The Biggest Ideas in the Universe | Q&A 8 - Entanglement

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll today we're doing the Q&A video associated with I think it was the 8th idea of entanglement quantum entanglement the seventh idea we did was quantum mechanics and we gave you basically some of the textbook stuff about wave functions and how they collapse when you measure them in video number eight we went a little bit deeper into the idea of entanglement which is something that truly separates quantum mechanics from classical mechanics and it let us talk about some of the more modern formulations of quantum mechanics formulations that are purportedly a little bit more rigorous and well-defined than the Copenhagen interpretation that you read about in textbooks opinions differ about whether or not that is true but I'll in this video answer a couple questions about entanglement generally and then go in a little bit to more detail about how things work in the many-worlds interpretation which is my favorite one I didn't do this in the original video because I've done things like this on videos online elsewhere but there's a completist aspect that indicates that we should probably do it here so it I think it thought about the different questions that were asked and I think that it will be useful before diving into the answers to talk a little bit more about this example of quantum mechanical spins because spins really when you talk about entanglement and the wave function of different parts of the system interacting with each other in different ways spins provide a very very nice explicit example that you can control much better that you sort of know what all the different options are in a way that we talk about the position of something or the momentum of something that's harder to do for the basic reason that there are an infinite number of possible positions you could measure but there are only two outcomes you can get when you measure the spin let's say an electron there's there are higher spin particles before an electron there are only two possible outcomes you could get which is why spins are characterized by qubits quantum mechanical versions of bits a bit is either 0 or 1 classically quantum mechanically it's some superposition of 0 or 1 and spins are represented quantum mechanically by qubits the spin is a physical thing a qubit is an abstract notion but they line up in this particular way of thinking about things so let's talk a little bit about how you find out about spins in the first place and what this teaches us about spins because again I don't like to keep anything in suspense we're going to use spins to think about the uncertainty principle in quantum mechanics in a slightly different way what super positions mean what basis states are which I mentioned in the previous Q&A video and then we can use that technology to talk about entanglement in a more detailed way so we measure the spin of a particle by something called well one way to do it something called the stern-gerlach experiment and I'm not gonna go into great detail about this I'm gonna give sort of a theorists version of the stern-gerlach experiment this is an experiment that was actually done by monsieurs Stern and Gerlach but we're not gonna go into the experimental details the basic idea is that if you have a magnetic field and it is an inhomogeneous magnetic field so it's sort of more pointy at one end than the other so it's like a line a set of lines of magnetic field that sort or originally spaced out and they come together okay and you pass an electron through it the spin of the electron the electron is little spinning particle that has an electric charge so it's a little magnet and will be deflected either in the direction of the magnetic field or opposite to the direction of the magnetic field if it were spinning such that its axis or pointing along or against the magnetic field so I'm not gonna try to draw an accurate depiction of the magnetic field because that's hard so here's my magnetic field B Z so we're thinking that in a little coordinate system here you'll see Y X Y Z our electron our spin let's how do I draw the electron not to say II - the electron is going in along moving along the y axis and it encounters this magnetic field and the point is that you might think I'm not sure if you should think this or not but you might think that if the electron is a little spinning magnet it has a magnetic pole and it could be pointed anywhere right that pole is aligned in some direction and therefore when it goes through it could be deflected up or down or any intermediate value right but quantum mechanics comes along and says in fact the electron only goes slightly up or slightly down and this is what we would call spin-up or spin-down okay there are two possibilities that's why it's a cubit it's the so it's a quantum mechanical version of 0 or 1 0 is spin up one is spin down now there's actually subtleties here whether or not you need quantum mechanics to explain this to valued Mis rather than infinite number of continuum of values but we're just going to take it as a fact that this is we're not comparing classical mechanics to quantum mechanics we're just saying this is the quantum mechanical answer so you represent then the state of the particle sy by some combination alpha times spin up plus beta times spin down this is the wave function of the spin this is a slight modification to look less scary of the Dirac notation that Paul Dirac invented he would have written alpha up and then so there's a line and a right angle bracket plus beta down angle bracket and this is called a ket this particular notation because this is half of a bracket so there are both bras and ket's a bra looks like this oops that's a bra for physicists this is a ket we're not going to use any of that notation okay but I'm just going to use this simplified parenthesis notation it's more than good enough for what we want to do okay and you know that these numbers alpha and beta these are two complex numbers these are the amplitudes for observing the particle to be spin up or spin down so the probability of in this experiment measuring spin up is magnitude of alpha squared probability of measuring spin down in this experiment is the magnitude of beta squared and therefore you know that alpha squared plus beta squared equals 1 because the sum of the probabilities has to be equal 1 and you only have two choices good so I think all that you know I sort of glossed over that but I mentioned things like this before but now let's do something a little bit more complicate we measured there the spin in the Z direction so we talked about spin up we always imagined the Z direction is pointing towards the sky so we talked about spin up versus spin down by measuring the spin in the Z direction but we could also measure it in other directions we could measure it in the X direction it's hard to measure it in the y direction because the electron is moving in the Y direction but X is easy enough so let's do the following experiment let's send the electron in okay let's measure it in the Z direction so it do what we just did and I'm not gonna I'm gonna run out of room if I keep drawing all the different options here so let's just imagine that what we got was spin up okay I could do better than that we got a spin up outcome good but now and again remember X Y Z now let's measure again the spin but this time with respect to I suppose I should draw it the other way around to be completely fair the X magnetic field oriented in the X direction okay so then again it's going to split and in fact you you could get just be consistent and draw it as a solid line you could get either yeah you could get either spin right if you want or spin left so let's call this spin let me see right all depends it's yeah it's not the notation is not gonna look very good but like all that's been writing spin left okay you get the point it turns out that if you do this if you measure the spin in the Z direction and you get spin up then every time that you put it in the X Direction spin measurement device you will get either spin left or spin right that's been oriented along the x axis and you cannot predict which one you are going to get okay just keep that fact in mind again and then let's say that you measure spin left okay let's say that's what you measure and then you decide to do it again the same electron and measure it in the zero okay so he had the electron he measured in the Z direction we found it was spin up we measured it in the X direction we found it was spin left now we're gonna measure in the Z direction again you could potentially think I don't know if you would think this or not but not trying to put ideas into your head but you could potentially think that there were two separate things one called the spin along the X direction and another called the spin along the z direction in that case you would think okay I've measured the spin along the z direction it's up I've measured the spin along the X direction it's left therefore if I measure the Z spin again it will be up again because if I didn't do this X Direction thing if I just did this we just had an electron come in measure it along Z and I find that it is spin up then if I measure it along Z again I will always get spin up again this is a reflection of the fact that this measurement in some sense measures this this going through well let's be SuperDuper clear about this because we're trying to be accurate here going through the electric field the going through the magnetic field does not measure the spin of the electron it evolves the wave function of the electron into two components that are separated in space what we imagine is that we then measure it okay so it's not the splitting that measures it cuz that's not entangle it with anything but when we actually measure it we need to entangle it with the environment so we have some screen or some electric field monitor that has an arrow that says oh one left went up down whatever it did okay so we're imagining that we both put it through the magnetic field and measured the spin of the electron so just according to our usual our old-fashioned Copenhagen way of thinking when we measured it and we saw that it was spin up the wavefunction collapsed and it's now spin up that's what it is so when it goes through B Z again is always gonna measure spin up it's not gonna be a superposition of both where as well we find here that so that might be what you think that we measured spin up and that's done with it well we actually find here if we go through the Z spin a Z magnetic field again is an equal superposition of spin up and spin down okay so putting the electron through the x-direction magnetic field and measuring it to be one thing or the other thing turned it back into a superposition of spin-up and spin-down in the z direction so what that means is and maybe this is not a surprise at all again I'm not telling you should be surprised what it means is there are not two separate things that spin along the X direction and the spin along the z direction there's only one thing to spin of the particle and when you measure it you collapse it into some state that is either spin up or spin down with respect to the direction in which you're measuring it and the way that we expressed that is the state that is spin up that we've written like this we can write as 1 over the square root of 2 spin right plus 1 over the square root of 2 spin down spin left so what this notation means is that this is a wave function this is sy you might want to call it sy for a spin up spin electron and you can rewrite it as in in terms of the left and right bases and if you had gotten spin down it would be 1 over square root of 2 these are conventions that we work out and I'm skipping steps spin right minus 1 over square root of 2 is been left so in either one of these two cases if it was spin up or spin down when we measure it along the x direction there's a 50-50 chance that we're going to get spin left or spin right in the X direction because these are two the coefficients are 1 over square root of 2 with a plus or minus out front squared in either case you get 1/2 so there's a half of the chance of getting it left or right so there's yet another way of thinking about this if you look at if you stare at these equations remember Hilbert space the space of all wave functions is a vector space ok you can add them scale them by numbers etc so what this is telling us is we can write the vector space for Hilbert space of the qubit of the single spin in terms of basis directions spin up and spin down and we can express any wave function sy let me give it a color any wave function sy is just a vector in this vector space so it has if you did this this would be if you wrote psy as alpha spin-up plus beta spin down this component would be alpha this component would be beta okay there you go and these equations telling you that spin up is a superposition a combination a sum of spin right and spin left and likewise for spin down can be thought of as saying I hope I'm gonna get the signs right here you know I'm again making this up as I go along but basically it's saying that spin left and spin right are two different axes in this same vector space two axes that are rotated with respect to what the old axes were so the axis that we label spin right might be like that and the access we label spin left might be like that you can actually work out what they are there's a 50-50 chance that I got the labels right in the directions right maybe there's only 25% chance but you get the point the point is that because you're in a two dimensional vector space what you're doing when you're measuring spin up versus spin down or spin left versus spin right is you're making two different choices about how to talk about the vector that you have likewise this this vector sigh I can still write it in terms of components here this is a different two sets of components in the different axes spin left and spin right all of which is to say the difference between vertical spin and horizontal spin is not two separate quantities it's two different ways of talking about the same underlying state that the that the thing could be in and the reveal here the reason why this is worth chatting about is this is the origin of the uncertainty principle okay remember we talked about the uncertainty principle and in the context of position and momentum and we said if you were completely localized in position you had no idea what your momentum is and we said if you're completely localized in momentum you have no idea what your position is the same thing is true for X spin and Z spin if you know what your X spin is so let's say you know what your x been let's say you are purely spinrite okay let's draw you this way here is Sai spinrite okay well then it's clear that you are maximally uncertain about whether you are spin up or spin down in the Z direction right and likewise if you are pure spin left you would be maximally uncertain as well and vice versa if you are maximally certain about whether you are spin up or spin down in the Z direction you'd have no idea what your spin was in the X direction so that's a reflection of the uncertainty principle it's saying that if you if you believe that the you know wave that I suggested you think about it there is no such thing as the X bin or Z spin it as an inherent property of the particle these are outcomes of possible observations you can make okay they are not pre-existing things and I'm emphasizing all this for a very good reason because now let's start talking about entanglement okay keep these keep these equations in mind the fact that up and down are related to left and right by this simple transformation okay so the questions come in because I think the biggest question I really want to absolutely get around is isn't entanglement just kind of like classical correlations in some sense so I'm gonna talk about entanglement versus correlation the example that was given what if like you buy a packet of two gloves okay and you know that there's a left-handed glove and a right-handed glove in there so you know that if you open one and pick one out of random if it's the left-handed glove you instantly know the other one is right-handed isn't entanglement just like that so the answer is no the entanglement is not just like that entanglement is more than that and almost exactly that example was given by John Bell he gave an example that he called brittle men's socks I'm not again completely clear that I'm spelling Bertil man's name right so John Bell wrote you know John Bell when he was inventing the Bell inequalities he's incredibly influenced by the einstein-podolsky-rosen argument to EPR argument with these entangled spins you can measure once he would happen somewhere else and so what he was trying to say is that entanglement is more than just classical correlation sober tlemen socks is the idea that this guy who was a friend of bells another physicist at CERN I think Reinhold Reinhard Bertil Minh would he had this cork of his personal style where when he wore socks which was every day but he would always wear two socks of different colors okay they were never the same color so Bell says if you see Bertil Minh coming down the hallway and you know he walks around a door and one of his legs is sticking out first and you see Oh on that leg he has a pink sock then you instantly know that whatever is going on on the other leg it's not pink because he always wears two different colored socks okay so that's an example of a classical correlation do you measure one thing and you know something just like it you know something from it about something from that you know something about something completely different okay but Bell's point is that that's not what entanglement is for a few reasons let me just give you the most important reason here we talked about let's here's Alice and Bob with their spins so here's Alice she has a spin here's Bob planet Bob over here there's another spin and the state of both of them by the way these are called Bell States this way of writing these spin States there's a state that is one over square root of two spin up for Alice spin down for Bob just as an example no actually let me it'll be easier for the subsequent math if I do both spins the same plus one over square root of two it's been down for Alice spin down for ball remember it doesn't matter whether or not they're aligned or anti aligned what matters is that they're related what matters is if you measure one then you instantly know the other one okay in either case they would be entangled one way or the other and so someone said I think there was one question that said you know isn't entanglement just the same as superposition or at least what is the difference between them so superposition is just possible with one particle with one qubit right here is a superposition this is a expressing the spin up state as a superposition of spin right and spin left and this is crucially important you know by the way and we'll skip ahead a little bit note that people want to say sometimes you know how many worlds are there in the many worlds theory of quantum mechanics or they want to say you know when the electron goes through the double slits is there a world in which he goes through the right-hand slit in a world in which he goes to the left-hand slit and that is not my point of view at all I know some people have points of view like that David Deutsch has a point of view kind of like that I don't want to try to say exactly what this point of view is but it's related to that it's sort of a fine-grained version of what the worlds are my statement is that the world's only come into existence when you experience decoherence and we'll talk about that a little bit just writing it as a superposition doesn't mean it's two worlds at all and that should be perfectly obvious because this equation right here is saying that the same state can be written either as spin up 100% or an equal superposition of spin right and spin left so is it one world or two worlds right it depends on how you write it but that's silly the number of worlds shouldn't depend on how you write it and my way of figuring that out is just say this is just one world because this spin is not entangled with anything at all entanglement is a relationship between two different parts of the quantum system and when you only have one part there's no such things entanglement but there is superposition okay so here is Ellis involved and if Alice measures her state and she finds it spin up then Bob will measure his and he finds it spin up but so I tried to get this across when I talk about the EPR experiment but I think I didn't necessarily do a very good job so let me try to do a little bit better sure this is like Behrman socks in the sense that if I measure the spin along the X along the Z Direction I know what Bob will get along but I can use these equations okay for the state of one spin to rewrite this state of Alice involve in terms of spin left and spin right in other words Alice could choose to measure her spin along the Z direction and then we know what Bob will get along the Z direction but Alice could also choose independently rather than measuring along the Z direction she could measure it along the x direction and it turns out go through the math again 50-50 chance I get the plus minus signs right here but this is equivalent to spin right for Alice spin right for Bob plus one over square root of two spin left for Alice spend left for Bob which means again modulo minus signs which means that had Alice chosen to measure the spin along the x axis instead of the z axis it would still be true that she knows what bob is going to get and the problem with that the reason why that is profound and and and weird is that well what Einstein in Podolsky and Rosen said is look I don't know what alice is gonna measure I don't know what answer she's gonna get but I can imagine that she measures something and she's gonna get something okay let's say she gets spin up when she measures in the Z Direction then if I refuse to countenance the possibility of spooky action at a distance I instantly know what Bob's gonna get he's gonna get spin up okay therefore EPR say it must have always been the case that bob was going to get spin up if he chose to measure the Z spin of his particle because Alice reveals that you know there must be some pre-existing reality is what EP are saying about Bob's sorry but the Z component of Bob spin they needed it was spin up if we were in the world where when Alice measured she was going to get spin up and then they also say look Alice could have chosen to measure the X spin she would have gotten some answer let's say it's spin right okay she gets an answer is all the point it doesn't matter what the answer is let's say it's been right then instantly we know what Bob is gonna get no signal can travel to Bob therefore it must have always been the case the bob was going to get spin right so EPR say the whole point of their argument which is often missed by people who talk about the argument the whole point of their argument is it must be the case if you don't have spooky action at a distance that it was simultaneously true that bob was always gonna get spin up if he measured in the Z direction and he was always gonna get spin right if he measured in the X direction but quantum mechanics says that can never happen that's the uncertainty principle you cannot have a state of your particle in which with a hundred percent certainty you will always get spin up if you measure along Z and spin right if you measure along X ok so that's why they say quantum mechanics must be incomplete and you see that this relationship is deeper than simple classical correlation because the whole point of the socks is that there is some fact of the matter about what the socks are right you don't know what color Bertil men's socks are but he knows I mean it's known someone could know if you did know you wouldn't be changing the condition of the world you wouldn't be collapsing the wavefunction or anything like that quantum mechanical entanglement is a deeper and more profound phenomenon than that there is no fact of the matter about whether Alice's spin is up or down before she measures it there's also no fact of the matter about whether it's been right or it's been left before she measures it and when she makes a measurement we instantly know something new about what Bob is going to do he doesn't know it so he didn't send him a signal but we instantly know something about it again this is not something that has any analog in classical physics okay that's why you need that's why Bell's Theorem says that unless you do something like super determinism or many-worlds you need some non locality in your theory of physics and there's no non locality needed to understand Bertil Minh socks so there is a difference between quantum entanglement and classical correlation that's important okay another question was how do we entangle particles and this is a very important question actually you should have talked more about this because this gets into the question of you know is there a difference between entanglement the process of entangling versus simply interacting is it true that every time two quantum systems interact with each other they become entangled or not right so no it is not true that every time two quantum systems interact with each other they become entangled the point is that you entangle only when the two quantum systems interact in such a way that if you think about either one of those quantum systems as a superposition of different possibilities then the two terms in the superposition interact differently with what you are interacting with okay so the classic example not control the whole thing again but Schrodinger's cat you know we had the alive cat there's the alive cat and then we had the sleeping cat let sleeping cats lie here and the point was that a photon in the environment could come along in such a way that if the cat were alive the photon would hit the cat and be absorbed whereas if the cat were asleep the photon would just go on its way that is why the cat becomes entangled with the environment if the cat is purely alive it's still a quantum mechanical system but every single I say alive and alive in a sleep awake if the cat is purely awake it's still a quantum mechanical system but every photon that interacts with it interacts with it in the same way so no entanglement happens the photon is either absorbed or not is only when the cat is in a superposition that the photon interacts differently with one term in the superposition versus another one that's when entanglement happens so if you like think of an electron falling in a gravitational field okay rather than a magnetic field the electron can be used superposition spin up and spin down it's clearly interacting with the gravitational field right because it's falling it's being pulled down but both the spin up part and the spin down part fall exactly in the same way therefore there is no entanglement between the gravitational field and the electron okay that's the difference between entanglement and interaction interaction can interact with the whole thing all the terms in the superposition in the same way no entanglement or it can interact differentially with different terms in the superposition and that's when you get entanglement so how do you actually get it how do you do it well there's many different ways to do it if you have two electrons you can zap lasers at them using entangled photons and sort of transfer the entanglement to the electrons let me think of my favorite example which is the decay of the Higgs boson I think I didn't talk about this to you folks I give this example all the time I give talks on quantum mechanics so the Higgs boson is a spin zero particle so that means that it has no spin at all it just sits there right it's there it is not spinning at all we discovered the Higgs boson in nineteen 2012 I wrote a book about it I'm not gonna show you the book but there it is it's a particle at the end of the universe one of the things about the Higgs is it can decay it can decay into let's say an electron and a positron a plus and a minus okay and of course they're gonna go off in momentum back to back already that's entangled right this was the original example we gave of two entangled particles we had them come in and scatter so that they were not entangled when they came in but they scattered they were instantly entangled because different ways in which they scatter matter right if it one scatters along one direction the other scatters along the opposite direction boom they're entangled but this example is a little bit more clear so the spin of this is spin zero there's no spin at all and let's say these like the Higgs boson is just sitting there so it decays into electron and positron they both have spin but angular momentum is conserved okay so it must be the case that whatever the spins are of the electron and the positron they have to be anti aligned they have to be going different directions so it could be and there's a 1 over square root of 2 out here it could be that the electron that the positron is spin up and the electron has spin down or it could be that the positron is spin down and the electron has spin up but it is some entangled superposition of both so it is trivially easy to make entangled particles just let a Higgs boson decay there are easier ways to do it than that actually pi ons or other particles that would decay into two spinning particles also in fact pi ons are probably a very easy way to do it because a pi on is easier to make then Higgs boson they're much lighter easier to produce and they became two photons moving back to back and photons are polarized either spinning clockwise or counterclockwise along the axis they're going on not perpendicular to it but the same exact kind of thing holds the polarizations the spins of the PI ons have to add to zero the spins of the photons have to add to zero since the original PI on zero spin so it's not at all difficult to make electron to make entangled particles finally along this line you might say what about the double slit experiment the double slit experiment remember I'm not gonna do the thick lines again sorry but you had the source and then it shot waves of electrons and they either went through one or the other then they saw an interference pattern here like that so there's a it's very common question why doesn't the electron going through the slits count as a measurement all by itself we said if you added a measurement device to look to see which slit the electron went through then you would collapse the wavefunction you break the interference pattern but why doesn't the why doesn't this list themselves detector why don't they count well the answer is because the electron does not become entangled and to be perfectly honest in the usual way of talking about it that's just assumed the assumption is the slits are the barrier whatever is a big heavy thing and the electron disclaims is off of it without becoming entangled okay now you can do better than that there's this there's a wonderful little book by yuckier Aronoff is a famous physicist our own of and oh no I'm gonna forget the other person's name well he has a co-author I'm very sorry co-author roar lick I think it's world like our own oven or lick a book I think it's called quantum paradoxes and they ask all these crazy questions about quantum mechanics they resolve the paradox is it's a way to teach you some of the more advanced features of the foundations of quantum theory and one thing they do is say well what if we put our slits on wheels right like what do we put the double slit experiment we put it on wheels so that when the electron goes through it could actually nudge the momentum of the slits like the slits are a quantum mechanical system they have a wave function they have a center of mass the momentum the whole bit and then while you can actually go through the measurement and you can figure out you can go through the calculation rather I think it'd be very hard to do it in a real experiment but you can go through the calculation and guess what you know it depends on how heavy the barrier is if it's a big heavy thing you Tron's not going to become dad and tangled with it it's gonna be exactly the double slit experiment as we describe it here if it's a light very easily disturbed barrier then the electron will become entangled with it because the momentum of the electron going one way will push the barrier in one direction going the other way will push in the other one so all this is just to say there's a little bit of entanglement that might happen but you can ignore it if you just do the experiment correctly and if you don't want to ignore it you can do the calculation correctly you can put all those factors back in get the right answer you're in good shape ok one tiny question which I'm not quite sure how to answer well let me say I'm quite sure how to answer a question I'm not sure is decorated to the question that it was being asked and the question is what space is the wave function in and what is meant by this of course I said wave functions are elements of Hilbert space this vector space this is set of all quantum states but if you think about this experiment we just did one electron going through the double slits being detected by the detector it looks like the wave function sy is just a function of space and maybe of time also right so that just looks like a function of space why is it in it looks like it the wave function is in space there it is at every point there's a value for the wave function why do you say it's in this big infinite dimensional vector space so two answers to that one is of course it depends on what you need by the word in in is not necessarily a technical term in physics or mathematics element of is a technical term so this idea if you only have one electron and you ignore its spin so all it has is a position then indeed the wave function sy can be thought of as a map as we talked about before the wave function at one moment of time is a map from space r3 to the complex numbers okay for every location in space there is a complex number the wave function of the electron but this does not mean that size in trees that in space it means that it's a function on space that's a different thing this sigh is an element of hilbert space which is the space of all possible wave functions which is infinite dimensional there's an infinite number of distinct perpendicular to each other wave functions that you can imagine writing down so that's the most trivial answer to this question is the difference between being a function on a space and being an element other space but there's more sophisticated construal of what's meant by here when you have two electrons two particles that's saying I'll say particles because I don't want to ignore spins now after having made a big deal of them and previously if we have two particles the wave function as we know is a function of the positions of both of them okay that's the origin of the possibility of entanglement so for that wave function it's a map from two copies of space to the complex numbers right and two copies of space space is just three dimensional Euclidean space r3 that's equal to r6 and indeed for n particles SCI works out to be a map from r3 n to the complex numbers and this is called configuration space for n particles so with his saying is you have some collection of nparticles you tell me the set of all possible configurations they can be in that is the space that a wavefunction is on that is the space the wavefunction maps from that space to the complex numbers and this is why wave functions are not classical fields classical fields you know electric field magnetic field gravitational field all these fields are independently functions of space you tell me where you are in space and I can tell you the gravitational field magnetic field electric field etc the wave function is not once you're beyond one particle you can't just say Here I am at a point in space what is the wave function that's not how it works you have to tell me the configuration of all the possible particles and of course as we discussed sorry trust me as we will discuss very very soon when you go to field theory it becomes much more complicated ok then you have a field all over space and you're talking about the configuration space of the field the space of all possible field configurations is already a big infinite dimensional space and the wave function maps that space into the complex numbers ok so it's a mathematically a little bit more abstract nonsensical but it makes perfect sense you can actually do all the math in it so even at this level of saying what what space is the wave function a function of it's not space ok it's the configuration space for whatever thing you're looking at and we also said by the way that you could equally well think of the wave function of a particle as a wave function of momentum so you can't do both so it's not a function of position and momentum its position of position or momentum and you can do that for n particles also and the that only makes sense because just like the set of positions is three-dimensional the set of momenta is three-dimensional in fact in any space of D dimensions if you have a D dimensional configuration space you're gonna have a d-dimensional momentum space also so that's why you can possibly work out that the wave function lives equally well in either space ok people I I do I did want to save this because people argue a lot about whether configuration space whether this whole thing means the configuration space is real like do we really live in configuration space this big space if you have 20 particles this is a 60 dimensional space already and if you have 10 to the 88th particles you know you have about 10 to the 88th dimensions roughly speaking three times in today's dimensions you really have an infinite number dimensions once you have a field and maybe you have less than that with quantum gravity but anyway I would say you live in Hilbert space that is what our universe has a wave function and that lives in it is an element of Hilbert space it's a different thing than being a function Hall in space okay good good so I think that that's what I wanted to say about entanglement in the nature of entanglement and then we can get into the fun topic of probability in the many-worlds interpretation again this is something that I have talked about before in lectures but we should mention it here probability not I'm not going to add the general notion about the probability with probability in many worlds many worlds interpretation I said that interpretation is not the right word to use for these theories but it's what people use so there you go this is the single biggest challenge well this is one of the two biggest challenges to many worlds I think one challenge is the way in which the empirical world around us which has three dimensional space-time you know planets and tables and chairs and people interacting in almost classical way the emergence of the classical world or many copies of the classical world from this big abstract wavefunction I think that's one big problem and the other big problem is the origin of probability I think both these problems are very solvable that's why I'm pro many worlds but they're absolutely out there these are the respectable problems these are the issues where if you say I don't believe in many worlds interpretation because of this problem I will go well okay you have a right to do that okay if you say I don't leaving the many-worlds interpretation because it's too many worlds that I'm just gonna go alright whatever but probability is a really big problem because in the textbook formulation mechanics when you say I have a state of a qubit that is alpha spin up plus beta spin down and I say I have a probability of observing of observing spin up which is just given by alpha squared and likewise for beta squared I can't write and talk at the same time what that means is something will happen in the conventional textbook to interpretation if something will happen I don't know what will happen I'll get spin up or spin down but I know that if I do it over and over again if I'm a good frequentist about my probability I can define the probability of something as what the fraction of outcomes would be were I to do the experiment an infinite number of times okay that's a frequentist notion of probability that makes perfect sense with respect to the conventional Copenhagen interpretation of quantum mechanics you can imagine measuring a system in this state over and over over again every time you measure it the wavefunction collapses but you can reap repair it in this state so you can manipulate your electron with magnetic fields or lasers or whatever measure it again do that many many times okay that's not what happens in many worlds in many worlds there's no collapse so what I should do is be very explicit here in the conventional Copenhagen interpretation you measure it and then it becomes this is sy 1 + sy 2 is just up and that's it with the probability of alpha squared or it becomes down with probability down is beta square so it becomes one or the other whereas in many worlds so this is textbook in many worlds sigh one equals alpha and then you have some apparatus and an environment can't through the whole thing right so this is the spin then you have an apparatus which is in zero state for the apparatus and zero state for the environment let me actually move this out over here you have the spin in a superposition as before but need to keep track of everything else there you go and then time of evolution just time evolution just means the Schrodinger equation there's no mystical collapses or anything like that and you get to situ we're skipping some steps but it becomes alpha spin is up apparatus measured the spin to be up environment is whatever it was having seen the apparatus measure it to be spin up plus beta spin is down apparatus saw it down environment is whatever the environment would be in the world where the apparatus measured the spin to be down and that's it you're done this is supposed to be the whole many-worlds interpretation there's no words in there about probability of observing anything but empirically our experimental history is that when we measure things we get different probabilities with alpha squared and beta squared being the different possibilities we get so here in this wavefunction both the only thing to say that has the word probability in it is with probability one both spin up and spin down will be observed by somebody okay will be observed by somebody on one branch or the other and so many different attitudes have been proposed for doing this people have said well can't you just use as a postulate that you find yourself in the spin up branch or the spin down branch with probability alpha squared or beta squared that doesn't work there's no such thing as finding yourself in that since there is a copy of you that is in the spin-up branch and there is a copy of you that is in the spin-down branch and you know that will be true with probability 1 ok to the people in these branches these numbers alpha and beta they're just going along for the ride what do they have to do with anything they didn't touch anything if I live here if I think that I'm the apparatus ok so if this is me let me call this the spin up version to me and this is the spin down version of me well I don't know anything about alpha and beta they're just outside you know multiplying my world in the wave function of the universe but who cares why should I defect my epistemic state my state about what I know about the universe ok so the origin of probability at all in many worlds interpretation is a common conundrum and what you might say so one thing is can you say it as a postulate the answer is no you can't because these are the postulates are done the whole idea of many worlds was there are wave functions there in Hilbert space they obey the Schrodinger equation that's it everything else you're supposed to derive ok no room for extra postulates the other thing you could do is maybe each branch has equal probability right there are two you know think about if you were if you lived in a big universe and the universe was so big much much bigger than the universe we actually observe outside the universe was so big that with very high probability there's an exact copy of you living somewhere else so it's a very big universe and someplace bajillions of light-years away here's you and here's an exact copy of you or is this one you and this is the copy which one is you and which one is the copy they're identical which one is you well you don't know maybe you know everything there is no about the universe but you don't know which of these two people you are this is called self locating uncertainty you know everything there is to know about the universe except where you are in it also sometimes called indexical uncertainty so one way of dealing with self locating a certain needs to say well look these are two copies of me in the big universe case they're completely indistinguishable they're identical locally you know there's not like any wristband that says I'm the one on the left okay so I should give fifty-fifty I should give equal credence as we say the credence is your degree of personal belief that you are one or the other you're completely ignorant there's two identical situations give them equal credence therefore equal probability in a Bayesian way of thinking about probability as opposed to a frequentist way a Bayesian thing waiving the probability says probabilities just expressions of our ignorance there is some fact of the matter or there might be but we don't know okay so when we talk about the probability that the Philadelphia 76ers are gonna win the NBA championship whenever the NBA championship is next held okay that's not a frequentist probability they're not going to have the NBA championship an infinite number of times at least not the 2020 NBA championship I'm using this example intentionally because in my book something deeply hidden I said we know that there will be 2020 NBA championship but we don't know who will win it I did not anticipate that we don't even know whether there will be a 2020 NBA championship this makes me very sad sorry about that the point is though we do we use probability talk to discuss things where we don't know what the answer is even though there is only one answer and there's not multiple different copies there's not multiple different ways of doing the experiment over and over again so we talk about assigning credence 'as in these cases Bayesian notions of probability are exactly fitted to that situation where what probability is is just a measure of your ignorance and as you gain more information you can update what your credence is and then therefore you can change the cretan PSA's and the probabilities you assigned to things okay so this doesn't work this equal probability this you know let's give equal credence to all the alternatives doesn't work in quantum mechanics it certainly doesn't work in this example I gave unless alpha and beta or equal to each other or at least equal to each other in magnitude right because that is not the born rule the born rule says the probability is alpha squared or beta squared not 50/50 if alpha was the square root of 0.99 and beta is the square root of 0.01 you don't want to give these equal probabilities also it shouldn't work right we shouldn't be surprised that it doesn't work and let me explain to you why so consider imagine that you have let's say what's the right way to do this yeah let's say that you have two spins okay and they're both in two spins and they're both once again I should really plan out these videos more than I do but they're both in let's call it spin ah one and spin too so spin one is an equal superposition of up and down and so is spin - and I'll explain why in a second 1 over square root of 2 spin up plus 1 over square 2 spin down I think this is gonna work I'm gonna have to change the numbers in a second but okay the point is you arrange the following situation you measure the span along the z direction of spin 1 ok so here you are with two spins psy 1 psy 2 and then you're gonna measure a spin 1 and you're either gonna get spin up or spin down for a spin 1 okay now there are two copies of you two branches of the wavefunction right and you're tempted to say well I give them equal probability right that's that maybe you're tempted to do that let's imagine you're tempted to do that but then you make an arrangement with yourself that if you get spin down you will measure again you will measure spin too if you don't get spin down you don't measure spin 2 so spin 1 is just up here and spin 2 is just remaining the same but now you measure spin 2 and you can get spin up or spin down for spin 2 and now there are three branches to the wave function so if you're really consistent here you want to assign probabilities 1/2 1/2 because there are two branches and you said I'm gonna assign equal weight to all the branches but here there are three branches so you should assign weight 1/3 1/3 1/3 but what in the world happened here like if I'm on the spin up branch and I start by assigning it probability spin 1/2 and then something happens on another branch of the wave function suddenly I should assign a different probability to my branch that makes no sense I'm in a different world my world does not change its probability just cuz someone else is doing something in another world ok so this idea that you weight the branches equally doesn't work it's just not a starter it's there is no reason to do it in the first place and it also doesn't actually pay okay so let's do a slightly better example an example that does work so the thing that you can do that does work is instead of saying I assign equal branches to all equal probability to each branch say to yourself I assign equal probability to branches that have the same amplitude okay I assign equal probability if and only if the wave function assigns an amplitude to that branch which is equal to the other branch okay so now we're seeing that the amplitudes could matter and that's going to be important so let me change my experiment a little bit here no actually I can keep that I can keep spin to as it is oops white good so spin two is still an equal superposition of spin up and spin down but imagine that spin one is an unequal superposition so that I say this is square root of one third spin up square root of two thirds spin down let's just say I'm picking my experiment here okay and now I start with psy 1 psy two I measure spin 1 I get either it's up or it's down I do the same thing if it's up it's just up if it's down I'm going to measure spin 2 and I can find that that's either up or down and when I'm here either here here before I've measured spin to the amplitude here is square root of 1/3 and the amplitude here is square root of 2/3 so those are not equal and therefore I say I cannot my rule which says I have equal probability if I have equal amplitudes doesn't help me doesn't tell me what the probability is okay just doesn't give me any information also I can't say anything yet but when I measure spin 2 then this square root of 2/3 gets multiplied by that square root of 1/2 and now this is 1/3 and this is 1/3 and therefore and this is well it's the square root of 1/3 the amplitude is the square root of 1/3 and this is still the square root of 1/3 and so now these three branches have the same amplitude square root of 1/3 so now I can assign them equal probability and there are three of them and I assign an equal probability so the equal probability is 1/3 1/3 1/3 and if I do that then I know what the probability was before I measured spin-2 because I want to make it consistent so if the top branch has probability 1/3 then it always did so I put should put a 1/3 here and if the two branches where I measured spin 2 if they both have probability 1/3 then the combination of them better have probability 2/3 right but you notice that now the probabilities have assigned are exactly the amplitude squared that's the born rule ok so in many worlds if you make the argument that branches with equal amplitudes should be given equal credence --is that you're on them you should resolve yourself locating uncertainty by saying not that I give every branch equal likelihood but I get branches equal likelihood when they have the same wave function when they have the same amplitude then you derive that the born rule is the one consistent way of assigning probabilities in many worlds now you might argue well even when the wave function even the amplitudes are equal why should I get equal Cretans and that's a longer story actually that's the whole part I mean if I had started with saying that you probably would've said sure I'll let you do that now you're more a little skeptical but there are very good reasons to do that also so this is a version of an argument that's been given in different forms by Floyd Tec Zurich chip cbins my colleague at Caltech and I wrote a version of this argument that explicitly uses self locating uncertainty which Zurich didn't do and there's another way of proving many-worlds gets the Warren rule from decision theory from David Deutsch and David Wallace so in my mind we mostly understand there's some fiddly things here that I think you can still pick at in terms of the logical structure of the argument but I'm 98% happy that the born rule is the right way to think about probability in many worlds you can disagree that's okay there are principled ways of disagreeing because as I said before you know many worlds is different the metaphysics of all these worlds and all these copies of you is something that we didn't grow up thinking about okay something that we're not we haven't developed the mental technology to really deal with and this what we're trying to do in experiments like this in papers like this like Chip Stevens and I wrote et cetera we're trying to expand our human logic and way of thinking to a world where the wavefunction branches and there many copies of us we will undoubtedly make mistakes along the way and maybe it's all a waste of time because it's not the right interpretation of quantum mechanics but it is science we are doing science we are making progress we can do better than just say well this makes me uncomfortable or I don't like it we actually have to sort of address these problems head-on and figure them out and whatever it is as I said in something deeply hidden many times I am not invested in many worlds being the correct interpretation of quantum mechanics I happen to think it is but if it turns out to be something else I'm fine what I'm invested in is people caring about the many about the correct interpretation of quantum mechanics whatever it is we need to figure it out we there's many reasons we should think it figured out not only in principle it's our job to figure things out but because it will be helpful understanding physics moving forward there's a lot of questions in physics we don't know the answer to understanding quantum mechanics would help us quite a bit

Channel: Sean Carroll

Views: 62,859

Rating: 4.9361534 out of 5

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Id: IB50S4J4c8k

Length: 59min 37sec (3577 seconds)

Published: Sat May 16 2020

Reddit Comments

[u/ddollarsign]

I watched this a few days ago, so apologies if I don't remember this exactly, but: Position and momentum in one dimension are a vector in a 2d space. Complex numbers, which feature in wave function equations, are vectors in a 2d space. Are these facts connected?

Is Quantum Mechanics just trionometry?