# Quantum Entanglement - What's the big deal?

Bearing in mind I am a layman - with no background in physics - please could someone explain what the "big deal" is with quantum entanglement?

I used to think I understood it - that 2 particles, say a light-year apart spatially, could affect each other physically, instantly. Here I would understand the "big deal".

On further reading I've come to understand (maybe incorrectly) that the spatially separated particles may not affect each other, but in knowing one's properties you can infer the other's.

If that it the case, I don't see what the big deal is... 2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties...?

What am I missing? Is it that the particles properties are in an "un-set" state, and only when measured do they get set? (i.e. the wave-function collapses). If this is true - why do we think this instead of the more intuitive thought that the properties were set at an earlier time?

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Possible duplicate of physics.stackexchange.com/q/3158/4066 – EnergyNumbers Feb 24 '13 at 18:46
I don't personally think this is a duplicate of that question, although they are, I agree, closely related. This question is asking for more of an intuition behind he strangeness of entanglement which I don't think the other question and its answers fully addresses. – joshphysics Feb 24 '13 at 18:54
Dear @Pete, your basic reasoning is quite correct. Entanglement is nothing else than correlation between two objects ("subsystems") and this correlation is always a consequence of their mutual contact or common origin in the past. Entanglement is the most general type of correlation that may be described using the formalism of QM (none of the properties is determined) but it's still correlation, leads and requires no "action at a distance", and everyone who is seeing something totally mysterious behind the entanglement is overlooking the forest - that it's just correlation - over some trees. – Luboš Motl Feb 24 '13 at 19:02
For a good, yet simplified, explanation of the fundaments of all this, see this Leonard Susskind lectures. They are intended for people with only some basic understanding of differential calculus. – Eduardo Guerras Valera Feb 24 '13 at 21:09
@LubošMotl, my jaw dropped at your answer because I know you are very aware of Bell's inequality and the math behind it. Sure, it's correlation, but not correlation that can be done with hidden variables (no "action at a distance"). Am I totally missing your intent somehow? How can you reconcile what you just said with the experimental evidence (and actual devices based on) Bell's inequality? – Terry Bollinger Feb 25 '13 at 3:24

I understand your confusion, but here's why people often feel that quantum entanglement is rather strange. Let's first consider the following statement you make:

2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties

A classical (non-quantum) version of this statement would go something like this. Imagine that you take two marbles and paint one of them black, and one of them white. Then, you put each in its own opaque box and to send the white marble to Los Angeles, and the black marble to New York. Next, you arrange for person L in Los Angeles and person N in New York to open each box at precisely 5:00 PM and record the color of the ball in his box. If you tell each of person L and person N how you have prepared the marbles, then they will know that when they open their respective boxes, there will be a 50% chance of having a white marble, and a 50% chance of having a black marble, but they don't know which is in the box until they make the measurement. Moreover, once they see what color they have, they know instantaneously what the other person must have measured because of the way the system of marbles was initially prepared.

However, because you painted the marbles, you know with certainty that person L will have the white marble, and person N will have the black marble.

In the case of quantum entanglement, the state preparation procedure is analogous. Instead of marbles, we imagine having electrons which have two possible spin states which we will call "up" denoted $|1\rangle$ and "down" denoted $|0\rangle$. We imagine preparing a two-electron system in such a way that the state $|\psi\rangle$ of the composite system is in what's called a superposition of the states "up-down" and "down-up" by which I mean $$|\psi\rangle = \frac{1}{\sqrt 2}|1\rangle|0\rangle + \frac{1}{\sqrt{2}}|0\rangle|1\rangle$$ All this mathematical expression means is that if we were to make a measurement of the spin state of the composite system, then there is a 50% probability of finding electron A in the spin up state and electron B in the spin down state and a 50% probability of finding the reverse.

Now me imagine sending electron $A$ to Los Angeles and electron B to New York, and we tell people in Los Angeles and New York to measure and record the spin state of his electron at the same time and to record his measurement, just as in the case of the marbles. Then, just as in the case of the marbles, these observers will only know the probability (50%) of finding either a spin up or a spin down electron after the measurement. In addition, because of the state preparation procedure, the observers can be sure of what the other observer will record once he makes his own observation, but there is a crucial difference between this case and the marbles.

In electron case, even the person who prepared the state will not know what the outcome of the measurement will be. In fact, no one can know with certainty what the outcome will be; there is an inherent probabilistic nature to the outcome of the measurement that is built into the state of the system. It's not as though there is someone who can have some hidden knowledge, like in the case of the marbles, about what the spin states of the electrons "actually" are.

Given this fact, I think most people find it strange that once one observer makes his measurement, he knows with certainty what the other observer will measure. In the case of the marbles, there's no analogous strangeness because each marble was either white or black, and certainly no communication was necessary for each observed to know what the other would see upon measurement. But in the case of the electrons, there is a sort of intrinsic probability to the nature of the state of the electron. The electron truly has not "decided" on a state until right when the measurement happens, so how is it possible that the electrons always "choose" to be in opposite states given that they didn't make this "decision" right until the moment of measurement. How will they "know" what the other electron picked? Strangely enough, they do, in fact, somehow "know."

Addendum. Certainly, as Lubos points out in his comment, there is nothing actually physically paradoxical or contradictory in entanglement, and it is just a form of correlation, but I personally think it's fair to call it a "strange" or "unintuitive" form of correlation.

IMPORTANT DISCLAIMER I put a lot of things in quotes because I wanted to convey the intuition behind the strangeness of entanglement by using analogies; these descriptions are not meant to be scientifically precise. In particular, any anthropomorphisations of electrons should be taken with a large grain of conceptual salt.

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A nice answer. One of the experimenters can even have the facility to change the orientation of his/her aparattus quite randomly and very quickly, and yet the measurements do show correlations that violate Bell's inequality!! – JKL Feb 24 '13 at 19:07
So how is it known that the electron's wave function hasn't collapsed until it has been measured? Is that what's confirmed in the double-split experiment? – Pete Oakey Feb 24 '13 at 19:45
Actually I'm asking how do we know it's in a superposition in the first place? – Pete Oakey Feb 24 '13 at 22:37
I think he is asking how do you know that the superposition isn't just a classical lack of information. The answer lies in Bell's inequality, which I think he needs an intuitive explanation of. – Mew Feb 25 '13 at 0:18
This nice analogy just claims you can't have hidden variables ("no one know in advance" the otcome with electrons, in contrast to marbles). But this argument does not distiugish classical mixture from proper entanlement, and thus, in my tastes, does not realy address the "weirdness". – Slaviks Feb 25 '13 at 8:11

Rather than repeat some very good standard answers, I want to discuss this issue from the perspective as to why classical systems should be viewed as strange.

If we accept quantum mechanics as being fundamental, then in some sense we shouldn't really find things like entanglement to be strange at all. As pointed out by the answer given by joshphysics, as well as the answer given by Lubos Motl in the similar question, entanglement is really just correlation. The strangeness enters because we are accustomed to the the idea of classical locality and separability of systems.

Locality is best understood as the concept prohibiting action-at-a-distance, and is closely tied to Newton's Third Law of Motion. Newton's third law is the statement,

Every action has an equal and opposite reaction

which basically tells us that forces on an object are the result of the interaction by another object. Action-at-a-distance is a situation where two objects separated in space share perfect correlation in their motion, implying that one object is directly responsible for the other objects activities. In Newtonian mechanics, there is no limit on velocity, so action-at-a-distance, while seemingly unbelievable, is not prohibited.

This situation changed when it was realized that there is an ultimate speed limit to how fast two objects can communicate, or rather influence each other via the third law. This is the speed of light, as enshrined in the theories of special relativity and general relativity. This ultimate speed limit on the transfer of real information between two spatially separated regions is where our "classical intuition" fails (which is not a statement about human intuition, it is a statement about an apparent contradiction that arises in the logical statements one can make in the context of a particular theory).

So really the question isn't so much,

"Why is quantum mechanics weird?"

it's

"Why does our classical intuition fail?"

Much of this failure in our intuition is related to the separability of states which is an inherent feature of classical mechanics.

Separability of states is possible when one is able to describe composite states as direct products of subsystem state vectors.

To explain this a little better, there is a postulate of quantum mechanics that states

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems

This is written mathematically as $$\mathcal{H}_{AB} = \mathcal{H}_{A} \otimes \mathcal{H}_{B}$$ This can be imagined as just an abstract infinite dimensional space (just a really big space). The direct product $\otimes$ tells us to take every component of the each space times every component of the other space (e.g. if I can provide a basis for one space as $x$,$y$,$z$ and the basis for the second space as $a$,$b$,$c$; the direct product space would be $xa$,$xb$,$xc$,$ya$,$yb$,$yc$,$za$,$zb$,$zc$ )

As implied above, the component subspace can be given a basis that spans the space (span = provide a complete coordinate system that can describe every point):

$$\mathcal{H}_{A} \rightarrow \{ |a_i \rangle \}$$ and $$\mathcal{H}_{B} \rightarrow \{ |b_j \rangle \}$$

with our basis chosen, the pure state of the composite system can be defined as:

$$|\psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j \rangle$$

As discussed in the wikipedia article, if the state $$|\psi\rangle \in \mathcal{H}_{A} \otimes \mathcal{H}_{B}$$ can be written as $$|\psi_A\rangle \otimes |\psi_B\rangle$$ and $$|\psi_i\rangle$$ is a pure subsystem (e.g. also has an independent Hilbert space), then the system is described as separable. If it is not separable, it is entangled, and therefore:

$$|\psi\rangle = \sum_{i,j} c_{ij} |a_i\rangle \otimes |b_j \rangle \neq |\psi_A\rangle \otimes |\psi_B\rangle$$

(Update Example borrowed from Marcini and Severini: Let $|\psi_{A1}\rangle$, $|\psi_{A\perp}\rangle$ be orthogonal states in $\mathcal{H_A}$, and $|\psi_{B1}\rangle$, $|\psi_{B\perp}\rangle$ be orthogonal states in $\mathcal{H_B}$. Then $$|\psi_{A1}\rangle \otimes |\psi_{B1}\rangle \in \mathcal{H_A} \otimes \mathcal{H_B}$$ as well as $$a|\psi_{A1}\rangle \otimes |\psi_{B1}\rangle + b|\psi_{A\perp}\rangle \otimes |\psi_{B\perp}\rangle \in \mathcal{H_A} \otimes \mathcal{H_B}$$ with $a$,$b$ $\in \mathbb{C}$. The first can be factorized into states of the subsystems, the second cannot. The existence of this second state would result in the above inequality.)

In our classical intuition, systems are separable, and it is only through some direct classical mechanical coupling that they show any correlation. So in the marble examples, there is some mechanical process that is involved in mixing marbles together. The marbles are still separable systems, and the correlation of one person finding a white marble, and one finding a black marble is still rooted in classical statistical mechanics, simply by the fact that the marbles have a definite color associated with them before they are put in the box. This means that the state of color for either marble is known and is not correlated with the state of the other marble. It makes sense for one to talk about the marbles being in a black or white state in classical mechanics. This is not a typical state in quantum mechanics, and systems having a definite state prior to observation is the root cause for the failure of our classical intuition

We must understand that the full state space in the entangled system is much larger than the space of separable systems. There is a good analogy in understanding the different size of state spaces in the context of the Born Oppenheimer approximation (and Emilio Pisanty does a good job explaining the derivation in his answer to this SE question). The Born Oppenheimer approximation provides a justification for allowing for the separation of the nuclear and electronic states of a molecular system:

$$\Psi_{Total} = \psi_{electronic} \times \psi_{nuclear}$$

Similarly in our "classical intuition" we can ignore many terms that describe the state of the system simply because their effects are too small to be considered. As systems become smaller, these effects are harder to ignore, and the notion of a quantum object being able to have a definite state (e.g. being a definitely black or white marble) prior to our observation is not possible. However, the correlation of the outcomes is not removable from the system, in this sense the correlation must be viewed as more fundamental then the definiteness of the state. This is a very different state of affairs than what we find in classical mechanics, where definiteness of state is viewed as more fundamental.

So hopefully this gives a little more clarity as to why we think quantum entanglement is a "big deal". It requires a fundamental change in our understanding and approach to physics.

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What happened here? I am in complete shock that so many respondents are answering "yeah, no big deal, nothing really spooky going on, yeah it's just correlation..." What in the world is everyone talking about?

Peter Oakey, forget all the math for a minute. This will take a few minutes of detailed but entirely non-mathematical setup, but if you can bear with me I can explain to you in a very pointed way why entanglement is spooky and cannot be explained by classical correlations alone.

First, we need something easily visualized with which to set up the situation. A clock with only one hand, an hour hand, works nicely.

Did I mention that the hands on these clocks are a bit weird? Well, actually... a lot weird.

Instead of being sharp pointy lines, the hands are painted onto a disk... badly. They are severely smeared, to the point that they are fully black only in the exact direction of the time they represent, e.g. 3 o'clock. From that direction they fade off into gray as you go around the disk on which the hand is painted. In fact, the disk remains pure white only on the exact opposite side from the pure black direction. So, if the pure black is pointing at 3 o'clock, the pure white is pointing the opposite direction at 9 o'clock. (I may add some simple graphics for this tomorrow, but it's too late tonight.)

Through a Slot Darkly

Um, did I mention that reading these clocks is also a bit weird? Well... a lot weird.

That's because you are only permitted to read them by looking through a single slot that you can dial into any position you want, such as 12 o'clock. Now, you might think that would make seeing the hand impossible most of the time, but don't forget: the hands on these clocks are so badly smeared out that in most cases when you look through the slot you will see some kind of gray, probably at least 50 different shades of it.

Occasionally, though, you will happen to see pure black or pure white. That means you got lucky and set your analyzer to one of the two positions from which you can read the clock with 100% certainty. Thus if you set the slot to 3 o'clock and see pure black, it means the clock was set to that same time, 3 o'clock. But notice that if you had instead set the slot to 9 o'clock, you would have seen the pure white that is always opposite to the pure black, and you again would have known with certainty that the time was 3 o'clock. Alas, if you had instead chosen any other setting for the slot, you would have seen only some shade of gray. Darker grays would have meant you were "closer" to the time on the clock, while lighter grays would have meant you are farther away from it. But for any of the gray shades you can only make a guess about the exact time.

Game, Reset, Match

Which leads to a final but very important oddity about these clocks: Every time you read one, the hand resets itself to match the orientation of your reading slot. Now that's really weird! How does this final twist work?

It's somewhat random, actually, but in a way that is strongly guided by how gray the disk is at the point where you read it. If you happened to read pure black or pure white, there is no problem: The hand simply stays exactly where it was, on black or white. If instead you happened to see the shade of gray that lies $90^{\circ}$ away from pure black or pure white, e.g. 12 or 6 for a hand pointing at 3, then the dial resets in a fully random fashion, with a 50/50 chance of moving either pure black or pure white into the reading slot position after. Everything in between becomes a probability that is more in favor of black or white. Thus a very dark shade of gray will almost always cause the clock dial to rotate pure black into the reading slot position... but not quite every time. As long as the disk has some white mixed in with the black, the pure white side of the dial disk will occasionally get rotated into the reading position.

Incidentally, in case you were wondering how to translate some weird shade of gray into a specific reading of the clock, this gray-based resetting feature provides the answer. What happens is that you final answer always is based on how your set your analyzing slot, specifically on the value that gets rotated into that position after you read the original value on the clock. So for example, if you set your analyzer to 12 o'clock, you will always get an answer of either 12 o'clock (pure black rotated into the 12 o'clock position) or 6 o'clock (pure white rotated into the 12 o'clock position). The original clock dial position no longer matters at that point, since the very act of reading the clock resets it and makes the new value into the only one that matters.

Strange Times

Strange times (and time pieces) indeed! But if you are wondering why I am putting in so many seemingly pointless restrictions, I assure you they are not as arbitrary as they seem. What I am doing it translating large chunks of quantum mechanics into a physical model that helps visualize certain types of quantum relationships. Because quantum mechanics deals with small systems that contain very little information, it is all about understanding these odd constraints that do not allow the huge freedoms to which we are so accustomed from the classical world.

I'll call these constructions fuzzy clocks due to all the probabilities going on in reading them.

Igor, Pull the Lever!

Next comes the experimental arrangement using these clocks, one that is the same for both classical correlation and quantum entanglement:

1. Set two fuzzy clocks to have exactly opposite but randomly selected times, e.g. 1-and-7 or 10-and-4. Keep these times secret from everyone in the universe.

2. Place the fuzzy clocks in two spaceships A and B and fly them to locations very far apart. For example, you could fly them so far away from each other that light takes an hour just to get from one spaceship to the other.

3. Have your observers in each spaceship read their clocks. There are multiple ways to do this, but in this case we'll prearrange for the observers to use identical orientations of their slot readers. For the rest of this discussion, we'll assume their slot readers are set to 3 o'clock.

Recall that when a slot reader is set to 3 o'clock, the final reading will always be either 3 o'clock (pure black) or 9 o'clock (pure white). That's because reading the clock causes it to reset (rotate) based on how much gray is seen through the slot. It is those new pure black or pure white values that become the final readings of the clocks.

Workin' 12 to 6, What A Way to Make a Livin'

Now let's focus on a particular subset of correlated fuzzy clocks, which is the ones that were set originally to either 12 or 6. What happens to these clocks when they are read by the 3 o'clock slot readers on both spaceships?

Recall that any clock value initially set to 12 or 6 will for a 3 o'clock slot reader show the shade of gray that results in a 50/50 toss-up. So, half of these clocks will end up with pure black at the slot position (3 o'clock), and the other half with pure white (9 o'clock).

Let's assume that ship A reads one of these 12-or-6 clocks and gets a value of pure black, meaning that it has been reset to 3 o'clock. What can the observer say then about what the other spaceship will see when looking at the correlated clock in the same way?

Losing It

Well... nothing, really. From the perspective of the observer, this worst-case scenario of 50/50 random reassignment has completely erased any information that would have been available about the time on the other fuzzy clock. So, all the observer on ship A can say for this group of clocks is "since this is the 12-or-6 clock group, ship B will have a 50/50 chance of reading black or white." Which is exactly correct: Ship B will get just as random a result in this case as ship A did. The correlation that potentially existed was in effectively erased by the nature of the reading procedure, so neither ship can say anything about what the other would have seen.

That's the classical case: No correlation -- no predictability -- is possible between ships for the 12-or-6 clock pairs analyzed using 3 o'clock slots.

Finding It

So, what if the clocks are quantum entangled instead of just sharing a correlated past?

Easy: When the observer on ship A sees pure black at 3 o'clock for a 12-or-6 clock pair, she knows that the observer on ship B will see pure white. Always. 100%.

Oops.

Um... how exactly did that happen?

Spooky Is as Spooky Does

Spooky action at a distance remains a pretty good name for it, because I guarantee you are not going to be able to construct a meaningful explanation for it in terms of actual experimentally accessible parameters. Nor is it a hypothetical effect. Real examples of this effect are always more complicated than the intentionally extreme version I've used here, but it doesn't get any less weird. John Bell was the fellow who first figured out that this effect was real and testable, decades after minds as great as Einstein and Bohr came very close to it but missed seeing the opportunity.

The fuzzy clocks provide a quite physical image of what has to happen. When one of the two spaceships A or B analyzes their clock, they cause it to reset (rotate) to the new time enforced by their slot position, e.g. from 12-or-6 to 3-or-9.

In classical physics, that's the end of it. Each disk rotates into its new position locally and without any connection to the other disk.

In entangled physics, the act of resetting the disk in A or B disturbs a very unforgiving conservation law, in this case the conservation of angular momentum (but other laws can also be used). It turns out that the universe is so unforgiving for such absolute conservation rules that issues such as the speed of light become secondary to ensuring that the quantity is absolutely conserved. So, spooky-style, the universe as a whole does not allow you to reset just one of the entangled disks, which would cause a slight non-conservation of angular momentum. Instead, when you must reset both.

So, when A analyzes her 12-or-6 clock with a 3 o'clock slot analyzer, she ends up resetting both disks to the new 9-or-3 orientation. All of this happens "instantly," even across light-years, whatever "instantly" means in such cases. (It doesn't really mean much when entanglement is involved, which is why I usually avoid such terminology.)

The Bottom (Entangled) Line

There are many ways to get lost in the weeds in all of this. However, entanglement in terms of a "something" that instantly resets the options available to distant events, even while prohibiting the conveyance of conventional information (a point I've skipped over), is both quite real experimentally and quite weird conceptually. It's one of those little mysteries of the universe that is still worth contemplating from time to time.

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Your clocks not have conservation laws in them. It is conservation laws in the quantum mechanical system that assure if you find a -1/2 spin at -infinity the partner will have +1/2 at +infinity. It momentum conservation that says one has gone to - infinity and angular momentum conservation that tags the other once one is seen. Unless you consider conservation laws spooky? – anna v Feb 25 '13 at 9:10
Anna, thanks: The clocks are hokey constructions intended only to capture in physical form the implications of an entangled pair of spin $\frac{1}{2}$ particles, e.g. an electron and positron created by a perfect two-gamma collision. The clock itself exhibits no entanglement, any more than a bra-ket notation does. And I think I agree completely and enthusiastically with you that "it is conservation laws that assure ... etc."? Gugg, thanks, I'll look this evening. – Terry Bollinger Feb 25 '13 at 17:20
Link mangling is my forte, it's sort of like an exercise in Internet entanglement don't you know... :) – Terry Bollinger Feb 25 '13 at 17:38
But when two distant detectors are aligned, you'll have 100% (anti)correlation even with classical physics. You had two entangled particles, you measured them the same way, you got (classical) correlation. I don't think the case you're describing (the 100% case) is what's "spooky" about QM. It is these other cases where readings match at a certain probability that don't match with classical predictions. – Lousy Coder Sep 5 '14 at 21:03
@LousyCoder I agree — it's the Bell Inequality that emerges out of the difference between classical alignment and spooky alignment that counts, and that cannot be seen except at angles that mix the two states... and I did not even get to that part. If I get the chance I promise to address that. Meanwhile, John Bell explains his own Inequality best in Speakable and Unspeakable in Quantum Mechanics. – Terry Bollinger Sep 5 '14 at 23:39

Here is the answer that made me realise what the big deal is. The description below is basically an expanded version of this blog post, which I came across a long time ago.

Imagine we are going to play a game. It's a cooperative game, so we'll either both win or both lose. If we win, we get lots of money, but if we lose we both die, so we should do our best to win.

The game is played as followed: you will be taken on a spacecraft to Pluto, whereas I will stay here on Earth. When you arrive at Pluto, someone will flip a fair coin. Depending on its result, they will ask one of the following two questions:

1. Do you like dogs?
2. Do you like cats?

You will then have to answer "yes" or "no". At the same moment, someone on Earth will flip a different fair coin and ask me one of the same two questions based on its result.

The rules of the game are slightly strange. They are as follows: we win the game if we each give a different answer from the other, unless we're both asked about cats, in which case we have to give the same answer as each other in order to avoid losing.

Since we're several light-hours apart there's no way we can communicate with each other during the game, but we can spend as long as we like discussing strategies before we go, and each of us can take anything we want along with us to help us answer the questions.

Now, with a little bit of thought you should be able to convince yourself that in a classical world, the best we can do is to have a $75\%$ chance of winning the game. To do this, we just agree that no matter which question we're asked, you'll say "yes" and I'll say "no". If we do this, we'll win unless we both get asked about cats, and the probability of that happening is 1 in 4. It doesn't matter what we take with us - as long as it behaves according to the familiar rules of classical mechanics, it can't help us do any better than this simple strategy. In particular, it doesn't make any difference if we each take a hidden object with us, which we later measure in some way.

However, in a quantum world things are slightly different: we can win the game $85.3\%$ of the time. I'm not going to go into the details of exactly how we achieve this, but it involves creating an entangled pair of particles, of which you take one and I take the other. Depending on whether you're asked about cats or dogs, you make one of two different measurements on your particle, and I do something similar. It just works out according to the rules of quantum mechanics that if we follow this procedure correctly, we'll win this game with a probability of $\cos^2(\pi/8)$ , or $85.3\%$. Many experiments that are equivalent to this game have been performed (they're called Bell test experiments) and the game is indeed won $85\%$ of the time.

There are other games that can be constructed, which are slightly more complicated to explain, where using entanglement allows you to win $100\%$ of the time, even though in the classical world you can't avoid losing some of the time. A paper describing such a game (among other examples of such quantum games) can be found here.

This is why entanglement is a big deal. It allows us to make things be correlated in this sort of way slightly more than they can be correlated in the classical world. It allows us to do something that wouldn't possible if entanglement didn't exist.

As an aside, there's another reason why entanglement is a bit weird: in the cats and dogs game, why does entanglement only allow us to win $85\%$ of the time and not $100\%$? It turns out that you can invent universes with "alternative physics" in which this game can be won $100\%$ of the time, while still not letting information be transmitted faster than light, but in our universe, $85.3\%$ is the maximum possible score. The reason why entanglement should be limited in this way is an open question in the foundations of quantum mechanics.

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Can you describe in a bit more detail how exactly you would improve you chances of winning? If you measure the entangled particle at both ends using aligned detectors, you will obviously measure opposite direction 100% of the time, but you gain nothing from that. It's like looking at a synchronized clock and saying "ok, so the other guy is now reading the same value". And if you align them at a different angle, you will measure opposite directions with lower probability (according to quantum mechanics laws, but still lower than 1). So how does this possibly help you decide what answer to give? – Lousy Coder Sep 5 '14 at 20:25
I use detectors aligned at 0 and 90 degrees, depending on the question I'm asked, and you use detectors aligned at 45 and 135 degrees. If you work through it, you should find that gives the expected score I quoted. – Nathaniel Sep 6 '14 at 1:27

The "big deal" seems to be that, due to Bell's theorem* and "given" quantum mechanics, we can only choose between non-locality ("spooky action at a distance") being true and/or counterfactual definiteness being violated (possibly implying no "free will", whatever that means), if we want to choose at all. The first is "unintuitive" and (the possible implication of) the second is, well, a "big deal" for many people (including at least some scientists who argue that implicitly science relies on "free will").

*"You can't argue with a theorem."

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There are two things that make the quantum version of "put a white ball in one box, a black ball in another, shuffle the boxes, then open a box and learn what the other contains" surprising.

1. We can play with the relationship between the two boxes. In ways that wouldn't be possible if each ball was actually in one box or the other.

2. You can do experiments where the results agree too often. More than would be possible if each ball was actually in one box or the other.

Part of what makes the second point especially confusing is that the extra agreement still doesn't allow communication. Isolated experiments can get some after-the-fact correspondences, but can't signal each other to change the procedure they were already going to follow.

I'll try to illustrate with a game. This is a game where classical strategies can't guarantee a win 100% of the time, but quantum strategies where you share some entangled stuff ahead of time can. Here's the rules:

• You and a partner are isolated from each other.
• You get two tokens.
• A referee will pick a row from a 3x3 board at random, and tell it to you.
• You can either not use your tokens, or use them to cover two cells of the row. You are not allowed to cover only a single cell. Just 0 or 2.
• Your friend goes through a similar process, but they play on a column instead of a row.
• Once you've both chosen how to play, you get back together and compare results.
• If the single cell that is both in your row and your friend's column is covered by exactly one token, then you win. Otherwise you lose.

Here's a picture illustrating a few outcomes of some people winning, losing, and failing to follow the rules:

You can see that this is a sort of coordination game. You and your friend need to cooperate so that one of you covers the common cell, but not the other. It's hard because during the game you each don't know which cell it is exactly, and can't communicate.

Try to come up with ways to win the game. I bet you can't do better than expecting to win 8/9ths of the time. You can even try involving colored marbles in boxes. It won't matter. The best you can do is 8/9.

An example of such an optimal strategy is: you always play in the left and center cell of your row, while your friend only plays if they get the right-most column. You only lose if the right-most column is picked and your friend gets unlucky, which happens 1/9th of the time.

But, like I said, there's a quantum strategy that wins 100% of the time. It massages the after-the-fact correspondence into cooperation on covering the common cell exactly once, despite the isolation.

It would be a bit long-winded to explain the strategy here, but that's why I wrote this handy but slightly technical blog post. Here's a picture showing the quantum circuits used as part of winning the game. It's pretty, but not too informative out of context:

There's also a wikipedia article, although it uses a slight variant of the game I described.

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I like to use the following analogies when thinking about quantum entanglement. If the world worked like @joshphysics describes in the first part of his answer where two different colored marbles are placed in two separate boxes and then separated, then upon opening one of the boxes, the person looking in will know with certainty what color is in the other box because the color of the marbles in each box was determined during the 'entanglement process', but there is nothing intuitively 'weird' about that. But, as @joshphysics showed, this is not equivalent to quantum entanglement.

Entanglement is more like this: imagine you bring two coins together in such a way that they become 'entangled' (maybe you cast some kind of spell or something). One person takes one coin to the moon and another person takes the other coin to Mars. From the perspective of each person, the probability of them getting heads or tails after flipping their coin is 50%. What quantum entanglement implies is that once one of the people flips his or her coin and finds that it comes up, say, heads, then no matter what (assuming the coins remain entangled), when the other coin is later flipped (it can be flipped any time after the first), it will have to turn up tails. From the second flipper's perspective, he will still find that he has a 50% chance of getting heads or tails, but once the first coin was flipped and turned up heads, the second coin will have a 100% chance of being tails, no matter when it is actually flipped after the first. That's weird.

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(Just adding one more answer to the clear ones given above) The "weirdness" of the entanglement of two spin 1/2 particles is the following: If you have two observers Alice and Bob, separated by any distance, and tell them to measure the spins of the entangled pair, in such a way that the analyzers are rotated randomically so that when a measurement on one particle of the pair is made it is not in causal contact with the other particle. That is, the event "Alice measuring the spin of a particle of the pair" is not in the past light cone of "Bob measuring the other spin of the pair" and viceversa. Even so the spins are correlated. This is, in essence, the Alain Aspect experiment and the spirit of the EPR correlation.

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Your mistake in interpreting entanglement as mere correlation is a very common one. In fact, Einstein's whole argument against quantum mechanics in EPR was to restore causality to quantum mechanics by interpreting entanglement as nothing more than a pre-existing correlation. However, Bell showed that this fails. Unfortunately, many people, Lubos Motl for example, have failed to understand this insight half a century after Bell's discovery. The entangled particles have to get their instructions on how to behave somehow, and this has to happen instantaneously.

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@Gugg The Consistent histories authors (Griffiths) humiliate themselves and demonstrate that they have no clue at any level what they are talking about in the foundations of quantum mechanics. They give an example of different colored strips of paper, when you see one strip, you know the strip of the other paper. They haven't the slightest clue what Bell's theorem means, because Bell's theorem exactly means that this analogy is completely wrong. They don't understand Bertlemann's socks at all. Nature has spooky action at a distance. – user7348 Feb 26 '13 at 19:14
@Gugg, I would like to mention that I am not the professor in that video. – user7348 Feb 26 '13 at 19:16
Thanks. Is there something in particular wrong with the approach (leaving interpretational issues aside for now), that you might point out to me, that's on its description on Wikipedia? – Keep these mind Feb 26 '13 at 19:31
@Gugg Well, the interpretation says there is no spooky action at a distance, but this is in disagreement with Bell's theorem which showed that non-locality is a feature which will never go away. Honestly, I'm not a physicist, but I enjoy reading about foundations of QM. The best author on this is Tim Maudlin. You might buy his book Non-locality and relativity. Or, you can read any of his articles. Try this(SECTION 3) ... bslps.be/meaningWF.pdf – user7348 Feb 26 '13 at 19:47
Consider this. Bell's theorem depends on the assumption of counterfactual definiteness (CFD). This CH doesn't have CFD, so Bell's theorem doesn't rule it out. Bell's theorem actually proves that every type of quantum theory must necessarily violate either locality or CFD. It's not that it's invalid, it simply doesn't apply. How about that? – Keep these mind Feb 26 '13 at 20:52

there is no big deal. Usually people who don't uderstand it will tell you that it is a big deal...

Let's say,you have 2 objects and observable with only two eigenvalues. One object is in state "+1" and the other "-1". The world that these objects live in has a rule that the sum of all these values is constant (zero in this case). Let's imagine that these object collide (interact in a manner that this observable can be changed).

Now, the best guess (if you don't know any details) is just to assume that the system is in state "+1"×"-1" or "-1"×"+1". And that's it.

If you look at the one object and determine the state, then you immediatelly know the state of the other, because of the conservation rule.

What is kind of a big deal (but I'd rather say that it's just 'cool' and not a big deal), that there are states that preserve corelations for multiple observables ("+-"-"-+" spin state if measured along any axis will always produce correlated results).

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This is just plain, empirically wrong. The whole point of Bell's theorem is that you cannot reproduce quantum mechanical predictions with a local hidden variable model like the one you described! It doesn't suffice to look at how measurement results are correlated for any given, arbitrary measurement axis as you have suggested. You have to look at correlations when the two observers vary their measurement axes relative to each other. Please read the Wiki link, you will learn something cool; you might even consider it a 'big deal' after all! – Mark Mitchison Feb 26 '13 at 22:41
But this doesn't change the fact, that allowed pure states that will make together the mixed state after collision must follow symmetries of the world... If you have two eletrons, one spin up, one spin down, than total angular momentum is zero and there is nothing you can do about it... I might not have expressed myself clearly - the conserved quantity determines allowed states. – asdf Feb 27 '13 at 16:07
Sorry, but this answer demonstrates a complete failure to understand the difference between entanglement and classical correlations. This is made all the worse by the statement: "Usually people who don't uderstand it will tell you that it is a big deal...", when you clearly don't understand it at all. There is much more to entanglement that simply conservation laws being respected on-shell. Please read up some more about Bell's theorem, I'd recommend Bell's book "Speakable and Unspeakable...". – Mark Mitchison Feb 27 '13 at 16:26
Bell's theorem will tell you what will be measured when you HAVE the state to begin with. It doesn't tell you how the state is chosen in the first place... – asdf Feb 27 '13 at 17:14
Just to show what I meant - example: Let's assume that we have 2 particle system with Hamiltonian(Ss are spin operators)... H=Sx×Sx+Sy×Sy. Comutators [H,Sz×I] and [H,I×Sz] are non-zero, but [H,Sz×I+I×Sz]=0, that means that any unitary process will preserve sum of the spins along z axis, but individuals spins are not conserved. If there is a colision and we don't know any details, than we must assume the state with the maximum entropy, BUT we have to take into account our knowledge of the conserved quatity. – asdf Feb 27 '13 at 18:04

## protected by Qmechanic♦Sep 24 '13 at 18:16

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