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20min chapter
"There is No Quantum Multiverse" | Jacob Barandes
Theories of Everything with Curt Jaimungal
Unraveling Quantum Entanglement
This chapter explores the complex concept of quantum entanglement, distinguishing between separable and entangled states through mathematical analysis. It discusses the contributions of notable figures like Schrödinger and Einstein, examining the implications of entanglement on measurement correlations and the challenges posed by the EPR paradox. The chapter culminates in an exploration of Bell's theorem, emphasizing the non-locality inherent in quantum mechanics and the misconceptions surrounding it.
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Episode notes
Speaker 1
This is what entanglement is about. Entanglement is what happens when you take superposition of states and extend it to composite systems, systems where you've got two systems, not one system anymore that can be the position of two states, but two systems. So suppose that I have system A, and it's the state one, and I've got system B, and it's in state one prime, and that's all I have. Well, then we would say, okay, the composite system is in the state one and one prime. System A is in state one, system B is in state one prime. That's all there is to say. I could also imagine that system A is in state 2 and system B is in state 2 prime. And the composite system is in the state 2, 2 prime. Perfectly fine. I could also imagine that system A alone is in a superposition of 1 and 1 prime. Let's say 1 over root 2 times 1, plus 1 over root 2 times 1 prime. Because in quantum mechanics, when we superpose, we put a number in front, and that number when you square it is supposed to be related to a measurement probability. The 1 over root 2s have the property that you square them, they become halves, and you add them, you get 1. That's probabilities adding up. You can imagine the system A is in the state 1 over root 2, 1, plus 1 over root 2, 1 prime. You can imagine system B is in the state 1 over root 2, 2 plus 1 over root 2, 2 prime. And you can imagine that those are the states of the two systems. Now the composite system is also in a, so the composite system is in the state, well, it's hard to say. Let me call the first state psi the greek letter psi psi is the state one over root two one plus one over two and let's i'm sorry i did my numbering wrong it's one plus two and two and and sorry because a can be in the state one or one prime uh one one or two and state and system b can be in state uh um uh one prime or two prime i got i got it wrong my apologies okay yes so uh so psi will say the greek letter psi yeah trident symbol psi will be one over root two state one plus one over root two state two and psi prime which corresponds system b is psi prime is the state that represents one over root two one prime plus one over root two, two prime. And I can say that the composite system is in the state psi comma psi prime. If I multiply everything out, I'll get four terms. There'll be a term that's one half, one one prime, plus one half, one two prime, plus one half, two one prime, plus 1 half, 2, 2 prime, okay? We would say this is not an entangled state because it's factorizable. I can factorize it into psi next to psi prime. Psi for system A, psi prime for system B, I would say these are not entangled, okay? And you can show that when they're not entangled, they also have statistical independence. If you do measurements on them and compute measurement probabilities, you'll find that they are statistically uncorrelated. But now let me propose a different quantum state. This quantum state is going to be
Speaker 3
1 over root 2, 1, 1 prime, plus 1 over root 2, 2, 2 prime, and that's it. Just
Speaker 1
those two terms. Notice this is a superposition, but over both systems. And now I've got 1, 1 prime in one term, and 2, 2 prime in the other term. And I don't have all those. I don't have the 1, 2 prime term. I don't have the 2, 1. They're not there. I only have one one prime plus two two prime. That's it. I cannot factorize that into two different states. There's no psi for the first system and psi prime for the second that would let me describe them both as having their own states. We would now say those are entangled. Just a quick
Speaker 2
question here. So people who are driving and they're listening to this or people maybe they have a pen and paper and they're thinking, OK, well, I'm going to try to multiply some states to get that. And then they don't. So then they wonder, okay, but just because I tried some, I didn't get to it. Is there a way that I can look at this and then prove that there exists no factorizable component? Yeah, there is. Another way to think about this is
Speaker 1
just it's forgetting to foil when you do arithmetic. So if someone gives you, for example, I've got X plus Y over here and I've got W plus Z over here and I multiply X plus Y as a quantity times W plus Z as a quantity, I get four terms. I get XW plus XZ plus YW plus Y, I get four terms. If I see those four terms, I know I can refactorize them and write them as a thing, X plus Y times other thing, Z plus W. But
Speaker 3
if I only give you XY plus, sorry, not XY, XW plus ZY, I only give you those two things.
Speaker 1
You can't factorize them. They don't factorize into a thing times a thing. This is like an arithmetic example of entanglement, basically. Now, entanglement is usually phrased as something that has no classical correspondence. There is nothing like entanglement classically. In fact, in a 1935 paper, Schrodinger wrote that entanglement was not A, but the feature of quantum mechanics that enforced its distinction from the classical case. You can also link to that paper. I'll send you a link to it. Now, you might go, well, there are certainly some things that are like entanglement. For example, John Bell has this paper Bertelman socks. He talks about this guy, Bertelman, who's got socks, and the socks are always different. If you know what color one sock is, you'll know the other color is not the same. There are systems in which, for example, if I have someone preparing coins, and they always prepare the coins so that when one is heads, the other is always tails. Always. And you discover one is heads, and you know the other is tails. They're correlated. Even if the coins are very far apart when you look at them, if you prepare the coins and send them far apart, and you look at one coin, it's heads, you know the other one even is very far away as tails. This is called correlation. And if you do this over many coins, and the coins are flipped, you don't always know what you you'll get heads or tails, but you know the results will be correlated. Statistically correlated. So statistical correlations certainly happen classically, but entanglement is stronger than that. And that's one of the things that Ayan Sempidolski and Rosen and Bell, they were trying to get at this feature of entanglement that is somehow stronger. You get correlations that are stronger than you would think could be possible on normal, how we usually reason about classical probability theory for systems that are widely separated from each other. To explain the Bell inequality, I have to start with where it came from. So Bell's theorem in 1964 is in a paper called On the Einstein-Podolsky Paradox. He's referring to a 1935 paper by Einstein, Podolsky, and Rosen. So I have to talk about that paper and what they did and then what Bell was supposed to do. You should link to a copy of that paper. People should read it. I don't know how many physicists have actually sat down and read that paper really carefully. But it's – and even Einstein wasn't super happy with it. He was a little upset about how it finally came out. But it is a very important paper to read. You mean the EPR paper? This is the famous EPR paper. Not Bell's paper. EPR paper. Yep. Yeah. It's a very subtle argument, but it basically boils down to this. If I've got
Speaker 3
two quantum systems and they're entangled, I prepare them. I prepare them in some
Speaker 1
state that's entangled. And to get them entangled, something has to be local between them. Either they have to be together initially or you have to send something from one to the other. But some kind of, at some point, local thing should happen in order to get them entangled with each other. And then you send one of the systems very, very far away. This is a weird thing about entanglement. When I measure the first system, usually people do these thought experiments, they imagine Alice and Bob. Alice has the first system, and Bob is very far away with the second system. Alice does a measurement on her system, and she could measure a variety of different observable features. She could measure some observable feature, and when she does it, she will know if you have the right kind of entanglement. She'll know exactly what Bob will get when he does his measurement. She'll measure observable A. She'll get some answer. And then she'll know, I got this answer because of the entanglement. I know what Bob will get. Bob will definitely get this other answer. But Alice didn't have to measure that thing. She could instead have measured a different observable. She could have measured observable A prime, a different observable that is not compatible with A in the same way that position is not compatible with momentum, which is what they originally used in the EPR paper. The original EPR paper was written in terms of position momentum, but these are incompatible observables. They obey an uncertainty principle. If you know one, you don't know the other with certainty. Sure. And so she measures A prime. She can make Bob's system collapse, have a definite answer for a different observable. Okay. So she can steer Bob's system. This is called quantum steering. The word steering was introduced by Schrodinger shortly after the EPR paper. Because it feels like Alice's choice of measurement, she measures A or A prime, is like steering Bob's system. Now, the steering does not send signals. Again, there's this theorem called the no signaling and no communication theorem that shows that Alice cannot deliberately send controllable messages this way. The steering is something more subtle and can't be used to send signals or communication. This is rigorously establishable because of this theorem. Nonetheless, there is some sense in which she is somehow steering Bob's system. She'll measure observable A. She doesn't control what answer she gets. A is uncertain. She could get this. She could get that. Depending on what she gets, Bob will get a certain corresponding thing. But because she can't control what she gets, she can't control what Bob gets. She just knows that once she's done her measurement and gets a certain result she knows that bob if he decided to measure the same thing would she know exactly what he would get if
Speaker 3
allison
Speaker 1
said measures a prime she'll collapse bob's system to a different basis and whatever result she gets she'll know bob if he measured that corresponding observable she'd know exactly what he would get. Now, there's two possibilities as far as EPR, Einstein, Bielskine, Rosenberg, concerned. Either Alice's decision is really changing Bob's system, and Bob's system, when they do the experiments, could be a light year away, and that would seem to be something superluminal, unacceptable happening, faster than light happening. But if not, Bob's system must already have known what answer it would get if he measured the first observable and what answer he'd get if measured the second observable. Because Alice could measure either of hers and depending on which she measures, she can make Bob's system have a definite value of one, measures everyone has a definite value of the other. And if Alice is not really changing Bob's system Bob's system must have known all along what it was going to have they call their paper can and they leave out the the but can the quantum mechanical description of reality be considered it complete they're saying that unless you allow something faster than light to be happening Bob's system must already know the answers it should yield for all of his measurements because Alice cannot possibly, by her choice of measurement, be doing the steering. So what EPR basically establishes that there is a logical fork, either you allow faster than light influences of some kind, causal influences of some kind, or there are hidden extra parameters and the wave function, the
Speaker 3
standard approach to quantum theory is incomplete. There's more to the story than just the wave function. That's where Bell
Speaker 1
starts. In 1964, he says, well, so here's what they said. They said that either you have non-local or some kind of causal influence happening that's going from Alice to Bob, or there's more to the story than just the wave function. There are some hidden variables. Bob's system already knew what answers it would yield. What Bell wanted to do was show that that fork was actually not really there. That there wasn't an escape from the non-local causation. That if you tried to escape the non-local causation the way that EPR argued you should, assume there's more to the story, hidden variables, extra things, that that actually doesn't let you escape. And what Bell did in his 1964 paper was prove his theorem, Bell's theorem. It's an inequality satisfied by, in his view, any theory of hidden variables that is purported to be a local theory. And then write down a simple example of a quantum mechanical system that violates it, that you can go out and do an experiment and check that it violates it. So in other words, Bell is trying to close a possible way out of the non-local causation. EPR says there's either non-local causation or hidden variables. And Bell is saying, well, even with hidden variables, you still get non-local causation. Therefore, quantum theory is simply a non-local theory. And that's the end of the story. That's
Speaker 3
what he did.
Speaker 1
This theorem has gone through a giant game of telephone. People have, so first of all, I should say that the paper was like published in an – I mean, Bell was not – I mean, he was a particle physicist doing this foundational work on the side. And he would caution people against doing foundational work because it was considered very bad for your career, which is really shameful. I mean, physics is supposed to be an intellectual enterprise. And closing down avenues of intellectual investigation, of exploration, is just anti-intellectual. That's a shame. But
Speaker 3
his paper somehow
Speaker 1
eventually became more widely known. And it's like through a game of telephone. And eventually people began thinking that what he did was prove there couldn't be hidden variables. And people would say, oh, you have a hidden variables theory? That's ruled out. Bell said there can't be hidden variables. In fact, the Nobel Prize was given for experimental tests of the violation of the Bell inequality, right? There's this Nobel Prize that was given to Clauser and Zeilinger and was Aspe?
Speaker 3
I think it was Aspe also. And
Speaker 1
it says, if you look at the press release for the Nobel Prize, it says that Bell proved there couldn't be hidden variables. And this Nobel Prize is given because they proved hidden variables are impossible. That's not what Bell showed at all. In fact, not only did Bell not show that, but he said in the paper that's not what he was showing. In fact, he begins the paper by talking about Bohmian mechanics. He says, Bohmian mechanics is, at least for systems of fixed numbers of finitely many non-relativist particles, a perfectly empirically adequate theory of quantum mechanics. It is grossly non-local. That's the words he used for it. Could there be a hidden variables theory that is better behaved than Bohmian mechanics when it comes to locality? And what he was showing was that there wasn't. But he wasn't, his argument wasn't that, okay, well, as long as you get rid of hidden variables, you can keep locality. He thought EPR had shown that if you don't have hidden variables, then you definitely have non-locality. So it wasn't like he was saying, well, it's hidden variables or locality. He was saying EPR said it was non-locality or hidden variables. And in fact, hidden variables, still non-locality. Non-locality is just all you get. That's what he thought he was doing. And this paper has been widely misinterpreted. Bell himself in later writings complained about how people kept misinterpreting his paper, either not reading it carefully or getting it secondhand or I guess like in the opening of what we talked about the textbook that said, oh, Bell showed that the orthodox approach is the only approach, right? I mean, that's not what Bell said. So, okay.
Speaker 3
But then what do we go from here? Bell claimed that he'd shown that quantum mechanics was just non-local full stop. But the EPR paper, the original EPR paper, and Bell's 1964 paper, these are arguments. They're mathematical arguments, especially Bell's paper, which is a theorem. And you have to be very careful
Speaker 1
when you talk about theorems in a physical context. these different arguments. In pure mathematics, a theorem begins with premises. The premises should be rooted ultimately if you have to in whatever the axioms are of the field you're working. And maybe they go back to the axioms of set theory. Who knows? And then you go through a sequence of logically valid mathematical arguments culminate in some conclusion. That's a math proof. And once you've proved it, as long as you have premises that are good, correct premises, and your logic was valid, you have a sound proof. You have a sound deductive argument, and you're done. And if anyone wants to claim there's something wrong, they're going to have to either challenge your premises or challenge your reasoning. And if they're both good, then you're just good. So Euclid proves the infinitude of the prime numbers. That's a great example. You begin with certain premises about how the natural numbers work, and then you have this logical argument that leads to the conjecture that there cannot be a biggest prime number, as long as you're willing to take on the axioms, the standard axioms that we use for arithmetic. But physical theorems, like Bell's theorem, theorems that are about physics, the Cauchin-Specker theorem, the PBR theorem, that's the Pusey-Barrett theorem, there are all these other theorems that are these so-called physical theorems. And these can suffer from another problem. They can succeed as mathematical theorems. They begin with mathematically formulated ingredients that you use in the premises, and then you proceed through rigorous, logical, deductive reasoning, and you arrive at a conclusion. That's the theorem you claim to prove. And that can all be fine. But your theorem is just floating out in math world. Unless it connects to something in the physical world. And that connection is where there can be a problem. So your mathematical ingredients aren't just supposed to be pure math anymore. They're supposed to have physical reference. And I'm sorry, the way a singular is referent. Referent is singular. Reference is plural. They're supposed to have things out in the world that they are representing. And the things they're representing need to be sufficiently rigorously defined. And the connection between those reference and the mathematical representations, the connections, have to be sufficiently rigorous. And if either of those two things breaks down, we have a connection problem. I call it the connection problem. So let's take Bell's theorem. Bell's 1964 theorem is a good example of this, okay? Well, let's even go back to EPR. Let's go back to the EPR paper. EPR paper is a good example. So the EPR paper has premises There are premises to the EPR paper.
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