WEBVTT 1 00:01:07.560 --> 00:01:12.084 Hey, good afternoon. People again, I'm not certain who can hear me. 2 00:01:22.375 --> 00:01:23.605 Okay, beautiful Thank you. 3 00:01:24.450 --> 00:01:31.734 Okay, so this is class three of quantum computer programming. I put a stuff up on the. 4 00:01:32.819 --> 00:01:41.905 Web, and if I can share it, then look at it. 5 00:01:56.245 --> 00:01:56.935 Sure, sure. 6 00:02:12.294 --> 00:02:16.764 Okay, cool. So. 7 00:02:18.330 --> 00:02:23.844 First, I put up a homework from to test some basic mathematics. 8 00:02:24.509 --> 00:02:38.455 Down here do on great scope at next week and number one, 9 00:02:39.175 --> 00:02:40.405 simple complex page, 10 00:02:40.405 --> 00:02:44.245 and just review from almost high school just a little matrix stuff. 11 00:02:44.395 --> 00:02:46.824 What are the value of that Matrix? 12 00:02:48.025 --> 00:03:01.284 Again, basic complex number thing, there's a nice a morphism between is a nice morphism between complex numbers and points in the two D plane. 13 00:03:02.069 --> 00:03:04.914 So that the second. 14 00:03:10.224 --> 00:03:24.444 What I'm trying to do well, for chat. Okay. For chat if you have input, then unmute your microphone and speak it because I've got trouble showing the chat window at the same time. So I'm sharing the screen. 15 00:03:25.104 --> 00:03:34.555 Okay. So this is a morphism question. Three of the homework between points in the two D plane and geometric operations. 16 00:03:34.555 --> 00:03:42.025 So multiplying any point by that complex number is tantamount to what geometric operation. 17 00:03:43.435 --> 00:03:51.264 And then application of the exponential form, Polar polar coordinate form for complex numbers, say to find hours. 18 00:03:52.439 --> 00:04:05.485 Groups question six is the, there's a common a function in complex numbers. Probably won't be transformation. I give it right there for parameters. 19 00:04:06.449 --> 00:04:09.775 In vertable, so I just asked you to give me the inverse and you could also. 20 00:04:10.590 --> 00:04:14.305 Or you could research it and inverting a had Martin matrix. 21 00:04:15.180 --> 00:04:22.225 Quick exercise now I mentioned that the difference between quantum gates in classical Gates is that. 22 00:04:23.339 --> 00:04:37.375 Quantum Gates preserve information classical Gates may not. So the question is, could could an or gate be a quantum gate yes. Or no. And justify your answer. 23 00:04:38.399 --> 00:04:42.024 Now, what questions nine and ten refer to. 24 00:04:43.290 --> 00:04:57.264 Is we might extend a quantum gate a classic okay to a quantum gate? Why the only computing something but by caring through one of the inputs to be the output. So, question nine. We've got two inputs a, and B, the output. 25 00:04:57.264 --> 00:05:06.204 A prime is a, or B. you know, it could be prime is just be not does that make it a legal quantum gate and then with, or with exclusive or. 26 00:05:07.410 --> 00:05:11.904 So that's see if there's any. 27 00:05:12.629 --> 00:05:25.495 Questions great, it's any questions from that. I don't know how you ended up on being shown. Okay so that was that was the homework. 28 00:05:25.495 --> 00:05:37.824 Now, what I want to do today is first I recommended some videos last weekend, two, ten minute videos here. So, if there were any questions on the videos, if so. 29 00:05:39.089 --> 00:05:44.964 You can speak up and so on the I may put a homework question on them. So I'll give you more videos later. 30 00:05:46.944 --> 00:05:49.764 What I want to do today is. 31 00:05:53.454 --> 00:05:53.995 the sec 32 00:05:55.404 --> 00:06:08.394 I want to review I want to review some mathematics so, at this point, I'm working from quantum computing for computer scientists textbook and chapter one is basically complex. Some complex numbers. 33 00:06:08.935 --> 00:06:11.814 You should know it and. 34 00:06:13.560 --> 00:06:16.644 Here bring up something else. 35 00:06:18.720 --> 00:06:32.425 So here okay. 36 00:06:33.295 --> 00:06:33.925 Okay. 37 00:06:37.649 --> 00:06:50.545 Okay, so I have your chapter one? Just basic quantum, basic complex math Chapter two is getting into some stuff from linear algebra, which I don't know what your background is, because you buried. 38 00:06:50.545 --> 00:07:02.485 Some of you may have had some of it or telling you not all of it. So, what I want to do is to go through some of it and it's just some terminology it starts off simple and gets on simple in a while. 39 00:07:02.485 --> 00:07:05.904 So, let me okay. 40 00:07:07.259 --> 00:07:08.154 This then. 41 00:07:16.194 --> 00:07:20.035 Okay, so we can pull this over here. 42 00:07:23.069 --> 00:07:32.725 To this this over here. 43 00:07:36.930 --> 00:07:37.829 Okay, 44 00:07:45.654 --> 00:07:55.014 okay so this is okay, 45 00:07:55.014 --> 00:07:55.824 so complex. 46 00:07:58.014 --> 00:08:00.144 What a complex sector spaces. 47 00:08:01.259 --> 00:08:05.845 It's okay, the mass petitions, they have various things called extraction to look at different. 48 00:08:07.045 --> 00:08:18.925 Applications and they find these different applications may have some common properties and so they idealize the abstract and give a name to the set of common properties. 49 00:08:19.314 --> 00:08:30.295 And then this lets if they can prove something about this abstract set, then they can, wherever they prove is useful in all the different applications. 50 00:08:31.134 --> 00:08:38.815 And one thing that they have is something called a complex factor space. 51 00:08:42.059 --> 00:08:44.875 So the idea is, you got some sort of factor. 52 00:08:47.245 --> 00:08:52.434 Zero C, one or whatever could be to whatever something like that. 53 00:08:53.549 --> 00:09:06.085 Four dimensions, and it's got certain properties we all do examples with, with, with say, in two dimensions. So we have, you can add them. So if we have a vector, which is one plus two, I. 54 00:09:07.139 --> 00:09:11.904 For us one vector, we can add it to a second vector. 55 00:09:13.409 --> 00:09:28.254 Six size, seven, plus eight, and we get the result six plus eight and ten plus twelve by. Okay. 56 00:09:29.215 --> 00:09:36.985 So we can add we can multiply. We can say seven times one plus two, two plus four I. 57 00:09:37.860 --> 00:09:42.054 And we'll get seven plus fourteen. I and twenty one plus twenty eight. 58 00:09:44.220 --> 00:09:56.394 okay. and then just some other lower. there's some other. less important operations we can find and. we can find the negative or something there's a zero in something and 59 00:09:57.360 --> 00:10:01.794 and if we have this it's called a complex. it's called a complex sector space 60 00:10:02.934 --> 00:10:15.985 So far, we cannot multiply two factors that will come on there can add them and we can scale them up. The scaler could be complex also. And and an example of. 61 00:10:16.945 --> 00:10:20.004 An example of a. 62 00:10:20.879 --> 00:10:26.845 Complex sector space, a complex matrices are complex sectors to be so matrices. 63 00:10:28.980 --> 00:10:38.815 Our complex sector space sounds sort of weird, but if I have some sort of. 64 00:10:41.664 --> 00:10:49.225 Two by two matrix or something the zero zero a zero one ninety one zero one one. 65 00:10:50.370 --> 00:10:53.304 That's it that's so for D vector space. 66 00:10:54.059 --> 00:11:06.330 So that could be useful. So we've got that another example of a complex sector space is yeah. 67 00:11:06.325 --> 00:11:11.784 Getting a little less critical is say, polynomials. 68 00:11:15.299 --> 00:11:18.355 All meals up degree in. 69 00:11:20.549 --> 00:11:24.024 that's a complex sector space. so if i something like 70 00:11:25.049 --> 00:11:36.445 Squared plus two X plus one, that's one factor and two X squared plus three X plus four I could add two of them three X squared plus five X plus five. 71 00:11:36.720 --> 00:11:48.504 So, I can add and I could multiply by three, you know, I could get something like ten X squared was twenty X plus ten. So I can add them I can scale them. So all meals are that. 72 00:11:50.784 --> 00:11:58.284 So now with matrices. So now back to matrices and we can represent some, something, let's say, see. 73 00:11:59.549 --> 00:12:03.264 I don't know, I'll put actual numbers in here or something. 74 00:12:06.539 --> 00:12:11.965 So, yeah, I don't know two by two so these are two by two complex matrices. 75 00:12:16.764 --> 00:12:31.615 This is a, this is a complex sector space, but now, what I'm going to do is if I add some more operations, I, I can make the thing more complicated. Let's say, and is it out there? 76 00:12:33.504 --> 00:12:34.914 We can add things like. 77 00:12:36.629 --> 00:12:39.715 So, we've got some matrix here. I don't know. 78 00:12:42.929 --> 00:12:52.644 One, I, I, two or something like that now we can transpose that and that would give us. 79 00:12:54.059 --> 00:12:57.205 They want to be the same thing. Let me make it. 80 00:12:57.899 --> 00:13:11.424 Two minus side, so one minus I two, we couldn't transpose it. We can take things like a conjugate of it. 81 00:13:15.565 --> 00:13:17.784 That would be one minus. I. 82 00:13:18.870 --> 00:13:24.865 To, and, and we do an ad joint, which we do both together. 83 00:13:27.955 --> 00:13:36.955 So one minor side too, or something, we've got some other operating things which turned out to be useful. So, and. 84 00:13:39.144 --> 00:13:43.315 And we have made, and then we add matrix modification. We get a better. 85 00:13:44.970 --> 00:13:45.389 Okay. 86 00:13:51.504 --> 00:13:52.615 And was dying. 87 00:13:53.909 --> 00:14:00.804 And then we add modification and we get analogy. 88 00:14:03.779 --> 00:14:05.995 And just some terminology here. 89 00:14:11.215 --> 00:14:15.835 And my screen just froze for summaries. 90 00:14:17.340 --> 00:14:18.174 Just a second. 91 00:17:05.339 --> 00:17:05.940 Professor, 92 00:17:05.934 --> 00:17:07.315 I can't hear you talking, 93 00:21:51.295 --> 00:21:52.704 so we're still having trouble hearing. 94 00:22:56.910 --> 00:22:59.664 Professor muting says. 95 00:25:01.559 --> 00:25:12.150 How about now? Okay how about now? No. 96 00:25:15.954 --> 00:25:28.015 Yeah, we can hear you now right? But I'm getting a call. Okay. Can you hear me now? And no echo? 97 00:25:29.694 --> 00:25:42.115 Yes, thank you. Okay. So, if you're wondering what I'm doing, I'm using my phone for the audio when I'm my last call for the hover cam. 98 00:25:43.434 --> 00:25:48.174 So, if the image key so that's why I'm signed in. 99 00:25:51.119 --> 00:25:56.664 One concern to get annoyed hardware after a while and. 100 00:26:02.515 --> 00:26:07.555 Okay, phone is also useful for I can see what I can see sorted what you're looking at. 101 00:26:10.049 --> 00:26:24.775 And let's speak up if you can't hear me, because I'm not always watching the look. Okay. 102 00:26:28.795 --> 00:26:34.825 Is see if I can do this here. 103 00:26:37.015 --> 00:26:45.505 Oh, okay. So okay. I'm watching the chat window in my phone and using my phone for the audio. 104 00:26:46.440 --> 00:26:58.884 Up for the hover cam and video and projecting the web page. I've got a second laptop up to my left that I'm reading from. 105 00:26:59.849 --> 00:27:12.954 Okay, so where you were talking about, he missed the last few minutes. Let's go back. What I wrote here. We've got complex, active space. Well, it makes it a complex actor. 106 00:27:12.954 --> 00:27:17.154 Space is back to center. There's some fixed dimension like. 107 00:27:17.880 --> 00:27:31.674 Four dimensional vector, four complex numbers in it. You can add vectors and you can multiply vectors via scaler. Those are the main properties. It's got some other minor things. You can take the indication and the. 108 00:27:33.059 --> 00:27:41.545 Okay, now, oh, as an aside you could also combine this with calculus. 109 00:27:41.545 --> 00:27:53.875 You could start talking about vector spaces with an infinite number of dimensions and for you series or something. We're not gonna talk about that in this course. 110 00:27:53.875 --> 00:27:59.275 So you don't have to worry about convergence. Some limits and stuff like that. 111 00:28:00.690 --> 00:28:06.474 Okay, so vector spaces, you can combine to vector spaces to make a third. 112 00:28:07.230 --> 00:28:12.654 Complicated vector space. The first method is totally direct. Some are. 113 00:28:13.500 --> 00:28:28.285 Cartesian product just an order pair. Sorry I got the one from the perspective space speakers in the second vector space. What I can do is that then the parent, I can add two factors, and I can and vice versa. 114 00:28:28.285 --> 00:28:31.555 That's the simple way to do. 115 00:28:34.829 --> 00:28:38.035 To do it what I'm going to see. 116 00:28:39.329 --> 00:28:43.615 complicated way. the one which causes and tangle states and so on 117 00:28:44.609 --> 00:28:47.845 Before I do that what I, what I want to do is. 118 00:28:50.365 --> 00:28:55.704 Second here, I wanna talk about base systems because this pops up. 119 00:28:56.970 --> 00:29:04.045 A lot. Okay. 120 00:29:04.980 --> 00:29:10.134 Again, this is a review for many of you, but a basis. 121 00:29:12.420 --> 00:29:22.615 That third space so so basically, if it's if it's an in dimensional space, then there's a set. 122 00:29:24.450 --> 00:29:31.974 And basis factors, and every factor is so called the. 123 00:29:32.910 --> 00:29:37.555 The one to the end, let's say, and every vector is a. 124 00:29:40.380 --> 00:29:45.954 It's a linear combo, the base factors. Okay your combo. 125 00:29:48.630 --> 00:29:58.105 I wouldn't call this being of the and it's so totally obvious. I mean, the natural base. 126 00:29:58.769 --> 00:30:03.835 Two to three D for fun. Okay. So the natural basis. 127 00:30:06.269 --> 00:30:16.285 And so I'm gonna make it easier. So then that's the basis. This one zero and zero one and it's well, totally trivial. 128 00:30:17.190 --> 00:30:20.815 Back here two, three, that's two times. One zero. 129 00:30:22.170 --> 00:30:34.795 There a one and well, you started wonder this looking a little silly, but we could have a more complicated. We could have a more complicated basis. 130 00:30:42.025 --> 00:30:42.414 Oh. 131 00:30:47.940 --> 00:30:53.394 But let's say, Here's another basis. 132 00:30:58.765 --> 00:31:01.555 So that would be say one one. 133 00:31:02.430 --> 00:31:11.125 And maybe one minus one. So now if I have two, three, we'd represented in that and. 134 00:31:12.569 --> 00:31:17.305 We get a little harder, so that's going to be something, you know, a times one one. 135 00:31:19.170 --> 00:31:31.404 one minus one. requirements for the basis is only that the. sort of basis factors be independent spend the whole space. but okay so for some a and b and we could figure out what is that 136 00:31:32.220 --> 00:31:42.265 two equals. a plus be. three calls. a minus be 137 00:31:47.095 --> 00:31:56.394 So, a is tabs and vehicles minus one half. 138 00:31:57.420 --> 00:32:09.355 Three equals five, have one one minus one, half one line, one, two different basis sets of basis factors. 139 00:32:10.289 --> 00:32:14.184 You know, it's to the space and you get a different representation. 140 00:32:16.255 --> 00:32:25.704 Another way to look at that is more geometrically. Oh, having something like this we could have a set of basis factors like this, and this. We have a. 141 00:32:26.670 --> 00:32:32.184 Some point here is a point here it would be the basis actor in this direction next direction. 142 00:32:33.599 --> 00:32:40.734 Plus a scale basis factor in that direction. However, what I could do is. 143 00:32:42.450 --> 00:32:48.805 Let's see here, you know, I, I could have another. 144 00:32:49.769 --> 00:32:57.954 Basis like this and which cases point is here this much here in this much here. 145 00:32:58.974 --> 00:33:08.515 Okay, so, and you convert between the representation. So, the point is that the. 146 00:33:09.660 --> 00:33:17.214 I vector to three different representations, suspected different basis factors and effectively we're doing rotations and transformation. 147 00:33:23.005 --> 00:33:24.144 Which is. 148 00:33:26.099 --> 00:33:30.835 Going to be happening a lot. Okay. Now. 149 00:33:31.589 --> 00:33:38.664 And we have the way to your probably Paris, why it's a half or whatever. Okay. And you can change your basis. Now. 150 00:33:39.930 --> 00:33:54.355 I'll give you a real world example actually, just for lightning the course up a little between difficulties, converting basis factors. If you take elevation above sea level. 151 00:33:54.930 --> 00:33:58.585 So I'll just scaler with one dimensional vector and. 152 00:33:59.160 --> 00:34:03.085 Could have one basis, I can say hi to above whatever. 153 00:34:04.049 --> 00:34:13.795 So, your something and surprising thing is that there are many different basis factors. He might say many different origins actually, four C level. 154 00:34:14.760 --> 00:34:18.414 And converting between different systems, give it two big ones here. 155 00:34:19.105 --> 00:34:30.804 Something called North American data of nineteen, twenty seven, and the word Geodetic system of nineteen, eighty four, and you can convert, you can convert between them and some people might get it wrong. 156 00:34:31.559 --> 00:34:40.554 A cool example here actually, Switzerland and Germany, these are countries that have certain reputation for comprehensive and engineering. 157 00:34:41.699 --> 00:34:49.824 Well, twenty years ago, or so, they're building a bridge across the river convert and it's not a particularly wide at that point. 158 00:34:50.875 --> 00:35:03.025 And turned out to be a different elevations so, if they hadn't done, like, correction after they started the bridge, the two and the one, the two, they're building from both shorts towards the middle, there were a half meter error. 159 00:35:04.050 --> 00:35:11.514 Because they were using different basis systems effectively, and they calculated wrong and I get a link there. You can Google. 160 00:35:14.400 --> 00:35:27.085 So, it gets even funnier these differences definitions are so you buy a handheld they're encoded in the firmware. You know, they're not state secrets. I completely public and. 161 00:35:29.219 --> 00:35:33.295 And it is known that Germany C level. 162 00:35:33.960 --> 00:35:48.565 Is calculated with respect to the Northeast, but some suspect the Mediterranean is a quarter needed front trouble. So I talked to the bridge and a half meter at her. Somebody added instead of subtracting. Okay. Real world with systems and coordinate systems. 163 00:35:48.565 --> 00:35:50.760 So okay. 164 00:35:59.755 --> 00:36:05.454 Well, yeah, you can have fun with that. Okay. 165 00:36:07.619 --> 00:36:16.375 On the causes other errors, these different systems. Well, I think there was a plane crash in Russia some years ago. 166 00:36:17.070 --> 00:36:28.375 Because the one of these types of errors and horizontally, not vertically. Okay. So another example changing, I'm changing basis. 167 00:36:28.375 --> 00:36:33.565 Systems is effectively the matrix, which pops up all over. 168 00:36:34.230 --> 00:36:44.755 For got it. 169 00:36:46.110 --> 00:36:50.875 One more one on over to. 170 00:36:52.019 --> 00:37:03.715 So normalize and that is, I mean, that is effectively doing a, and it's basically sort of doing a forty five degree. 171 00:37:04.829 --> 00:37:07.014 So that's a change of basis. 172 00:37:07.735 --> 00:37:22.704 Okay so I'm working my way to things like section up through section two point three and so on in the book thing in the book is the book 173 00:37:23.034 --> 00:37:24.775 matrix multiplication. 174 00:37:27.780 --> 00:37:32.784 Let's make this a multiplication with the star just so it's not ambiguous. 175 00:37:33.565 --> 00:37:34.344 Okay. 176 00:37:41.099 --> 00:37:44.755 And the book talks about linear maps and functions, and it's all on. 177 00:37:46.260 --> 00:37:53.394 Section two point three is basis dimension and you can convert between basis factors. 178 00:37:57.150 --> 00:38:03.985 Okay, now you might ask yourself why do you use different basis systems in the natural basis system? 179 00:38:05.670 --> 00:38:09.385 So, obvious, well, there may be cases. 180 00:38:10.320 --> 00:38:14.065 Or another basis system, let you solve the problem more easily. 181 00:38:14.820 --> 00:38:23.364 So, to give an example on the book, so you want to have the basis factors, sort of, setting the problem you're trying to solve and that's why. 182 00:38:24.750 --> 00:38:28.255 Okay, so. 183 00:38:29.244 --> 00:38:33.144 and you can. enter product and so on will i'll cover later 184 00:38:34.440 --> 00:38:37.974 it has. limits to how much mathematics you can take in one 185 00:38:38.909 --> 00:38:42.085 you can take in one day. but what i wanted to know 186 00:38:42.750 --> 00:38:47.965 is the second. way to combine factors so this is the second way 187 00:38:50.039 --> 00:38:58.434 combining doctor spaces. places. it's called the tens of product 188 00:39:02.280 --> 00:39:05.994 I sometimes call it exterior product also. Okay. 189 00:39:07.469 --> 00:39:12.684 And that give an example what the, what what this is. And this is. 190 00:39:13.980 --> 00:39:19.945 In the book key into the book here. 191 00:39:26.789 --> 00:39:39.894 See, I again, value section two point five do next time I may not want to hit you with. 192 00:39:40.650 --> 00:39:52.525 Too much. Okay, this is section. Two point seven page. Sixty six. So six. Okay. 193 00:39:54.210 --> 00:40:03.054 So, what we do here is what I was showing you last time. So what we do if I've got. 194 00:40:04.739 --> 00:40:07.764 Suppose I've got two, two D vectors basis. 195 00:40:12.659 --> 00:40:14.574 So, we've got a vector from one. 196 00:40:21.420 --> 00:40:26.394 From the first say a zero one. 197 00:40:29.369 --> 00:40:37.764 And the zero one, and then the tenser product. 198 00:40:39.989 --> 00:40:44.425 The Tensor product is they do this combo zero V zero. 199 00:40:45.719 --> 00:40:56.994 The one to zero there one be zero one. The one. So you, every Columbo, every element of the first combine multiplied with every element. 200 00:40:58.500 --> 00:41:10.945 Okay, and and what happens and now this is a, this is now, how, how would you prove it's a vector space. 201 00:41:15.420 --> 00:41:16.614 You might ask yourself. 202 00:41:24.025 --> 00:41:25.045 Factor space. 203 00:41:28.320 --> 00:41:32.635 Well, what you have to show is you have to show you that you can add. 204 00:41:35.369 --> 00:41:36.264 And scale. 205 00:41:40.590 --> 00:41:53.755 Well, if I go back and look up here, well, I could certainly scale I can on time, but scale or the front into scales all of those. 206 00:41:55.764 --> 00:42:06.835 And I could if I want see if I could add things, then I take a different color here. 207 00:42:08.789 --> 00:42:12.204 Close this plus see zero C one. 208 00:42:13.110 --> 00:42:16.554 Then then the result would be. 209 00:42:18.449 --> 00:42:22.014 Phase zero plus zero times the zero. 210 00:42:22.619 --> 00:42:25.974 All the way down, which would be zero. 211 00:42:26.940 --> 00:42:30.684 All the way down zero all the way down. 212 00:42:31.619 --> 00:42:38.454 So, we can, I, I'm being a little sloppy here basically if we define a combo, two active spaces. 213 00:42:39.300 --> 00:42:47.574 This then we can, we can add factors and we can scale thing. So that actually is a vector space. 214 00:42:51.000 --> 00:42:57.894 Now, the thing is that. 215 00:43:04.525 --> 00:43:14.394 If let's say, let's say dimension, random work there dimension of the one them and dimension of V two then. 216 00:43:15.360 --> 00:43:19.614 That a dimension of the one that's assemble the. 217 00:43:20.789 --> 00:43:28.945 M, time the first thing, the direct combo dimensions added this. 218 00:43:30.239 --> 00:43:40.644 Scarier tense their combo, the dimensions multiply. Okay. And three. This happens to be the way compact bottoms. 219 00:43:41.639 --> 00:43:49.255 Combine and there's some nice examples in the book, which you can look at. I won't, I won't hand right. 220 00:43:51.960 --> 00:44:00.894 Now, so this is how we create, you know, how do we create one of these sensor product factors? 221 00:44:09.445 --> 00:44:10.014 Cray. 222 00:44:13.409 --> 00:44:19.224 Product sector there's various ways. 223 00:44:23.519 --> 00:44:27.864 You know, one, we might say, combine. 224 00:44:29.130 --> 00:44:35.364 You know, two factors from the components space. 225 00:44:39.239 --> 00:44:42.985 On the spaces, or you might, you know. 226 00:44:43.679 --> 00:44:50.394 So, on et cetera, now, the thing is that once you do this. 227 00:44:51.809 --> 00:44:55.434 You may get a resulting factor, which cannot be split back. 228 00:44:57.420 --> 00:45:05.695 So the result might not be separate. 229 00:45:10.320 --> 00:45:15.355 What would you might be noticed things? I'm leading back into entanglement from another direction. 230 00:45:16.139 --> 00:45:22.764 And and the, the book has and has an example. 231 00:45:28.980 --> 00:45:34.164 So, I'll work on way through so you want to see the book. 232 00:45:36.000 --> 00:45:40.824 Page seventy example two seventy two. 233 00:45:43.559 --> 00:45:48.054 Page seventy, it's example two point seven point two. 234 00:45:49.019 --> 00:45:59.605 And what they're doing is combining a two dimensional space with the three dimensional vector say, so it will give us six dimensional. 235 00:46:02.005 --> 00:46:13.945 The extra space, I mean, the types of product, it's vector space, it's just form. This is how you have them to format, and they give some examples up here that. 236 00:46:18.210 --> 00:46:32.664 You know, suppose I've got a vector eight, twelve, six, Kofi I know. Well, that that's the Tensor product of. 237 00:46:34.559 --> 00:46:40.224 Because you look at the second, three numbers or fifty percent more than three numbers. 238 00:46:40.735 --> 00:46:51.989 So we can represent this as two, three, serial product center product from two, three and. 239 00:46:58.050 --> 00:47:08.394 In four, four, six, three. Okay, because two times four is a two time. Six is twelve, two times. 240 00:47:09.000 --> 00:47:19.914 Okay, so this vector here, it can be separated back again. Give you another one say this. 241 00:47:24.480 --> 00:47:28.644 This cannot be, I mean, if I say, okay, well, that's going to be. 242 00:47:31.045 --> 00:47:33.565 A B times. 243 00:47:34.349 --> 00:47:37.735 The de, then. 244 00:47:38.969 --> 00:47:46.585 We're going to get eight times C, equals a times. You know, we were gonna get out that this. 245 00:47:51.269 --> 00:47:53.664 If you tried to if you tried to work it out. 246 00:47:57.210 --> 00:48:05.724 You could you've got five unknown variables here we've got sixty equation and you set them out that Tom. 247 00:48:07.440 --> 00:48:11.905 it just absorbs. the. eight times the 248 00:48:13.045 --> 00:48:23.485 zero so. like if a time c is a an eight times the zero. then. the has to be zero for example on a. z is zero 249 00:48:25.014 --> 00:48:31.344 he has to be zero. and then you can go the times. it just doesn't work out 250 00:48:32.610 --> 00:48:35.875 no solution. okay 251 00:48:40.710 --> 00:48:44.545 but you could. split it to. do it as the sum of two 252 00:48:45.780 --> 00:48:54.985 or you could say something like this. plus 253 00:49:01.019 --> 00:49:04.764 we could do something like that it's the sum. up to these types of products 254 00:49:07.764 --> 00:49:11.184 and again anticipating and tanglement and so on 255 00:49:11.969 --> 00:49:23.005 You might be able to take this and wrote apply a six by six matrix to it. And who knows, you know, produce something, which can be separated. 256 00:49:24.360 --> 00:49:28.675 Okay, so so, in this case, you'd say that the there. 257 00:49:29.369 --> 00:49:36.505 So the first thing is separate, and this is not. 258 00:49:38.130 --> 00:49:42.445 So, it's not separable and that means untangle. 259 00:49:48.179 --> 00:49:51.864 Now, getting back to the quantum systems. 260 00:49:53.639 --> 00:50:08.184 The quantum states, a separate it means you can work on one Cupid without messing with the other queue. But you could say, do a measure at all operation on one to bed. And that does not affect the other queue. 261 00:50:08.184 --> 00:50:14.724 But if they're not separable, then if you do something to the one queue, but you work constrain. 262 00:50:16.380 --> 00:50:20.275 The other Cuba. Oh, okay. 263 00:50:22.079 --> 00:50:35.460 So that's what, and they have another example. Yeah the example on page seventy. So one basically. Okay so what I'll do Thursday I've taken you up through. 264 00:50:38.875 --> 00:50:45.505 Well, then then they go into in more detail on the next few pages, but I gave you the full content of that. So. 265 00:50:47.400 --> 00:51:02.244 We'll talk about transforming them, but basically that was Chapter two, except for that stuff in the middle, which I'll do Thursday stuff in the middle. Well, more on some basis systems and stuff. Okay. 266 00:51:04.949 --> 00:51:09.114 So, and then Thursday, all get into. 267 00:51:09.750 --> 00:51:12.625 Three, I'll give you a little teaser of it. 268 00:51:14.130 --> 00:51:17.844 Now, it's related, but first, I want to come back to the. 269 00:51:19.260 --> 00:51:24.025 Entanglement thing again. Okay. We're talking about it last time. 270 00:51:25.710 --> 00:51:29.545 I just want to hit it, come at it from another direction. 271 00:51:30.539 --> 00:51:43.704 So suppose we've got to cube it, so each, each single cubic, it's a two factor. We combine them against their product, the results support vector and we combine them. 272 00:51:44.369 --> 00:51:55.945 So, Tuesday, two separate one cubic systems. We can we decide to consider them as a group because maybe the, I mean, you want to do that because maybe the cube to interact with each other. 273 00:51:56.760 --> 00:52:03.445 Okay, so it's now that state of that two cubic combo is the port actor. 274 00:52:04.945 --> 00:52:12.655 And you operate on it with four by four rotations in four dimensional space and. 275 00:52:13.409 --> 00:52:15.295 The only thing you can do with quantum systems, 276 00:52:15.295 --> 00:52:15.505 I mean, 277 00:52:15.505 --> 00:52:18.804 here on the paper short adding and so on, 278 00:52:18.804 --> 00:52:18.985 like, 279 00:52:18.985 --> 00:52:19.255 there, 280 00:52:19.255 --> 00:52:22.170 but the quantum system factors, 281 00:52:22.224 --> 00:52:25.644 the only thing you can do is a four by four makes modified BC, 282 00:52:25.889 --> 00:52:28.855 rotation matrix and rotation in a forty. 283 00:52:30.684 --> 00:52:35.304 Okay, so after you do your rotations and so on. 284 00:52:36.989 --> 00:52:42.025 Sometimes you can separate it back out again and to the tens of product or two factors. And sometimes, you. 285 00:52:43.079 --> 00:52:47.844 A separate bull, or it's not and I mentioned if it's separate measuring. 286 00:52:48.869 --> 00:52:57.594 The other, if it's not separable, then the to bits are are entangled because measuring. 287 00:52:59.190 --> 00:53:01.735 Constraints what you'll see if you work on the other. 288 00:53:05.820 --> 00:53:14.065 And you could rotate it again and and disentangle it. So we've mentioned Thursday, you know, all this stuff goes both ways. 289 00:53:17.369 --> 00:53:25.135 Now, and I'm talking about it more down here, but there was this paper that was asked about Thursday. 290 00:53:26.250 --> 00:53:29.335 Talking about some of this and. 291 00:53:33.840 --> 00:53:40.315 Let me B*** up here, I'm assuming. 292 00:53:41.550 --> 00:53:44.965 This works yeah. 293 00:53:50.670 --> 00:53:59.934 What I really need to stuff one more computer in front of me so I can look at what you're seeing and also look at the chat window. Okay. So just paper that was. 294 00:54:00.900 --> 00:54:05.724 Thursday then just talks about examples where. 295 00:54:07.440 --> 00:54:11.184 Well, you're doing your rotations and time in the same time things. So. 296 00:54:12.929 --> 00:54:20.485 A nice descriptions and talked about a little last time and that and if you want be a good reading exercise with people in the class, 297 00:54:20.485 --> 00:54:23.125 I think I'm going to slowly and, 298 00:54:27.175 --> 00:54:27.565 you know, 299 00:54:27.565 --> 00:54:29.215 on something a little deeper for some of you. 300 00:54:30.869 --> 00:54:33.264 Free to deal with that. 301 00:54:35.610 --> 00:54:40.914 Getting into other things, like quantum error correction and stuff like that. Okay. 302 00:54:58.590 --> 00:55:01.735 Okay, so this is a rehash of I won't read outcome. 303 00:55:03.570 --> 00:55:16.465 This day just repeated the same thing. Okay. Now, what I want to do now is gave you a teaser of chapter three of the book. I haven't typed in now. 304 00:55:24.744 --> 00:55:25.164 So. 305 00:55:26.880 --> 00:55:30.864 What else we should do at some point is prove that there's nothing. 306 00:55:35.489 --> 00:55:45.625 Three teaser yeah, these things get related to. 307 00:55:52.980 --> 00:55:56.994 Models et cetera, part of okay. 308 00:55:59.010 --> 00:56:01.344 And just a second here. 309 00:56:06.719 --> 00:56:14.184 Okay, good. And the idea. 310 00:56:14.909 --> 00:56:22.405 This is oh, you're using terminology terminology from. 311 00:56:24.989 --> 00:56:28.795 The book I'm going to do is iterate through a couple of different. 312 00:56:30.000 --> 00:56:33.804 Versions of a system so this is like version one. 313 00:56:36.235 --> 00:56:40.525 We have we had something like this. 314 00:56:41.309 --> 00:56:50.275 Darius parts with marbles box with marbles. 315 00:56:53.280 --> 00:57:05.275 Step at each step, they move so we've got to park here with five marble. 316 00:57:06.570 --> 00:57:21.565 Here with three here with one talk here with two, we have some, we have some rules perhaps and it might be something like all the marbles here. Go there. 317 00:57:21.625 --> 00:57:26.155 All the marbles here to go there. Marbles here. Don't do anything or something. 318 00:57:28.079 --> 00:57:35.094 So this is at time, zero time one, all the five marbles went there and all those ones here we had zero. 319 00:57:36.510 --> 00:57:36.989 Here, 320 00:57:36.985 --> 00:57:39.385 we had those two went up there, 321 00:57:39.385 --> 00:57:43.675 but the five down here you went up here we had five up here also, 322 00:57:44.065 --> 00:57:53.369 and this one marble and then we do this again time to the zero all ten marble here. 323 00:57:54.054 --> 00:58:04.855 This marble stays the same and we get zero down here so we have these transitions. Okay. Each time is quantitized. We start off with we've got two things here. 324 00:58:05.184 --> 00:58:14.125 We've got four talks with marbles and we've got these arrows showing how the marbles each step. Okay. And. 325 00:58:15.480 --> 00:58:18.625 And so the clock ticks and bang the. 326 00:58:20.309 --> 00:58:25.735 Follow the arrow circles here to here and here, and this is tied to the losses original three, but got five. 327 00:58:26.519 --> 00:58:29.545 Okay, that's so simple. It looks silly. 328 00:58:31.045 --> 00:58:34.195 What I'm doing is loading into a false sense of security. 329 00:58:35.130 --> 00:58:38.905 And what we can do here, is that the state. 330 00:58:42.780 --> 00:58:47.275 It's effective. Okay. Just go do to do a vector five three. 331 00:58:50.550 --> 00:58:54.085 One two. Okay. And the transition. 332 00:58:56.369 --> 00:59:09.804 It's a matrix. Okay, so I've got the stake back. You're here. I can by, you know, I can apply. I won't do it. I can apply to the sector and we get the next day Times Square. Do we get that state and so on? 333 00:59:10.315 --> 00:59:13.224 So, we've got the States, which are the factors giving. 334 00:59:14.579 --> 00:59:16.525 Place and we've got. 335 00:59:19.710 --> 00:59:25.045 And transition make mistakes. Okay so far so good. 336 00:59:29.969 --> 00:59:42.744 But what I'm going to do next, so this is version one version to this version. One. This is deterministic version. 337 00:59:42.744 --> 00:59:56.034 Two is gonna be probabilistic by the way if I page through, and, you know, tear the paper off and go to the next page too quickly. Let me know, and I'll oh, okay. 338 00:59:57.775 --> 01:00:03.809 Deterministic version to probabilistic. 339 01:00:09.360 --> 01:00:12.804 Okay, Jeremy do it again. 340 01:00:13.440 --> 01:00:20.065 for. five and. thirty. one and. two 341 01:00:24.780 --> 01:00:37.824 Okay, but now, let's say that we have probabilistic. So we have this one here might be just with one half and there with one, half or something. And then up here, it might go with. 342 01:00:41.130 --> 01:00:55.675 Say one half and be there with one half for something in here. I don't know. Go to their one third states where it is two. Okay. Okay. So that's fine. Zero. 343 01:00:56.519 --> 01:01:11.514 Happens by one needed to do. Oh, okay. Arrow. He won this thing, got a third of that. 344 01:01:11.514 --> 01:01:12.175 So we got. 345 01:01:14.579 --> 01:01:25.014 This one got the one in the bottom left, got a half of the five. That's fine. Has and a half of the two. That's one. So it's seven top, right? 346 01:01:25.980 --> 01:01:29.155 Got two thirds of it's old value that. 347 01:01:30.210 --> 01:01:35.155 Two to three bottom, right? And a half the pie that's five has. 348 01:01:40.139 --> 01:01:47.755 Just for fun initially we had a plus three, which is eleven marbles here. We had four plus. 349 01:01:48.690 --> 01:01:52.494 ten marbles. a marble disbanded two d material 350 01:01:53.730 --> 01:01:55.135 So, what happened here. 351 01:01:59.425 --> 01:02:01.255 Half and a half conserve. 352 01:02:02.789 --> 01:02:08.394 On half and a half or a third two thirds. 353 01:02:12.630 --> 01:02:15.565 On here, it gets a half of the five. That's fine. 354 01:02:16.920 --> 01:02:21.175 One here, just the half of the two and a half of the five seven has. 355 01:02:23.875 --> 01:02:36.175 It's a third of the screen that one, this one talk, right? It's a half of the two plus two thirds of the three. I don't know. 356 01:02:36.809 --> 01:02:43.704 Went wrong, you know, you find your way. Okay. 357 01:02:53.010 --> 01:03:00.264 Could be a homework question. Okay. Yes, I suppose the problem. 358 01:03:01.559 --> 01:03:10.375 For the top, right? Oh, thank you. One third. The three goes to the left with one third and stays where it is with. 359 01:03:16.375 --> 01:03:20.905 So, that ends up. So the one third, we get a one going to the left of here. That's the only things. 360 01:03:23.760 --> 01:03:33.985 The two thirds of the three keeps two, and it gets a half of a two, which is one two and one sweepstakes are the three top, right? Looks. Okay. Well, what. 361 01:03:36.750 --> 01:03:39.985 Well, we have a half to half leaving the top left. 362 01:03:40.675 --> 01:03:44.005 Oh, the one down here, nothing is leaving the one down here. 363 01:03:44.905 --> 01:03:56.965 Oh, so so the one down here, that's the fourth one so it's a half of that plus a half of that all of that nine have. Okay so this is four and a half here. 364 01:03:58.284 --> 01:04:02.364 Okay that was here now it adds up conservation. Okay. 365 01:04:03.719 --> 01:04:11.875 So that's probabilistic. Let me give you a slide more complicated probabilistic one. Not again. 366 01:04:12.719 --> 01:04:17.335 Location oh, okay. So okay. So this. 367 01:04:18.059 --> 01:04:23.695 Is make a small duplication, so. 368 01:04:26.965 --> 01:04:31.375 Let me go up and I won't do the matrix. 369 01:04:33.210 --> 01:04:37.644 And I'm paging, so the work. Okay. 370 01:04:40.170 --> 01:04:43.974 And let me start to sign a new page. 371 01:04:45.210 --> 01:04:57.264 Another example is, so what I'm gonna do now, this example so it'd be a, to select example. We've got a source here. 372 01:04:59.094 --> 01:05:08.215 And we've got a wall and we have five targets behind the wall. 373 01:05:11.335 --> 01:05:25.945 We currently can't see the top of the page there. Okay. Okay so we got a source here. Qualcomm slips. 374 01:05:25.945 --> 01:05:28.885 They pick a historical example and. 375 01:05:29.820 --> 01:05:33.775 Then they go to I stay here and then they go to. 376 01:05:35.190 --> 01:05:40.315 And the way the probabilities work is this one, half and one path. 377 01:05:41.159 --> 01:05:55.164 And then from here, if I go to the top slip, it's one, third, third, one, third and one, third, third, one, third. Okay. Now what's gonna happen. 378 01:05:55.164 --> 01:06:06.925 So there is one photon here after time one, there's a photon here and a half a folks on there after time to probabilities are going to be. 379 01:06:08.940 --> 01:06:13.434 The one six one, six on third. 380 01:06:19.409 --> 01:06:24.085 So, and so this is actually this is a classical. 381 01:06:26.969 --> 01:06:40.255 Since late exempt now, here, the probabilities at these add. Okay if you look up there, one, six, plus one, six simple form. 382 01:06:42.210 --> 01:06:46.195 Abilities add and the probabilities they're real. 383 01:06:48.780 --> 01:06:53.755 And then your probabilities, you know, these are all real numbers. 384 01:06:57.929 --> 01:07:01.795 Okay, so and this is. 385 01:07:03.630 --> 01:07:09.864 You witness a neighborhood of page eighty six and so the book okay. 386 01:07:12.000 --> 01:07:20.605 Second here right? You were right here. 387 01:07:30.449 --> 01:07:40.135 So, why am I wasting your time or things? So simple because. 388 01:07:42.360 --> 01:07:49.315 They probabilities or complex. 389 01:07:55.590 --> 01:08:01.315 Then the probabilities at and get bigger. 390 01:08:04.320 --> 01:08:07.344 We have the six plus the six becomes a third then. 391 01:08:10.980 --> 01:08:13.764 They like, and so. 392 01:08:16.770 --> 01:08:28.675 And gets smaller, something, qualitatively different with complex. My real example. Here we add probability. So they always get bigger. 393 01:08:29.880 --> 01:08:37.494 If I start putting in complex numbers here, they don't always get bigger, qualitatively different. 394 01:08:38.725 --> 01:08:44.005 And which is what. 395 01:08:44.760 --> 01:08:57.895 What really happens and at this point, I'll leave it for Thursday and so basically the section. 396 01:08:59.664 --> 01:09:08.484 Very. Yeah. So what are we coming in with complex numbers is we're getting you can get interference because. 397 01:09:12.630 --> 01:09:24.505 Does everyone again you see, your backgrounds are quite diverse I'm curious how many of you know about the two slit experiment. 398 01:09:25.380 --> 01:09:34.045 Physics is this totally familiar he had it and Greg can or is it you've never heard of the term two slit experiment in physics two we want to. 399 01:09:35.784 --> 01:09:39.114 Cool. Okay. Thank you. Remember physics to. 400 01:09:40.529 --> 01:09:44.274 Okay, thank you. Thank you corner. Okay. 401 01:09:45.659 --> 01:09:48.265 So, the thing is just as a rehash. 402 01:09:50.100 --> 01:09:50.340 Yeah, 403 01:09:50.335 --> 01:09:51.625 that's why I got a physics courses, 404 01:09:51.625 --> 01:09:55.914 a Pre work for this course if, 405 01:09:56.935 --> 01:09:57.625 you know, 406 01:09:57.625 --> 01:09:59.515 if you had waves going through, 407 01:09:59.609 --> 01:10:08.994 then the two waves going to the two slits would interfere with each other and right in the middle instead of getting one third you might get zero or something. 408 01:10:10.944 --> 01:10:24.175 That was done by young Michael. I think now, Chris, the fun thing with this is you can lower the intensity as a light good lower down to the point where there's one quantum I'm going through. 409 01:10:24.864 --> 01:10:35.154 It doesn't have to be a photon could be an electron also. I think they've done it with you got one photon or one electron is a time going through this apparatus. 410 01:10:35.789 --> 01:10:39.534 And you still get an interference batter. Okay. 411 01:10:42.329 --> 01:10:51.444 So, this was an argument for the way theory of, like, I guess Way's theory of matter to, I guess, since, I think would be an electron, not just the photon. 412 01:10:53.970 --> 01:11:02.965 Going to the math of that so any case so I'll get the quantum version of that on Thursday. That's enough. New stuff. 413 01:11:03.720 --> 01:11:17.635 Today, I'm watching the chat window on my iPhone if you said as well as looking at the book on one laptop and running WebEx on a second laptop what's going on now? 414 01:11:19.170 --> 01:11:22.734 And having the hover cab, so let me just review what. 415 01:11:23.430 --> 01:11:27.204 And this time, I hope I'm doing a recording sorry about. 416 01:11:28.710 --> 01:11:43.045 No, any case I have what I on the blog so what are we doing today? Introducing some mathematics and it's nice to attach names the things. 417 01:11:43.074 --> 01:11:53.725 It's unifying, a generalizing concept. So we call this thing complex vector space that factors in elements of complex numbers. 418 01:11:53.755 --> 01:11:58.074 And you can two factors, and you can, you can multiply that by scale factor, which is a. 419 01:11:59.699 --> 01:12:03.685 Right. We haven't talked about multiplying doctors yet. That'll come in. 420 01:12:04.800 --> 01:12:14.305 See, I guess, okay, so a matrix is also a complex sector space. You know, just they're in the square instead of laid out, but you can add make receipts and you can scale them. 421 01:12:15.810 --> 01:12:19.555 Okay, all males are a. 422 01:12:20.699 --> 01:12:24.114 Price to the factor of the coefficient, you can add them, you can scale them. 423 01:12:26.670 --> 01:12:30.534 And we also were talking about. 424 01:12:31.680 --> 01:12:35.005 Buying a dimensional space, and if they are infinite dimensional. 425 01:12:36.539 --> 01:12:50.515 For a series, like, say, then we'd have to worry about considerations of convergence on whatever historically, two hundred, odd years ago. The problem was when they went to independent dimensional things. 426 01:12:51.210 --> 01:13:00.685 They started getting paradoxes and and contradictions, and that's why they have to develop some analysis analysis means like, calculus. 427 01:13:03.625 --> 01:13:08.305 Okay, so and then if it is a matrix, you can transpose it. 428 01:13:09.300 --> 01:13:11.755 Complex conjugate mind to get an ad drawing. 429 01:13:13.529 --> 01:13:25.284 And you might get a cell by joint matrix where you had dropped the same as a matrix, and they make this easy multiply them traditional things. So, this makes things the complex active space it makes it into and out something calling out. 430 01:13:27.145 --> 01:13:38.604 And this is relevant, because the quantum system is a complex sector space. Okay. And now we have to simple back your spaces. We can two ways. 431 01:13:38.635 --> 01:13:53.034 I'm gonna show you to combine them to make a Messier bigger effector space, more degrees of freedom. We did direct some drifts. I'm just in ordered care. Okay, so getting back to you spaces X coordinates and other vector space X. Y, coordinate. 432 01:13:54.385 --> 01:14:00.805 And the dimensions are just one is the two dimensional V two, three dimensional the. 433 01:14:03.359 --> 01:14:06.625 Order there's nothing it's, there's no trick here or anything like. 434 01:14:07.890 --> 01:14:22.255 Okay, second here and you could add actually I'm going live on proofs in this course I can go heavy on proofs. 435 01:14:22.255 --> 01:14:23.095 If you want. 436 01:14:24.689 --> 01:14:28.494 Some hand waving just because we can say this. 437 01:14:29.069 --> 01:14:34.164 Some is a vacuous space. We can stay many things. You might, you know, somebody had to prove it. 438 01:14:35.310 --> 01:14:43.555 Much basis you've seen that before and dimensional vector spaces and factors on a basis. He represents the vector. 439 01:14:45.060 --> 01:14:57.534 Some of the basis factors, but the point is, you could have many basis sets of basin factors and another basis, but it's not an actual basis factor that you can do that. And on the way to different. 440 01:14:57.534 --> 01:15:07.314 So intrinsically it's the same factor. But it's got two different representations change of basis factors. The representation changes, but you'd argue it's the same vector. 441 01:15:08.784 --> 01:15:13.795 Example, I would like to use since I work with cryptography somewhat. 442 01:15:14.694 --> 01:15:18.534 And you've got a point on the surface, like, right here in Albany. 443 01:15:19.739 --> 01:15:26.845 Then, you know, I could express it in well, they did have importances. 444 01:15:26.845 --> 01:15:35.814 I mean, I could I could express my coordinates in terms of, say, not twenty seven coordinates, or could express it in terms of W. 445 01:15:36.659 --> 01:15:50.215 Coordinates and the numbers at different I didn't move, but the representation of my location move what I went to a different coordinates, which is like a different basis that I'm being a touch sloppy here. But you get the idea. 446 01:15:51.060 --> 01:15:56.215 Okay oh, if you curious historically, why do. 447 01:15:56.970 --> 01:16:11.725 Different coordinates systems, it's because of the nineteen th, century each country and alive. So which part of the surface? So, United States, but it on the flipside to the United dates, the best fit prints fit in the left side the best fit France and so on. 448 01:16:13.289 --> 01:16:17.364 They were slightly different, but no one cared until GPS came along. 449 01:16:21.029 --> 01:16:33.895 Inventing a common system and s, W eighty four and of course, the United States, you can buy Maps today, new maps that are still in that twenty seven of us. Eighty four. It's been out for. 450 01:16:35.430 --> 01:16:38.814 Thirty six years now it's still not used in some new map. 451 01:16:39.449 --> 01:16:42.444 Well, okay, so. 452 01:16:44.364 --> 01:16:50.305 Why I'm spending time on that in this course, is that the transformation make me see sorry. 453 01:16:51.869 --> 01:16:59.515 Well, the head of our Matrix, it rotates the cube bits effectively two different basis system. So. 454 01:17:00.270 --> 01:17:05.875 Can be convenient just say the hazard Mark basis some metrics a sudden changing basis. 455 01:17:06.659 --> 01:17:10.614 Yeah, let's add a more matrix. 456 01:17:11.789 --> 01:17:23.845 And you made its own application also, it turns the factory space into analogy where you can modify factors, not necessarily able to mobilize actors, but if you can, we just call it algebra another way. 457 01:17:23.935 --> 01:17:27.444 The complicated way to combine to have active space is called defensive product. 458 01:17:28.375 --> 01:17:38.364 Just the exterior product. Same thing I tied into the book and what this does it, so it multiplies dimensions of the two vector spaces. 459 01:17:38.364 --> 01:17:43.914 If we do tenser product space in the three D space the result is of six dimensional space. 460 01:17:44.640 --> 01:17:50.635 And specifically, cancer product, two factors, I forgot a vector to the annuity to make last Friday. 461 01:17:52.765 --> 01:17:57.805 Me talking yeah. 462 01:18:00.060 --> 01:18:08.310 See, the combo for doctor, with every element of the first factor multiplied scale multiplication why? 463 01:18:08.305 --> 01:18:22.045 Every element of the second okay and we can also add we can do we can distribute that addition a cancer product over addition. So, high dispatchers, this plus this, then we can do that. 464 01:18:22.104 --> 01:18:24.385 So, the obvious things like distribution and the. 465 01:18:29.904 --> 01:18:43.829 Okay, of course, you'll have to somebody has to prove that the result is, you know, I could say, let's divide them or whatever, but that would not be effective space Dr space you can add, and you can scale report. 466 01:18:45.984 --> 01:19:00.895 Okay. That's one way we can create attentive product vector. We do a tenth of productive, two component factors, but then once we've got some types of product factors, we can add them in so on and once or rotate the multiply. 467 01:19:00.895 --> 01:19:03.805 But once we do, the thing is, the result might not be separable anymore. 468 01:19:06.600 --> 01:19:18.925 And the book gives an example, you're doing page seventy. Here's an example. You can separate. Here's an example you cannot separate because it was created by adding two separable factors. 469 01:19:19.164 --> 01:19:21.805 So okay. 470 01:19:22.619 --> 01:19:31.765 Okay, so then I went into chap, I skipped parts Chapter two later chapter three is getting the stops and starts looking like Mark off models. 471 01:19:32.460 --> 01:19:36.475 Had that we got to take transitions. 472 01:19:37.914 --> 01:19:45.595 We have these different for different parts here just the marbles in it. We have some transition roles as time steps times. quantitized. 473 01:19:46.350 --> 01:19:49.645 The narrow say how the marbles move. So if time zero that. 474 01:19:51.479 --> 01:19:54.805 That and just transition can be represented. 475 01:19:55.710 --> 01:20:01.314 It's a factor of the transition is a matrix modification. Okay this was. 476 01:20:02.039 --> 01:20:06.475 Deterministic transition. Here's probabilistic. Transitions. 477 01:20:08.545 --> 01:20:15.175 The marvels are a probability of going from one to the next one and so times there was that time on. We finally got that. 478 01:20:16.135 --> 01:20:19.524 And classical. 479 01:20:20.789 --> 01:20:35.694 Two slit experiment, classical and time zero the quantums here time one, it's fifty fifty going to one of the two slits and time to hit one targets and the middle one probability. 480 01:20:35.694 --> 01:20:36.145 A third. 481 01:20:38.159 --> 01:20:44.064 The thing is, the probability is configured so I look at that middle one right here. Okay. That's one. Third. 482 01:20:44.064 --> 01:20:55.765 It got a six of its probability from a quantum photon going to the upper slide and another six going the lower slit and each time without another contribution. It's probability. 483 01:20:57.420 --> 01:21:10.585 Classical things as they get bigger section. Three, three, the Pro build is going to be complex numbers. They might not at all. They are, but. 484 01:21:11.520 --> 01:21:15.505 The magnitude don't get bigger. I'm adding. 485 01:21:16.800 --> 01:21:22.045 Finding sloppy and they might get it might cancel. Okay so that's. 486 01:21:24.390 --> 01:21:30.354 New stuff for today is the homework online there is. 487 01:21:34.109 --> 01:21:37.465 And there's the, you know, the blog that I typed in and. 488 01:21:38.579 --> 01:21:45.449 I encourage you to see this go back. Yeah. 489 01:21:50.369 --> 01:22:00.835 I encourage you to read the book. It's a whole sixty dollars online so, you know, I unconscious of getting expensive. 490 01:22:00.835 --> 01:22:08.965 So okay, so you an idea how the course is progressing I'm open for questions. 491 01:22:09.689 --> 01:22:14.244 And I'll stay around as long as people have questions. So, floor is open. 492 01:22:23.574 --> 01:22:29.005 And if anyone wants any of these cards that I showed you last time. 493 01:22:30.779 --> 01:22:34.045 I worked with these as an undergrad like I said, you know. 494 01:22:35.130 --> 01:22:43.135 Punch these cards, like, so this is one line of the program. colum is one character in the program he told is one that punch them. 495 01:22:45.810 --> 01:22:48.925 Was his large is a table and. 496 01:22:51.359 --> 01:22:55.345 Make a deck, so two thousand, nine programs that. 497 01:22:57.000 --> 01:23:09.534 Whatever wouldn't have long read it into the computer. Hard reader might read a thousand cards a minute. The cards is damaged on edge. There was on the jabs. 498 01:23:11.609 --> 01:23:18.444 And then we print out the result on tenfold paper. Look at that. That is a card deck, send it in again. 499 01:23:19.289 --> 01:23:21.505 Get a couple of runs a day. Okay. 500 01:23:26.310 --> 01:23:30.475 Programming environments, better now, but I do maintain the software is more. 501 01:23:31.770 --> 01:23:37.345 Okay, this reading, but a major. 502 01:23:37.890 --> 01:23:41.515 Break in caused by a sequel injection error. 503 01:23:42.210 --> 01:23:52.854 The sequel injection error, you know, he takes some input from a hostile user and sticking in the middle of it programming parts and executed. The input may have unbalanced quotes and things. 504 01:23:52.854 --> 01:24:00.175 So, high tech analog to reading a number and using it as an index in the Ray without checking that it's within the bounds. Okay. 505 01:24:01.170 --> 01:24:13.614 Still allowed to someone so have a good week and if there's no questions, then let's see. 506 01:24:22.074 --> 01:24:33.595 Otherwise, no. Okay. 507 01:24:39.270 --> 01:24:39.659 No.