University of California - Qubits & Uncertainty Principle lyrics

UMESH VAZIRANI: In this lecture, we'll talk about the basic unit of quantum information--qubit. So let's start with a thought experiment. Let's suppose that we want to represent a bit using the state of an electron. So how would we do this? So let's recall something. The energy of an electron in an atom is quantized. Well, what does this mean? What it means is that, let's say you have a hydrogen atom. Then the electron in the hydrogen atom is not allowed to take on any old energy. So it's in one of several discrete orbitals. And it has one of several discrete energy levels. So for example, it's allowed to be in the ground state or the first excited state or the second excited state and so on, depending upon how high its energy is allowed to be. So now if you wanted to represent a bit of information, what we could do is we could make sure that the energy of this electron is allowed to be high enough so that it could be in the ground or the first excited state. But it's not high enough to be in any higher energy state. So here's our situation. So we have our hydrogen atom. The electron is allowed to either be in the ground state or in the first excited state, which we'll now call the excited state. So we could have this --we could encode the bit by saying that the ground state encodes for 0 and the excited state encodes for 1. ------------------------------------------------------------------------------------------ So now it's at the end of the story. Well, no. Because you see, the electron in an atom doesn't quite make up its mind which state it's in. So it ends up being partly in the ground state and partly in the excited state. Now what would we mean by saying that it doesn't make up its mind? So one thing we could mean is that it's in the ground state with probability, say 1/3. And it's in the excited state with probability 2/3. So that might be the kind of thing that might go on. But in fact, quantum mechanics tells us that something else happens. And so what happens is that in fact, the electron ends up not making up its mind whether it's in the ground or excited state, but it ends up in a superposition of ground and excited state, where it has some complex amplitude. Say, alpha, of being in the ground state, and some complex amplitude beta, of being in the excited state. So this is ground and this is excited. Now what do we mean by this? What does it means that there's a complex amplitude of being in the ground or excited state? This is exactly the problem that we encountered last time with the double slit experiment. We can only describe this phenomenon. It's hard to picture what the electron could be doing. But let me just say a few more constraints about what these complex numbers might look like. So it could be, for example, that--so the way we'll write the state of the electron is just by writing it in this notation. We'll say it's in the ground state with amplitude alpha, excited state with amplitude beta, 0, 1, et cetera. --------------------------------------------------------------------------------------------- So let's just do a few examples. So it could be that you were in the ground state with amplitude 1 over square root of 2, and in the excited state amplitude 1 over square root 2. You could also be in the ground state with amplitude 1 over square root 2. And then the excited state with amplitude minus 1 over square root 2. Now, there's something going on here. Why did I choose 1 over square root 2? Well, there's a condition that must be satisfied, which is that the state has to be normalized. So the absolute square of the magnitude of alpha plus the magnitude of beta must be 1. And so you can see that's the case here. 1 over square root 2 squared is just 1/2, so it's 1/2 plus 1/2 regardless of the sign. But now as I said, the amplitude is allowed to be complex. So for example, you might have the state 1/2 plus 1/2i 0 minus 1 over square root 2, 1. So is this state normalized? Well, what's the square of the magnitude?If this is alpha, what's the square of the magnitude of alpha? So remember. Let me remind you about complex numbers. If alpha equal 2, a plus b times i, where a and b are reals, then the magnitude of alpha is just the square root of a squared plus b squared. So in this case, alpha squared would be 1 over 2 squared plus 1 over 2 squared, which is 1/2. And of course magnitude of beta squared is also 1/2, so it's a normalized state. So let's get back to what this is saying about the electron. ------------------------------------------------------------------------------------------------ So here's what quantum mechanics says about the state of the electron. What it says is the electron doesn't make up its mind whether it's in the ground or excited state. It's in some sort of a superposition of the two, where it's in each with some complex amplitude. The complex amplitudes are normalized. However, if you look at it--so if it's in the state alpha 0 plus beta 1, as soon as you look at it, what's called a measurement, then it quickly makes up its mind whether it's in ground or excited state. So it goes into the ground state 0, with probability magnitude of alpha squared, and into the state 1 with probability magnitude of beta squared. And that's what the result of the measurement is. So when you measure the system, you disturb the state. So when you're not looking, the electron is in the superposition of ground and excited. But as soon as you measure it, it quickly makes up its mind and it goes into either ground or excited with certain probabilities. And as you can see, this is the reason why we wanted the state to be normalized, because these probabilities must add up to 1. So we have the condition that alpha magnitude squared plus beta magnitude squared equal to 1. ------------------------------------------------------------------------------------------------- Now, there are a number of questions you could be asking at this point. Of course, the primary one is how could the electron possibly be doing this? Or what does it mean for it to be in a linear superposition? You could say, well, I can imagine what it means for it to have certain probabilities of each outcome. That's just a statement of uncertainty. The fact that we--or lack of knowledge about the system. What could it mean to say that it's in the ground state with amplitude 1/2 plus I over 2? Or that it's in the excited state with amplitude minus 1 over square root 2? Well, there's a lot of very smart people have tried for a very long time to interpret what this means. And they have many ideas about the subject, but nothing that I can say I found very convincing. But on the other hand, what's very easy to say is here's the mathematics of it, and we'll start developing an intuition for this peculiar state of affairs. The second thing that's worth noting here is that the electron has kind of complicated state --the superposition state-- only as long as you don't look at it. As soon as you look at it, it's either in the ground state or the excited state. So think about it. The amount of information it takes to specify the state of the electron when you're not looking, you have to specify two complex numbers. That's an infinite number of bits of information. So it's a very complicated state that describes what the state of this electron is when you're not looking at it. But then nature presents a very simple face when you actually try to look at it. When you look at it, it just pretends to be this very simple system --either ground or excited. It can be represented by a single bit.