to make this quantum, one is red and the other is blue only if you're looking for a red ball when you stick your hand in the bag. If you're looking for a purple ball when you stick your hand in the bag, then one is purple and the other is green.
agree. The issue that is confusing in explanations of entanglement is the use of a statistical measurement to a model of an individual event. Entanglement is a state made after the observations of many events. Before there is a red ball and a blue ball, there is a precursor, purple ball. That purple ball splits and two balls wrapped in paper are created. They can travel great distances over long times to separate locations, A and B. When some humans start to unwrap them, they are amazed at the correlation-one blue is always matched by one red. So until the balls are unwrapped, how are the humans to describe them? They use the word superposition-unfortunately, many interpret that as a SIMULTANEOUS existence of both states in each ball. Whereas in reality there are many events - the humans at location A see both colors as do the humans at location B. That does not mean that individual particles assume both states. A similar confounding statistic was taught in grade school. The average family has 2.5 children. Yet half a live child was never born.
But aren’t these “informations” just representations of (something abstract) reflected in a bunch of quantum states of your neurons? And we humans decide there are homomorphisms between mine and yours and thus they are representing the “same informations”. But really they were fundamentally different. There are no copying. Only some kind of lossy compression mimicking.
At that point you would need to decide what ‘copying’ is, exactly. Making a terrible VHS recording of a TV show would still be considered copying by most, even if none of the relative pixels ever matched.
The difference is that we can agree upon a set of measurements and a procedure for comparison (eg using difference of Gaussians) to determine how much of “copy” the recorded is to the original. We can repeatedly conduct this experiment (copy->measure->compare) and with high confidence we’d obtain a numerical value that can act as a “proof” it is a “copy”.
My argument is that not only that we do not know a way to conduct such category of copy->measure->compare experiments for human subjects, even with advancement in BCI, etc, it is perhaps impossible to conduct such experiment due to some nature of consciousness that we do not yet understand concerning “information”.
I used the word “mimicking” earlier as apparently when concerning “information” with humans, a (somewhat) “conscious” act has to be performed for the whole phenomenology to be interpreted as that of “copying”. We are encoding “information” in an extremely none-“traditional” way as information is studied and made sense of in computer science.
Similarly, the notion of “semantics” opens up two categorically different sets of paths for inquisition in programming language theory vs linguistics. There is something mysterious and trippy about what “meaning” and “information” really are (eg in regards to qualia).
