How to make sense of quantum entanglement

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25:56Okay, so at the end of our last session, I dropped the bombshell that all of physics is wrong. And Daniel is going to let me know why he thinks I'm wrong. So, all right, Daniel, quantum entanglement. How can that coin that you flipped and it's in your hand, but you've covered it, how can it be both heads and tails superimposed at the same time, but doesn't actually end up as one until you move your hand away? Yeah, it's bizarre. And to really probe this, we're going to have to make this setup one step more complicated and even more counterintuitive. And so to explain this, we're going to have to make this setup a little bit more complicated.

26:33We're going to need two objects and we're going to need them to be connected in an important way. We need their fates to be connected. And so instead of having a coin, let's imagine that we have two bags. One has a red ball in it and one has a blue ball in it. OK, and these are just classical normal balls. It's no big deal. And I have one and Kelly has the other. and we pick bags and we go back to our homes. I'm in California and you're in Virginia. And I look at my bag and I see that I have the blue ball in my bag. Now I instantly know that you have the red ball, right? Because we knew there was one blue and one red.

27:05And I know you have the red ball because I'm applying this constraint, this condition, this requirement that there was only one blue and one red. And therefore, if I had the blue, you have to have the red, no magic at all. Right. And also no instantaneous communication of information, right? Like I know instantly that you have the red ball. You don't know that, right? I know that. I know something about a ball that's really far away. And I know that instantly. But again, there's been no instantaneous communication of information, right? I mean, I'll be honest, Daniel, I probably peaked. You are a cheater, aren't you?

27:39I knew that about you. Okay. But I totally understand the scenario you've laid out. All right. Now let's imagine the quantum version. In the quantum version, things are different. Okay. So number one, it's not determined who has the blue and the red. Like in the classical version, I had the blue one the whole time. I just didn't know it, right? In the quantum version, it's not determined. I could have the blue or the red. And so the ball is in this undetermined state. It has a probability of being red and a probability of being blue. And your ball also has a probability of being red and a probability of being blue.

28:15But because we know there's only one blue ball, if I look inside my bag and I see the blue ball, now I know you have a red ball, right? And so then something amazing happens. My ball goes from both possibilities to being blue. And at the same time, your ball goes from both possibilities to only one possibility of being red. Okay, so is the ball both blue and red, or it's just neither until you look, what is the right way to be thinking about this? Yeah, so people like to say that it's both things at the same time. Electrons can be in multiple places at the same time. I think that's confusing in a way that doesn't educate and isn't accurate because it's not true that it's in both locations at the same time or that it has both colors.

29:01It just has the probability to have both colors and it's not yet determined, right? So it doesn't make any sense to say the ball is blue and it's red or it has both colors. It's just that it has both possibilities and we don't know yet. And the universe has not yet decided, right? And the weird thing about this is like, how can you tell, right? All I'm doing is I'm picking a bag. I'm going to California, I'm opening up. I'm seeing it's blue. How do I know the universe didn't actually just decide when I picked the ball which one I had and which one you had? How can I tell that it was really uncertain and there really is random?

29:36And that's the crux of the question. That's what we want to know. like what's really going on inside these bags. And of course, when we do these experiments, there are no quantum versions of balls and bags. And so we do it with particles and we do with particles that are constrained by laws of physics to have some opposite characteristic. So for example, you create two electrons in such a way that one has to be spin up and the other one has to be spin down to conserve angular momentum. And so when you measure one, you'd learn that it's spin up, you know the other one has to be spin down. And that's just because we need a quantum property of a quantum object.

30:11And we've been talking about bags and balls, but there aren't actually quantum bags and balls. And so in the real world, we do this with spin and with particles. So I think when we were talking about quantum internet, you explained some of this there. And I asked how this differs than the Schrodinger's cat example. Is the cat both alive and dead? And I think you told me that that's not even almost about what we're talking about. But I've forgotten because my memory is great. So how does this – is this the same thing? This is the same concept, yes. Okay. The Schrodinger's cat experiment says you have a cat in a box and it's going to be killed based on some quantum process, which is unpredictable, right?

30:56It's truly random. and it does this thing where it tries to connect that quantum process to something classical and intuitive, which is a cat. And so, you know, before you open the box, the cat has a probability being alive and a probability of being dead. And people like to say it's both dead and alive, which I don't think makes any sense. I think it has a probability of being dead and a probability of being alive. And the quantum version of the story is that before you open the box, it has both probabilities and the universe has not yet decided. And the classical view of that, but it says, no, no, no, it's determined.

31:28You just don't know until you open the box. And the question, the deep question is, can we tell the difference? Can we really know if the universe is playing this undetermined random game, which would be really, really strange? Or is there some way in which the universe decides all this stuff in advance and it's all predetermined? We just don't know how it does it. It's just some hidden detail that we're missing that tells one particle to go up and one particle to go down. And so those are the two questions. Is it actually random and quantum mechanical and undetermined? Or is there some hidden variable, some detail which is controlling this that we just are missing or not understanding?

32:09All right. So how could you possibly tell the difference? What experiments do we need to do? Well, the best thing to do is the most obvious. It's like, well, just repeat the experiment multiple times. Start it exactly the same way. Set it up exactly the same way with the balls or the electrons or whatever. And see, do you get different outcomes? Because if you do it the same way, starting with exactly the same initial conditions multiple times, and you always get the same outcome, then you know it's determined. And if you use the same initial conditions and you get different outcomes, then you know that it's not.

32:39That sounds great, right? What a clean experiment. Just test it. The problem is, how do you do the same experiment exactly the same way twice, right? Like, you know, you can never step in the same river twice. If you repeat an experiment the next day, the Earth is in a different place around the sun, and the temperature is slightly different, and you had a different breakfast, and there's like a zillion things that you could never control for. And the quantum philosophy nerds are like really nerdy about all these loopholes. And so that's just impossible. Even at the particle accelerator, like we smash protons together millions of times a second, but it's never exactly the same collision.

33:16The angles are slightly different. The energies are slightly different. And so that's essentially impossible. So we need something more clever. You can't just run the same experiment twice, which is a bummer because, man, that would be awesome. That would be awesome. Okay, but I'm hoping physicists haven't just thrown in the towel. But maybe you all did because you were like, oh, shoot, we're probably wrong. No. Oh, okay. No, we did not throw in the towel. There was a very clever guy named John Bell who came up with an experiment that could tell us the difference between these two hypotheses. One, that the universe has somehow figured this out in advance and we just are missing the information.

33:51And two, that no, it's actually random and undetermined until you look. That these particles, even when they're separated by great distance, somehow decide together at the same moment which one is up and which one is down. And it sounds like impossible to tell the difference, but he came up with this really clever way. And unfortunately, there's no like smoking gun individual experiment where you can say, I'm looking at the outcome and it proves A versus B, right? This is not like, do unicorns exist? Oh, I found one. Therefore, we know there are unicorns. The results are a subtle statistical correlation across many experiments.

34:30So is it a quantum result? There's a constant probability that you're wrong? and uh briefly you take measurements of these two distant entangled particles and you look at how often you get the same result and if there are hidden variables you can't get the same result more than two-thirds of the time but quantum mechanics allows you to get the same result on these two particles more than two-thirds of the time it breaks that restriction by not determining the result in advance but it's a correlation right it's not like any individual experiment proves it. It's like a pattern among many, many, many runs of the experiment.

35:07So it's a little frustratingly indirect, but it's also mathematically very crisp. And I'm going to try to walk you through as you can get an intuition for what's going on here. Okay. So how does this Bell's experiment work and why is two thirds an important threshold? So imagine I have an electron here in California and there's an electron in Virginia and we've entangled them. So we know if one is up, the other one is down. So what are the possibilities? If I measure up, then Virginia is down. If California measures down, Virginia is up. Okay. So we get opposite spins a hundred percent of the time.

35:35So far, this is not evidence of anything. This just says our particles are constrained. They're entangled, right? And it's important for people to understand really what this means, because a lot of people don't get the simplicity of what entanglement means. The entanglement just removes some possible outcomes. Like without entanglement, I could get up and you could get up. I could get down and you could get down. Entanglement just says, no, those possibilities are zero. So entanglement just removes possibilities and only leaves the ones that satisfy, in this case, like angular momentum. All right, so we know that they have to be opposite.

36:09So far, we haven't learned anything. But remember that also, I don't have to measure spin in the same direction as you do. We have three dimensions of space, X, Y, and Z, right? And I could measure spin along like a Z axis, and you can measure it along a Y axis that's perpendicular. Or, for example, I can measure it in one direction and you could flip your machine upside down. What happens if you flip your machine upside down? Then we expect that we always get the same answer. If I read up, then you're also going to read up. I would have read down on your particle, but your machine is upside down.

36:41So I get up, you get up. If I get down, you get down. So in that scenario, we get the same results 100 % of the time, right? Where our machines are flipped, but also the particles are entangled. So they have opposite spins. Yep. Right? You with me still? Okay. Bell's experiment says, let's get even weirder, folks. Let's pick three axes in advance. So I'm going to pick three directions in space, maybe up and then left, and then also some weird angle in between. Okay? So we pick that in advance. We have our electrons, one in California and one in Virginia. Now I randomly choose which of those three axes I'm going to measure my electron on.

37:24Is it the Z? Is it the Y? Is it the in-between? You're also going to do that. You're going to randomly choose an axis, okay? How do I randomly choose an axis if randomness is so hard to, I can't use the random number generator on my computer, you've told me. That's chaotic. No, you're being persnickety about it, but in a really actually fascinating way that quantum theorists get really nerdy about. And we're going to talk about that later on. And people use like lava lamps and cosmic rays to try to be like really, truly random to make sure they're not being like weirdly influenced by something.

37:56Because that randomness is absolutely essential for this whole argument. So we're going to come back to that point. Okay. And it's going to involve like scripts of Gilligan's Island. It's really weird. That's great. I was hoping that's where this episode would end up. All right. So we each have an electron and we each have three axes and we randomly pick which axis we're going to measure this electron on, right? So imagine that these things actually are determined, that some hidden variable on this electron makes mine be up and yours be down along some axis, right? It's not random. It's not quantum mechanical.

38:28Let's imagine that hidden variables are really at work here. Well, then what would happen? Well, it's all determined in advance, right? I have my three axes. I have my particle. You have your particle. But my particle is actually pointing in some direction. Your particle is pointing in some direction. And so it's all determined in advance. and you can actually enumerate all the possibilities, right? And we have three axes. And so one third of the time, we're gonna be choosing the same axis, right? Like if I choose Z, you're gonna choose Z because we have three axes and we're both choosing randomly.

38:58So a third of the time we choose the same axis, which means that we'll get the opposite results. Like we'll choose the same axis, I'll get up and you'll get down, right? So at least a third of the time, we get the opposite results, which means that we get the same result less than two thirds of the time, right? Okay. So that's what we expect for hidden variables. And that just comes out of having three axes and choosing them randomly. Okay. So what if there aren't hidden variables? What if there's quantum mechanics going on? Gasp. Okay. So what happens if there's quantum mechanics going on? So if there's no hidden variables, if quantum mechanics is at play, then there's something sneaky going on here, which is that then Heisenberg uncertainty principle sneaks in the door.

39:46Heisenberg says that there's some things you can't know simultaneously about the universe, like you can't know the speed and the location of a particle at the same time, right? Well, it also applies to spins of a particle in different directions. So for example, if I measure the spin of a particle on one axis, I can't know it in the other ones. Or if I measure it in some axis, I can't know it on my first one. So there is no like true spin direction of these particles in the quantum mechanical view, right? It's not like there is a true spin and we're measuring along some axis so we get up or down.

40:21It's like scrambled in this weird way. And in the quantum mechanical view, you represent the probability of these particles being spin up or spin down using these complex numbers. It comes out of the Schrodinger equation and it's all determined by these complex amplitudes. And like in many quantum effects, these complex amplitudes can interfere with each other. And so quantum mechanics allows these particles to interfere with each other. It's not like all set up in advance. They dynamically respond to the situation. And quantum mechanics predicts that you should get the same spin around three-fourths of the time.

40:56Now, remember, the hidden variables prediction says you cannot get the same spin more than two-thirds of the time. Absolutely not. Totally impossible. That break logic. Quantum mechanics says, no, no, no. At these weird angles, then three-fourths of the time, you can get the same spin. If you arrange things right, depending on the angles between your axes, you can get the same spin more than two-thirds of the time, up to three-fourths of the time. Okay. So I was totally following you, but adding the Heisenberg uncertainty principle feels like cheating because I don't really understand why that works.

41:29And it's like, okay, but also now we're playing by a totally different set of rules that you don't understand And it makes sense. So could we take a quick break and talk about Heisenberg Uncertainty Principle just for a second? Yeah, sure. So, you know, Heisenberg uncertainty principle is just a way of thinking about the spread of possible outcomes and what's allowed. And the fact that measurements are connected, that you can't measure one thing independently, put that in a box, know it, and then move on to measure something else. It's just a way of thinking about how like the truth isn't totally determined.

42:01And so, and that way it's kind of a shorthand. We can actually do without introducing the Heisenberg uncertainty principle entirely if we could just think about the nature of quantum measurements. And so really all you need to know is that the quantum mechanical prediction for whether my California particle is spin up or spin down along some axis is probabilistic, right? That's the quantum mechanical nature of it. It predicts some probability of this and some probability of that. And yours predicts the opposite. And that's very simple. But if you rotate your axis, so you're not measuring along the same axis as I am, there's a relationship between those probabilities.

42:35But that relationship is different in the quantum version than in the hidden variables version because we have these amplitudes, because we have these complex numbers that are interfering with each other. And Bell realized that as that rotates, the number changes differently for the quantum version than it does for the true everything is determined hidden variables version. And it's because of those complex amplitudes and the way that calculation happens. And it's really fascinating because normally these complex amplitudes aren't things that you can see. But when there's interference, then those results are apparent.

43:09It's sort of like in the double slit experiment. You can't see the probabilities, but you can see them interfering with each other. And so that's roughly what's happening here is that the probability of measuring along one axis is interfering with the probability of measuring along a different axis in a way that gives us a different dependence as the angle changes. And so that comes up with a different prediction. And that's why it's not an individual experiment where you're like, I ran it, I got up and down, and therefore chronomechanics is correct and hidden variables is wrong. It's like I ran it a thousand times and I got the same direction for both particles 72 % of the time, which is impossible in the hidden variables theory.

43:47So it's an average over many experiments. Okay. So at the end of this experiment, we can say that what's happening is definitely random, not chaotic. Yeah. And so there are no hidden variables. That's right. And it's incredibly powerful and broad result. It's saying it cannot be a hidden variable, right? You don't even have to know what the hidden variable is. You can imagine some additional dimension of space, and these particles have some features in that space, and that's what's determining it. No, we don't even have to discover those dimensions. This proves that that cannot be happening. No hidden variable theory satisfies these experiments.

44:23It's really incredible. The consequences are huge. But there's a very important caveat. We've been talking about hidden variables, And Bell's experiment actually only rules out local hidden variables, information that's connected to the particle, a little detail that the electron has tucked into its pocket that's determining whether it's going to be plus or minus, right? Local hidden variables. And this was actually misunderstood for decades. And when we get back from the break, we'll find out why the answer was obscured for so long.

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47:48all right we're back and bell's experiments were misunderstood for decades and daniel's going to explain to us why so this very smart guy john von neumann he's like widely considered one of the smartest dudes in history and he's the guy who showed that heisenberg's matrix quantum mechanics was the same thing mathematically as Schrodinger's wave quantum mechanics, even though those two guys hated each other. Really like a towering figure. And he did this proof, this conceptual proof before Bell's experiments that showed that quantum mechanics couldn't have any hidden variables at all. But it turns out there was a mistake in it.

48:24And people actually argue about like, was it von Neumann's mistake or did people misinterpret what von Neumann was saying? And he really understood it. They don't like pointing out mistakes in genius's work. But for a long time, the lore was that quantum mechanics was inconsistent with any kind of hidden variable. And it was Bell who came up with these experiments that showed, actually what they do is they show no local hidden variables. His experiments can't disprove a different kind of hidden variables, non-local, right? Like global hidden variables. And as a result, they're like interpretations of quantum mechanics like Bohmian mechanics that have these global guiding functions, this pilot wave that tells this particle to go positive and that particle to go negative.

49:07So they are deterministic, but they require this weird non-local coordination between all the particles in the universe in a strange way that's very counterintuitive. But the big picture result for Bell's experiment is not that it shows no hidden variables, but no local hidden variables, which means that quantum mechanics is weirdly non-local, right? Like what's happening here is my particle is collapsing and your particle is collapsing at the same time. It's instantaneous across time and space. Quantum mechanics is non-local. Okay, so Bell proved that there was no local hidden variable. But did we ever convince ourselves that there's no Bohmian global guiding function?

49:54Or could there still, that's still an open question. That's still an open question. Now, these Bell's experiments can't tell us whether Bohmian mechanics is correct and there are hidden variables, but they're global, or there are no hidden variables at all, right? So people often interpret Bell's experiments too broadly. They say, well, there's no hidden variables, but actually they just show no local hidden variables. So if your theory has like weird global hidden variables, yeah, that's still cool. And that's what Bohmian mechanics is. And Bell actually was a strong proponent of hidden variables, right?

50:26And he thought that global hidden variables were the way the universe worked. So it's sort of weird because he's famous for demolishing hidden variables, but he actually believed in them. But he believed in the global version of it. And people have actually done these experiments. These are not just thought experiments. The first ones were done in the 70s. And then they do them in fancier and fancier ways. And they keep the particles further and further apart to test this question of like, is this really happening instantaneously across time and space? And the way they do this is they entangle the particles, and they really do separate them in vast macroscopic distances and then make their measurements at the same time.

51:04So there's not enough time for light to go from California to Virginia to inform my California particle what happened to your Virginia particle. So we know that this really does have to happen at the same time. Well, wouldn't that have to mean that the information is traveling faster than light? Yeah, and so this was a question from a listener who wrote in and asked something similar. Here's Muhammad asking his question. Hi, Kelly and Daniel. I would like to understand that if information travels at the speed of light, then how do we know that quantum entanglement is instantaneous? How do we know that the particles influence each other faster than the speed of light when the measurement itself is kept at the speed of light?

51:44How can we measure them precisely at the same time? when the particles have moved far apart after they are entangled, they are subjected to change in gravitational field. And that seems to me like it is enough to throw everything out of whack. So what kind of witchcraft allows us to ignore GR? All right. And so, yeah, Muhammad is asking, how do we know it's faster than light? And Kelly is asking, doesn't that violate everything I thought I know about physics? I've been listening. And so to answer Muhammad's question, what they do is they bring these things really far apart and they make the measurement simultaneously as they can.

52:21So if they know that they make the measurement within a microsecond, then as long as they're further apart than a light microsecond, further apart than light can go in a microsecond, then they know that the collapse is faster than light. You can't prove it's literally instantaneous, but you can prove that it's faster than light because the particles are separated by more distance than light could go in the intervening time. And so how does that not violate relativity? Well, relativity tells us no information can be transmitted from California to Virginia faster than light. But no information is being transmitted.

52:54Like if I measure my particle and it goes from undetermined to up, then Kelly's particle goes from undetermined to down, but she doesn't know that. There's no information she's gathered there. She just has her particle. She hasn't measured it yet. When she goes to measure it, she sees, oh, it's down, she doesn't know whether it was collapsed or not. People often write in, they're like, what if Daniel collapses his particle? Then Kelly can see that it's collapsed and that's a way to communicate information faster than light. But Kelly has no way to know whether her particle is collapsed or not.

53:25All she can do is measure it and say, oh, I got a minus. She doesn't know she's collapsing it to get that minus or if it was already collapsed. So yes, the quantum wave function is non-local. It extends from California to Virginia and It collapses simultaneously across space, which is really weird. But you can't use it to transmit any information from California to Virginia faster than light. And so it doesn't break special relativity, which feels like a really lawyerly loophole. But it's true. All right. All right. Let's clear up Kelly's misconception here. So my first thought was, well, the information between those two electrons has already been transmitted when they were entangled.

54:07And so no information is being transmitted when one is actually observed because they already were tied, but they don't know if they're up or down yet until they're observed. And so there is, okay, I get it. That's really insightful. Yeah. So what's happening when you're entangling them is you're removing some possibilities, right? You're removing the up, up, and the down, down possibilities. You're only leaving the up, down, and the down, up. However, we're leaving those possibilities undetermined. undetermined. And so when I make my measurement, it crosses one more off your list, leaving you with only one possibility.

54:42So that is happening across space and time. But you're right, the entanglement itself is local, and it was made when the particles were created in these initial states. Hey, Daniel and Kelly, this is Matt. You may remember me from editing audio on your show. I was in the middle of editing this episode when some of the stuff that Daniel is saying started to hurt my brain much in the way seemingly that it hurt Kelly's brain. And I have a question for you. One thing in particular that I'm having an especially hard time understanding is the observational element of quantum physics. How and why does it matter if the state of a particle is or isn't being observed?

55:29Do quantum particles have awareness? Are they shy? What is it about observation that determines or sets a quantum object in place, so to speak? Doesn't this mean that consciousness is somehow related to the state of things, that consciousness somehow dictates or influences reality? And if so, doesn't this indirectly mean that before consciousness evolved in the universe, the universe didn't fully exist? If you could shed some light on this matter, I'd really appreciate it. But I'm fully prepared to be confused about this for the rest of my life. Thanks. All right, great question, Matt. So first, remember that observation is not passive, it's active.

56:17It requires some kind of interaction with the thing you're studying. You want to see a particle, you have to bounce a photon, a probe off of it in order to see where it is. You want to measure the spin of your electron, you have to put it through a magnetic field that acts like a probe and interacts with it. So observation requires interaction. Now, if that interaction yields information, that means that your probe is now entangled with the system. That's because if the way your probe particle behaves afterwards depends on the electron's spin, then your probe is entangled with the quantum system because the spin of the electron determines what the probe does.

56:56So now your probe is part of the system. But if, for example, your probe was badly designed so it doesn't depend on the electron's spin, it doesn't extract information, then it's decoupled and it's not entangled. So it's not about consciousness or shyness. It's about interacting in a way that yields information about the state of the system. Now, whether there's like wave function collapse or not is another question of philosophy. Copenhagen interpretation says interaction collapses the wave function if one of the objects involved is classical, like if the probe is classical. But it also doesn't define what classical means and why quantum objects, when they come together, somehow become classical.

57:38The major alternative, many-world hypothesis, says the interaction entangles you and you become part of the system. So now you only see one outcome. There's no collapse. You're just along one of the branches. But philosophically, the whole thing is kind of a mess. All right, great question. But Kelly, let's go back to your other fun loophole, which is this experiment requires people to choose axes at random and then measure how often something happens over a bunch of random trials. how do we know that those are random? We're testing randomness. It relies on randomness. Is there some circularity there?

58:11Are we on firm ground? And there's a really fun kind of paranoid theory of quantum mechanics called super determinism. And it says, look, the outcome of these experiments relies on those things being random, but what if they weren't? What if Kelly and Daniel were manipulated somehow into choosing axes that give this result, right? Because if they're not random, then the result can't be relied on. And, you know, what if it all is actually determined by something that happened a billion years ago and set the series in motion? And so to try to get around this loophole, they've done really hilarious things like they've made the choice super duper chaotic.

58:49Like they randomly sample scripts from television shows and, you know, like if the letter is greater than K, then they choose this axis and if it's less than G, you know, on the third line of page two, this kind of stuff. And they combine it with lava lamps and cosmic rays to try to get like as random as possible. But in the end, super determinism is something you can never really totally knock down because there's always some crazy paranoid theory you could have that these things are just being like orchestrated by folks in the simulation or super intelligent aliens or something. All right. So all of our best efforts at getting random numbers say the same thing, that Bell's experiment works.

59:32Yeah. Okay. Yeah. And so broadly, the consensus in the community is, yes, the universe really is random at the microscopic level and really is undetermined. And these extraordinarily subtle but very clever experiments reveal that that's how the universe works at the lowest level. And it relies not just on the universe being random and the results being undetermined, but having these tests with particles that are distant from each other, yet quantum mechanics having their fates connected to each other. Okay. And so now nobody should ever be confused again about entangled particles. This stuff entangles your brain for sure.

1:00:09Yeah, it does. Follow-up questions welcome. But it's a moment where you should be skeptical, where you should ask, hmm, how do we know that's really true? I hear this all the time. People are telling me quantum mechanics is random. What experiment really proves it? And you should demand an answer and you should demand an intuitive explanation that satisfies you. because this is something that's true about the whole universe and the philosophical implications are huge and far-reaching. And so before you update your priors and change the way you look at the universe, make sure it makes sense to you.

1:00:41I think my new favorite conspiracy theory is super determinism. Because it includes all the other conspiracy theories. That's right. That's right. All right. Well, thanks very much, everyone, who wrote in asking for an explanation of quantum entanglement. I hope that helped. It sure helped me. Have a good day, everyone.

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