That's the definition of the word. There's Hadamard. And the photons, who knows what they're doing with each other, right? The crucial move is observing that you can entangle degrees of freedom, quantum mechanically. But they're not very big in a useful sense. So it's going to take a long time to get any sort of statistics. So here's a question I want to ask, what do we do with this? 9? Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. This is where we apply our function evaluation. Now what's the problem? It is not a super hard topic. That's a nontrivial-- Ok. Deutsch says actually, I tell you what, give me a million and a half, and I'll do the computation give you the answer. I could be talking about all sorts of things. And then we send them into a color box, and what happens? And I want to spend the first good chunk of the class talking about what entanglement can do, and also what it can't. And the way it works is this. So this is an expensive--. And a, again, up at 0 and not c is down at 2 theta. But we'll talk about that in some detail later on in the course. No questions? Yeah, in a very useful sense, a very real sense. And this is going to be millions of miles away from this guy. And so my input is going to be some wave function, psi n, for these n qubits. Then the electron would have demonstrably not go along the soft path. So let's talk about what that means. Questions? So we're going to have to deal with that on a case by case basis. Any of those combinations, any of these ports would have given the same results, 50-50. Thank you. Imagine we really had this equipment in front of us. It's just a weird thing about electrons. But it only came out hard 50% of the time because we sent in initially white electron. Send to friends and colleagues. Quantum mechanics, this is a course in quantam mechanics. But, from last time, you know that's just some stupid matrix. Every once in a while, I'll post auxiliary readings, and they'll be available on the Stellar website. So this is my system. Now if we measure 0, we know they're the same, and if we measure 1, we know they're different. This becomes minus i times cosine of pi over 2 times the identity, the 2 by 2 identity, up plus i sine of pi over 2 times 0, 1, 1, 0. So here is what's now called the Deutsch-- and it's really the one bit version of the Deutsch-Josza algorithm. Yeah? Are you guys old enough for you can't do this on television? And e to the minus i pi over 2, is minus i. Here's something you can't do. And then we take the first qubit, so there's our two bits, we take the first qubit, we send it somewhere faraway. Sign in. And different books emphasize different aspects and use different languages. And, in fact, it's wildly contrived. If we measure the spin in the z-direction, we will measure either up or down. And the language there is linear algebra. It has three apertures, an in port and two out ports, one which sends out black electrons and one which sends out white electrons. Well, I say neither. Something else is going on here. So remember, we're going to go through a few experiments first where it's going to be very easy to predict the results. And 1, 1-- I perform a NOT on this bit, which gives me 1, 0. In fact, a vector of operators, Sx, Sy, and Sz, that satisfy the same [? Lecture 10: Clicker Bonanza and Dirac Notation. Bob's state will be up, because I chose this one. Maybe electrons are sneaky little devils that split in two, and part of it goes one way and part of it goes the other. And those probabilities cannot be explained through some underlying classical dynamics. So at this point, this isn't very interesting. It's a persistent property. This is an idea which is enshrined in physics with a term which comes with capital letters, the Uncertainty Principle. OK. Right? One chance. Gusto, I like it. This leads me into the next experiment. So before you say something silly, like, oh, it's just electrons, it's 20 kilo mirrors. Oh, lord. They say, let me give you precise experiment that embodies all the weirdness of this. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. And I'll represent that not with some unitary transformation U sub NOT. But thank you, that's a great question. So here's what we'll do. So that's an excellent observation. So if you take a white electron and you send it into the hardness box, 50% of the time it will come out the hard aperture, and 50% of the time it will come out the soft aperture. It's a little bit funny to give these a special name and call them entangled, because the generic state is entangled. So how to compute. And how much we juice you can get out of actually building such a quantum computer. ยป Because I do know several other lecture series that I like very much, but I don't know if they're available on YouTube or publicly. Look, I'm not a professor of spelling. But what's cosine of pi upon 2? Can use, in particular, the quantum evolution of the system to perform calculations that you care about, rather than classical computation? We implement that with our boxes, or however we want to implement it. Yeah? Good. Quantum mechanics. But not every electron is hard or soft. So that the full system, psi, is a superposition of sum over all possible-- sorry, I didn't mean to write that. When we compute the probabilities in general, when we have many terms in our superposition, we're going to get interference effects from different terms in the superposition. Well, it violates them both, but for unitarity, you manage to take a linear combination of these guys, where the two states y are orthogonal, and you take the norm squared. And he's like, give me a million bucks. So for a Stern-Gerlach experiment, we turn on a magnetic field that had a gradient. No. And this one's a little sneaky. So here's the first natural move. Anyway, the U Hadamard does this. And so people have spent a tremendous amount of time and energy looking at these initial electrons and looking with great care to see whether there's any sort of feature of these incident electrons which determines which port they come out of. Or indeed, whether it's running on vacuum tubes? One electron through the apparatus. Did I erase Hadamard? Then they're the same. We really do wait. Is anything better? Doot do do. And similarly, send these black electrons into the hardness box, and here's hard and black, and here's soft and back. AUDIENCE: And this also works if you go one electron at a time? There you go. 6? Better have good funding. If we take this 1 of root 2-- so I'm going to rewrite this in a slightly simpler form-- this is 1/2, 0, times-- 0 times 0 minus 1. So with one call to Matt, to my Oracle, with one call to Matt, which cost me one million dollars, we get the answer to whether it's the same or different. Instead of having bits, let's take quantum mechanical bits. And I'm going to measure at the end, and in particular at the output, the hardness. So it's tempting to think that this is a philosophical question, but it turns out not to be in a way that will be made sharp in about 10 minutes with Bell's Inequality. So this operation has a name because it's going to turn out to be very useful for us. And this is discussed on the Stellar web page. Pyewww. You can also focus on special problems. If we had the barriers out, if the barrier was out, what do we get? And what I mean by measure the hardness is I throw these electrons into a hardness box and see what comes out. I don't know. That's not random. Similar, if we measure here. Again, think through the logic, follow the electrons, come up with a prediction. Controlled-NOT does a really cool thing. [INAUDIBLE]. And here's the quantity we wanted to measure. Behaving like particles in a viscous fluid can help bunches of electrons squeeze through a tight space. Massachusetts Institute of Technology. Similarly, we can build a hardness box, which again has three apertures, an in port. But it may be, more generally, we'll be in some superposition of the states we want to measure. They're never squishy. This is one of over 2,200 courses on OCW. Because it's saying that the statistics, the random statistics for Bob and Alice, are correlated by a classical dynamics rather than independent. ALLAN ADAMS: With equal probability. The first is by the end of the day, we'll talk about EPR that we've picked up on the last couple of lectures. Matt's new to the department, so welcome him. All we've done here is tease out something that was existing in the experiment, something that was disturbing. Cool? MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. And here we get 1 plus 1, which is 0 minus 0, which is equal to minus 0 minus 1. Physics They do not have to correspond to a definite state in the 1, 0 basis. This is so sad. First qubit is 0, in the state 0. That means inside the apparatus, if it takes a pico-second to cross, triumph, right? So the set up is that Matt needs to be able to give me-- Matt provides-- a unitary transformation, a unitary operation, use of f that takes two qubits, x and y to x and f of x plus y. And I want to know, does white determine hardness. And you still don't know if they're all the same, until you get to the very last one.