qm1.eng.srt

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Stanford University

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In the last quarter we studied classical mechanics

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and Classical mechanics is fairly intuitive 

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The basic ideas are things that you can visualize

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You can picture in your head the motion of the particles the motion of the objects

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It gets a little hard when you start thinking of ten particles all at the same time

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But, nevertheless, the whole thing is based on the concepts

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drawn from pretty much everyday work

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Of course it gets more abstract as you go along

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The crazy French who are obsessed with elegance

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kept making it more and more mathematically elegant

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Lagrangians, Poisson Brackets, blah blah blah

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turn out all to be tremendously important

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The abstraction is not all that difficult to understand

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And the basic concept is the concept of updating 

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Well before that, the concept of a space of state

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What is a state of a system?

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Even before that, the concept of a system

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The closed, isolated system that doesn't interact with anything

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and whose motion and whose evolution whose history you wish to predict

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Classical mechanics is predictive it's deterministic

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In the first concept as I said, 

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after the concept of a system is the concept of a state of a system

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A system can have a collection of different state

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We've talked about very simple examples,

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The first state of the quarter last time

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I just describe for you the most elementary simple system 

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consisting nothing more for example than the coin

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Heads and tails, the only states of the systems are heads and tails 

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They form a mathematical set of states

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Two of them for heads and tails, two points, one state, two states

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And we can draw them as dots 

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This one stands for head, this one stands for tails 

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And, you know, it's easy to picture in your head 

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From there, we talked about equations of motion

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Equations of motion or laws of evolution 

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for the very simple system of a couple of points

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Heads, tails, we formulated some very very simple possible laws of motion

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The simplest being if nothing happens

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If you start with heads you stay with heads, heads, heads, heads

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If you start with tails you stay with tails, tails, tails, tails

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That was a very simple law of motion

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More complicated motion begin with heads you get tails

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If you begin with tails, you go to heads, and just oscillate back and forth

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The basic ideas is the idea of updating 

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If you know everything there is to know about a system in any instant of time

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Then classical mechanics, everything there is to know 

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essentially mean everything that is needed in order to predict the future

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If you know everything there is to know 

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then classical mechanics consist of a set of equations 

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which tell you how to update the state given it's value at given time 

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and go from one time to the next

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That basic idea, for example, well, 

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I don't think I need to describe the example again

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We describe it at length the last time 

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And it's extremely intuitive in going from this very discrete example 

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to a world of continuous motion, indeed we have to abstract, 

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we have to replace the ideas of a small number of state

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a finite number of configuration by an infinite number of configuration 

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For example the motion of a particle along a line 	 

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We have to label the state by points along the line as well as their velocities

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So it does get more complicated but nevertheless

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Classical physics consist of equation which tell you 

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how to update the states in any given time

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And it's completely deterministic

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And as I said, easy to visualize, at least by comparison easy to visualize 

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Because it makes use of the concept of the ordinary world 

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which out brain is hardwired to be able to understand

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Now I always give a little sermon of this point and give it again

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Probably you have heard it a dozen times

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about the way one has to think of physics beyond classical physics

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So we move past classical physics 

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I don't think it can be said enough, it's so rarely said 

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It's that once physics move past the realm of parameters 

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that has to do with ordinary experience

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either objects which are so small, 

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they are far beyond what ordinary physics is all about

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Or velocity which are so fast that far beyond ordinary velocities

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When everyone move out of the range of parameters 

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that we were familiar with in ordinary experience 

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We inevitably run into things we cannot visualize 

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Nobody can visualize the motion of an electron 

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It's just you are not wired for it

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Nobody can visualize four dimensional space or four dimensional space time

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let along ten or eleven dimensions 

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It's not the way physicists have done

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I often read some of the stuff that goes back and forth 

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between some people especially people in this class

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about trying to explain to themselves to each other 

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or ask questions to each other about a certain abstract physical questions 

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And I know very much that a lot of the mistakes a lot of confusion

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happens because of trying to visualize things 

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using old fashion or standard intuition you are hardwired for

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You may think physicists good physicists  

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are especially good for visualizing things

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They are not, I cannot visualize five, six dimension any better than you can

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But I know how to use abstract mathematics to describe it 

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There is no substitutes, there is no substitutes for the process of abstracting 

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and using mathematics to describe the thing 

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which is beyond the ability to directly visualize

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The example that I gave, I'm gonna do it again tonight, 

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I just want you to really focus on it and realize

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you are not going to understand quantum mechanics by trying to visualize it 

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by some funny form of classical mechanics

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What's more, you will always get it wrong

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But just the idea of visualizing space of dimension of different three dimensions 

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In particular I always use the same, okay, 

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how many people here can visualize five dimensions?

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And everybody would say, no

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We can't, of course there will be a few freak() or clown in the class 

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who would say yeah I can do it I can do it

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They couldn't , I know they couldn't 

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They can't visualize five dimensions

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But they I began to realize although you have probably heard it before

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that it's equally hard to directly visualize two dimensions

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Okay I will say to the class how many people now can visualize two dimensions?

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Almost everybody will raise their hands 

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And I would say, no you are wrong 

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Close your eyes and try to visualize two dimensions

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They would say okay I see it I see two dimensional of curved space 

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What they are seeing is not two-dimensional curved space 

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What they are seeing is a two-dimensional surface embedded in three dimension

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Even trying to see one dimensional space 

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One-dimensional space, abstract one dimensional space is just a line

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You cannot see that line in your head 

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without seeing it as a line drawing in a plane

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And the plane you can't see without seeing it as embedded in three dimensional 

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What's going on? I can do it, 

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I hate to tell you this but 

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you are a prisoner of your own neurological architecture

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Your neural architecture is built for three dimension 

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Alright how you get around it?

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You don't get around it by training yourself to see four or five dimension 

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You get around it by training yourself to think abstractly about it

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If three dimensional space there is a point x y and z

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In four dimensional space it's x y z and w

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And you learn to manipulate the symbols 

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It's not to say that after a while 

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you don't start to gain a new kind of intuition for things

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You do, but the intuition is not the process of physical visualization 

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as the same kind as you visualize a wave as the wave in the ocean 

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So I have warn you about that

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And I will tell you quantum mechanics is as abstract as anything can be

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Well perhaps things are getting a little bit more abstract now 

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But again, quantum mechanics is about things that 

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you evolutionarily developed neural structure 

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is not prepared to dealt with directly

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How do you deal with it? Abstract mathematics 

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And relativity, this is also true with relativity

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And it's worse in quantum mechanics   >> end o

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Okay let's begin quantum mechanics 

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with a very very simplistic primitive question you can ask

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What is the state, I think I can ask the most primitive question is

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what is a system?

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Now I don't think I can answer it,

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it's sort of indefinable you thought of the idea of a system

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and I don't think I would try to give it a definition

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but in particular closed system

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A closed system is a system at least temporarily 

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is not interacting with anything else 

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Isolated, it's not interacting with anything else

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And so it can be describe by itself 

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as a lonesome entity with nothing else involved

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In classical mechanics, or it's not even a question of classical mechanics 

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It's a question of classical logic

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The whole logic of quantum mechanics 

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is different than the logic of classical mechanics

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The logic of classical mechanics again begin 

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of course with a system I just describe

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But the next step is the space of state

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The collection of the possible states of the system 

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If the system is a discrete system like a point 

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Or a die, you know the six-faced die

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It can have six states, of course a real die

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a real genuine die in three-dimensional space of course 

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can be in all sort of orientation in infinitely many states

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But what I mean by a die now was I mean in an abstract mathematical die

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which whose state is consist of one of six numbers

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The space of states is a set, one two three four five six 

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This is the space of states of a die 

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This is the space of states of a coin

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Okay again the abstract coin which only has two states heads and tails

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This will be heads this will be tails 

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The abstract space of a point particle moving along a line

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What is that? Anybody know what that is?

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What is the abstract space of a particle moving along a line?

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Well, I will remind you the term of physics space 

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The space consist of the possible position of the particle 

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and the possible momentums, the possible velocities

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And so for the particles moving on a line 

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The possible states of a system form a plane 

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And the plane has a position sometimes call q 

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and the momentum sometimes call p

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And every point on that plane is a possible state of 

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a point particle moving along a line

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What is a set? In this case it's the continuously infinite set	 of two dimension

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In this case it's just two points, in this case it consist of six points

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And now we can start building some logical concepts for classical physics 

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For example, concepts such as OR and AND what do they mean?

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OR and AND are concept which apply to a collection of states

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Let's consider a proposition, proposition such and such

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The proposition could be, the die it's locate, has a value which is even 	 

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It's even value okay?

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That's a subset of the state of space 

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In fact it consist of three of them if I'm not mistaken 

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Three are one, one three and five 

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And another proposition would be the die is odd 

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No did I say odd, odd or even whatever it is 

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Another overlapping set of configuration 

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Let's take another case, let's take two proposition,

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one of which again is the die is odd

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And another proposition is that the die is less than or equal to three

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So let's see, it's one, three, five, it's two four six

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And less than or equal to three is this set over here

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So now there are overlapping sets 

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And we can ask concepts of AND and OR

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For example the proposition that it is that 

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the die is both less than or equal to three

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And odd is the intersection of these two subsets

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One and three are odd are less than or equal to three

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The concept of AND, AND in this subsets, the mathematical concept

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is the intersection of two sets

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It's the intersection of two sets, so it satisfied two things

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That's the concept of AND

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Given two proposition, each proposition is a collection of states

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The concept of AND is intersection 

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What about the concept of OR?

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OR becomes a concept of union 

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What does it say here, either it's less than or equal to three OR odd

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It's the inclusive OR incidentally 

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Inclusive OR means it can be both

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Alright that is not the intersection of two sets 	 

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It's the union of two sets, and union meaning everything in two sets 

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Everything here is either equal or less than three	OR it is odd, inclusively

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both being allowed

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So the inclusive OR translated into union 

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And the concept of AND is the intersection of sets

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That's the logic of a classical reasoning to a logical stance 

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Here's a concept, two sets are completely none overlapping, 

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this one and that one

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That simply means the AND statement of false

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There is no state in there for which both statements are true

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So the whole logic that it's connected with the space of state being a set 

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And it's the logic of set theory, we are not going through the logic of set theory

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But it's important for me to tell you right now that 

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the logic of quantum mechanic is different 

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It's not based on the idea that the states of a system is set 

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The state of a system is something entirely different and only in certain limits

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only in certain limits where system behaves classically 

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does whatever a space of states of quantum mechanics

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Whatever it is, 

252
00:19:48,047 --> 00:19:52,642
does it approximately reduced to the concept of a mathematical set

253
00:19:54,106 --> 00:19:56,653
In fact if I just tell you what it is, it's vector space 

254
00:19:57,258 --> 00:19:58,778
And we will talk about that 

255
00:20:00,358 --> 00:20:01,831
Alright but before we do, 

256
00:20:01,863 --> 00:20:10,814
before we start to talk about the abstract mathematics of abstract spaces 

257
00:20:10,927 --> 00:20:17,785
And spaces of states in quantum systems, let's talk about experiments 

258
00:20:17,875 --> 00:20:23,028
And describe the simplest possible system in quantum mechanics 

259
00:20:23,097 --> 00:20:30,251
It's the coin of quantum mechanics, a system with two states, heads and tails

260
00:20:35,523 --> 00:20:38,931
We have to completely erase the blackboard to start over again 

261
00:20:42,539 --> 00:20:46,643
So we have some systems, whenever we measure that system 

262
00:20:46,747 --> 00:20:54,315
We measure something about the system, namely analog it's heads or tails 

263
00:20:54,387 --> 00:20:56,971
We either get heads or tails

264
00:20:57,059 --> 00:21:03,979
Alright let's give that a mathematical notation 

265
00:21:06,019 --> 00:21:10,452
I'm not going to call it a coin, I'm going to call it a qubit

266
00:21:11,307 --> 00:21:17,299
A qubit is a system which has two states, and qu stands for quantum

267
00:21:17,389 --> 00:21:24,051
The analog for classical physics is call the c-bit, classical bit

268
00:21:24,147 --> 00:21:27,035
And it's the simple idea of a coin, heads or tails

269
00:21:27,150 --> 00:21:34,267
We gonna see that a qubit is a much more interesting well complicated for sure

270
00:21:34,342 --> 00:21:39,763
But a much richer idea even though it has some sense as two states

271
00:21:39,875 --> 00:21:46,297
Okay so the qubit, the qubit on it's measure - 

272
00:21:46,378 --> 00:21:50,874
I'm gonna talk about measuring in a moment - is either heads or tails

273
00:21:53,298 --> 00:21:54,872
But let's describe it in another way 

274
00:21:54,944 --> 00:21:58,154
And give it a freedom the mathematical degree of freedom 

275
00:21:58,224 --> 00:22:01,592
And I'm gonna call that mathematical degree of freedom sigma

276
00:22:02,288 --> 00:22:09,503
Sigma is the traditional name for the degree of freedom describe in a qubit

277
00:22:10,231 --> 00:22:15,456
And sigma can either take on the value one, or it can take on the value of -1

278
00:22:15,560 --> 00:22:21,439
It's one when qubit is heads and minus one when qubit is tails

279
00:22:24,111 --> 00:22:27,175
And there is no more content in it - other than heads and tails

280
00:22:27,279 --> 00:22:30,655
But it allow us start being able to write equations for something 

281
00:22:33,495 --> 00:22:37,376
You can also describe it some other way by a picture

282
00:22:37,503 --> 00:22:42,303
I'm gonna describe sigma as one by an arrow pointing up

283
00:22:42,391 --> 00:22:45,471
Sigma equals minus one by and arrow pointing down

284
00:22:45,543 --> 00:22:51,479
No implication here, not anything geometric here, just notation

285
00:22:51,583 --> 00:22:54,426
Notations, three different notations, heads and tails, 

286
00:22:54,520 --> 00:22:58,035
sigma equals one or sigma equals minus one 

287
00:22:58,103 --> 00:23:01,111
And arrow pointing up and arrow pointing down 

288
00:23:03,783 --> 00:23:12,279
Okay let's imagine an experiment, an experiment involve more than one system

289
00:23:12,991 --> 00:23:17,479
It involve a second system, the second system is call the apparatus

290
00:23:18,047 --> 00:23:23,559
In classical mechanics we never bother thinking too much about details

291
00:23:24,568 --> 00:23:26,752
Unless we happen to be experimental physicists 

292
00:23:27,281 --> 00:23:30,447
and really going to the laboratories and doing experiments

293
00:23:30,535 --> 00:23:37,023
We sort of don't worry very much about, at least in a deep sense 

294
00:23:37,991 --> 00:23:41,527
We think the system can be describe completely without worrying

295
00:23:41,583 --> 00:23:44,223
how you detect it how you measure it or anything else

296
00:23:44,823 --> 00:23:46,342
But in quantum mechanics, 

297
00:23:46,542 --> 00:23:55,022
one can not get away ignoring the concept of a apparatus

298
00:23:55,702 --> 00:23:59,963
So let's introduce an apparatus, and I've tell you what an apparatus

299
00:24:00,051 --> 00:24:02,742
We are going to make a mathematically abstract apparatus

300
00:24:02,830 --> 00:24:05,529
I'm not gonna tell you how it works, black box

301
00:24:06,465 --> 00:24:12,745
It's a black box I'm gonna call it A, A for apparatus 

302
00:24:15,113 --> 00:24:19,810
It's a box that has a little window, there is the window

303
00:24:22,688 --> 00:24:26,536
It also, it really is a box, it's like a carten(), 

304
00:24:26,681 --> 00:24:34,025
a carten getting mail which says this side up, this side up

305
00:24:36,864 --> 00:24:38,856
You want to stand up right?

306
00:24:39,922 --> 00:24:43,512
Here is a little window, by window I mean screen 

307
00:24:44,080 --> 00:24:45,744
And numbers appears on that screen 

308
00:24:47,008 --> 00:24:51,328
And it has some sort of detector on the end of a wire 

309
00:24:51,792 --> 00:24:57,380
that senses the qubit

310
00:24:57,492 --> 00:25:01,584
It senses the qubit is plus one or minus one 

311
00:25:02,895 --> 00:25:08,149
So you start with your qubit and you don't know what state it's in 

312
00:25:08,191 --> 00:25:11,556
The experiment is to determine what state it's in

313
00:25:11,612 --> 00:25:21,044
So the qubit begins with an unknown state, unknown

314
00:25:21,716 --> 00:25:24,044
We don't know it's plus one or minus one 

315
00:25:24,100 --> 00:25:27,892
We are gonna do an experiment , the experiment

316
00:25:27,924 --> 00:25:30,732
As I said, I'm not gonna tell you how the detector work 

317
00:25:30,820 --> 00:25:34,440
It's not important for us, it's only important the detector do exists

318
00:25:34,559 --> 00:25:39,688
We are gonna, we will talk later about the process of detection, but not for now

319
00:25:40,546 --> 00:25:44,305
Let us central over here, central over the qubit

320
00:25:46,388 --> 00:25:47,738
What does it do?

321
00:25:47,851 --> 00:25:51,641
A number lights up in here, if it's plus one

322
00:25:51,754 --> 00:25:59,039
Oh, first of all, before we do the detection, the apparatus is in the neutral state

323
00:25:59,153 --> 00:26:02,287
The blank state, no number there at all

324
00:26:02,401 --> 00:26:05,539
So let's just call that a blank configuration of the apparatus 

325
00:26:05,651 --> 00:26:11,891
We take the apparatus over, close enough to the qubit to react with it 

326
00:26:12,004 --> 00:26:17,114
And it's a good faithful apparatus it will register a number here

327
00:26:17,224 --> 00:26:19,879
And the number would either be plus one or minus one 

328
00:26:21,264 --> 00:26:27,418
Plus one or minus one, depending on which way, whether it's heads or tails

329
00:26:29,162 --> 00:26:30,892
That's the act of the measurement 

330
00:26:32,263 --> 00:26:36,576
Now you may or may not be there to look at the apparatus this is a secondary 

331
00:26:36,692 --> 00:26:42,748
The apparatus has detected and recorded what is going on over here

332
00:26:42,899 --> 00:26:49,073
So that's something that, in classical mechanics of course we do the same thing

333
00:26:49,186 --> 00:26:51,399
There is nothing in particular quantum mechanical about this

334
00:26:51,511 --> 00:26:53,607
But we tend never to think about it, 

335
00:26:53,753 --> 00:26:57,723
then formulate the basic principle of quantum mechanics, oh, classic mechanics

336
00:26:58,673 --> 00:27:04,307
Okay so that's the experiment that measures the state of 

337
00:27:04,455 --> 00:27:07,628
Or it measure whether sigma is plus one or minus one 

338
00:27:07,741 --> 00:27:10,267
But we can also think of it as another way

339
00:27:10,344 --> 00:27:15,034
We can think of it not so much as a measurement but as a preparation

340
00:27:15,149 --> 00:27:18,127
The preparation of the system

341
00:27:19,055 --> 00:27:25,095
Here's the idea, supposing we in fact do measured sigma to be plus one 

342
00:27:27,421 --> 00:27:30,002
Supposing that is the outcome of the experiment 

343
00:27:30,117 --> 00:27:34,899
That we can erase the one that it wasn't, and start over 

344
00:27:35,014 --> 00:27:37,855
And then say let's back off 

345
00:27:41,885 --> 00:27:47,216
We set the apparatus, we set it means start it over in the blank state 

346
00:27:47,330 --> 00:27:51,279
And then we do the experiment again 

347
00:27:52,267 --> 00:27:55,044
Let's say we do it fast enough so that 

348
00:27:55,157 --> 00:27:57,062
there wasn't many times for anything to happen

349
00:27:57,173 --> 00:28:04,046
What happens? What happen is that the detector records again

350
00:28:04,126 --> 00:28:09,862
And if it's a good detector, and the system hasn't had time to change very much 

351
00:28:09,973 --> 00:28:12,619
It will simply record the same number again 

352
00:28:12,732 --> 00:28:15,541
It will confirm the previous experiment 

353
00:28:15,650 --> 00:28:20,672
This is important, the experiment can be confirmed, again

354
00:28:20,787 --> 00:28:24,526
No difference between classical and quantum mechanics in this respect 

355
00:28:24,603 --> 00:28:27,226
We just emphasize it in quantum mechanics 

356
00:28:27,339 --> 00:28:29,636
Experiment can be confirmed

357
00:28:30,696 --> 00:28:33,951
We can do it again and again, and again, and again

358
00:28:34,065 --> 00:28:38,343
And we will continue to get the same answer that we got in the first place 

359
00:28:38,459 --> 00:28:42,564
So we can think a little bit more differently the first experiment

360
00:28:42,712 --> 00:28:50,793
Instead of saying that it determines whether it is heads or tails 

361
00:28:50,899 --> 00:28:55,699
We can say it prepare it with a state which was in this case heads

362
00:28:55,804 --> 00:29:00,314
After the first detection now known to be heads

363
00:29:00,429 --> 00:29:02,615
it's been prepared in states heads

364
00:29:02,728 --> 00:29:05,413
We can do the experiments over and over and over again

365
00:29:05,512 --> 00:29:06,949
And always get the same answer 

366
00:29:07,779 --> 00:29:12,267
So the devices that measures

367
00:29:12,411 --> 00:29:16,512
is also device that prepares the system in a given state

368
00:29:24,186 --> 00:29:25,954
Of course the same thing would be true 

369
00:29:26,075 --> 00:29:31,442
had the device determines in the first round the signal was minus one

370
00:29:31,555 --> 00:29:35,259
Then it would continue to detcet the same thing

371
00:29:35,383 --> 00:29:41,165
Of course there can be bad detectors, not all detectors are well made

372
00:29:41,278 --> 00:29:44,437
Not all detectors work the way you want them to be work

373
00:29:45,648 --> 00:29:47,295
In which case it's just wasn't a good detector 

374
00:29:47,407 --> 00:29:48,826
It didn't do what you wanted it to do

375
00:29:48,938 --> 00:29:53,499
>> would this heads and tails have to do with this point

376
00:29:53,608 --> 00:29:56,456
>> what's the purpose of having a heads and tails 

377
00:29:56,565 --> 00:29:58,378
It's just two possibilities

378
00:29:58,456 --> 00:30:03,791
You wanna know what actually has to do with a heads and a tails?

379
00:30:06,961 --> 00:30:08,459
It's just two possibilities

380
00:30:08,587 --> 00:30:11,913
I just used that for familiarity

381
00:30:14,388 --> 00:30:16,178
The marines() do heads and tails

382
00:30:22,450 --> 00:30:24,800
Okay now we are gonna do something new

383
00:30:24,922 --> 00:30:26,531
We are gonna do a new experiment 

384
00:30:27,880 --> 00:30:30,934
First we are gonna start with a preparation

385
00:30:31,890 --> 00:30:39,833
The preparation determines the spin was, what was a spin, I call it a spin

386
00:30:39,946 --> 00:30:42,361
I'm giving away, it is a spin

387
00:30:42,473 --> 00:30:46,851
The qubit is in the state plus one alright?

388
00:30:46,961 --> 00:30:51,848
So the qubit in the other notation is an arrow pointing up

389
00:30:51,966 --> 00:30:55,446
We've prepared, we now know, we can do a few times to check

390
00:30:55,558 --> 00:30:59,567
But once we check it we now know that this qubit is up

391
00:30:59,678 --> 00:31:03,230
And if we continue to do it, we will continue to get the same answer

392
00:31:03,345 --> 00:31:05,365
Now we are gonna do something funny

393
00:31:05,474 --> 00:31:09,084
After we've determine or prepared the qubit

394
00:31:09,161 --> 00:31:13,610
We are now gonna take the detector and turn it over

395
00:31:15,687 --> 00:31:30,020
A... I don't know, this side up?

396
00:31:32,121 --> 00:31:44,215
We turn the detector over and now we are going to probe the qubit again

397
00:31:44,328 --> 00:31:46,568
And what have we learn in this case?

398
00:31:46,680 --> 00:31:54,596
We discover, we discover that instead of getting plus one we get minus one

399
00:31:54,745 --> 00:31:59,845
By turning the detector over we get the opposite answer

400
00:31:59,990 --> 00:32:02,369
What is this telling us?

401
00:32:02,515 --> 00:32:09,636
What is this telling us about the nature of this system?

402
00:32:09,753 --> 00:32:18,803
What is it telling us? Of course then if we take our upside-down detector 

403
00:32:18,915 --> 00:32:21,649
and continue to do the experiment over and over 

404
00:32:21,760 --> 00:32:23,855
We will continue to get minus one 	  

405
00:32:23,926 --> 00:32:27,169
Of course what happen let say if we do it thirty five times 

406
00:32:27,278 --> 00:32:29,958
We turn the detector back up right?

407
00:32:30,034 --> 00:32:31,842
Then we get plus one 	 

408
00:32:31,954 --> 00:32:35,926
What we were learning is whatever this system is  	 

409
00:32:38,696 --> 00:32:43,503
This particular qubit has a sense of directionality to it

410
00:32:44,589 --> 00:32:48,033
Then when I drew it as up versus down

411
00:32:48,182 --> 00:32:56,936
There really was a sense in which the orientation is being distinguished

412
00:32:57,056 --> 00:33:01,625
By turning the detector upside down we interchange up and down

413
00:33:01,734 --> 00:33:04,547
Well that's what I normally mean by up and down

414
00:33:04,659 --> 00:33:07,517
Turn it over, we got it upside down

415
00:33:07,625 --> 00:33:11,213
And it changes the sign of the qubit

416
00:33:11,319 --> 00:33:16,035
That's the indication the qubit has some directionality in space

417
00:33:16,149 --> 00:33:18,360
Spatial direction associate with it

418
00:33:18,502 --> 00:33:22,945
And not every qubit has spatial directionality with it

419
00:33:23,049 --> 00:33:26,562
But by doing this experiment we discover that whatever this qubit is

420
00:33:26,676 --> 00:33:29,982
It really does have a sense of orientation

421
00:33:30,089 --> 00:33:33,028
An upness and downness in space 

422
00:33:34,969 --> 00:33:37,695
So that's a piece of interesting information

423
00:33:37,843 --> 00:33:41,297
We might begin to suspect that it's a vector 

424
00:33:42,571 --> 00:33:46,576
We might begin to suspect since it has a directionality in space 

425
00:33:47,404 --> 00:33:51,454
It's a vector, the one that is pointing up 

426
00:33:51,569 --> 00:33:53,279
is different to the one that is pointing down 	

427
00:33:54,811 --> 00:33:57,433
We can make... well yes alright... 

428
00:33:57,545 --> 00:34:03,989
so that's the basic idea of a qubit which also have a sense of directionality

429
00:34:04,098 --> 00:34:11,783
Like this pen then our detector when we measure it up then we measure it down

430
00:34:11,894 --> 00:34:14,439
But now we are gonna do something else

431
00:34:14,553 --> 00:34:18,672
Oh we might, here is what we might believe, here is what we might believe

432
00:34:18,780 --> 00:34:22,447
Given that the detector has an orientation to it 

433
00:34:22,517 --> 00:34:24,124
and it does have an orientation to it

434
00:34:24,198 --> 00:34:27,376
There is right side up and wrong side up

435
00:34:28,436 --> 00:34:30,618
In fact in the detector we might even think 

436
00:34:30,727 --> 00:34:35,969
that something which itself is an orientation

437
00:34:36,076 --> 00:34:40,843
Something built into the detector which has an orientation in it	

438
00:34:40,952 --> 00:34:44,292
And you know what we might think, we might think that 

439
00:34:44,367 --> 00:34:49,486
this detector is detecting the value of a component of a vector

440
00:34:50,289 --> 00:34:55,057
along the direction of the detector itself

441
00:34:56,172 --> 00:35:00,882
So when the detector is right side up

442
00:35:00,996 --> 00:35:02,697
And it's detecting the things here 

443
00:35:02,807 --> 00:35:08,579
it's detecting the component of some vector in the plus direction 

444
00:35:08,687 --> 00:35:11,292
When you turn the detector over, 

445
00:35:11,405 --> 00:35:14,673
it's measuring the component of the same object

446
00:35:14,778 --> 00:35:17,944
except with respect to an axis which has been flipped

447
00:35:18,053 --> 00:35:20,641
So you might start to think this thing might be a vector, this qubit

448
00:35:21,302 --> 00:35:25,156
So now we do something, let's say alright, do something else now 

449
00:35:25,232 --> 00:35:29,883
Let's turn the detector on it's side, let's turn the detector on it's side

450
00:35:33,669 --> 00:35:38,252
We have initially determine that the qubit was up

451
00:35:38,397 --> 00:35:43,459
We definitely make sure it's up, we've done it a thousand time 

452
00:35:43,572 --> 00:35:47,696
with the right side up detector and know for sure it's up

453
00:35:47,767 --> 00:35:48,347
It's not down

454
00:35:50,460 --> 00:35:53,826
And now we detect, take the side way detector

455
00:35:54,699 --> 00:35:57,678
A this side up over right here 

456
00:35:59,980 --> 00:36:03,485
With a internal arrow pointing in this direction over here

457
00:36:04,474 --> 00:36:08,687
And what should we expect ordinariliy

458
00:36:09,466 --> 00:36:16,167
Ordinarily we say, well if this thing is going to measure now 

459
00:36:16,244 --> 00:36:19,791
not the vertical component of some vector 

460
00:36:19,899 --> 00:36:22,272
but the horizontal component of some vector 	 

461
00:36:22,385 --> 00:36:26,822
we know that the vector was pointing upward by the initial preparation 

462
00:36:26,930 --> 00:36:32,492
which you do get here, well what's the component along horizontal axis?

463
00:36:32,613 --> 00:36:37,268
Zero, so the answer is we should get zero

464
00:36:37,414 --> 00:36:41,241
If this were classical physics and this were classical little vector 

465
00:36:41,388 --> 00:36:43,123
That's exactly what we will get

466
00:36:43,271 --> 00:36:52,952
So we are going to be experiment, probe it, and we get not zero

467
00:36:53,062 --> 00:36:55,218
But either one or minus one again

468
00:36:55,328 --> 00:37:00,879
Which one do we get? One or minus one? One or minus one?

469
00:37:00,956 --> 00:37:06,222
So it seem to be symmetrically located relative to the axis of the detector

470
00:37:06,336 --> 00:37:12,888
So we say oh I got one, let's do this experiment a whole bunch of time

471
00:37:13,001 --> 00:37:18,849
Let's go back, get us some new qubits

472
00:37:18,961 --> 00:37:21,126
Throw this qubit away, we've finished with it

473
00:37:21,245 --> 00:37:25,032
Get us a new qubit, make sure it's pointing up

474
00:37:25,148 --> 00:37:27,264
And do the same experiment again 

475
00:37:28,384 --> 00:37:32,638
Well, we might get plus one again, but again we might get minus one

476
00:37:32,743 --> 00:37:35,528
We do it many many times, 

477
00:37:35,640 --> 00:37:41,492
exactly the same experiment with a qubit known to be pointing up

478
00:37:41,563 --> 00:37:44,182
And we measure what you would've like to think 

479
00:37:44,294 --> 00:37:45,947
with the horizontal component of an object

480
00:37:46,061 --> 00:37:51,247
And we always get plus one or minus one, randomly

481
00:37:51,358 --> 00:37:56,622
Randomly but in such a way that if you do it a great many times

482
00:37:56,735 --> 00:38:02,593
Then the average value is zero

483
00:38:02,705 --> 00:38:06,114
There is many z... as many plus one as minus ones

484
00:38:06,220 --> 00:38:08,940
If you do it a great many times 

485
00:38:09,055 --> 00:38:14,196
And you will find every single one of them is either plus one or minus one 

486
00:38:15,356 --> 00:38:17,713
But on the average it add up to zero

487
00:38:17,827 --> 00:38:20,851
>> you said you threw the qubit away ()

488
00:38:23,471 --> 00:38:28,463
well if you reoriented it, if you reoriented it and make sure it's up again

489
00:38:28,577 --> 00:38:31,445
then it would not matter if it's a new qubit or not

490
00:38:31,611 --> 00:38:33,926
>> if you look at the same qubit at the same place 	 

491
00:38:34,039 --> 00:38:37,602
Okay good question, alright good, good, good

492
00:38:37,715 --> 00:38:40,367
So let's go back ,good question

493
00:38:40,479 --> 00:38:42,393
We start with the qubit known to be up

494
00:38:42,504 --> 00:38:47,111
We made a detection and we got plus one

495
00:38:47,223 --> 00:38:54,405
The implication of that is the qubit is lying in the horizontal direction

496
00:38:54,492 --> 00:39:00,470
Somehow, somehow we've detected a plus one 

497
00:39:00,616 --> 00:39:03,141
but a plus one relative to a new axis

498
00:39:03,253 --> 00:39:05,636
Not the original axis but a new axis

499
00:39:06,152 --> 00:39:11,394
Supposing we now take that same qubit don't modify it

500
00:39:11,509 --> 00:39:14,771
But do the same experiment over and over and over again

501
00:39:14,885 --> 00:39:17,440
We will always get the same plus one

502
00:39:21,227 --> 00:39:25,756
Once that qubit has registered plus one and we don't disturb it

503
00:39:25,868 --> 00:39:29,563
Do the experiment again and again we will continue to get plus one 

504
00:39:29,637 --> 00:39:32,880
Another words we will confirm we will confirm the experiment 

505
00:39:33,904 --> 00:39:39,828
>>difference between two actions you projected from 

506
00:39:39,940 --> 00:39:42,915
>>going from top side to the bottom side, 

507
00:39:43,025 --> 00:39:44,889
>>that's a hundred and eighty degrees 

508
00:39:45,001 --> 00:39:47,685
>>whereas the other is only ninety degree

509
00:39:51,295 --> 00:39:54,108
When I turn the detector for ninety degrees 

510
00:39:54,218 --> 00:39:57,350
it no longer have the definiteness about the answer 

511
00:39:57,424 --> 00:40:00,442
The answer is random but only the first time 

512
00:40:00,519 --> 00:40:03,325
Once I do it, once I do it

513
00:40:03,431 --> 00:40:09,694
and I determine whether the horizontal component is plus one or minus one

514
00:40:09,771 --> 00:40:11,022
Then it's determined, 

515
00:40:11,132 --> 00:40:14,861
and if I do the same horizontal experiment over and over again

516
00:40:14,975 --> 00:40:17,384
I will continue to get the same answer 

517
00:40:20,334 --> 00:40:23,610
Well, then I could come back and say okay, 

518
00:40:23,727 --> 00:40:26,650
now having done that and pretty sure 

519
00:40:26,729 --> 00:40:32,222
the horizontal components of it is in the plus direction 

520
00:40:32,299 --> 00:40:34,634
I can come back and turn the detector again 

521
00:40:35,957 --> 00:40:37,315
Guess what will happen? 

522
00:40:37,419 --> 00:40:44,279
I will find the random result after time plus after time minus

523
00:40:44,430 --> 00:40:49,658
But again, once it's determined and I do it over and over and over again 

524
00:40:49,771 --> 00:40:50,939
It stays the same, ya?

525
00:40:51,053 --> 00:40:56,631
>> this is probably not important but on your side way detector, 

526
00:40:56,745 --> 00:40:59,214
>>your arrows are tickle in the wrong way

527
00:41:03,900 --> 00:41:04,076
How so?

528
00:41:04,258 --> 00:41:06,390
>> the point from A 

529
00:41:06,503 --> 00:41:09,928
Oh oh oh, maybe there was an A here

530
00:41:12,588 --> 00:41:18,158
This side up I think, okay yes you are right 

531
00:41:24,157 --> 00:41:27,354
It's interesting okay, so therefore, 

532
00:41:27,467 --> 00:41:33,722
having done this experiment and found out that I get a plus one

533
00:41:33,834 --> 00:41:36,782
I do it over and over and over a few times 

534
00:41:36,819 --> 00:41:38,693
I get a plus one each time 

535
00:41:38,767 --> 00:41:42,992
Let's say I take the detector and turn it around 180 degrees

536
00:41:43,102 --> 00:41:43,946
What do I get?

537
00:41:45,211 --> 00:41:48,952
Minus one, or what happen if I turn it 90 degrees?

538
00:41:49,064 --> 00:41:58,555
Random, random means a particular kind of randomness 

539
00:41:58,626 --> 00:42:01,484
equal probability for up and down

540
00:42:01,551 --> 00:42:07,286
equal probability for plus one and minus one 

541
00:42:07,395 --> 00:42:12,418
And that means if I would do the same experiment, 

542
00:42:12,567 --> 00:42:13,855
the whole thing over and over 

543
00:42:14,216 --> 00:42:17,773
Perhaps with a whole bunch of different qubits 

544
00:42:17,888 --> 00:42:22,176
I have whole() lots of qubits over here, a lots of them

545
00:42:22,286 --> 00:42:24,426
I don't even know what their directions are 

546
00:42:24,538 --> 00:42:26,324
They are in, whatever they are doing 	 

547
00:42:26,438 --> 00:42:28,586
And I take them one at a time 

548
00:42:28,658 --> 00:42:31,743
I subject them to the first experiment 

549
00:42:32,924 --> 00:42:34,747
If I get up I keep them 

550
00:42:35,209 --> 00:42:37,978
If I get down I throw them in the trash

551
00:42:38,756 --> 00:42:42,296
At the end of the day, what I would have gotten myself 

552
00:42:42,410 --> 00:42:45,608
with a whole bunch of qubit known to be pointing up

553
00:42:49,570 --> 00:42:54,481
Now I take one of them , and subject it to the side way experiments

554
00:42:55,815 --> 00:42:58,422
I may get one, or I may get minus ones

555
00:43:00,004 --> 00:43:03,722
I record what I got for all the way

556
00:43:04,507 --> 00:43:07,573
Go to the next one do the same experiment 

557
00:43:07,683 --> 00:43:10,572
and make it plus one and minus one record what I get

558
00:43:10,644 --> 00:43:14,950
Take this whole collection, large collection of them

559
00:43:15,019 --> 00:43:18,005
And do the side way experiment for all of them 

560
00:43:18,116 --> 00:43:21,029
What would be the average result?

561
00:43:24,890 --> 00:43:32,362
Zero, zero, I will have as many plus ones as minus ones

562
00:43:32,434 --> 00:43:35,214
And so the average would be zero 

563
00:43:36,371 --> 00:43:38,590
Let's try that in this case here 

564
00:43:38,661 --> 00:43:44,190
The same that the average value, we need a symbol for the average value 

565
00:43:44,260 --> 00:43:50,643
But after the experiment here we will discover the average value of the x

566
00:43:50,789 --> 00:43:56,585
Or the horizontal component of the qubit here is zero

567
00:43:56,691 --> 00:44:03,316
Classically if we would have created a little vector in the up direction 

568
00:44:03,384 --> 00:44:07,010
And we measure the horizontal component we simply get zero

569
00:44:08,039 --> 00:44:13,322
In the real world with quantum mechanics we always get plus or minus one 

570
00:44:13,391 --> 00:44:17,326
But in such a way that the average is zero

571
00:44:19,573 --> 00:44:22,311
>> Is that true (even if that you don't throw away the qubit)

572
00:44:22,382 --> 00:44:24,216
>> which measured plus one or minus one?

573
00:44:24,285 --> 00:44:30,410
Yeah, actually it's still true, that is correct, the answer is yes

574
00:44:32,288 --> 00:44:38,950
>> In your description either the detector () seems to have some type of memory

575
00:44:39,063 --> 00:44:50,584
Some type of, yeah ,we were assuming that the law of the motion of the qubit 

576
00:44:50,689 --> 00:44:54,512
is that nothing happens to it so we remember when you detect it 	 

577
00:44:54,580 --> 00:44:59,191
And then you detect it again instantly after , and it hasn't changed

578
00:44:59,263 --> 00:45:00,999
Right that's correct 

579
00:45:01,779 --> 00:45:04,838
Now later on, we are gonna consider possibilities

580
00:45:04,917 --> 00:45:08,199
where during the intervening time between the measurement

581
00:45:08,312 --> 00:45:09,882
some interesting things might happen

582
00:45:10,028 --> 00:45:12,756
For the moment we supposing nothing interesting happen in between

583
00:45:12,869 --> 00:45:16,051
It's analog of the classical coin

584
00:45:16,150 --> 00:45:19,420
with law of physics there's just nothing happens

585
00:45:21,123 --> 00:45:25,153
>> we don't talk about the collapsing of a wave function

586
00:45:27,079 --> 00:45:29,675
that's of course up till now but 

587
00:45:31,490 --> 00:45:40,140
No, let's not talk about it, let's talk about the operational things that

588
00:45:40,250 --> 00:45:45,389
we might do to discover in the clause of quantum mechanics

589
00:45:46,723 --> 00:45:53,536
Then we will puts some words on them such as collapsible wave package 

590
00:45:55,657 --> 00:46:02,113
Okay, so things are funny in some average sense 	 

591
00:46:04,242 --> 00:46:11,138
The horizontal component of something zero within each individual instance 

592
00:46:11,249 --> 00:46:14,035
You only get plus one or minus one 

593
00:46:15,030 --> 00:46:16,646
Now let's go a little further 

594
00:46:18,419 --> 00:46:21,417
Let's, or instantly the same thing would be true 

595
00:46:21,529 --> 00:46:26,688
if instead of turning the detector this way we turn it this way 

596
00:46:26,801 --> 00:46:29,745
so the internal vector was pointing outward

597
00:46:29,894 --> 00:46:31,416
It's to be true, same thing

598
00:46:32,485 --> 00:46:35,092
Okay now let's do a little different experiment 

599
00:46:37,040 --> 00:46:39,454
Same set up to begin with  

600
00:46:39,530 --> 00:46:44,111
But instead of rotating the detector by 90 degrees

601
00:46:44,975 --> 00:46:52,079
Let's rotate it by 45 degrees, in fact any angle, let's not do 45 degrees

602
00:46:58,043 --> 00:47:06,150
Let's rotate it by an arbitrary angle so that the internal detector ()

603
00:47:06,317 --> 00:47:11,727
is now characterize by an angle theta relative to the vertical 

604
00:47:17,779 --> 00:47:23,365
It's been rotated by an angle theta and do the same experiment now

605
00:47:23,478 --> 00:47:25,945
We take the first qubit

606
00:47:26,056 --> 00:47:40,008
subject it now to the detector oriented to the angle theta

607
00:47:40,077 --> 00:47:43,571
What do we get , plus one or minus one?

608
00:47:44,306 --> 00:47:48,312
do it again what do we get , plus one or minus one?

609
00:47:48,432 --> 00:47:51,300
What do we get do it again , plus one or minus one?

610
00:47:51,403 --> 00:47:53,201
Never do anything in between, 

611
00:47:53,270 --> 00:48:02,708
but in this case the average value of the experimental value, 

612
00:48:03,822 --> 00:48:08,113
or what comes out here is not equal to zero

613
00:48:09,880 --> 00:48:12,658
It's equal to, what do you expect it to equal to?

614
00:48:13,375 --> 00:48:16,392
Cosine of theta, cosine of the angle 

615
00:48:16,505 --> 00:48:21,164
Exactly like the x component of a vector would be

616
00:48:21,277 --> 00:48:29,029
Or exactly like yes, exactly like the components of these vectors 

617
00:48:29,102 --> 00:48:31,759
along the detector axis

618
00:48:31,868 --> 00:48:36,265
The component of these vector along the detector axis()

619
00:48:36,378 --> 00:48:41,549
would be smaller by an amount cosine theta, classically,

620
00:48:41,620 --> 00:48:46,285
quantum mechanically, we simply get plus one minus one

621
00:48:46,355 --> 00:48:49,105
Plus one plus one minus one plus plus plus whatever 

622
00:48:50,568 --> 00:48:56,083
In such a way as to averages to the cosine of the angle 

623
00:48:57,540 --> 00:49:01,086
As the detector gets more and more up right

624
00:49:01,196 --> 00:49:04,479
Cosine theta gets closer and closer to one 

625
00:49:05,238 --> 00:49:11,757
And that just mean it gets closer and closer to the value to ()

626
00:49:11,871 --> 00:49:15,596
the answer if the detector is perfectly up right

627
00:49:16,697 --> 00:49:21,885
Now and then you might get a minus sign, but mostly plus sign

628
00:49:21,997 --> 00:49:24,688
And as soon as the detector get perfectly right side up

629
00:49:24,757 --> 00:49:26,339
All plus ones

630
00:49:28,099 --> 00:49:31,749
What about the detector is almost upside down?

631
00:49:32,345 --> 00:49:36,332
Well then you get mostly minus one, a few in there are plus ones

632
00:49:37,350 --> 00:49:44,652
And so it seems that the average values of sigma 

633
00:49:44,720 --> 00:49:49,147
are behaving as if sigma was a component of a vector

634
00:49:51,283 --> 00:49:54,628
this is something you know, really different 

635
00:49:56,538 --> 00:50:05,446
>>() if you have prepared these as up, if you just took more qubits with there

636
00:50:07,734 --> 00:50:08,896
if you what?

637
00:50:09,027 --> 00:50:11,109
>> it stay up because you started with up

638
00:50:11,235 --> 00:50:12,891
yes, yes

639
00:50:15,497 --> 00:50:17,817
>> why do we need a collection of qubits?

640
00:50:17,930 --> 00:50:19,024
Why do we need what?

641
00:50:19,119 --> 00:50:22,310
>> a collection of qubits? Since nothing happening to the qubit right?()

642
00:50:24,983 --> 00:50:27,759
If you take the same one and do it over and over and over again 

643
00:50:27,834 --> 00:50:30,920
with the same detector, you will just get the same answer over and over 

644
00:50:33,007 --> 00:50:37,670
You just confirm the previous measurement 

645
00:50:39,122 --> 00:50:44,411
So we take one qubit known to be up because you made it you know that it's up

646
00:50:44,488 --> 00:50:50,582
And you subject it to the angularly ()rotated experiment 

647
00:50:50,692 --> 00:50:54,214
You either get plus one or minus one but if you do it a second time

648
00:50:55,030 --> 00:50:57,241
You will just get the same answer you got the first time	 

649
00:50:58,423 --> 00:50:59,743
You won't learn anything new

650
00:51:02,085 --> 00:51:06,769
Incidentally if that weren't the case, there will be no way to confirm

651
00:51:07,429 --> 00:51:11,461
That the answer you got the first time was in any sense the right answer 

652
00:51:12,022 --> 00:51:15,384
The only way you confirm an answer is right is by checking it

653
00:51:15,499 --> 00:51:18,424
And one way to check it it's just

654
00:51:20,307 --> 00:51:23,531
If I close my eyes and look at you see you up right

655
00:51:23,640 --> 00:51:26,912
And then I close my eyes again and look at you again 

656
00:51:27,025 --> 00:51:29,324
I've confirm the first time you are up right

657
00:51:30,192 --> 00:51:32,850
>> but it also gonna be the same stick?

658
00:51:36,599 --> 00:51:39,003
This one here? Second one?

659
00:51:39,894 --> 00:51:42,479
Ya , I'm saying if you only have one of them

660
00:51:44,215 --> 00:51:48,746
Then you do the experiment you will either get plus one or minus one

661
00:51:48,859 --> 00:51:53,323
And then if you do the same experiment again on the same one over and over

662
00:51:53,764 --> 00:51:54,588
You just get plus one

663
00:51:54,731 --> 00:51:57,506
But if you have a sequence of them

664
00:51:57,575 --> 00:52:00,232
and you do the experiment with a sequence of them

665
00:52:00,304 --> 00:52:02,474
All having them prepared in the same way

666
00:52:03,119 --> 00:52:06,500
Then you will get a statistical distribution 

667
00:52:06,978 --> 00:52:10,082
>>So if you turn off the detector and turn it on again 

668
00:52:10,154 --> 00:52:12,376
>> will you still have the memory of the same 

669
00:52:14,032 --> 00:52:16,608
That's what I mean by switch it off again 

670
00:52:16,645 --> 00:52:18,977
Switch the detector off and turn if back on 

671
00:52:19,056 --> 00:52:21,427
>>So does the detector seem to store memory or 

672
00:52:21,501 --> 00:52:26,573
the detector has an orientation to it, and when I say you turn

673
00:52:26,653 --> 00:52:29,751
>> and what you say you change or did not change 

674
00:52:29,820 --> 00:52:31,837
change the qubit? Say it again?

675
00:52:31,910 --> 00:52:34,958
>> how does you know that you changed or did not change the qubit

676
00:52:36,561 --> 00:52:40,389
No, they didn't change the detector 

677
00:52:41,298 --> 00:52:44,239
>> But when you switch the qubit and now it's reset

678
00:52:44,349 --> 00:52:49,221
>> but if you ()qubit and it gives you the same reading?

679
00:52:56,401 --> 00:52:59,467
There was only one qubit you did the same 

680
00:52:59,542 --> 00:53:05,392
>> isn't that the detector is not remembering things but the qubit has changed?

681
00:53:05,462 --> 00:53:07,457
I would like to say the students and audience 

682
00:53:07,594 --> 00:53:08,939
Some of them are sitting upside-down and 

683
00:53:09,009 --> 00:53:10,387
some of them are sitting rightside-up

684
00:53:12,430 --> 00:53:14,868
If I randomly open my eyes to the students 

685
00:53:15,176 --> 00:53:18,330
50% are standing rightside-up or not 

686
00:53:18,403 --> 00:53:22,398
I close my eyes and look at another student, 50% I don't know

687
00:53:22,509 --> 00:53:27,122
But I if I open my eyes, and then close them, open them, close them

688
00:53:27,194 --> 00:53:29,586
And look at the same student it's always the same way

689
00:53:31,185 --> 00:53:33,859
How do my eyes know the student was gonna that

690
00:53:34,239 --> 00:53:36,017
I wasn't looking at the different student?

691
00:53:37,576 --> 00:53:43,262
Answer is just look at the same student okay?

692
00:53:43,845 --> 00:53:48,230
So the whole thing is not that weird that 

693
00:53:48,299 --> 00:53:52,110
we can't keep track and remember which of these things we were looking at

694
00:53:53,001 --> 00:53:56,741
>> so basically we can think of it as the act of observing

695
00:53:56,890 --> 00:54:00,172
>> ()prepare that particular qubit

696
00:54:00,287 --> 00:54:02,973
>> to let you observe that one you prepare it 

697
00:54:03,042 --> 00:54:05,175
>> so now you always get that answer 

698
00:54:05,289 --> 00:54:09,355
until you intervened by something else

699
00:54:11,690 --> 00:54:15,024
>> if you change the qubit but let the measuring apparatus 

700
00:54:15,096 --> 00:54:18,233
>> in the same position, you still get plus ones?

701
00:54:20,775 --> 00:54:25,079
>> you are not gonna change the orientation of the measurement device?

702
00:54:25,118 --> 00:54:28,849
>> but you just measure a sequence of different qubits?

703
00:54:28,929 --> 00:54:32,026
>> are they really gonna be plus ones or minus ones?

704
00:54:32,097 --> 00:54:34,524
It depends, it depends on how they were prepared 

705
00:54:35,322 --> 00:54:38,141
Somebody might have given you a sequence of qubit 

706
00:54:38,255 --> 00:54:40,611
which happens to be up down up down up down up down

707
00:54:41,349 --> 00:54:43,560
You might have gotten them from some place 

708
00:54:43,672 --> 00:54:46,631
You got them from a , you know , qubit store

709
00:54:46,753 --> 00:54:49,045
If they didn't have a very good quality control

710
00:54:50,643 --> 00:54:58,020
But if all of these qubits are prepared in the same way and identical way

711
00:54:58,090 --> 00:55:01,495
And then you send them to here they will continue to be identical

712
00:55:05,489 --> 00:55:07,520
Okay, what this is telling us, 

713
00:55:07,595 --> 00:55:12,122
is there something funny about the notion of the status of a system 

714
00:55:12,574 --> 00:55:14,786
is not as clear cut in another symbol as 

715
00:55:14,863 --> 00:55:21,387
a state of either a coin which is either heads or tails

716
00:55:22,534 --> 00:55:27,754
Or a little vector which has components in different directions

717
00:55:27,866 --> 00:55:31,300
But for example when you measure a component along the axis

718
00:55:31,377 --> 00:55:36,143
perpendicular to the known direction of the vector, you get zero

719
00:55:36,955 --> 00:55:40,242
Not so, so there is something intrinsically different 	 

720
00:55:42,529 --> 00:55:47,337
And the difference does trace to the big difference between the notion

721
00:55:47,413 --> 00:55:51,036
of a state in quantum mechanics and the state of classical mechanics

722
00:55:52,583 --> 00:55:55,894
>> but you have everything there up plus one?

723
00:55:55,971 --> 00:56:00,439
>> and one by one you put it plus one or minus one?

724
00:56:00,476 --> 00:56:04,472
>> and each one you bring up that was just minus one 

725
00:56:05,074 --> 00:56:07,607
>> then becomes the minus one of the original

726
00:56:08,411 --> 00:56:09,175
if you do the experiment

727
00:56:09,244 --> 00:56:15,923
>> If you bring one down, let's say it's minus one down here

728
00:56:16,496 --> 00:56:20,274
>> then you turn the apparatus up again 

729
00:56:20,565 --> 00:56:23,771
>> and now you say it will be minus one from now on

730
00:56:23,915 --> 00:56:25,576
No no, none of you turn the apparatus

731
00:56:29,479 --> 00:56:34,630
>> I thought you said if you turn the apparatus up if it measure minus() 

732
00:56:34,701 --> 00:56:37,268
>> it would stay minus if you turn the apparatus ()

733
00:56:38,483 --> 00:56:45,484
If you turn the apparatus, then you get a statistical distribution

734
00:56:46,396 --> 00:56:48,152
Okay when you turn an apparatus

735
00:56:49,378 --> 00:56:54,580
Once you have the qubits that comes out of this apparatus

736
00:56:55,489 --> 00:56:59,681
Let's say it was plus one, let's say it comes out plus one 

737
00:57:00,263 --> 00:57:02,416
And now you take that qubit over here 

738
00:57:02,486 --> 00:57:05,303
Then you turn the apparatus back up again 

739
00:57:05,380 --> 00:57:09,609
Then you get a random plus or minus

740
00:57:10,280 --> 00:57:12,869
>> then if will be random if you turn back up 

741
00:57:12,938 --> 00:57:14,261
>> so in that way then

742
00:57:15,391 --> 00:57:18,252
every time you turn it, it will be random

743
00:57:21,013 --> 00:57:22,873
you will end up in a random bunch of qubits

744
00:57:23,997 --> 00:57:28,748
now it depends on the angle, if the angle is not too large, if the angle is small

745
00:57:29,459 --> 00:57:31,048
then mostly you will get pluses

746
00:57:31,118 --> 00:57:33,214
Right, mostly you will get pluses 

747
00:57:35,782 --> 00:57:38,361
>> two state system and that with a detector, 

748
00:57:38,430 --> 00:57:43,088
>> what would be the difference in large, when the signal can be variable

749
00:57:43,162 --> 00:57:48,103
>> and the detector and two state system, see if I could get the same out

750
00:57:49,435 --> 00:57:50,705
Say it again, say it again

751
00:57:50,813 --> 00:57:54,055
>> you postulated two state system plus and minus one

752
00:57:54,127 --> 00:57:59,277
>> but then you have a detector that always produces plus and minus one

753
00:57:59,355 --> 00:58:03,260
>> if you postulated with variable to start with the detector 

754
00:58:03,372 --> 00:58:06,157
>> will still like to be what you 

755
00:58:06,265 --> 00:58:07,586
The detector will still be what?

756
00:58:08,456 --> 00:58:15,949
>> It seems like you are really saying the detector is ()

757
00:58:16,063 --> 00:58:18,511
>> and it doesn't seem to imply that

758
00:58:20,970 --> 00:58:25,621
You are absolutely right that at some point we will come back to this

759
00:58:25,699 --> 00:58:30,828
And say the detector is a system and it had states

760
00:58:31,695 --> 00:58:34,218
And understand the combine system as 

761
00:58:34,288 --> 00:58:37,569
a quantum system composed of two quantum systems 

762
00:58:37,608 --> 00:58:47,014
But let's not do that yet, let's divide the worlds into detectors and systems

763
00:58:47,094 --> 00:58:49,453
And then come back later and then say, look

764
00:58:49,533 --> 00:58:54,806
A detector really is a system and we have to describe it quantum mechanically

765
00:58:54,876 --> 00:59:00,364
Also as a system and to understand the entire thing as 

766
00:59:00,436 --> 00:59:03,885
the interplay between the two quantum systems, 

767
00:59:03,956 --> 00:59:08,240
but that would take a, that would take some steps before we get there

768
00:59:10,987 --> 00:59:14,413
>> the axes of rotation doesn't matter?

769
00:59:15,525 --> 00:59:19,343
You mean if we rotate this way or, right it doesn't matter

770
00:59:20,731 --> 00:59:25,647
>> I mean if we do rotation on one axis and then on a different axis

771
00:59:29,180 --> 00:59:34,995
Right, you are right and guessing the whole story is much more complex

772
00:59:35,068 --> 00:59:42,734
But still, the story is more complex but not inconsistent with what I said

773
00:59:42,803 --> 00:59:47,041
So don't try to guess too far ahead, don't try to guess to far ahead

774
00:59:47,113 --> 00:59:49,362
Let's try the simple version of the first

775
00:59:49,401 --> 00:59:53,161
And then we will eventually come to a much more rich understanding of it

776
00:59:53,563 --> 00:59:56,699
>> so if it is correct to think the detector is something 

777
00:59:56,747 --> 01:00:00,211
>> could repetitiously continue value of plus one or minus ones

778
01:00:00,638 --> 01:00:04,594
>> the system would only be, the key would only be 

779
01:00:04,675 --> 01:00:10,154
For now, yes, if you like, you could think of,

780
01:00:10,224 --> 01:00:13,724
For now you can think of the detector as the thing 

781
01:00:13,803 --> 01:00:15,573
which can be oriented in many direction

782
01:00:15,649 --> 01:00:19,338
And when we measure it, it behaves like a classic system

783
01:00:21,428 --> 01:00:23,874
Eventually we will have to reconcile that 

784
01:00:23,953 --> 01:00:27,075
with idea of the detector themselves are quantum systems

785
01:00:31,534 --> 01:00:35,999
And have to respect the same law as the system itself 

786
01:00:45,167 --> 01:00:53,688
Okay, now let's, I leave you to ponder now

787
01:00:53,802 --> 01:00:59,301
And we are going to separate from this for a while and discuss mathematics

788
01:00:59,369 --> 01:01:03,495
We are gonna describe the mathematics of a state space 

789
01:01:03,568 --> 01:01:06,846
Or the space of state of a quantum system

790
01:01:07,520 --> 01:01:09,782
And as I said , it's not a set, 

791
01:01:09,858 --> 01:01:15,038
not a set in the same sense of set of configuration of a die

792
01:01:17,210 --> 01:01:19,240
It's not a set of six configurations

793
01:01:19,352 --> 01:01:20,512
It's more complicated

794
01:01:21,617 --> 01:01:23,381
#Alright but we need some mathematics

795
01:01:23,495 --> 01:01:26,122
We need some mathematics, we need some mathematics abstract mathematics

796
01:01:26,202 --> 01:01:31,461
And the mathematics as I said it's not the mathematics of set theory

797
01:01:31,535 --> 01:01:34,370
The mathematics of vector spaces 

798
01:01:35,029 --> 01:01:38,854
Now let's just discuss the word vector for a moment

799
01:01:40,352 --> 01:01:43,364
You are familiar of course with the idea of a vector

800
01:01:44,536 --> 01:01:48,669
A vector is an arrow pointing to certain direction in ordinary space

801
01:01:50,297 --> 01:01:53,321
And it's one concept of a vector but 

802
01:01:53,398 --> 01:01:57,051
there is more abstract notion of a mathematical vector space 

803
01:01:57,120 --> 01:02:00,538
A mathematical vector space can accomplice 

804
01:02:00,588 --> 01:02:02,630
many many kinds of mathematical objects

805
01:02:03,124 --> 01:02:09,092
Which are not necessarily pointers along an axis in space

806
01:02:10,562 --> 01:02:15,851
I generally, only I get to this point I generally make a new name

807
01:02:15,924 --> 01:02:19,391
An un(non?)conventional new name nobody else I known of users

808
01:02:19,706 --> 01:02:23,275
For the notion of an ordinary vector in an ordinary space 

809
01:02:23,969 --> 01:02:25,953
Another words a thing pointing at ordinary space

810
01:02:26,028 --> 01:02:29,572
I call it a pointer that distinguish it from 

811
01:02:30,311 --> 01:02:33,856
the abstract mathematical concept of a vector

812
01:02:34,373 --> 01:02:38,876
We are gonna talk about the abstract mathematical concept of a vector

813
01:02:40,495 --> 01:02:43,355
Don't think of it as pointing in three dimensional space 

814
01:02:43,465 --> 01:02:45,247
Ordinary three dimensional space 	

815
01:02:46,089 --> 01:02:48,357
We will discuss what space it points 

816
01:02:49,855 --> 01:02:53,923
Alright so let's talk about vector spaces 

817
01:02:54,636 --> 01:02:58,873
A vector space is a collection of mathematical objects 

818
01:02:58,949 --> 01:03:01,603
Like numbers is a collection of mathematical objects

819
01:03:02,203 --> 01:03:06,559
And numbers, just the real number axis for example, is vector space

820
01:03:06,596 --> 01:03:08,240
A special case of a vector space 

821
01:03:08,928 --> 01:03:10,350
It's a one dimensional vector space 

822
01:03:11,212 --> 01:03:13,787
Complex numbers are two dimensional vector space 

823
01:03:14,539 --> 01:03:18,088
We are going to be talking about vector spaces

824
01:03:18,164 --> 01:03:20,300
For the moment I'm ()looking on dimensionality

825
01:03:20,376 --> 01:03:24,503
I'm just gonna write down the mathematical notation 

826
01:03:24,615 --> 01:03:30,684
for an object in the vector space, and as I said, vectors space is just a name, 

827
01:03:30,762 --> 01:03:33,101
for a collection of mathematical objects

828
01:03:34,304 --> 01:03:40,329
Completely abstract, a is a vector

829
01:03:40,442 --> 01:03:43,896
And that's a notation for an abstract vector 

830
01:03:44,671 --> 01:03:52,584
It's Dirac notation, it consists of a vertical line a cricket line

831
01:03:52,662 --> 01:03:56,664
and a letter in between, and that stands for the vector a 

832
01:03:57,223 --> 01:03:58,791
Whatever the vector a is 

833
01:04:00,712 --> 01:04:02,489
What can you do with vectors?

834
01:04:03,279 --> 01:04:04,929
Well you can add them

835
01:04:06,155 --> 01:04:08,127
()I'm not gonna tell you how to add them, I just tell you

836
01:04:08,237 --> 01:04:13,303
given any two vectors a and b you can form this sum

837
01:04:15,572 --> 01:04:17,332
And the sum is another vector 

838
01:04:17,406 --> 01:04:20,854
Let's call that the vector c

839
01:04:22,158 --> 01:04:25,424
So a vector space consists of a collection of objects

840
01:04:25,500 --> 01:04:26,804
which can be added

841
01:04:29,462 --> 01:04:33,184
>> I'm sorry, vector versus vector space?

842
01:04:33,904 --> 01:04:36,124
Vector space is a collection of vectors

843
01:04:38,741 --> 01:04:40,794
Vector space is a collection of vectors

844
01:04:43,731 --> 01:04:46,989
The same distinction between number and numbers

845
01:04:47,101 --> 01:04:51,553
Numbers are all the numbers, number is a specific number

846
01:04:53,411 --> 01:04:56,853
So as I said, numbers is a special case for this you can add them

847
01:04:58,298 --> 01:05:00,974
You could do something else with numbers

848
01:05:01,695 --> 01:05:03,330
You can multiply them by another numbers

849
01:05:03,506 --> 01:05:07,904
Well, eventually we will be talking about multiplying vectors , but not yet

850
01:05:08,914 --> 01:05:14,487
What we can do is multiply vectors by numbers	 

851
01:05:14,560 --> 01:05:17,535
Ordinary numbers or complex numbers

852
01:05:17,603 --> 01:05:19,897
And we are going to talk about complex numbers

853
01:05:19,967 --> 01:05:23,283
I hoped everybody is comfortable with complex numbers

854
01:05:24,029 --> 01:05:28,066
If not, you got to quickly get to speed on complex numbers

855
01:05:29,605 --> 01:05:31,765
But we do not, we don't need to know too much

856
01:05:33,281 --> 01:05:36,033
You need to know that a complex number 

857
01:05:36,111 --> 01:05:39,520
is a combination of real number and imaginary number

858
01:05:41,441 --> 01:05:45,492
And how to multiply, add complex numbers 

859
01:05:45,607 --> 01:05:48,989
And you need the notion of complex conjugate

860
01:05:49,064 --> 01:05:52,611
How many people here are familiar with complex conjugate?

861
01:05:53,456 --> 01:05:57,260
Most people, almost everybody, so I'm not going to go into it now

862
01:05:58,228 --> 01:06:01,140
But basically for every complex number 

863
01:06:01,255 --> 01:06:05,634
There is another complex number for it's complex conjugate

864
01:06:05,710 --> 01:06:09,193
It's a mapping between numbers and their complex conjugate

865
01:06:09,265 --> 01:06:12,120
So it's an important notion, keep that in mind

866
01:06:12,771 --> 01:06:13,966
We are just gonna come back

867
01:06:14,004 --> 01:06:20,555
But for the moment, ordinary vectors let's call it, pointers

868
01:06:20,595 --> 01:06:26,013
Pointers in space, ordinary pointers you can multiply by ordinary numbers

869
01:06:26,083 --> 01:06:32,469
Positive or negative, a pointer with a certain length on that direction

870
01:06:32,541 --> 01:06:36,142
If you multiply it by two, it's just twice as long in the same direction

871
01:06:36,905 --> 01:06:41,421
Multiply it by minus one, it just became the same direction (length) 

872
01:06:41,535 --> 01:06:44,064
except on the opposite direction

873
01:06:44,212 --> 01:06:48,529
Alright so yes you can multiply vectors by numbers

874
01:06:49,360 --> 01:06:52,136
And in a complex vector space, 

875
01:06:52,208 --> 01:06:55,453
we are now gonna talk about complex vector space

876
01:06:56,220 --> 01:07:00,176
you can multiply vectors by complex number

877
01:07:00,249 --> 01:07:02,166
Now do not try to picture this in your head 

878
01:07:02,279 --> 01:07:04,923
as any kind of pointers in some direction

879
01:07:05,659 --> 01:07:08,828
Just accept we are talking about a mathematical abstraction

880
01:07:09,717 --> 01:07:16,589
Where a vector, any given vector can be multiply by any complex numbers

881
01:07:16,702 --> 01:07:19,696
I'm gonna use, let's see what shall I use for complex number?

882
01:07:20,408 --> 01:07:23,695
In my notes I use c, but I don't wanna use c

883
01:07:24,316 --> 01:07:30,224
z, you know z is the standard notation, you know x + yi is equal to z

884
01:07:31,294 --> 01:07:35,466
so you can take any complex number, sorry, any vector

885
01:07:35,505 --> 01:07:37,931
multiply it by any complex number

886
01:07:38,003 --> 01:07:41,680
And it's still some, I'm not trying to draw it for you

887
01:07:41,748 --> 01:07:45,780
Stop trying to think about drawing

888
01:07:45,850 --> 01:07:53,062
Vectors if there was a vector there was a complex number you multiply them 

889
01:07:53,133 --> 01:08:00,178
and it gives you some new vector, let's call it, let's call it 'a prime'

890
01:08:01,442 --> 01:08:06,684
So you can multiply vectors by numbers, complex numbers

891
01:08:11,345 --> 01:08:15,151
And that's about it, that's all there is to a vector space

892
01:08:15,220 --> 01:08:20,429
Any set of an object for which you can do this is a vector space 

893
01:08:20,500 --> 01:08:24,140
Give you some examples of vector spaces, if a-s and b-s

894
01:08:27,624 --> 01:08:30,502
If a-s and b-s are just real numbers 

895
01:08:31,367 --> 01:08:33,521
Then you can add them to make real numbers 

896
01:08:34,279 --> 01:08:37,527
But you cannot multiply them by complex numbers 

897
01:08:37,641 --> 01:08:39,530
and still expect to get real numbers 

898
01:08:41,115 --> 01:08:43,744
You can multiply them by real numbers

899
01:08:44,173 --> 01:08:45,818
And get all the real numbers

900
01:08:46,453 --> 01:08:49,200
So the real numbers are real vector space

901
01:08:50,368 --> 01:08:51,897
How about the complex numbers?

902
01:08:51,970 --> 01:08:55,805
A complex number a complex vector space ?

903
01:08:56,444 --> 01:09:01,317
Yes , you can add them and get other complex numbers 

904
01:09:01,390 --> 01:09:02,983
Any two complex number you can add

905
01:09:03,337 --> 01:09:06,826
And any complex number you can multiply by another complex number

906
01:09:06,896 --> 01:09:08,234
And back to complex number

907
01:09:08,943 --> 01:09:14,096
So the complex numbers are the simplest example of a complex vector space

908
01:09:15,021 --> 01:09:19,010
Let me give you another example of complex vector space 

909
01:09:19,083 --> 01:09:23,151
This is incredibly complicated but still, pretty easy to write down

910
01:09:23,411 --> 01:09:28,909
Take any function of a variable, let's call it phi of x

911
01:09:28,978 --> 01:09:34,522
Now x maybe an ordinary variable, a function on a line

912
01:09:38,428 --> 01:09:41,208
A function on a line, but it's a complex function on a line

913
01:09:41,312 --> 01:09:46,649
It has a real and imaginary part, complex function on a line

914
01:09:47,292 --> 01:09:52,721
Can you add two functions and get another function? Certainly

915
01:09:53,586 --> 01:09:59,713
So functions, just functions are things you can add to get other functions

916
01:10:00,373 --> 01:10:04,621
Can you multiply a complex function 

917
01:10:04,690 --> 01:10:07,193
by a complex number to get a complex function?

918
01:10:07,261 --> 01:10:11,996
Yes, take any complex function multiply it by a complex number

919
01:10:12,071 --> 01:10:13,667
You get another complex function 

920
01:10:14,221 --> 01:10:17,769
So functions form a vector space 

921
01:10:18,548 --> 01:10:21,595
Now that's a complicated idea, I don't wanna that complicated yet

922
01:10:22,790 --> 01:10:24,838
So let's erase that from the blackboard 

923
01:10:26,689 --> 01:10:29,264
I will give you another vector space 

924
01:10:30,760 --> 01:10:32,758
Another complex vector space 

925
01:10:34,055 --> 01:10:42,706
Let's just make some symbols, a bracket

926
01:10:43,332 --> 01:10:47,442
And put in entries, two entries into the bracket 

927
01:10:47,996 --> 01:10:50,793
And the two entries are themselves complex numbers

928
01:10:51,653 --> 01:10:54,468
Alpha one and alpha two

929
01:10:54,534 --> 01:10:57,350
Alpha equals a complex number

930
01:10:59,387 --> 01:11:00,927
Alpha one and alpha two

931
01:11:01,000 --> 01:11:03,692
And alpha one and alpha two can be anything

932
01:11:04,788 --> 01:11:10,957
The collection of such symbols is also a complex vector space

933
01:11:11,029 --> 01:11:12,671
Here's the rule

934
01:11:12,743 --> 01:11:14,727
Supposing you have two of them

935
01:11:14,765 --> 01:11:19,245
Here's one and here's another one, beta one beta two

936
01:11:20,035 --> 01:11:24,785
And you add them, what does that mean? Definition 

937
01:11:24,853 --> 01:11:29,995
With definition of the sum of two is called column

938
01:11:30,063 --> 01:11:34,384
The sum of two columns like that is just to add the entries

939
01:11:36,238 --> 01:11:39,728
Alpha one plus beta one, alpha two plus beta two

940
01:11:41,603 --> 01:11:46,731
So given any two abstract symbols 

941
01:11:46,803 --> 01:11:49,853
in which the entries are filled in with complex numbers

942
01:11:49,923 --> 01:11:52,056
Any two of them alpha and beta 

943
01:11:53,156 --> 01:11:55,312
We can add them and get a third one

944
01:11:56,645 --> 01:12:03,751
So column vectors like this, column of two entries like this

945
01:12:04,355 --> 01:12:09,563
satisfied the first rule, and the second rule is that 

946
01:12:09,635 --> 01:12:14,680
if you want to take a column and multiply it by a complex number

947
01:12:15,390 --> 01:12:21,043
All you do is multiply each entry by the complex number 

948
01:12:23,669 --> 01:12:28,264
Multiply each entry separately and then you construct something

949
01:12:29,637 --> 01:12:30,681
which is just z

950
01:12:32,406 --> 01:12:37,288
So just pairs of complex number like this form a vector space 

951
01:12:37,935 --> 01:12:40,116
So very abstract concept but not very hard

952
01:12:41,031 --> 01:12:43,841
How about triples of them?

953
01:12:45,141 --> 01:12:51,371
Sure, from any columns like that 

954
01:12:51,452 --> 01:12:55,031
Now what you don't wanna do is add a column of two 

955
01:12:55,074 --> 01:12:57,039
to a column of three, there's no kind of rule for that 

956
01:12:59,173 --> 01:13:03,817
So column of different lengths form different vector spaces

957
01:13:03,887 --> 01:13:06,083
Completely different abstract vector spaces 

958
01:13:11,156 --> 01:13:16,966
Any questions about the meaning, this is not hard, but it is abstract

959
01:13:18,002 --> 01:13:20,321
>> are there an () part of a vector space?

960
01:13:20,967 --> 01:13:24,079
Yeah I think it's also important to have a zero vector	 

961
01:13:24,777 --> 01:13:27,248
The zero vector is THE unique vector 

962
01:13:27,361 --> 01:13:29,932
which you add to any other vector you get the same

963
01:13:30,007 --> 01:13:32,894
the same vector

964
01:13:33,656 --> 01:13:37,555
Right, I didn't feel like writing it down but they 

965
01:13:39,143 --> 01:13:41,970
>> the general closure requirement that they are all close 

966
01:13:42,040 --> 01:13:44,086
>> in a particular space?

967
01:13:44,197 --> 01:13:45,246
Close under addition?

968
01:13:45,356 --> 01:13:46,159
>> I think, right

969
01:13:46,270 --> 01:13:47,828
And close up at the multiplication

970
01:13:47,866 --> 01:13:48,811
>> I think you have illustrated

971
01:13:50,653 --> 01:13:51,206
That's right

972
01:13:52,047 --> 01:13:56,507
>> is this same as the matrices ?

973
01:13:59,327 --> 01:14:08,987
Matrices, ya, this is a, usually the term matrices in this class in any case

974
01:14:09,609 --> 01:14:12,060
Will be reserved for square matrices 

975
01:14:12,210 --> 01:14:17,454
Square matrices are collection of numbers which form square ways

976
01:14:17,522 --> 01:14:22,753
Many rows and columns, but the general notion of a matrix can have 

977
01:14:22,823 --> 01:14:25,975
different numbers of rows and columns

978
01:14:26,711 --> 01:14:32,204
And the vector would be a matrix with only one column

979
01:14:32,273 --> 01:14:33,861
in this case two rows

980
01:14:34,686 --> 01:14:37,851
So yes this will be a special case of a matrix

981
01:14:38,458 --> 01:14:45,188
But you know in many context one reserve the term matrix for a square matrix

982
01:14:45,909 --> 01:14:49,215
And I will use it that way, I will use the term matrix to mean a square matrix

983
01:14:51,160 --> 01:14:54,227
So I didn't want to use that terminology

984
01:14:56,976 --> 01:15:04,757
Okay are you thoroughly saturated with abstract mathematics?

985
01:15:04,828 --> 01:15:08,357
This should be straightforward for you

986
01:15:11,892 --> 01:15:15,007
#okay now we introduce another abstraction

987
01:15:16,785 --> 01:15:18,555
Another abstraction, 

988
01:15:20,655 --> 01:15:29,678
basically it's the abstraction of given a complex vector space

989
01:15:30,600 --> 01:15:33,915
The complex conjugate of a vector space	

990
01:15:34,664 --> 01:15:36,840
The complex conjugate of a vector space	

991
01:15:37,513 --> 01:15:42,259
is sort of the space of complex conjugate vectors

992
01:15:42,370 --> 01:15:47,768
just like you can have any number a complex conjugate any complex number

993
01:15:48,450 --> 01:15:50,547
There is a complex conjugate to go with it

994
01:15:52,220 --> 01:15:54,839
So in that sense the complex number system

995
01:15:56,506 --> 01:16:05,205
has a dual version which is the complex conjugate of the original numbers

996
01:16:05,859 --> 01:16:08,285
So do the complex conjugate vector spaces

997
01:16:08,361 --> 01:16:12,246
and roughly speaking, the complex conjugate vector spaces

998
01:16:12,320 --> 01:16:15,025
The dual vector space, usually call the dual

999
01:16:15,099 --> 01:16:19,339
Is roughly speaking the complex conjugate of the original vectors

1000
01:16:19,409 --> 01:16:21,276
We will make it more precise as it goes along

1001
01:16:21,387 --> 01:16:25,791
But what it is is a separate space

1002
01:16:27,706 --> 01:16:32,494
whose elements are in one to one correspondent 

1003
01:16:32,532 --> 01:16:35,520
to the element of the origin vector space 

1004
01:16:35,562 --> 01:16:39,913
So if there is an vector a in the original space

1005
01:16:40,740 --> 01:16:41,620
Let's write it this way

1006
01:16:44,150 --> 01:16:49,741
then there is a vector in a dual way

1007
01:16:49,817 --> 01:16:50,964
Called the dual vector space 

1008
01:16:51,075 --> 01:16:53,773
But as I said it's more or less complex conjugate

1009
01:16:53,843 --> 01:16:55,024
And it's written this way

1010
01:16:57,249 --> 01:17:00,705
It's just written as a backward Dirac symbol

1011
01:17:04,348 --> 01:17:07,737
We will learn this things as bra vectors and ket vectors

1012
01:17:07,806 --> 01:17:10,371
No this is the bra vectors this is the ket vectors

1013
01:17:10,447 --> 01:17:12,885
For the moment they are just symbols	 

1014
01:17:14,145 --> 01:17:19,789
An example, if the vector space is just the space of complex numbers

1015
01:17:20,755 --> 01:17:22,717
Then the dual vector space is just

1016
01:17:22,755 --> 01:17:25,082
the space of complex conjugate of the numbers

1017
01:17:28,633 --> 01:17:31,654
And so there is one to one correspondent 

1018
01:17:32,846 --> 01:17:35,090
between vectors and their complex conjugates

1019
01:17:35,676 --> 01:17:44,170
Let's write down first some postulate about the complex conjugate vector space

1020
01:17:44,208 --> 01:17:46,850
And test it out by saying

1021
01:17:46,965 --> 01:17:50,712
if we can find a concrete representation of the notion

1022
01:17:52,036 --> 01:17:54,661
Write down some abstract postulate and check

1023
01:17:54,723 --> 01:17:55,655
Just what we did here

1024
01:17:55,732 --> 01:17:58,049
We said here's a concrete representation 

1025
01:17:58,825 --> 01:18:02,722
of the notion that we wrote down over here

1026
01:18:02,761 --> 01:18:10,889
Let's write down the abstract postulate about the dual vector space

1027
01:18:11,968 --> 01:18:20,177
So first of all, for any vector there is a dual vector 

1028
01:18:21,213 --> 01:18:23,550
They are in one to one correspondent 

1029
01:18:24,240 --> 01:18:33,108
Next , if you take two vectors, and you add them together

1030
01:18:34,753 --> 01:18:40,920
Then the dual of the sum is just the sum of the dual

1031
01:18:46,306 --> 01:18:47,921
So if you take a vector which is 

1032
01:18:48,033 --> 01:18:49,785
made up out of the sum of two other vectors

1033
01:18:51,390 --> 01:18:55,551
Then it's reflection in the dual space

1034
01:18:55,663 --> 01:19:00,225
is just the sum of the individual dual vectors

1035
01:19:01,173 --> 01:19:03,871
These are postulates, these are postulates if you like

1036
01:19:04,780 --> 01:19:06,068
But they are not so obscure

1037
01:19:07,516 --> 01:19:12,253
The third assumption is that if you take any vector

1038
01:19:14,269 --> 01:19:19,650
And multiply it by a complex conjugate, ah sorry, by a complex number

1039
01:19:21,206 --> 01:19:25,859
Then it's reflection in the dual space now anybody wanna guess?

1040
01:19:28,328 --> 01:19:35,002
Of course it involve the dual vector a, but not multiplication by z

1041
01:19:35,663 --> 01:19:41,171
Or multiplication by complex conjugate of z, z star

1042
01:19:42,140 --> 01:19:45,963
z star is the complex conjugate of the z

1043
01:19:48,632 --> 01:19:52,363
This would be true for complex number if you take a complex number 

1044
01:19:52,474 --> 01:19:54,777
and multiply it by another complex number

1045
01:19:56,679 --> 01:19:58,823
Then the complex conjugate of the whole thing

1046
01:19:59,201 --> 01:20:02,193
is the product of complex conjugate, that's all it is saying 

1047
01:20:02,910 --> 01:20:06,987
So we think of the dual vector space 

1048
01:20:07,069 --> 01:20:11,568
as basically just the complex conjugate vector space in some sense

1049
01:20:12,424 --> 01:20:15,029
Okay let's see if we can make any sense out of this	 

1050
01:20:15,103 --> 01:20:19,737
In terms of this columns vectors here

1051
01:20:19,772 --> 01:20:24,320
>> Is a plus b the same as b plus a?

1052
01:20:24,430 --> 01:20:27,290
Yes, a plus b is the same as b plus a, good question

1053
01:20:27,405 --> 01:20:31,994
When you add vectors and you can see that from here

1054
01:20:33,362 --> 01:20:35,968
Alright should have, sorry, somebody else asked me 

1055
01:20:36,038 --> 01:20:38,849
if there is any other rules on the definition of the vector space

1056
01:20:38,917 --> 01:20:41,454
Indeed there are, among other ones

1057
01:20:42,883 --> 01:20:47,037
That maybe order-ness it doesn't matter which order you add

1058
01:20:48,779 --> 01:20:52,548
But you can see that here the sum of two vectors 

1059
01:20:53,186 --> 01:20:56,814
is just given by numerical adding the complex numbers here

1060
01:20:56,890 --> 01:20:59,403
And of course numerical addition of complex number

1061
01:20:59,512 --> 01:21:01,543
it doesn't mean out of which way you order them

1062
01:21:02,149 --> 01:21:03,864
So that's a good point 

1063
01:21:08,495 --> 01:21:13,145
Okay so let's come back now to, let's put this up on top

1064
01:21:13,751 --> 01:21:19,170
And come back to this concrete non abstract representation 

1065
01:21:19,239 --> 01:21:21,681
of the complex of vector space

1066
01:21:21,751 --> 01:21:23,826
Here we have a vector

1067
01:21:33,896 --> 01:21:39,749
This object is representing the vector a

1068
01:21:41,423 --> 01:21:46,234
The one that I erased with beta, that's representing vector b

1069
01:21:46,313 --> 01:21:47,711
But right now I don't need the vector b

1070
01:21:48,444 --> 01:21:54,819
What about the object if this object represents a 

1071
01:21:56,075 --> 01:22:01,660
Then what is the object which represents the dual vector

1072
01:22:01,738 --> 01:22:03,832
The dual vector the corresponding dual vector

1073
01:22:04,426 --> 01:22:09,450
And the answer is, it's the same pair of number except their complex conjugate

1074
01:22:10,110 --> 01:22:13,276
The same pair of numbers, not quite

1075
01:22:13,364 --> 01:22:17,645
Their complex conjugates, alpha one star, and alpha two star

1076
01:22:18,324 --> 01:22:23,433
But just to keep track, they were talking of a different vector space 

1077
01:22:24,812 --> 01:22:30,341
We write it in row form, not as a column, as a row

1078
01:22:30,412 --> 01:22:32,455
This is just a trick to remember 

1079
01:22:33,283 --> 01:22:36,518
that this object belongs to the dual vector space

1080
01:22:36,588 --> 01:22:39,395
That this object belongs to the original vector space

1081
01:22:39,465 --> 01:22:40,414
A bit different

1082
01:22:41,267 --> 01:22:47,246
Alright, so the dual vector space image of alpha one and alpha two

1083
01:22:47,360 --> 01:22:51,866
is alpha one star, alpha two star

1084
01:22:57,297 --> 01:23:03,826
That's the dual vector, I know you can check, you can check

1085
01:23:05,062 --> 01:23:13,383
With this definition, are the postulates namely if you add the two vectors

1086
01:23:14,213 --> 01:23:19,608
and then take the dual, that whether or not you get the sum of duals

1087
01:23:20,597 --> 01:23:24,410
Then you can check whether you multiply this by a complex number

1088
01:23:24,486 --> 01:23:28,870
And then mul- , and then take it to, or just multiply by it's complex conjugate

1089
01:23:29,916 --> 01:23:30,936
The answer is yes

1090
01:23:31,799 --> 01:23:37,094
So just laying out a pair of complex numbers like this

1091
01:23:37,817 --> 01:23:40,410
makes a vector space, and the dual vector space

1092
01:23:40,524 --> 01:23:43,402
The same pair of numbers except complex conjugated

1093
01:23:44,014 --> 01:23:47,424
and lay out in a row, lay out in a row 

1094
01:23:47,502 --> 01:23:52,276
just to remember that it's not the same creature, 

1095
01:23:52,351 --> 01:23:54,209
doesn't live in the same space 

1096
01:23:56,674 --> 01:24:04,452
And that's the idea of a complex vector space 

1097
01:24:07,365 --> 01:24:10,293
As I said again, very abstract but very easy

1098
01:24:18,125 --> 01:24:19,374
#final concept 

1099
01:24:20,074 --> 01:24:23,362
Alright it's not the final concept but a (final concept)

1100
01:24:29,510 --> 01:24:33,103
The n plus frist concept, I'm not sure what n is

1101
01:24:36,522 --> 01:24:37,949
How many people find this hard?

1102
01:24:39,568 --> 01:24:41,905
It's not very hard, how many people find it familiar?

1103
01:24:43,498 --> 01:24:45,059
Quiet a lot of people find it familiar

1104
01:24:46,069 --> 01:24:48,510
Let's, those don't find it familiar apparently didn't find it hard

1105
01:24:48,622 --> 01:24:48,940
That's also good

1106
01:24:54,746 --> 01:24:56,460
The product of the two vectors

1107
01:24:57,139 --> 01:24:59,338
The inner product of the two vectors

1108
01:24:59,430 --> 01:25:02,222
The inner product of the two vectors is 

1109
01:25:02,300 --> 01:25:05,220
the analog of the dot product for pointers

1110
01:25:05,296 --> 01:25:08,382
The analog of dot product for pointers

1111
01:25:10,113 --> 01:25:13,942
The inner product between two vectors is quickly speaking

1112
01:25:14,021 --> 01:25:20,806
The inner product between a vector and a vector and the dual of the vector

1113
01:25:21,507 --> 01:25:24,973
Given two vectors let's call them a and b

1114
01:25:29,212 --> 01:25:33,562
One does not form the product with a or b

1115
01:25:33,637 --> 01:25:38,864
The inner product, instead, one takes b and 

1116
01:25:38,940 --> 01:25:42,559
construct it's image in the dual vector space 

1117
01:25:43,568 --> 01:25:50,336
And then takes the product of vector a to the dual vector b

1118
01:25:51,391 --> 01:25:58,301
In Dirac's language, you take a ket vector and you multiply if by a bar vector

1119
01:25:58,412 --> 01:26:06,591
And when you do so, you make a bra-ket, bracket

1120
01:26:12,143 --> 01:26:15,849
A bracket of course is this thing over here, and the two together

1121
01:26:17,317 --> 01:26:19,856
I think this one is call the ket, this one is called the bra

1122
01:26:19,934 --> 01:26:21,237
Bra, ket

1123
01:26:22,412 --> 01:26:30,440
So the inner product of the two vectors is really a product of two vectors

1124
01:26:30,549 --> 01:26:32,054
And the dual of the vector

1125
01:26:32,639 --> 01:26:42,820
Now is this the same as multiplying the bra vector a with the ket vector b

1126
01:26:43,438 --> 01:26:49,032
Question, is it the same to multiply the bra b to the ket a 

1127
01:26:49,109 --> 01:26:51,202
or a bra a to a ket b? 

1128
01:26:51,271 --> 01:26:51,849
The answer is no.

1129
01:26:53,716 --> 01:26:59,191
It is not the same operation, but in fact they are closely related

1130
01:27:01,571 --> 01:27:04,620
The axiom, we are still talking about the axiom

1131
01:27:05,385 --> 01:27:12,325
The axiom is that these two are related by complex conjugation

1132
01:27:12,948 --> 01:27:20,292
A complex conjugation a to go from the ket vector to the bra vector

1133
01:27:20,838 --> 01:27:25,143
A complex conjugated b to go from bra vector to the ket vector

1134
01:27:26,029 --> 01:27:29,686
So this is product of a, roughly speaking, times b star

1135
01:27:29,764 --> 01:27:32,080
This is the product of b star times a

1136
01:27:32,190 --> 01:27:34,089
This is the complex conjugate of this

1137
01:27:34,532 --> 01:27:42,503
So another postulate of an axiom of a complex vector space

1138
01:27:49,606 --> 01:27:53,129
is that the inner product of a with b 

1139
01:27:53,972 --> 01:28:00,989
is the complex conjugate of the inner product of b with a star

1140
01:28:03,604 --> 01:28:07,039
>> what do you mean in normal vector space 

1141
01:28:10,014 --> 01:28:12,576
>> ... quantum mechanics... for sake, I means

1142
01:28:12,645 --> 01:28:17,004
>> there will be inner product just all the same vector space

1143
01:28:17,084 --> 01:28:18,696
In ordinary vector space?

1144
01:28:18,769 --> 01:28:23,753
>> this is still called the inner product even though it's

1145
01:28:23,832 --> 01:28:26,489
>> one of the special thing with complex conjugate?

1146
01:28:28,640 --> 01:28:31,986
The usual vector spaces that you think about are real vector spaces

1147
01:28:32,066 --> 01:28:35,051
And for real vector space, for real numbers 

1148
01:28:35,128 --> 01:28:37,850
there is no difference between numbers and it's complex conjugate

1149
01:28:38,390 --> 01:28:43,112
So the real vector space satisfied the same postulate

1150
01:28:43,190 --> 01:28:47,088
Except that the complex conjugation is completely attribute operation

1151
01:28:47,158 --> 01:28:49,922
>> but in the other vector space if you take the two product

1152
01:28:49,992 --> 01:28:53,865
>> of two complex vector and it doesn't involve any conjugation

1153
01:28:53,935 --> 01:28:57,152
>> you just multiply the components up

1154
01:28:58,274 --> 01:29:00,980
Yes you could, you could invent such a thing

1155
01:29:01,092 --> 01:29:04,950
Then of course the inner product of a vector with itself would not be real

1156
01:29:05,577 --> 01:29:06,979
That's okay, who cares?

1157
01:29:08,275 --> 01:29:10,231
So you can invent such things 

1158
01:29:10,341 --> 01:29:13,728
turns out not to be terribly useful in quantum mechanics context 

1159
01:29:14,366 --> 01:29:18,000
So ,you know, it's largely a question of what's useful

1160
01:29:18,736 --> 01:29:22,269
And what's useful wines up being this concept

1161
01:29:23,454 --> 01:29:26,319
Alright, I guess you could, you could certainly define such a thing

1162
01:29:28,954 --> 01:29:31,020
It's not a commonly define thing 

1163
01:29:31,855 --> 01:29:33,603
Doesn't seem to have a lot of use

1164
01:29:36,714 --> 01:29:40,776
This structure occurs in all kinds of places, not only in quantum mechanics

1165
01:29:40,888 --> 01:29:44,946
So this complex vector spaces are very very common things in mathematics

1166
01:29:47,057 --> 01:29:57,238
And okay, let's check now, and see if we can postulate a construction

1167
01:29:57,904 --> 01:30:07,716
for the inner product of the dual vector beta

1168
01:30:09,325 --> 01:30:13,082
Remember when you complex conjugate, 

1169
01:30:13,151 --> 01:30:17,405
when you take times the original vector alpha

1170
01:30:21,536 --> 01:30:23,311
This is supposed to be now

1171
01:30:26,446 --> 01:30:29,993
I guess I call it b a 

1172
01:30:32,917 --> 01:30:36,641
And I haven't written down the notation for b

1173
01:30:36,710 --> 01:30:38,375
And the notation for a 

1174
01:30:38,488 --> 01:30:43,103
Column vector for a, row vector for b, complex conjugated

1175
01:30:44,808 --> 01:30:46,273
I will tell you now what the rule is

1176
01:30:47,215 --> 01:30:55,054
Again we are making up rules which will satisfy postulates up there

1177
01:30:55,787 --> 01:31:02,229
And it's very simple construction, it's just beta one star times alpha one

1178
01:31:02,336 --> 01:31:06,506
plus beta two star times alpha two

1179
01:31:07,490 --> 01:31:12,381
Another word you take the first entry of the dual vector

1180
01:31:12,491 --> 01:31:16,692
Times the first entry of the vector with alpha	 

1181
01:31:17,830 --> 01:31:21,606
Add it to the first entry and second entry times a second entry

1182
01:31:22,814 --> 01:31:27,419
So it's just the generalization of multiplying beta times alpha

1183
01:31:27,489 --> 01:31:30,964
But it's also a generalization of dot product

1184
01:31:31,712 --> 01:31:34,925
If you remember the dot product of ordinary vectors

1185
01:31:35,587 --> 01:31:39,990
You simply take up the sums of the product of the components

1186
01:31:40,678 --> 01:31:44,520
If you have two pointers, they are labeled by components 

1187
01:31:46,030 --> 01:31:47,716
Both of them, then the dot product 

1188
01:31:47,826 --> 01:31:51,087
is just sums of the products of the components

1189
01:31:52,816 --> 01:31:55,156
Does that satisfy the rules up here?

1190
01:31:57,648 --> 01:31:58,763
Where did I write the rules

1191
01:32:00,295 --> 01:32:02,505
I think I have erased the rules 

1192
01:32:06,871 --> 01:32:08,369
The rule was 

1193
01:32:14,083 --> 01:32:16,988
>> It's up there, the last one

1194
01:32:22,961 --> 01:32:24,584
ah I've lost it, okay

1195
01:32:27,171 --> 01:32:31,185
What happen if you interchange alpha and beta or a and b?

1196
01:32:32,039 --> 01:32:35,506
And this becomes alpha one star, alpha two star

1197
01:32:37,141 --> 01:32:39,518
Times beta one beta two

1198
01:32:39,587 --> 01:32:43,201
If I interchange them, and that's equal to 

1199
01:32:44,089 --> 01:32:49,750
alpha one star beta one plus alpha two star beta two

1200
01:32:50,087 --> 01:32:51,994
And that's just the complex conjugate of it

1201
01:32:52,065 --> 01:32:53,311
That really is the complex conjugate

1202
01:32:53,378 --> 01:32:56,500
So interchanging alpha and beta or a and b

1203
01:32:56,577 --> 01:33:02,275
Really does interchange, not the value of the product

1204
01:33:02,377 --> 01:33:03,587
But it's complex conjugate

1205
01:33:05,446 --> 01:33:08,829
That's the notion of a complex vector spaces

1206
01:33:08,898 --> 01:33:13,797
And an inner product on a complex vector space

1207
01:33:14,807 --> 01:33:20,978
Now let's just take a couple of special cases of inner products

1208
01:33:22,479 --> 01:33:25,352
What about the inner product with the vector itself?

1209
01:33:25,990 --> 01:33:29,244
Or the vector with it's image in the dual space?

1210
01:33:29,897 --> 01:33:33,112
Let's take the product of a with a

1211
01:33:36,314 --> 01:33:38,295
What do we know about that?

1212
01:33:39,281 --> 01:33:48,285
Well first of all let's, I'm setting b equals to a 

1213
01:33:51,130 --> 01:33:55,067
That tells me the inner product of a vector to itself 

1214
01:33:55,141 --> 01:33:57,267
is equal to it's own complex conjugate

1215
01:34:00,915 --> 01:34:04,248
I just substitute it for b a

1216
01:34:04,428 --> 01:34:06,502
The inner product of vector with itself 

1217
01:34:06,613 --> 01:34:09,312
was equal to it's own complex conjugate

1218
01:34:09,930 --> 01:34:12,452
What kind of number is equal to it's own complex conjugate?

1219
01:34:14,007 --> 01:34:18,457
A real number, so the inner product of a vector with itself

1220
01:34:18,527 --> 01:34:23,324
is always real, it's pretty obvious just from here

1221
01:34:23,435 --> 01:34:30,036
If we take alpha and multiply it by it's image

1222
01:34:30,111 --> 01:34:36,779
I get alpha star alpha, plus alpha two star alpha two

1223
01:34:38,987 --> 01:34:40,280
Each one of these is real

1224
01:34:40,346 --> 01:34:46,902
Not only is it real, it's also something else, positive

1225
01:34:47,706 --> 01:34:53,333
Alpha star alpha , a number times it's own complex conjugate is always positive

1226
01:34:54,015 --> 01:34:55,586
So it's real and positive

1227
01:34:56,382 --> 01:35:01,078
The inner product of a vector with itself is real and positive

1228
01:35:01,638 --> 01:35:05,809
And it's consider to be the square of the length of the vector

1229
01:35:06,554 --> 01:35:10,123
The square root of the inner product of a vector with itself

1230
01:35:10,199 --> 01:35:12,271
defines the length of the vector

1231
01:35:13,786 --> 01:35:17,220
So complex vectors have real lengths

1232
01:35:17,932 --> 01:35:22,581
Real and positive lengths, also the square of 

1233
01:35:24,335 --> 01:35:28,419
both the square and the length itself are defined to be positive

1234
01:35:29,898 --> 01:35:35,599
So that's not too obscure, every complex vector has a length

1235
01:35:35,712 --> 01:35:40,051
Which is the square root or the sum of the square of it's components

1236
01:35:40,122 --> 01:35:44,522
Square know mean multiply by complex conjugate

1237
01:35:47,619 --> 01:35:51,212
#next notion, important notion

1238
01:35:51,281 --> 01:35:54,664
The notion of orthogonality of vectors

1239
01:35:55,624 --> 01:36:01,457
The notion of orthogonality of vectors is a very important one

1240
01:36:02,447 --> 01:36:04,772
Now I give some examples in a few minutes

1241
01:36:05,901 --> 01:36:10,673
The notion of orthogonality between two vectors	 

1242
01:36:12,352 --> 01:36:16,191
Is exactly the same as it is for ordinary vectors

1243
01:36:16,264 --> 01:36:20,566
The inner product the rear-dot product if you like

1244
01:36:20,678 --> 01:36:22,601
The inner product is zero

1245
01:36:22,679 --> 01:36:25,433
If the inner product of two vectors are zero

1246
01:36:25,512 --> 01:36:27,431
It is set to be orthogonal	  

1247
01:36:29,055 --> 01:36:31,678
That defines, or perpendicular

1248
01:36:38,918 --> 01:36:41,309
Okay so two vectors are orthogonal 

1249
01:36:41,355 --> 01:36:43,555
if the inner product between them is zero

1250
01:36:45,594 --> 01:36:47,938
Let's see if we can make up some orthogonal vectors

1251
01:36:48,009 --> 01:36:48,522
It's easy

1252
01:36:51,823 --> 01:36:54,798
Oh, oh oh, what about reading, 

1253
01:36:54,870 --> 01:36:58,221
what about take the inner product the opposite order

1254
01:37:01,039 --> 01:37:03,474
It's just a complex conjugate of a zero

1255
01:37:03,544 --> 01:37:05,339
The complex conjugate of a zero

1256
01:37:05,377 --> 01:37:11,796
As zero is zero, so saying that the two vectors are orthogonal

1257
01:37:11,869 --> 01:37:14,523
it doesn't matter which order you put them in

1258
01:37:14,595 --> 01:37:18,038
The notion of orthogonality doesn't depend on order

1259
01:37:18,107 --> 01:37:20,960
Although the general idea of inner product does

1260
01:37:22,705 --> 01:37:24,427
So we now have a notion of orthogonal vectors

1261
01:37:27,998 --> 01:37:29,727
In a real vectors space, 

1262
01:37:31,723 --> 01:37:34,634
how can you define the dimensionality of real vector space?

1263
01:37:36,506 --> 01:37:39,169
Well give me a definition of dimensionality

1264
01:37:39,284 --> 01:37:41,110
There are many definitions of course

1265
01:37:41,189 --> 01:37:43,581
Anybody have one?

1266
01:37:46,894 --> 01:37:48,287
>> the () matrix of a vector

1267
01:37:48,359 --> 01:37:52,354
Yeah, but let's work it in a more abstract way without the

1268
01:37:52,424 --> 01:37:54,516
Yes, of course that's correct

1269
01:37:56,898 --> 01:38:01,110
The maximum numbers of orthogonal vectors if you can find in the space

1270
01:38:03,214 --> 01:38:05,323
So if a vector space is two dimensional

1271
01:38:05,401 --> 01:38:07,126
Ordinary two dimensional vector space	 

1272
01:38:07,195 --> 01:38:10,481
There are two mutually orthogonal vectors

1273
01:38:10,586 --> 01:38:14,345
You can not find a third vector which is orthogonal to both of them

1274
01:38:15,593 --> 01:38:19,372
So the maximum number of mutually orthogonal vectors

1275
01:38:19,450 --> 01:38:26,982
In three dimensions, I've got three mutually orthogonal vectors

1276
01:38:27,052 --> 01:38:30,809
Of course there are many sets of mutually orthogonal vectors

1277
01:38:30,882 --> 01:38:33,741
But given three I can not find the fourth one

1278
01:38:34,955 --> 01:38:37,109
So three dimensional vector space 

1279
01:38:37,180 --> 01:38:41,538
has a maximum number of three mutually orthogonal vectors 

1280
01:38:42,556 --> 01:38:45,540
and the same is true, or the same definition

1281
01:38:45,611 --> 01:38:49,926
it's definition is true for complex vector space

1282
01:38:50,746 --> 01:38:53,778
The maximum number of orthogonal vectors that you can find 

1283
01:38:53,851 --> 01:38:57,125
is equal to the dimensionality of the space

1284
01:38:59,507 --> 01:39:03,621
It does correspond to the number of entries here

1285
01:39:03,735 --> 01:39:07,189
The number of components , if it's the same you could prove that

1286
01:39:07,266 --> 01:39:11,630
As exercise, prove that the number of components of vectors

1287
01:39:11,701 --> 01:39:14,895
is the same as the number mutually orthogonal vectors

1288
01:39:15,413 --> 01:39:16,613
But let's just talk, 

1289
01:39:17,694 --> 01:39:20,263
Are there any mutually orthogonal vectors I can write down?

1290
01:39:20,336 --> 01:39:21,939
Sure, loads and loads of them

1291
01:39:22,017 --> 01:39:25,842
But the very simple case would be one zero

1292
01:39:26,755 --> 01:39:30,114
Now one and zero, are one and zero complex numbers?

1293
01:39:30,881 --> 01:39:34,289
Well real number are special cases of complex numbers

1294
01:39:35,151 --> 01:39:38,012
So real, complex number can be real

1295
01:39:38,870 --> 01:39:44,397
Here's one, and here's another one, zero one

1296
01:39:46,375 --> 01:39:48,402
This one has the one in the first place

1297
01:39:48,472 --> 01:39:52,196
This one has the zero in the first place, you get the idea

1298
01:39:52,268 --> 01:39:55,758
The inner product is zero times one plus one times zero

1299
01:39:57,110 --> 01:40:01,393
Okay, so the two vectors or 

1300
01:40:03,428 --> 01:40:05,906
Let's just write it in terms of column vectors

1301
01:40:06,876 --> 01:40:08,789
one zero and zero one

1302
01:40:08,900 --> 01:40:13,712
One zero and zero one are orthogonal vectors

1303
01:40:15,093 --> 01:40:18,909
The inner product of this one with the dual of this one is zero

1304
01:40:20,385 --> 01:40:24,018
That's because one times zero plus zero times one

1305
01:40:24,089 --> 01:40:25,545
This is zero

1306
01:40:26,349 --> 01:40:28,644
So there is plenty of orthogonal vectors around

1307
01:40:29,214 --> 01:40:32,651
But in two dimensions this is two dimensions	 

1308
01:40:33,214 --> 01:40:36,067
Once you found two of them, you can not find the third

1309
01:40:36,139 --> 01:40:40,633
So as exercise, prove that there can be no vector

1310
01:40:40,704 --> 01:40:43,707
which are both orthogonal to this and orthogonal to that

1311
01:40:45,474 --> 01:40:48,296
That's not hard to do, very easy to do

1312
01:40:53,927 --> 01:41:01,415
Oh ,year, "zero zero" is always orthogonal to everything

1313
01:41:04,012 --> 01:41:08,962
So one should be careful and say the maximum number of non zero

1314
01:41:09,039 --> 01:41:12,458
Orthogonal, mutually orthogonal vectors, good point

1315
01:41:13,230 --> 01:41:18,830
Alright, that's a mathematical interlude for tonight

1316
01:41:19,679 --> 01:41:21,561
Complex vector spaces

1317
01:41:21,633 --> 01:41:29,386
And the next time we will discuss how this space of states or what it means

1318
01:41:29,462 --> 01:41:32,277
Is the space of states of a simple qubit 

1319
01:41:32,954 --> 01:41:34,863
to be two dimensional vector spaces?

1320
01:41:35,916 --> 01:41:40,603
Very very different than saying they consist of a point and a set

1321
01:41:41,262 --> 01:41:44,604
The space of state is in fact a vector space 	

1322
01:41:44,676 --> 01:41:48,138
And try to make contact with the mathematics of vector spaces

1323
01:41:48,216 --> 01:41:50,679
And the strange things that are happening 

1324
01:41:50,751 --> 01:41:53,264
or seem to be happening over here with these

1325
01:41:58,357 --> 01:42:01,323
These funny properties of measurements

1326
01:42:01,433 --> 01:42:02,093
That's where we want to go

1327
01:42:03,496 --> 01:42:07,376
Learn some physics, some experimental physics blackboard up there, 

1328
01:42:07,461 --> 01:42:10,441
and we've done some abstract mathematics over here

1329
01:42:10,519 --> 01:42:14,576
And next time I want to connect them and show 

1330
01:42:14,613 --> 01:42:19,668
what the abstract mathematics how we represent or 

1331
01:42:19,739 --> 01:42:25,810
how it can be use to represent this unusual kind of logic of qubits

1332
01:42:27,080 --> 01:42:29,346
>> Is it restricted to a pattern, 

1333
01:42:29,457 --> 01:42:34,167
>> Is it going to be meaningful to define the multiplication operation

1334
01:42:34,238 --> 01:42:36,380
>> created two by two (raches)? 

1335
01:42:36,451 --> 01:42:39,255
>> It should be beta one alpha one 

1336
01:42:43,066 --> 01:42:48,809
You are asking? Yes the answer is yes

1337
01:42:48,921 --> 01:42:53,499
Yes absolutely, we will construct inner products outer products

1338
01:42:53,604 --> 01:42:54,936
so all sort of products

1339
01:42:56,631 --> 01:42:58,465
Ya, ya that's right

1340
01:42:59,288 --> 01:43:03,742
The outer products of the two matrix or two vectors as the matrix ya

1341
01:43:05,988 --> 01:43:10,492
Of course how do you represent it or just, well

1342
01:43:12,091 --> 01:43:16,739
The outer product of two vectors represented in this way

1343
01:43:18,354 --> 01:43:21,351
It is a matrix but we will come to that

1344
01:43:23,681 --> 01:43:28,035
>> all the postulates applies to the vectors and functions?

1345
01:43:28,947 --> 01:43:30,934
>> it's less obvious than

1346
01:43:31,848 --> 01:43:37,944
A vector space of function can be thought of as column vectors

1347
01:43:38,976 --> 01:43:41,824
but the entries are continuous

1348
01:43:43,407 --> 01:43:45,690
So we have alpha, alpha is the function

1349
01:43:45,771 --> 01:43:49,798
And the alpha of the first entry and alpha of the second entry

1350
01:43:49,869 --> 01:43:53,059
Out of that we have a continuous family of alpha

1351
01:43:53,134 --> 01:43:57,368
Alpha evaluated upon x 1, alpha evaluated upon x 2

1352
01:43:57,473 --> 01:43:59,316
Alpha evaluated upon x 3

1353
01:43:59,390 --> 01:44:02,442
And of course we can't really write it literally as

1354
01:44:05,048 --> 01:44:06,470
a discreet collection like that

1355
01:44:06,548 --> 01:44:10,913
But if you like think of these alphas here, alpha one alpha two

1356
01:44:11,025 --> 01:44:12,875
Is just the function of the two variables

1357
01:44:12,951 --> 01:44:16,763
A function of an entry one and two

1358
01:44:19,014 --> 01:44:20,776
Instead of a function of an x,

1359
01:44:20,889 --> 01:44:24,717
is a function of variable which would be either one or two

1360
01:44:26,166 --> 01:44:30,775
So yes, complex function forms complex vector spaces

1361
01:44:30,887 --> 01:44:34,270
The inner product, we will discuss that

1362
01:44:38,940 --> 01:44:43,458
>> () oriented apparatus turn at the angle of theta

1363
01:44:43,531 --> 01:44:48,316
>> If you consider the sequence of inputs going to outputs

1364
01:44:48,428 --> 01:44:52,565
>> the portion of minus one to one is given by close I tell you that

1365
01:44:52,607 --> 01:44:57,366
>> Is there anything interesting about the distribution of minus one 

1366
01:44:58,072 --> 01:45:00,163
What does it mean?

1367
01:45:00,311 --> 01:45:02,300
>> Well like high order statistics?

1368
01:45:03,954 --> 01:45:08,927
Oh oh oh, they are just completely random, completely random

1369
01:45:09,818 --> 01:45:11,309
These are completely random variables

1370
01:45:12,368 --> 01:45:21,191
You are talking about the relationship between different replications

1371
01:45:21,304 --> 01:45:25,226
And whole bunch of space you subject them all to the same, the question is

1372
01:45:25,300 --> 01:45:27,720
I think you are asking whether there are correlations between them

1373
01:45:28,414 --> 01:45:33,356
No, no, they are completely independent uncorrelated variables

1374
01:45:33,929 --> 01:45:37,926
>> So is it still the variable ()

1375
01:45:38,452 --> 01:45:43,257
>> Does it make sense to say that when you are measuring a qubit

1376
01:45:43,328 --> 01:45:47,774
>> you are affecting it, so you are actually preparing a new state

1377
01:45:47,850 --> 01:45:49,700
Yes yes yes, that's absolutely right

1378
01:45:49,777 --> 01:45:53,780
>> That's why first time you get something and the second time you just

1379
01:45:53,859 --> 01:45:54,885
>> keep to them

1380
01:45:54,959 --> 01:45:55,743
And there is a big difference 

1381
01:45:55,815 --> 01:45:58,346
between classical mechanics and quantum mechanics

1382
01:45:58,419 --> 01:46:02,209
In classical mechanics you can measure something 

1383
01:46:02,311 --> 01:46:06,511
sufficiently gently to not affect it, to not disturb it

1384
01:46:06,580 --> 01:46:10,020
In quantum mechanics whenever you measure something

1385
01:46:10,089 --> 01:46:11,591
you always disturb it

1386
01:46:12,503 --> 01:46:17,667
So this is an example of how measuring something disturb it 

1387
01:46:17,710 --> 01:46:19,027
and turn into a new state

1388
01:46:19,100 --> 01:46:21,559
And let's say that is a good point, yes

1389
01:46:25,509 --> 01:46:26,296
Okay

1390
01:46:28,251 --> 01:46:32,342
For more, please visit us at Stanford.edu