1 00:00:00,050 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:20,766 --> 00:00:24,950 PROFESSOR: OK, let me get going. 9 00:00:24,950 --> 00:00:29,700 Last time we were talking about multi-particle states 10 00:00:29,700 --> 00:00:31,840 and tensor products. 11 00:00:31,840 --> 00:00:38,110 And for that, we explained that if we 12 00:00:38,110 --> 00:00:42,390 have a system, a quantum mechanical system of one 13 00:00:42,390 --> 00:00:46,620 particle described by a vector space V, 14 00:00:46,620 --> 00:00:51,020 and the quantum mechanical system of another particle 15 00:00:51,020 --> 00:00:56,530 described with a vector space W, the quantum mechanics 16 00:00:56,530 --> 00:01:02,310 of the total system composed by the two particles 17 00:01:02,310 --> 00:01:08,910 is defined on a new vector space called the space V tensor W. 18 00:01:08,910 --> 00:01:12,920 And that was a construction that showed that in particular it 19 00:01:12,920 --> 00:01:17,230 was not true to say that, oh if you want 20 00:01:17,230 --> 00:01:20,770 to know the system of particle 1 and 2, 21 00:01:20,770 --> 00:01:23,780 you just tell me what state particle 1 is 22 00:01:23,780 --> 00:01:26,930 and what state particle 2 is, and that's 23 00:01:26,930 --> 00:01:28,050 the end of the story. 24 00:01:28,050 --> 00:01:32,520 No, the story is really more sophisticated than that. 25 00:01:32,520 --> 00:01:38,390 So the typical elements on this space 26 00:01:38,390 --> 00:01:47,230 were of the form aij vi cross wj. 27 00:01:49,780 --> 00:01:59,790 And it's a sum over i and j numbers times these vectors. 28 00:01:59,790 --> 00:02:02,050 So you pick a vector in the first vector 29 00:02:02,050 --> 00:02:04,120 space, a vector in the second vector space, 30 00:02:04,120 --> 00:02:09,580 you put them in here and take linear combinations of them. 31 00:02:09,580 --> 00:02:15,310 So that's the general state in the system. 32 00:02:15,310 --> 00:02:19,590 Now we said a few things about this. 33 00:02:19,590 --> 00:02:22,860 One thing I didn't say too much about 34 00:02:22,860 --> 00:02:29,630 was the issue of the vector 0 in this tensor space. 35 00:02:29,630 --> 00:02:34,650 And well, vector 0 is some element of any vector space 36 00:02:34,650 --> 00:02:36,700 is an important element. 37 00:02:36,700 --> 00:02:40,360 And we could get a little confused about how it looks. 38 00:02:40,360 --> 00:02:51,305 And here's for example, the vector 0 in v cross w. 39 00:02:55,120 --> 00:03:04,770 An example of the vector 0 is the vector 0 tensor wi. 40 00:03:04,770 --> 00:03:12,150 If you put in the first input, the vector 0, that's it. 41 00:03:12,150 --> 00:03:18,635 That is also the vector 0 in here. 42 00:03:21,680 --> 00:03:28,320 Vi tensor the vector 0 in w. 43 00:03:28,320 --> 00:03:31,790 Here is 0 in w. 44 00:03:31,790 --> 00:03:39,240 Here is 0 in v. This is also 0. 45 00:03:39,240 --> 00:03:41,330 It's maybe a little surprising. 46 00:03:41,330 --> 00:03:43,070 Now how do we see that? 47 00:03:43,070 --> 00:03:44,570 Well we had a property. 48 00:03:44,570 --> 00:03:55,610 For example, this one. av tensor w is equal to av tensor w, 49 00:03:55,610 --> 00:03:56,710 where a is a number. 50 00:03:56,710 --> 00:04:00,660 So pick a equals 0. 51 00:04:00,660 --> 00:04:05,270 Well 0 times any vector is the 0 vector. 52 00:04:05,270 --> 00:04:08,660 0 cross w. 53 00:04:08,660 --> 00:04:12,910 But 0 times any vector is also the vector 0. 54 00:04:12,910 --> 00:04:17,579 So this is the 0 in v cross w. 55 00:04:17,579 --> 00:04:24,110 So 0 cross w is the vector 0. 56 00:04:24,110 --> 00:04:28,320 Once you put 0 in one of the two inputs, you're there. 57 00:04:28,320 --> 00:04:29,700 You're at 0. 58 00:04:29,700 --> 00:04:32,570 You don't have more. 59 00:04:32,570 --> 00:04:37,820 So that's just a comment on the vector 0. 60 00:04:37,820 --> 00:04:40,280 Now we did a few things. 61 00:04:40,280 --> 00:04:43,530 And one thing I didn't do last time 62 00:04:43,530 --> 00:04:48,270 was to define an inner product on the new vector space. 63 00:04:48,270 --> 00:04:52,570 So let's define a way to get numbers 64 00:04:52,570 --> 00:04:55,080 from one vector in the tensor space 65 00:04:55,080 --> 00:04:56,980 and another vector in the tensor space. 66 00:04:56,980 --> 00:04:58,075 So inner product. 67 00:05:03,700 --> 00:05:06,470 And again, here you're supposed to define it 68 00:05:06,470 --> 00:05:10,511 to your best understanding and the hope 69 00:05:10,511 --> 00:05:12,260 that, once you make the right definitions, 70 00:05:12,260 --> 00:05:15,950 it has all that axiomatic properties it should have. 71 00:05:15,950 --> 00:05:17,970 So let me take the following thing. 72 00:05:17,970 --> 00:05:21,650 The inner product with this thing aij 73 00:05:21,650 --> 00:05:35,470 vi omega j with bpq vp wq. 74 00:05:39,560 --> 00:05:46,220 So I will define this by assuming 75 00:05:46,220 --> 00:05:53,100 the linearity in the inputs on the right inputs 76 00:05:53,100 --> 00:05:56,540 and the anti-linearity here on the left input. 77 00:05:56,540 --> 00:05:59,950 So this would be the sum over inj here. 78 00:05:59,950 --> 00:06:07,300 So I'll put sum over inj aij star sum 79 00:06:07,300 --> 00:06:22,160 over pq bpq and then vi wj comma vp wq. 80 00:06:22,160 --> 00:06:24,530 So by declaring that this is the case, 81 00:06:24,530 --> 00:06:31,700 I'm saying that the inner product in the tensor space 82 00:06:31,700 --> 00:06:36,560 has the-- I'm demanding it has the properties that we expect. 83 00:06:36,560 --> 00:06:40,030 If you have a vector plus another vector here, well 84 00:06:40,030 --> 00:06:43,350 you get this times the first plus this times the second. 85 00:06:43,350 --> 00:06:47,830 So you can take the sums out and arrange it this way. 86 00:06:47,830 --> 00:06:50,570 But I still haven't got a number, 87 00:06:50,570 --> 00:06:53,230 and the inner product is supposed to be a number. 88 00:06:53,230 --> 00:06:56,260 So how do we get a number at this stage? 89 00:06:56,260 --> 00:06:58,870 I have this thing, and nobody has told me 90 00:06:58,870 --> 00:07:02,130 what this is supposed to be. 91 00:07:02,130 --> 00:07:04,290 At this stage, the only thing you can say 92 00:07:04,290 --> 00:07:07,670 is, well, you know I suspect that, if I 93 00:07:07,670 --> 00:07:14,630 had an inner product in V and I had an inner product in w, 94 00:07:14,630 --> 00:07:16,910 I must have an inner product here, 95 00:07:16,910 --> 00:07:19,390 and somehow I should use that. 96 00:07:19,390 --> 00:07:29,700 So they still define to be ij pq aij bpq. 97 00:07:29,700 --> 00:07:34,050 And then what you do is use the inner product in v 98 00:07:34,050 --> 00:07:36,100 to get a number from these two vectors. 99 00:07:39,910 --> 00:07:42,460 This is going the v inner product. 100 00:07:42,460 --> 00:07:50,556 And use the inner product on w to get a number from the two w 101 00:07:50,556 --> 00:07:51,055 vectors. 102 00:07:57,320 --> 00:07:58,030 And that's it. 103 00:07:58,030 --> 00:07:59,175 The end of the definition. 104 00:08:02,900 --> 00:08:08,300 Now here maybe this is the sort of most interesting step, where 105 00:08:08,300 --> 00:08:10,785 this part was set equal to this. 106 00:08:16,890 --> 00:08:22,480 And consistent with what I was telling you about 0, 107 00:08:22,480 --> 00:08:27,880 suppose any of this vi was 0. 108 00:08:27,880 --> 00:08:31,520 If this vi was 0, we would have 0 with vp. 109 00:08:31,520 --> 00:08:35,600 That would be 0, so this whole number is 0. 110 00:08:35,600 --> 00:08:40,230 So the way this can happen is one 111 00:08:40,230 --> 00:08:42,830 of the vectors must be 0 here. 112 00:08:42,830 --> 00:08:46,560 And well, you have the 0 vector here, 113 00:08:46,560 --> 00:08:51,240 and the zero vector inner product with anything is 0. 114 00:08:51,240 --> 00:08:54,650 So it's, again, consistent to think 115 00:08:54,650 --> 00:08:58,030 that, once you put one of these entries to 0, 116 00:08:58,030 --> 00:08:59,570 you've got the 0 vector. 117 00:09:04,320 --> 00:09:06,490 So where are we going today? 118 00:09:06,490 --> 00:09:08,610 Well, we have now the inner product, 119 00:09:08,610 --> 00:09:12,950 and I want to go back to a state we had last time. 120 00:09:12,950 --> 00:09:16,590 What we're going to do today is define 121 00:09:16,590 --> 00:09:20,200 what we called an entangled state. 122 00:09:20,200 --> 00:09:24,710 Then we will consider basis of entangled states, 123 00:09:24,710 --> 00:09:28,830 and we will be able to discuss this sort of nice example 124 00:09:28,830 --> 00:09:33,430 of teleportation, quantum teleportation. 125 00:09:33,430 --> 00:09:36,050 So that's where we're going today. 126 00:09:36,050 --> 00:09:39,035 I wanted to remind you of a calculation 127 00:09:39,035 --> 00:09:40,680 we were doing last time. 128 00:09:40,680 --> 00:09:45,020 We had established that there was a state in the tensor 129 00:09:45,020 --> 00:09:47,470 product of 2 spin 1/2 particles. 130 00:09:53,300 --> 00:10:02,460 And the state was alpha plus tensor minus minus 131 00:10:02,460 --> 00:10:06,020 minus tensor plus. 132 00:10:06,020 --> 00:10:13,250 Now you can sometimes-- this is an example of a superposition 133 00:10:13,250 --> 00:10:17,740 of vectors of the from in the v cross w. 134 00:10:17,740 --> 00:10:19,790 So here is a vector of that form. 135 00:10:19,790 --> 00:10:21,430 There is a vector of this form. 136 00:10:21,430 --> 00:10:28,140 Sometimes we put here 1 and 2. 137 00:10:28,140 --> 00:10:33,190 And sometimes it will be useful to put those labels. 138 00:10:33,190 --> 00:10:35,230 Because if you don't put the labels, 139 00:10:35,230 --> 00:10:37,080 you better make sure that you're always 140 00:10:37,080 --> 00:10:40,290 talking that the first ket, is the one for the first vector 141 00:10:40,290 --> 00:10:43,200 space, and the second ket is the one for the second vector 142 00:10:43,200 --> 00:10:44,320 space. 143 00:10:44,320 --> 00:10:47,490 There's nothing really known commutative here. 144 00:10:47,490 --> 00:10:54,830 So if somebody would write for you 1 minus 2, 145 00:10:54,830 --> 00:11:02,900 or they would write minus 2 1, both of you 146 00:11:02,900 --> 00:11:04,990 would be talking about the same state. 147 00:11:04,990 --> 00:11:07,112 But if you don't put the labels, you 148 00:11:07,112 --> 00:11:09,070 know you're not something about the same state, 149 00:11:09,070 --> 00:11:11,530 because you assume always the first one goes 150 00:11:11,530 --> 00:11:13,390 to the first Hilbert space. 151 00:11:13,390 --> 00:11:17,130 The second one goes with the second vector space. 152 00:11:17,130 --> 00:11:20,520 So we considered an entangled state of two 153 00:11:20,520 --> 00:11:22,360 spin 1/2 particles. 154 00:11:22,360 --> 00:11:26,690 I'm not using-- it's not fair to use the word entangled yet, 155 00:11:26,690 --> 00:11:29,920 but we'll be able to say this very soon. 156 00:11:29,920 --> 00:11:33,990 So the one thing we can do now given the inner product 157 00:11:33,990 --> 00:11:37,000 is try to normalize this state. 158 00:11:37,000 --> 00:11:41,310 So how do we normalize this state? 159 00:11:41,310 --> 00:11:44,070 Well, we must take the inner product of this state 160 00:11:44,070 --> 00:11:44,970 with itself. 161 00:11:44,970 --> 00:11:48,110 So psi psi. 162 00:11:52,440 --> 00:11:54,560 So then what do we do? 163 00:11:54,560 --> 00:11:57,570 Well, given these rules, we're supposed 164 00:11:57,570 --> 00:12:01,480 to take all this vector here, all that vector there, 165 00:12:01,480 --> 00:12:04,360 1 alpha-- the alpha that is on the left 166 00:12:04,360 --> 00:12:06,440 goes out as an alpha star. 167 00:12:06,440 --> 00:12:10,320 The alpha that this on the right goes out as an alpha. 168 00:12:10,320 --> 00:12:21,900 And we have plus minus minus minus plus inner product 169 00:12:21,900 --> 00:12:32,315 with plus minus minus minus plus. 170 00:12:40,140 --> 00:12:44,460 Now this is easier than what it seems from what I'm writing. 171 00:12:44,460 --> 00:12:47,120 You will be able to do these things, I think. 172 00:12:47,120 --> 00:12:52,610 Or you can already maybe do them by inspection. 173 00:12:52,610 --> 00:12:54,970 Basically at this stage, you have 174 00:12:54,970 --> 00:12:58,640 to do each one with each one here. 175 00:12:58,640 --> 00:13:00,210 And let's see what we get. 176 00:13:00,210 --> 00:13:03,870 Well, what is the inner product of this with this? 177 00:13:03,870 --> 00:13:08,720 This works, because the inner product of plus with plus is 1 178 00:13:08,720 --> 00:13:11,750 and minus with minus is 1. 179 00:13:11,750 --> 00:13:16,270 This on the other hand, doesn't give any contribution, 180 00:13:16,270 --> 00:13:18,620 because the first one is a plus has 181 00:13:18,620 --> 00:13:20,770 0 inner product with a minus. 182 00:13:20,770 --> 00:13:22,900 A minus has 0 with a plus. 183 00:13:22,900 --> 00:13:23,810 That doesn't matter. 184 00:13:23,810 --> 00:13:25,250 It's an overkill. 185 00:13:25,250 --> 00:13:31,990 So this one couples with this, and this one couples with that. 186 00:13:31,990 --> 00:13:34,750 Another way people would do this is 187 00:13:34,750 --> 00:13:40,790 to say oh don't worry just take the bra here. 188 00:13:40,790 --> 00:13:43,766 So it's plus minus. 189 00:13:48,140 --> 00:13:50,300 Here is one. 190 00:13:50,300 --> 00:13:52,060 I'll put the labels too. 191 00:13:52,060 --> 00:13:56,690 Minus the bra of the minus is the minus like that. 192 00:13:56,690 --> 00:14:04,800 1 plus 2. 193 00:14:04,800 --> 00:14:08,880 And now you do this with this ket, 194 00:14:08,880 --> 00:14:16,090 the plus minus 1 2 minus the minus plus 1 2. 195 00:14:19,960 --> 00:14:24,020 And bras and kets, you know that this one goes with this one. 196 00:14:24,020 --> 00:14:27,450 Plus plus, minus minus, this one goes with this one. 197 00:14:27,450 --> 00:14:32,015 And here I put the labels, because when I form the bra, 198 00:14:32,015 --> 00:14:35,000 it's not obvious which one you would put first, 199 00:14:35,000 --> 00:14:36,630 but it doesn't really matter. 200 00:14:39,830 --> 00:14:47,070 So back here, we have norm of alpha squared. 201 00:14:47,070 --> 00:14:51,290 And this with this is 1. 202 00:14:51,290 --> 00:14:54,570 And minus is one, this is another one. 203 00:14:54,570 --> 00:14:57,360 So this is 2 alpha squared. 204 00:14:57,360 --> 00:15:03,850 So if I want it to be normalized, I take alpha 1 205 00:15:03,850 --> 00:15:05,500 over square root of 2. 206 00:15:05,500 --> 00:15:09,705 And this is the well normalized state. 207 00:15:23,110 --> 00:15:26,220 So this is the unit normalized state. 208 00:15:36,250 --> 00:15:40,550 So we have this state. 209 00:15:40,550 --> 00:15:45,580 This state is something you've played with over last week. 210 00:15:45,580 --> 00:15:49,530 Is that state that we started very fine 211 00:15:49,530 --> 00:15:54,790 in lecture that had 0 z component of angular momentum, 212 00:15:54,790 --> 00:15:58,690 0 x component of angular momentum, 213 00:15:58,690 --> 00:16:01,720 and 0 y component of angular momentum. 214 00:16:01,720 --> 00:16:05,730 Total angular momentum as we defined it. 215 00:16:05,730 --> 00:16:11,290 And this has a state with absolutely no angular momentum. 216 00:16:11,290 --> 00:16:14,040 And what you verified in the homework 217 00:16:14,040 --> 00:16:19,020 was that that state, in fact, is rotational invariant. 218 00:16:19,020 --> 00:16:21,810 You apply a rotation operator to that state 219 00:16:21,810 --> 00:16:27,080 by rotating in both spaces, and out comes the same state. 220 00:16:27,080 --> 00:16:29,180 The state is not changed. 221 00:16:29,180 --> 00:16:32,955 So it's a very interesting state that will be important for us 222 00:16:32,955 --> 00:16:33,455 later. 223 00:16:36,370 --> 00:16:41,520 All right, so having taken care of inner products 224 00:16:41,520 --> 00:16:46,700 and normalizations, let's talk a little about entangled states. 225 00:16:46,700 --> 00:16:49,800 So entangled states. 226 00:16:57,690 --> 00:17:02,260 So these are precisely those states in which you cannot say, 227 00:17:02,260 --> 00:17:06,060 or describe them by saying particle one is doing this, 228 00:17:06,060 --> 00:17:09,240 particle two is doing that. 229 00:17:09,240 --> 00:17:26,819 You've learned that v cross w includes a state superpositions 230 00:17:26,819 --> 00:17:33,660 alpha ij vi cross omega j. 231 00:17:33,660 --> 00:17:40,480 The question is, if somebody hands you a state like this, 232 00:17:40,480 --> 00:17:44,910 maybe you could do some algebra or some trickery. 233 00:17:44,910 --> 00:17:54,310 And is it equal, you ask, to some sort of vector u star 234 00:17:54,310 --> 00:18:02,290 tensor v star times some vector w star. 235 00:18:02,290 --> 00:18:02,920 Is it equal? 236 00:18:05,820 --> 00:18:14,530 Is there vectors v star and omega star belonging to v 237 00:18:14,530 --> 00:18:19,630 and belonging to w in such a way that this thing, the sum, 238 00:18:19,630 --> 00:18:22,280 can be written as a product of something and that. 239 00:18:22,280 --> 00:18:26,690 If you would have that, then you would be able to say look, 240 00:18:26,690 --> 00:18:29,470 yes, this is an interesting state, 241 00:18:29,470 --> 00:18:32,180 but actually it's all simple here. 242 00:18:32,180 --> 00:18:36,420 Particle one is to state v star. 243 00:18:36,420 --> 00:18:40,830 Particle two is in state w star. 244 00:18:40,830 --> 00:18:54,790 If this has happened, if so, this state of the two particles 245 00:18:54,790 --> 00:18:57,573 is not entangled. 246 00:19:02,350 --> 00:19:06,750 So if you can really factor it, it's not entangled. 247 00:19:06,750 --> 00:19:14,310 If there are no such vectors v star and w star, 248 00:19:14,310 --> 00:19:17,420 then it is entangled. 249 00:19:17,420 --> 00:19:22,060 So you can say, well, it's a complicated factorization 250 00:19:22,060 --> 00:19:22,580 problem. 251 00:19:22,580 --> 00:19:25,360 And indeed, it might take a little work 252 00:19:25,360 --> 00:19:28,610 to figure out if a state is entangled or not. 253 00:19:28,610 --> 00:19:34,030 It's not a basis dependence problem. 254 00:19:34,030 --> 00:19:36,930 It's not like it's entangled in one basis or not. 255 00:19:36,930 --> 00:19:40,400 Here is a state, and you find any two things 256 00:19:40,400 --> 00:19:43,650 that tensor this way give you the state. 257 00:19:43,650 --> 00:19:46,310 So the simplest example to illustrate 258 00:19:46,310 --> 00:19:52,680 this is two dimensional vector spaces, v and w. 259 00:19:52,680 --> 00:19:57,810 Two dimensional complex. 260 00:19:57,810 --> 00:20:03,570 So v will have a basis e1 and e2. 261 00:20:03,570 --> 00:20:07,615 w will have a basis f1 and f2. 262 00:20:10,650 --> 00:20:14,600 And the most general state you could write 263 00:20:14,600 --> 00:20:26,420 is a state, general state, is a number a11 e1 f1 264 00:20:26,420 --> 00:20:41,640 plus a2 e1 f2 plus a21 e2 f1 plus a22 e2 f2. 265 00:20:44,560 --> 00:20:46,520 That's it. 266 00:20:46,520 --> 00:20:52,310 There's two basis states in v, two basis state in w. 267 00:20:52,310 --> 00:20:58,440 v cross w is dimension for product of dimensions for basis 268 00:20:58,440 --> 00:21:01,660 states, the products of the e's with the f's. 269 00:21:01,660 --> 00:21:03,410 So that's it. 270 00:21:03,410 --> 00:21:07,900 That's the general vector. 271 00:21:07,900 --> 00:21:14,120 The question is if this is it equal to something 272 00:21:14,120 --> 00:21:19,670 like a1 e1 plus a2 e2. 273 00:21:19,670 --> 00:21:26,355 Some general vector, you write the most general vector in v, 274 00:21:26,355 --> 00:21:37,620 and you write the most general vector b1 f1 plus b2 f2 in w. 275 00:21:37,620 --> 00:21:41,940 And you ask is it equal to a product, tensor product, 276 00:21:41,940 --> 00:21:45,390 of some vector in v with some vector in w. 277 00:21:45,390 --> 00:21:52,125 So the question is really are their numbers a1, a2, b1, 278 00:21:52,125 --> 00:21:56,275 and b2 so that this whole thing gets factorized. 279 00:22:00,640 --> 00:22:06,170 So that's happily not a complicated problem. 280 00:22:06,170 --> 00:22:11,930 We could see if those number exist, 281 00:22:11,930 --> 00:22:18,170 if a1, a2, b1, b2 exist, then the state is not entangled. 282 00:22:18,170 --> 00:22:22,120 You've managed to factor it out. 283 00:22:22,120 --> 00:22:23,650 So let's see. 284 00:22:23,650 --> 00:22:27,120 Well, we know the distributive laws apply. 285 00:22:27,120 --> 00:22:32,830 So actually e1 f1 can only arise from this product. 286 00:22:32,830 --> 00:22:42,650 So to have a solution you must have 287 00:22:42,650 --> 00:22:48,390 that a11 is equal to a1 b1. 288 00:22:48,390 --> 00:22:54,470 a12 can only appear from the product of e1 with f2. 289 00:22:54,470 --> 00:23:01,428 So a12 must be equal to a1 b2. 290 00:23:01,428 --> 00:23:05,790 a21 must be equal to a2 b2. 291 00:23:05,790 --> 00:23:13,050 And a22 must be equal to a2 b2. 292 00:23:13,050 --> 00:23:16,515 And we must try to solve for these quantities. 293 00:23:20,350 --> 00:23:23,075 Actually, there is a consistency condition. 294 00:23:26,540 --> 00:23:31,820 You see these quantities repeat here in a funny way. 295 00:23:31,820 --> 00:23:43,510 If this holds from this, a11 a22 minus a12 a21 is equal to what? 296 00:23:43,510 --> 00:23:47,180 a11 a22 would be a1 b1 a2 b2. 297 00:23:51,680 --> 00:23:55,440 And a12 a21 also have the same things. 298 00:23:55,440 --> 00:23:58,570 a1 b2 a2 b1. 299 00:23:58,570 --> 00:24:06,090 Well, both terms have both a's and both b's, so this system 300 00:24:06,090 --> 00:24:13,690 only has a solution if this product is 0. 301 00:24:13,690 --> 00:24:18,420 So if you give me four numbers, if you hope to factorize it, 302 00:24:18,420 --> 00:24:23,240 you must have the determinant of this matrix-- 303 00:24:23,240 --> 00:24:30,670 if you collapse it into a matrix, a11 a12 a21 a22, 304 00:24:30,670 --> 00:24:35,230 if you encode the information about this state in a matrix, 305 00:24:35,230 --> 00:24:41,645 it's necessary that the determinant of the matrix a 306 00:24:41,645 --> 00:24:44,920 be equal 0. 307 00:24:44,920 --> 00:24:53,510 So the determinant of a is equal to 0 308 00:24:53,510 --> 00:24:59,210 is certainly necessary for the factorization to take place. 309 00:24:59,210 --> 00:25:03,330 But a very small argument that will be in the notes, 310 00:25:03,330 --> 00:25:06,530 or you can try to complete it, shows 311 00:25:06,530 --> 00:25:09,660 that the determinant equal to 0, in fact, 312 00:25:09,660 --> 00:25:13,270 guarantees that then you can solve this system. 313 00:25:13,270 --> 00:25:15,170 There's a solution. 314 00:25:15,170 --> 00:25:17,370 And this is not complicated. 315 00:25:17,370 --> 00:25:26,390 So determinant equals 0 is actually the same as not 316 00:25:26,390 --> 00:25:26,890 entangled. 317 00:25:35,440 --> 00:25:38,720 We've done not entangled. 318 00:25:38,720 --> 00:25:42,080 So there's a solution implies determinant a equals 0, 319 00:25:42,080 --> 00:25:46,140 but determinant of a equals 0 also implies not entangled. 320 00:25:46,140 --> 00:25:50,660 You do that by solving this. 321 00:25:50,660 --> 00:25:53,920 Let's not spend time doing that. 322 00:25:53,920 --> 00:25:56,700 The basic way to do it is to assume-- 323 00:25:56,700 --> 00:26:00,320 consider, say, a11 equals 0 and solve it. 324 00:26:00,320 --> 00:26:03,000 Then a11 different from 0, and then you 325 00:26:03,000 --> 00:26:06,690 can show that you can choose these quantities. 326 00:26:06,690 --> 00:26:08,540 So it can be factored. 327 00:26:08,540 --> 00:26:14,220 And you have that, if these numbers are 328 00:26:14,220 --> 00:26:18,400 such that the determinant is 0, then the state is entangled. 329 00:26:18,400 --> 00:26:22,530 And it's very easy to have a determinant of this non-zero. 330 00:26:22,530 --> 00:26:26,900 For example, you could have these two 0 and these two 331 00:26:26,900 --> 00:26:28,530 non-zero. 332 00:26:28,530 --> 00:26:32,930 That will be entangled because the determinant is non-zero. 333 00:26:32,930 --> 00:26:36,290 You can have this two that will be entangled. 334 00:26:36,290 --> 00:26:40,110 There are many ways of getting entangled states. 335 00:26:40,110 --> 00:26:44,970 So in fact, there's enough ways to get entangled states 336 00:26:44,970 --> 00:26:49,980 that we can construct a basis. 337 00:26:49,980 --> 00:26:56,290 We had a basis here of e1 f1 e2 f2. 338 00:26:56,290 --> 00:26:56,920 This thing. 339 00:26:56,920 --> 00:27:00,430 This four vector basis. 340 00:27:00,430 --> 00:27:05,550 We can construct a basis that is all the states, 341 00:27:05,550 --> 00:27:09,920 all the basis vectors are entangled states. 342 00:27:09,920 --> 00:27:11,430 That's what we're going to do next. 343 00:27:11,430 --> 00:27:14,850 But maybe it's about time for questions, 344 00:27:14,850 --> 00:27:19,905 things that have become a little unclear as I went along. 345 00:27:24,970 --> 00:27:25,620 Yes? 346 00:27:25,620 --> 00:27:29,500 AUDIENCE: So what exactly does an entangled state mean? 347 00:27:29,500 --> 00:27:32,895 What are the [INAUDIBLE] to give me an entangled state. 348 00:27:35,750 --> 00:27:39,010 PROFESSOR: Well, the main thing that it happens 349 00:27:39,010 --> 00:27:42,730 is that there will be interesting correlations when 350 00:27:42,730 --> 00:27:45,730 you have an entangled state. 351 00:27:45,730 --> 00:27:48,470 If you have an entangled state and you 352 00:27:48,470 --> 00:27:51,720 find a state that is not entangled, 353 00:27:51,720 --> 00:27:54,220 you can say particle one is doing this 354 00:27:54,220 --> 00:27:56,280 and particle two is doing that. 355 00:27:56,280 --> 00:27:58,860 And particle two is doing this independent 356 00:27:58,860 --> 00:28:00,780 of what particle one is doing. 357 00:28:00,780 --> 00:28:03,850 But when a state is entangled, whatever 358 00:28:03,850 --> 00:28:06,180 is happening with particle one is 359 00:28:06,180 --> 00:28:07,805 correlated with what is happening 360 00:28:07,805 --> 00:28:11,320 in particle two in a strange way. 361 00:28:11,320 --> 00:28:14,640 So if particle one is doing something, 362 00:28:14,640 --> 00:28:16,420 then particle two is doing another thing. 363 00:28:16,420 --> 00:28:18,480 But if particle one is doing another thing, 364 00:28:18,480 --> 00:28:20,450 then particle two is doing something. 365 00:28:20,450 --> 00:28:23,190 And these particles can be very far apart, 366 00:28:23,190 --> 00:28:26,110 and that's when it gets really interesting. 367 00:28:26,110 --> 00:28:30,950 So we're going to do a lot of things with entangled states. 368 00:28:30,950 --> 00:28:33,320 Today we're doing this teleportation 369 00:28:33,320 --> 00:28:37,870 using entangled state, and you will see how subtle it is. 370 00:28:37,870 --> 00:28:42,770 Next time we do EPR, these Einstein Podolsky Rosen 371 00:28:42,770 --> 00:28:45,320 arguments and the Bell inequalities 372 00:28:45,320 --> 00:28:48,360 that answered that with entangled states. 373 00:28:48,360 --> 00:28:51,300 There's a couple of problems in the homework set also 374 00:28:51,300 --> 00:28:55,770 developing entangled states in different directions. 375 00:28:55,770 --> 00:28:57,820 And I think by the time we're done, 376 00:28:57,820 --> 00:29:03,200 you'll feel very comfortable with this. 377 00:29:03,200 --> 00:29:06,590 So a basis of entangled states. 378 00:29:06,590 --> 00:29:08,840 Here are those. 379 00:29:08,840 --> 00:29:10,220 We're going to use spins. 380 00:29:10,220 --> 00:29:22,110 So we're going to use v is the state space of spin 1/2. 381 00:29:22,110 --> 00:29:26,110 And we're going to consider a v tensor 382 00:29:26,110 --> 00:29:30,900 v where this refers to the first particle and this 383 00:29:30,900 --> 00:29:32,195 this to the second particle. 384 00:29:35,330 --> 00:29:44,440 So let's take one state, phi 0, defined 385 00:29:44,440 --> 00:29:49,820 to be 1 over square root of 2, and I don't put indices. 386 00:29:49,820 --> 00:29:52,510 And probably at some stage, you also 387 00:29:52,510 --> 00:29:56,550 tend to drop the tensor product. 388 00:29:56,550 --> 00:29:59,680 I don't know if it's early enough to drop it. 389 00:29:59,680 --> 00:30:02,940 Probably we could drop it. 390 00:30:02,940 --> 00:30:09,640 We'll put plus plus minus minus. 391 00:30:09,640 --> 00:30:13,200 Of course, people eventually drop even the other ket 392 00:30:13,200 --> 00:30:16,320 and put it plus plus. 393 00:30:16,320 --> 00:30:19,630 So those are the evolutions of notation. 394 00:30:19,630 --> 00:30:23,970 As you get to more and more calculations, you write less, 395 00:30:23,970 --> 00:30:26,490 but hopefully, it's still clear. 396 00:30:26,490 --> 00:30:28,420 But I will not do this one. 397 00:30:28,420 --> 00:30:30,940 I will still keep that because many times, 398 00:30:30,940 --> 00:30:33,480 I will want to keep labels. 399 00:30:33,480 --> 00:30:36,110 Otherwise, it's a little more cumbersome. 400 00:30:36,110 --> 00:30:41,900 So this state is normalized. 401 00:30:41,900 --> 00:30:46,610 Phi0 phi0 is equal to 1. 402 00:30:46,610 --> 00:30:48,330 It's the state we built. 403 00:30:48,330 --> 00:30:50,280 Oh, in fact, I want it with a plus. 404 00:30:50,280 --> 00:30:52,070 Sorry. 405 00:30:52,070 --> 00:30:55,110 It's similar to the state we had there. 406 00:30:55,110 --> 00:30:59,040 And by now, you say, look, yes, it's normalized. 407 00:30:59,040 --> 00:31:01,070 Let's take the dual. 408 00:31:01,070 --> 00:31:03,680 Plus plus with plus plus will give me 1. 409 00:31:03,680 --> 00:31:06,780 The minus minus with minus minus will give me 1. 410 00:31:06,780 --> 00:31:07,990 This is 2. 411 00:31:07,990 --> 00:31:11,890 1 over square root of 2 squared, 1. 412 00:31:11,890 --> 00:31:14,921 It should become sort of easy by inspection 413 00:31:14,921 --> 00:31:15,920 that this is normalized. 414 00:31:19,310 --> 00:31:25,150 And this is entangled state because in the matrix 415 00:31:25,150 --> 00:31:29,190 representation, it's a 1 here and a 1 there. 416 00:31:29,190 --> 00:31:33,510 You have the 1 1 product and the 2 2 product. 417 00:31:33,510 --> 00:31:37,110 So 1 1, the determinant is non-zero. 418 00:31:37,110 --> 00:31:39,710 There's no way, we've proven, you 419 00:31:39,710 --> 00:31:42,640 can find how to factor this. 420 00:31:42,640 --> 00:31:45,860 There's no alpha. 421 00:31:45,860 --> 00:31:50,670 There's no way to write this as an alpha plus, plus beta minus, 422 00:31:50,670 --> 00:31:54,490 times a gamma plus, plus delta minus. 423 00:31:54,490 --> 00:31:55,660 Just impossible. 424 00:31:55,660 --> 00:31:56,450 We've proven it. 425 00:31:56,450 --> 00:31:58,650 It's entangled. 426 00:31:58,650 --> 00:32:05,340 So this is an entangled state, but the state space 427 00:32:05,340 --> 00:32:08,110 is four dimensional. 428 00:32:08,110 --> 00:32:12,700 So if it's four dimensional, we need three more basis states. 429 00:32:12,700 --> 00:32:13,930 So here they are. 430 00:32:13,930 --> 00:32:17,040 I'm going to write a formula for them. 431 00:32:17,040 --> 00:32:23,170 Phi i for i equals 1, 2, and 3 will 432 00:32:23,170 --> 00:32:26,640 be defined to be the following thing. 433 00:32:26,640 --> 00:32:34,820 You will act with the operator 1 tensor sigma i on phi 0. 434 00:32:42,400 --> 00:32:44,300 So three ways of doing. 435 00:32:44,300 --> 00:32:47,560 Let's do 1, for example, phi 1. 436 00:32:47,560 --> 00:32:48,410 What is it? 437 00:32:51,230 --> 00:32:59,950 Well, you would have 1 times sigma 1 acting on the state phi 438 00:32:59,950 --> 00:33:05,770 0, which is 1 over square root of 2 plus, plus, 439 00:33:05,770 --> 00:33:09,360 plus minus, minus. 440 00:33:09,360 --> 00:33:12,910 Well, the 1 acts on the first ket, the sigma 441 00:33:12,910 --> 00:33:16,240 acts on the second ket. 442 00:33:16,240 --> 00:33:18,200 So what do we get here? 443 00:33:18,200 --> 00:33:22,660 1 over square root of 2-- let me go a little slow-- plus 444 00:33:22,660 --> 00:33:32,620 sigma 1 plus, plus, minus sigma 1 minus. 445 00:33:36,690 --> 00:33:46,150 And this is phi 1 equals sigma 1 plus is the minus state, 446 00:33:46,150 --> 00:33:51,560 and sigma 1 minus is the plus state. 447 00:33:51,560 --> 00:33:53,480 1 over square root of 2. 448 00:33:53,480 --> 00:33:57,360 Those are things that you may just 449 00:33:57,360 --> 00:34:01,830 remember sigma 1 is this matrix. 450 00:34:01,830 --> 00:34:05,730 So you get 1 over square root of 2 plus, 451 00:34:05,730 --> 00:34:11,620 minus, plus, minus, plus. 452 00:34:11,620 --> 00:34:13,100 So that's phi 1. 453 00:34:18,580 --> 00:34:24,540 And phi 1 is orthogonal to phi 0. 454 00:34:24,540 --> 00:34:29,010 You can see that because plus minus cannot have an overlap 455 00:34:29,010 --> 00:34:32,130 with plus plus, nor with minus minus. 456 00:34:32,130 --> 00:34:34,050 Here minus plus, no. 457 00:34:34,050 --> 00:34:35,659 In order to get something, you would 458 00:34:35,659 --> 00:34:40,429 have to have the same label here and the same label here 459 00:34:40,429 --> 00:34:41,599 so that something matches. 460 00:34:45,469 --> 00:34:49,510 Well, we can do the other ones as well. 461 00:34:49,510 --> 00:34:55,090 I will not bother you too much writing them out. 462 00:35:01,540 --> 00:35:03,070 So what do they look like? 463 00:35:03,070 --> 00:35:13,520 Well, you have phi 2 would be 1 tensor sigma 2 on phi 0. 464 00:35:13,520 --> 00:35:23,310 And that would give you-- I will just copy it-- an i 465 00:35:23,310 --> 00:35:25,660 because sigma 2 has i's there. 466 00:35:25,660 --> 00:35:33,345 So i over square root of 2 plus, minus, minus, minus, plus. 467 00:35:36,850 --> 00:35:44,720 Finally, phi 3 is 1 tensor sigma 3 phi 0. 468 00:35:44,720 --> 00:35:48,340 And it's 1 over square root of 2 plus, 469 00:35:48,340 --> 00:35:52,815 plus, minus, minus, minus. 470 00:35:59,620 --> 00:36:02,400 We got the states here. 471 00:36:02,400 --> 00:36:04,712 Let's just check they're orthonormal. 472 00:36:08,140 --> 00:36:10,735 Well, here's one thing. 473 00:36:13,960 --> 00:36:27,370 If you take phi 0 with 1 tensor sigma i phi 0, 474 00:36:27,370 --> 00:36:32,230 which is phi 0 with phi i. 475 00:36:36,500 --> 00:36:39,550 Well, this is 0. 476 00:36:39,550 --> 00:36:42,120 You could say, well, how do you know? 477 00:36:42,120 --> 00:36:44,010 How do you prove it easily? 478 00:36:44,010 --> 00:36:50,320 Well, I think the best way is just inspection, 479 00:36:50,320 --> 00:36:52,140 so let's look at that. 480 00:36:52,140 --> 00:36:55,660 Phi 1, we said, is orthogonal to phi 0 481 00:36:55,660 --> 00:36:59,490 because it has plus minus and minus plus, 482 00:36:59,490 --> 00:37:02,440 and that can never do anything with that. 483 00:37:02,440 --> 00:37:06,490 Phi 2 also has plus minuses and minus pluses, 484 00:37:06,490 --> 00:37:09,980 so we can never have anything to do with phi 0. 485 00:37:09,980 --> 00:37:12,480 The only one that has a chance to have 486 00:37:12,480 --> 00:37:15,730 an inner product with phi 0 is phi 2 487 00:37:15,730 --> 00:37:19,290 because it has a plus plus and a minus minus. 488 00:37:19,290 --> 00:37:21,890 On the other hand, when you flip them, 489 00:37:21,890 --> 00:37:25,870 this term with a plus plus of phi 0 will give you 1, 490 00:37:25,870 --> 00:37:27,710 but here's a difference of sign. 491 00:37:27,710 --> 00:37:32,770 So this with the second term of phi 00 will give you a minus, 492 00:37:32,770 --> 00:37:35,370 and therefore, it will be 0. 493 00:37:35,370 --> 00:37:41,315 So these things are all 0 by inspection. 494 00:37:46,460 --> 00:37:50,070 You don't really have to do a calculation there. 495 00:37:50,070 --> 00:37:52,530 The one that takes a little more work 496 00:37:52,530 --> 00:37:58,550 is to try to understand what is the inner product of phi i 497 00:37:58,550 --> 00:38:00,750 with phi j. 498 00:38:00,750 --> 00:38:03,850 Now, you could say, OK, I'm going to do them by inspection. 499 00:38:03,850 --> 00:38:14,170 After all, there's just six things to check. 500 00:38:14,170 --> 00:38:19,250 But let's just do it a little more intelligently. 501 00:38:19,250 --> 00:38:24,470 Let's try to calculate this by saying, well, this is phi 0. 502 00:38:24,470 --> 00:38:27,770 Since the Pauli matrices are Hermitian, 503 00:38:27,770 --> 00:38:33,020 this phi i is also 1 tensor sigma i. 504 00:38:33,020 --> 00:38:37,440 They're Hermitian, so acting on the left, 505 00:38:37,440 --> 00:38:40,260 they're doing the right thing. 506 00:38:40,260 --> 00:38:44,090 Given our definition, here is a definition as well. 507 00:38:47,530 --> 00:38:52,560 So you take the bra and that's what it is. 508 00:38:52,560 --> 00:38:56,630 It would have been dagger here but it's not necessary. 509 00:38:56,630 --> 00:39:04,240 And then you have the phi j, which is 1 tensor sigma j. 510 00:39:04,240 --> 00:39:05,760 And that's phi 0 here. 511 00:39:08,640 --> 00:39:10,500 That sounds like the kind of thing 512 00:39:10,500 --> 00:39:15,450 that we can make progress using our Pauli identities. 513 00:39:15,450 --> 00:39:19,990 Indeed, first thing is that the product of operators, they 514 00:39:19,990 --> 00:39:24,120 multiply just in that order in the tensor product. 515 00:39:24,120 --> 00:39:32,470 So phi 0, you have 1 times 1, which is 1 tensor sigma i sigma 516 00:39:32,470 --> 00:39:34,340 j phi 0. 517 00:39:39,060 --> 00:39:47,090 And this is equal to phi 0 1 tensor. 518 00:39:47,090 --> 00:39:49,790 Now, the product of two Pauli matrices 519 00:39:49,790 --> 00:39:54,300 gives you an identity plus a Pauli matrix. 520 00:39:54,300 --> 00:39:58,700 You may or may not remember this formula, but it's 1 times 521 00:39:58,700 --> 00:40:09,635 delta ij plus i epsilon ijk sigma k phi 0. 522 00:40:17,420 --> 00:40:21,930 Now, what do we get? 523 00:40:21,930 --> 00:40:26,580 Look, the second term has a sigma k on phi 0, 524 00:40:26,580 --> 00:40:32,430 so it's some number with a psi k here, 525 00:40:32,430 --> 00:40:34,950 while the first term is very simple. 526 00:40:34,950 --> 00:40:37,270 What do we get from the first term? 527 00:40:37,270 --> 00:40:42,256 From the first term, we get-- well, 1 tensor 528 00:40:42,256 --> 00:40:46,520 1 between any two things is nothing because the 1 529 00:40:46,520 --> 00:40:50,640 acting on things and the 1 acting on another thing is 0. 530 00:40:50,640 --> 00:40:56,425 So the unit operator in the tensor product is 1 tensor 1. 531 00:40:56,425 --> 00:40:58,770 That's nothing whatsoever. 532 00:40:58,770 --> 00:41:00,590 So what do you get here? 533 00:41:00,590 --> 00:41:19,866 Delta ij times phi 0 phi 0 plus i epsilon ijk phi 0 phi k. 534 00:41:22,700 --> 00:41:24,190 But that is 0. 535 00:41:24,190 --> 00:41:32,140 We already showed that any phi i with phi 0 is 0. 536 00:41:32,140 --> 00:41:34,210 And this is 1. 537 00:41:34,210 --> 00:41:36,040 So what have we learned? 538 00:41:36,040 --> 00:41:40,040 That this whole thing is delta ij. 539 00:41:43,310 --> 00:41:47,810 And therefore, the basis is orthonormal. 540 00:41:47,810 --> 00:41:53,180 So we've got a basis of orthonormal states 541 00:41:53,180 --> 00:41:58,720 in the tensor product of two spin 1/2 particles. 542 00:41:58,720 --> 00:42:01,490 And the nice thing about this basis 543 00:42:01,490 --> 00:42:07,890 is that all of these basis states are entangled states. 544 00:42:07,890 --> 00:42:09,770 They're entangled because they fill 545 00:42:09,770 --> 00:42:11,360 different parts of the matrix. 546 00:42:11,360 --> 00:42:16,690 Here you have 1 and 1 and minus 1 here. 547 00:42:16,690 --> 00:42:23,210 This would be plus minus, would be an i here and a minus i 548 00:42:23,210 --> 00:42:23,730 there. 549 00:42:23,730 --> 00:42:28,040 The determinants are non-zero for all of them, 550 00:42:28,040 --> 00:42:30,100 and therefore, they can't be factored, 551 00:42:30,100 --> 00:42:32,590 and therefore, they're entangled. 552 00:42:32,590 --> 00:42:35,920 So the last thing I want to do with this 553 00:42:35,920 --> 00:42:39,510 is to record a formula for you, which 554 00:42:39,510 --> 00:42:46,550 is a formula of the basis states in the conventional way, 555 00:42:46,550 --> 00:42:50,900 written as superposition of entangled states. 556 00:42:50,900 --> 00:42:56,668 So for example, you say, what is plus plus? 557 00:42:56,668 --> 00:43:01,620 Well, plus plus, looking there, how would you solve it? 558 00:43:01,620 --> 00:43:05,420 You would solve it from phi 0 and phi 3. 559 00:43:05,420 --> 00:43:11,410 You would take the sum so that the minus minus states cancel. 560 00:43:11,410 --> 00:43:15,200 Phi 0 and phi 3, and therefore, this state 561 00:43:15,200 --> 00:43:24,160 must be 1 over square root of 2, phi 0 plus phi 3. 562 00:43:24,160 --> 00:43:25,720 A useful relation. 563 00:43:25,720 --> 00:43:28,150 Then we have plus minus. 564 00:43:28,150 --> 00:43:31,510 Then we have minus plus. 565 00:43:31,510 --> 00:43:36,440 And finally, minus minus. 566 00:43:36,440 --> 00:43:40,700 Well, minus minus would be done by 1 567 00:43:40,700 --> 00:43:46,440 over square root of 2 phi 0 minus phi 3. 568 00:43:52,360 --> 00:43:56,320 The other ones, well, they just leave complex numbers. 569 00:43:56,320 --> 00:44:05,180 Phi 1 has this plus minus, and this has a plus minus in phi 2. 570 00:44:05,180 --> 00:44:07,670 The only problem is it has an i, so you 571 00:44:07,670 --> 00:44:12,470 must take this state minus i times this state 572 00:44:12,470 --> 00:44:18,510 will produce this state twice and will cancel this term. 573 00:44:18,510 --> 00:44:19,780 That's what you want. 574 00:44:19,780 --> 00:44:29,880 So phi 1, this should be 1 over square root of 2 phi 1 minus i 575 00:44:29,880 --> 00:44:33,090 phi 2. 576 00:44:33,090 --> 00:44:40,870 And this one should be phi 1 plus i phi 2. 577 00:44:44,080 --> 00:44:47,190 And if this was a little quick, it's 578 00:44:47,190 --> 00:44:50,100 just algebra, one more line. 579 00:44:50,100 --> 00:44:54,070 You do it with patience in private. 580 00:44:57,100 --> 00:44:59,780 So here it is. 581 00:45:02,710 --> 00:45:08,025 It's the normal product, simple product basis expressed 582 00:45:08,025 --> 00:45:11,640 as a superposition of entangled states. 583 00:45:11,640 --> 00:45:18,330 This is called the bell basis, this phi 1 up to phi 4, 584 00:45:18,330 --> 00:45:19,350 the bell basis. 585 00:45:29,080 --> 00:45:33,620 And now, I have to say a couple more things 586 00:45:33,620 --> 00:45:38,230 and we're on our way to begin the teleportation thing. 587 00:45:38,230 --> 00:45:40,330 Are there questions? 588 00:45:40,330 --> 00:45:46,670 Any questions about bell basis or the basis we've introduced? 589 00:45:46,670 --> 00:45:47,910 Any confusion? 590 00:45:47,910 --> 00:45:49,245 Errors on the blackboard? 591 00:46:00,290 --> 00:46:08,510 So we have a basis, and I want to make two remarks before we 592 00:46:08,510 --> 00:46:13,840 get started with the teleportation. 593 00:46:13,840 --> 00:46:16,260 It's one remark about measurement 594 00:46:16,260 --> 00:46:20,960 and one remark about evolution of states. 595 00:46:20,960 --> 00:46:21,620 Two facts. 596 00:46:29,230 --> 00:46:32,010 The first fact has to do with measurement 597 00:46:32,010 --> 00:46:33,730 in orthonormal basis. 598 00:46:33,730 --> 00:46:38,360 If you have an orthonormal basis, 599 00:46:38,360 --> 00:46:42,320 the postulate of measurement of quantum mechanics 600 00:46:42,320 --> 00:46:44,640 can be stated as saying that you can 601 00:46:44,640 --> 00:46:47,550 do an experiment in which you find 602 00:46:47,550 --> 00:46:52,750 the probability of your state being along any of these basis 603 00:46:52,750 --> 00:46:56,320 states of the orthonormal basis. 604 00:46:56,320 --> 00:46:59,890 So you can do an experiment to detect 605 00:46:59,890 --> 00:47:03,590 in which of the basis states the state is. 606 00:47:03,590 --> 00:47:06,240 Now, the state, of course, is in a superposition 607 00:47:06,240 --> 00:47:11,230 of basis states, but it will collapse into one of them 608 00:47:11,230 --> 00:47:12,720 with some probability. 609 00:47:12,720 --> 00:47:17,010 So the Stern-Gerlach experiment was an example 610 00:47:17,010 --> 00:47:20,232 in which you pick two basis states, orthogonal, 611 00:47:20,232 --> 00:47:22,170 and there was a device that allowed 612 00:47:22,170 --> 00:47:26,190 you to collapse the state into one or the other. 613 00:47:26,190 --> 00:47:28,920 So this is a little more general, not just 614 00:47:28,920 --> 00:47:30,810 for two state systems. 615 00:47:30,810 --> 00:47:33,770 If there would be a particle with three states, 616 00:47:33,770 --> 00:47:36,310 well, orthonormal states, then there 617 00:47:36,310 --> 00:47:39,270 is in principle an operator in quantum mechanics 618 00:47:39,270 --> 00:47:43,640 that allows it to measure which of these basis states 619 00:47:43,640 --> 00:47:45,380 you go into. 620 00:47:45,380 --> 00:48:00,530 So let me state this as saying, given an orthonormal basis, e1 621 00:48:00,530 --> 00:48:16,225 up to en, we can measure a state, psi, 622 00:48:16,225 --> 00:48:31,640 and we get that the probability to be in ei is, as you know, 623 00:48:31,640 --> 00:48:36,336 ei overlapped with a state squared. 624 00:48:36,336 --> 00:48:42,090 And if you measure that this probability, 625 00:48:42,090 --> 00:48:45,150 the state will collapse into one of these states. 626 00:48:45,150 --> 00:49:05,500 So after the measurement, the state goes into some ek. 627 00:49:05,500 --> 00:49:07,050 There are different probabilities 628 00:49:07,050 --> 00:49:09,980 to be in each one of those basis states, 629 00:49:09,980 --> 00:49:12,000 but the particle will choose one. 630 00:49:16,580 --> 00:49:18,870 Now, the other thing I want to mention 631 00:49:18,870 --> 00:49:25,950 is that a fact that has seemed always a gift, 632 00:49:25,950 --> 00:49:30,820 the Pauli matrices are not only Hermitian, 633 00:49:30,820 --> 00:49:36,520 but they square to one, and therefore they're also unitary. 634 00:49:36,520 --> 00:49:40,910 So the Pauli matrices are unitary. 635 00:49:40,910 --> 00:49:47,480 So actually, they can be realized as time evolution. 636 00:49:47,480 --> 00:49:53,340 So you have a state and you want to multiply it by sigma 1. 637 00:49:53,340 --> 00:49:57,700 You say, OK, well, that's a very mathematical thing. 638 00:49:57,700 --> 00:50:01,720 Not so mathematical because it's a unitary operator, 639 00:50:01,720 --> 00:50:04,880 so it could respond to some time evolution. 640 00:50:04,880 --> 00:50:07,190 So we claim there is a Hamiltonian 641 00:50:07,190 --> 00:50:10,830 that you can construct that will evolve the state 642 00:50:10,830 --> 00:50:14,390 and multiply it by sigma 1. 643 00:50:14,390 --> 00:50:23,330 So all these Pauli matrices, sigma 1, sigma 2, and sigma 3 644 00:50:23,330 --> 00:50:32,370 are unitary as operators. 645 00:50:32,370 --> 00:50:45,590 They can be realized by time evolution 646 00:50:45,590 --> 00:50:51,400 with a suitable Hamiltonian. 647 00:50:51,400 --> 00:50:55,020 So if you're talking spin states, 648 00:50:55,020 --> 00:50:58,160 some magnetic field that lifts for some few 649 00:50:58,160 --> 00:51:01,890 picoseconds according to the dipole, and that's it. 650 00:51:01,890 --> 00:51:04,250 It will implement sigma one. 651 00:51:04,250 --> 00:51:06,990 Just in fact, you can check, for example, 652 00:51:06,990 --> 00:51:13,670 that e to the i pi over 2 minus 1 plus sigma i. 653 00:51:16,350 --> 00:51:19,260 This is i this, and this is Hermitian. 654 00:51:22,920 --> 00:51:25,350 Well, this is 1 and sigma i. 655 00:51:25,350 --> 00:51:28,040 1 and sigma i commute, so this is 656 00:51:28,040 --> 00:51:32,090 equal to e to the minus i pi over 2 times 657 00:51:32,090 --> 00:51:37,330 e to the i pi sigma 1 over 2. 658 00:51:37,330 --> 00:51:42,640 The first factor is a minus i, and the second factor 659 00:51:42,640 --> 00:51:54,990 is 1 times cosine of pi over 2 plus i sigma 1 sine 660 00:51:54,990 --> 00:51:57,800 of pi over 2. 661 00:51:57,800 --> 00:52:04,000 So this is minus i times-- this is 0-- times i sigma 1. 662 00:52:04,000 --> 00:52:06,960 So this is sigma 1. 663 00:52:06,960 --> 00:52:10,172 So we've written sigma 1 as the exponential 664 00:52:10,172 --> 00:52:13,730 of i times the Hermitian operator. 665 00:52:13,730 --> 00:52:16,960 And therefore, you could say that this 666 00:52:16,960 --> 00:52:23,500 must be equal to some time times some Hamiltonian over h bar. 667 00:52:23,500 --> 00:52:28,120 And you decide, you put the magnetic field 668 00:52:28,120 --> 00:52:30,260 in the x, y, z direction. 669 00:52:30,260 --> 00:52:33,270 You realize it. 670 00:52:33,270 --> 00:52:36,360 So sigmas can be realized by a machine. 671 00:52:39,480 --> 00:52:46,960 We're all done with our preliminary remarks, 672 00:52:46,960 --> 00:52:51,530 and it's now time to do the teleportation stuff. 673 00:53:00,020 --> 00:53:04,600 Quantum teleportation. 674 00:53:12,200 --> 00:53:17,040 So we all know this teleportation 675 00:53:17,040 --> 00:53:20,890 is the stuff of science fiction and movies 676 00:53:20,890 --> 00:53:28,180 and kind of stuff like that, and it's pretty much something that 677 00:53:28,180 --> 00:53:32,180 was, classically, essentially impossible. 678 00:53:32,180 --> 00:53:36,410 You have an object, you sort of dematerialize it and create it 679 00:53:36,410 --> 00:53:37,820 somewhere else. 680 00:53:37,820 --> 00:53:39,740 No basis for doing that. 681 00:53:39,740 --> 00:53:43,150 The interesting thing is that quantum mechanically, 682 00:53:43,150 --> 00:53:46,680 you seem to be able to do much better, 683 00:53:46,680 --> 00:53:49,720 and that's the idea that we want to explain now. 684 00:53:49,720 --> 00:53:52,400 So this is also not something that 685 00:53:52,400 --> 00:53:56,240 has been known for a long time. 686 00:53:56,240 --> 00:54:00,960 The big discovery that this could be done is from 1993. 687 00:54:00,960 --> 00:54:05,870 So it's just 20 years ago people realized finally 688 00:54:05,870 --> 00:54:08,536 that you could do something like that. 689 00:54:08,536 --> 00:54:12,330 In that way, quantum mechanics is, in a sense, 690 00:54:12,330 --> 00:54:14,960 having a renaissance because there's 691 00:54:14,960 --> 00:54:19,310 all kinds of marvelous experiments-- teleportation, 692 00:54:19,310 --> 00:54:23,060 entanglement, ideas that you could build one day a quantum 693 00:54:23,060 --> 00:54:23,750 computer. 694 00:54:23,750 --> 00:54:26,330 It's all stimulating thinking better 695 00:54:26,330 --> 00:54:31,170 about quantum mechanics more precisely, 696 00:54:31,170 --> 00:54:33,255 and the experiments are just amazing. 697 00:54:36,920 --> 00:54:39,540 This thing was done by the following people. 698 00:54:39,540 --> 00:54:41,330 We should mention them. 699 00:54:41,330 --> 00:54:57,780 Bennett at IBM, Brassard, Crepeau-- 700 00:54:57,780 --> 00:55:05,650 can't pronounce that-- Jozsa, all these people in Montreal. 701 00:55:12,700 --> 00:55:25,840 Peres, at Technion, and Wootters at Williams College. 702 00:55:31,600 --> 00:55:35,800 1993. 703 00:55:35,800 --> 00:55:41,580 So big collaboration all over the world. 704 00:55:41,580 --> 00:55:45,790 So what is the question that we want to discuss? 705 00:55:48,480 --> 00:55:53,900 In this game, always there's two people involved, 706 00:55:53,900 --> 00:55:57,680 and the canonical names are Alice and Bob. 707 00:55:57,680 --> 00:56:00,370 Everybody calls Alice and Bob. 708 00:56:00,370 --> 00:56:02,330 It's been lots of years that people 709 00:56:02,330 --> 00:56:04,670 talk about Alice and Bob. 710 00:56:04,670 --> 00:56:08,390 They use it also for black hole experiments. 711 00:56:08,390 --> 00:56:11,750 Depending on your taste, Alice stays out 712 00:56:11,750 --> 00:56:16,230 and Bob is sucked into the black hole, or Bob stays out, 713 00:56:16,230 --> 00:56:18,270 Alice goes down. 714 00:56:18,270 --> 00:56:21,010 But it's Alice and Bob all the time. 715 00:56:21,010 --> 00:56:23,700 So this time, the way we're going to do it, 716 00:56:23,700 --> 00:56:26,145 Alice has a quantum state. 717 00:56:36,650 --> 00:56:40,510 It has been handed to her, and it's 718 00:56:40,510 --> 00:56:44,670 a state of a spin 1/2 particle. 719 00:56:44,670 --> 00:56:48,050 Spin 1/2 is nice because you have discrete labels. 720 00:56:48,050 --> 00:56:50,460 So she has this state. 721 00:56:50,460 --> 00:56:55,010 It's alpha plus beta minus. 722 00:56:58,200 --> 00:57:02,340 And she has it carefully there in a box, just 723 00:57:02,340 --> 00:57:05,570 hoping that the state doesn't get entangled with anything 724 00:57:05,570 --> 00:57:08,940 and disappear, or doesn't get measured. 725 00:57:08,940 --> 00:57:15,870 And her goal is to send this state to Bob, who's far away. 726 00:57:18,860 --> 00:57:32,650 So Alice is sitting here and has this state, 727 00:57:32,650 --> 00:57:38,240 and Bob is sitting somewhere here and has no state. 728 00:57:38,240 --> 00:57:43,010 And she wants to send this state. 729 00:57:43,010 --> 00:57:46,270 This is the state to be teleported. 730 00:57:46,270 --> 00:57:49,540 Now, there's a couple of things you 731 00:57:49,540 --> 00:57:53,040 could try to do before even trying to teleport this. 732 00:57:53,040 --> 00:57:55,490 Why teleport it? 733 00:57:55,490 --> 00:58:01,600 Why don't you create a copy of this state and just 734 00:58:01,600 --> 00:58:07,950 put it in FedEx and send it to Bob, and he gets it? 735 00:58:07,950 --> 00:58:12,300 The problem is that there's something in quantum mechanics, 736 00:58:12,300 --> 00:58:14,550 something called no cloning, that you 737 00:58:14,550 --> 00:58:17,030 can't create a copy of a state, actually, 738 00:58:17,030 --> 00:58:19,720 with a quantum mechanical process. 739 00:58:19,720 --> 00:58:21,430 It's really a funny thing. 740 00:58:21,430 --> 00:58:27,560 You've got a qubit-- this is called a qubit-- a quantum bit. 741 00:58:27,560 --> 00:58:30,730 Bit is something that can be 0 or 1. 742 00:58:30,730 --> 00:58:32,630 Quantum, it can be two things. 743 00:58:32,630 --> 00:58:35,380 So instead of calling it a spin state, 744 00:58:35,380 --> 00:58:38,170 sometimes people call it a qubit. 745 00:58:38,170 --> 00:58:40,390 For us, it's a spin state. 746 00:58:40,390 --> 00:58:43,220 It has two numbers. 747 00:58:43,220 --> 00:58:46,040 And there's no cloning. 748 00:58:46,040 --> 00:58:47,700 We will not discuss it here. 749 00:58:47,700 --> 00:58:49,640 It's a nice topic for a recitation. 750 00:58:49,640 --> 00:58:51,130 It's a simple matter. 751 00:58:51,130 --> 00:58:52,540 You can't make a copy. 752 00:58:52,540 --> 00:58:57,410 So given that you can't make a copy, let's avoid that idea, 753 00:58:57,410 --> 00:59:05,340 save ourselves $15 of FedEx and just try to do something else. 754 00:59:05,340 --> 00:59:08,170 So the one thing Alice could do is 755 00:59:08,170 --> 00:59:10,590 that she could say, all right. 756 00:59:10,590 --> 00:59:12,810 Well here is alpha and beta. 757 00:59:12,810 --> 00:59:15,290 Let me measure the state. 758 00:59:15,290 --> 00:59:17,960 Find alpha and beta. 759 00:59:17,960 --> 00:59:22,460 And then I'll of send that information to Bob. 760 00:59:22,460 --> 00:59:23,680 OK. 761 00:59:23,680 --> 00:59:26,180 But she has one copy of the state. 762 00:59:26,180 --> 00:59:28,580 How is she going to measure alpha and beta 763 00:59:28,580 --> 00:59:31,130 with one copy of the state. 764 00:59:31,130 --> 00:59:34,130 She puts it through a Stern-Gerlach experiment, 765 00:59:34,130 --> 00:59:37,280 and the particle comes out the plus side. 766 00:59:37,280 --> 00:59:40,340 Now what? 767 00:59:40,340 --> 00:59:42,900 The probability that it went to the plus. 768 00:59:42,900 --> 00:59:45,370 You've got some information about the alpha squared. 769 00:59:45,370 --> 00:59:48,160 Not even because you just did the experiment once 770 00:59:48,160 --> 00:59:49,465 and your cubit is gone. 771 00:59:52,010 --> 01:00:00,760 So Alice actually can't figure out alpha and beta. 772 01:00:00,760 --> 01:00:06,950 So if she's handed the qubit, she better not measure it. 773 01:00:06,950 --> 01:00:10,360 Because if she measures it, she destroys the state, 774 01:00:10,360 --> 01:00:15,130 goes into a plus or a minus, and it's all over. 775 01:00:15,130 --> 01:00:19,450 The state is gone before she could do anything. 776 01:00:19,450 --> 01:00:23,090 So that doesn't work either. 777 01:00:23,090 --> 01:00:25,440 Now there's the third option. 778 01:00:25,440 --> 01:00:31,460 Maybe Alice cannot talk to Bob, and Alice created that state 779 01:00:31,460 --> 01:00:32,520 with some Hamiltonian. 780 01:00:32,520 --> 01:00:34,990 And she knows because she created it 781 01:00:34,990 --> 01:00:38,650 what alpha and beta is. 782 01:00:38,650 --> 01:00:43,750 So she could in principle tell Bob, OK. 783 01:00:43,750 --> 01:00:45,290 Here is alpha and here is beta. 784 01:00:45,290 --> 01:00:47,950 Create it again. 785 01:00:47,950 --> 01:00:51,680 That would be a fine strategy, but actually there's 786 01:00:51,680 --> 01:00:55,970 even plausibly a problem with that. 787 01:00:55,970 --> 01:01:02,100 Because maybe she knows this state, but alpha is a number. 788 01:01:02,100 --> 01:01:10,490 It is 0.53782106, never ends. 789 01:01:10,490 --> 01:01:11,670 Doesn't repeat. 790 01:01:11,670 --> 01:01:16,350 And she has to send that infinite string of information 791 01:01:16,350 --> 01:01:20,120 to Bob, which is not a good idea either. 792 01:01:20,120 --> 01:01:23,950 She's not going to manage to send the right state. 793 01:01:23,950 --> 01:01:27,750 So these are the things we speculate about 794 01:01:27,750 --> 01:01:29,940 because it's a natural thing to one wonder. 795 01:01:29,940 --> 01:01:33,900 So what we're going to try to do is somehow 796 01:01:33,900 --> 01:01:38,810 produce an experiment in which she'll take this state, 797 01:01:38,810 --> 01:01:42,700 get it in, and somehow Bob is going 798 01:01:42,700 --> 01:01:46,680 to create that state on his other side. 799 01:01:46,680 --> 01:01:51,340 That's the teleportation thing that we'll try to do. 800 01:01:51,340 --> 01:01:58,210 So let's do a little diagram of how we're going to do this. 801 01:01:58,210 --> 01:02:01,560 So here is going to be the state that is going to be teleported. 802 01:02:01,560 --> 01:02:07,240 We'll call it the state C. So I'll 803 01:02:07,240 --> 01:02:19,770 write it as psi alpha plus in the state space C sub particle 804 01:02:19,770 --> 01:02:24,700 plus beta minus in this state space C. And C is 805 01:02:24,700 --> 01:02:27,825 the state she is going to try to teleport. 806 01:02:31,720 --> 01:02:36,580 But now they're not going to be able to do it 807 01:02:36,580 --> 01:02:43,030 unless they use something different. 808 01:02:43,030 --> 01:02:44,810 They try something different. 809 01:02:44,810 --> 01:02:51,110 And the whole idea is going to be to use an entangled state. 810 01:02:51,110 --> 01:02:55,780 So basically what we're going to do 811 01:02:55,780 --> 01:03:08,615 is we're going to put the source here, entangled state source. 812 01:03:11,430 --> 01:03:13,940 And we're going to produce and an entangled 813 01:03:13,940 --> 01:03:18,180 state of two particles. 814 01:03:18,180 --> 01:03:25,110 And one particle is going to be given to A, to Alice. 815 01:03:25,110 --> 01:03:29,050 And one particle is going to be given to Bob. 816 01:03:29,050 --> 01:03:35,320 So particle B for Bob is going to be given to Bob. 817 01:03:35,320 --> 01:03:42,070 And particle A is going to be given to Alice. 818 01:03:45,760 --> 01:03:47,855 And this is an entangled pair. 819 01:03:54,050 --> 01:03:57,950 So there it is. 820 01:03:57,950 --> 01:03:59,750 Now what's going to happen? 821 01:03:59,750 --> 01:04:02,690 What are we going to do? 822 01:04:02,690 --> 01:04:06,140 Entanglement really correlates what 823 01:04:06,140 --> 01:04:09,420 goes here with what goes in there. 824 01:04:09,420 --> 01:04:14,220 Now entanglement happens instantaneously, 825 01:04:14,220 --> 01:04:16,820 and we can discuss this. 826 01:04:16,820 --> 01:04:19,450 You have no way of sending information 827 01:04:19,450 --> 01:04:22,570 through entanglement in general. 828 01:04:22,570 --> 01:04:26,950 There's no such thing as learning something about A when 829 01:04:26,950 --> 01:04:32,260 B doesn't measure, learning anything nontrivial about A. 830 01:04:32,260 --> 01:04:36,150 So the entangled state is there, and that's 831 01:04:36,150 --> 01:04:41,160 what we're going to try to use in order to do the teleporting. 832 01:04:41,160 --> 01:04:47,070 Now morally speaking, suppose I wanted to teleport myself 833 01:04:47,070 --> 01:04:49,860 from one place in this room to another. 834 01:04:49,860 --> 01:04:55,820 What I would have to do is create an enormous reservoir 835 01:04:55,820 --> 01:04:57,290 of entangled states. 836 01:04:57,290 --> 01:05:00,710 So here's my generator, and I create 837 01:05:00,710 --> 01:05:03,680 billions of entangled pairs. 838 01:05:03,680 --> 01:05:08,470 And I put them all here, all the ones here 839 01:05:08,470 --> 01:05:12,040 and all the corresponding pairs over there. 840 01:05:12,040 --> 01:05:17,650 And then I sort of-- somebody takes 841 01:05:17,650 --> 01:05:24,030 me and these billions of entangled pairs, one side 842 01:05:24,030 --> 01:05:26,480 of the pair, and does a measurement 843 01:05:26,480 --> 01:05:30,910 in which every atom or every quantum state in my body 844 01:05:30,910 --> 01:05:33,840 is measured with some entangled state. 845 01:05:33,840 --> 01:05:36,255 They've done the measurement, and boom. 846 01:05:36,255 --> 01:05:38,140 I reappear on the other side. 847 01:05:38,140 --> 01:05:39,840 That's what's going to happen. 848 01:05:39,840 --> 01:05:42,580 So we're going to do this. 849 01:05:42,580 --> 01:05:44,570 We're going to have this state, and now we're 850 01:05:44,570 --> 01:05:47,980 going to a measurement between this state and this state. 851 01:05:47,980 --> 01:05:50,030 Alice is going to do a measurement. 852 01:05:50,030 --> 01:05:53,370 That's going to force this particle 853 01:05:53,370 --> 01:05:56,480 to actually pretty much become the state you 854 01:05:56,480 --> 01:05:59,320 wanted to teleport. 855 01:05:59,320 --> 01:06:02,240 So that's the goal. 856 01:06:02,240 --> 01:06:07,150 So let me say a couple more things. 857 01:06:07,150 --> 01:06:10,260 Alice will have to send some information actually. 858 01:06:10,260 --> 01:06:13,660 Because she is going to have to do a measurement, and she 859 01:06:13,660 --> 01:06:22,140 has a console with four lights, zero, one, two, and three. 860 01:06:22,140 --> 01:06:24,960 Four lights. 861 01:06:24,960 --> 01:06:30,830 And when she will do her measurement, one of the lights 862 01:06:30,830 --> 01:06:33,790 will blink. 863 01:06:33,790 --> 01:06:38,280 And she will have to tell Bob which one blinked. 864 01:06:38,280 --> 01:06:42,680 So she will have to send the number and information of two 865 01:06:42,680 --> 01:06:43,370 bits. 866 01:06:43,370 --> 01:06:46,500 Because with two bits, you can represent 867 01:06:46,500 --> 01:06:50,390 any of four numbers, binary code. 868 01:06:50,390 --> 01:06:53,735 So she will send information of which clicked. 869 01:07:00,890 --> 01:07:08,100 And then Bob will have a machine with four entries here. 870 01:07:16,010 --> 01:07:20,240 And according to the information that he gets, 871 01:07:20,240 --> 01:07:27,000 he will make the state to go through one 872 01:07:27,000 --> 01:07:32,590 of those machines, the zero, the one, the two, or the three. 873 01:07:32,590 --> 01:07:37,600 So he will push B into one of them out, 874 01:07:37,600 --> 01:07:42,210 we claim, will come this teleported state. 875 01:07:52,260 --> 01:07:55,510 So that's the set up. 876 01:07:55,510 --> 01:07:58,480 You have to get a feel for the set up. 877 01:07:58,480 --> 01:08:02,170 So are there questions on what we're doing? 878 01:08:02,170 --> 01:08:03,998 AUDIENCE: So after teleportation would 879 01:08:03,998 --> 01:08:07,147 have some kind of copy [INAUDIBLE]? 880 01:08:07,147 --> 01:08:07,730 PROFESSOR: No. 881 01:08:07,730 --> 01:08:11,050 After the replication, this state 882 01:08:11,050 --> 01:08:15,330 will be destroyed beyond repair as you will see. 883 01:08:15,330 --> 01:08:19,520 So there will not be a copy created by this procedure. 884 01:08:19,520 --> 01:08:20,330 You destroy. 885 01:08:20,330 --> 01:08:23,319 It's really what teleportation was supposed to be. 886 01:08:23,319 --> 01:08:25,740 Not to create another copy of you there, 887 01:08:25,740 --> 01:08:28,075 but to take you there. 888 01:08:28,075 --> 01:08:31,130 Destroy you here and recreate you there. 889 01:08:31,130 --> 01:08:34,270 So no other copy. 890 01:08:34,270 --> 01:08:35,380 Other questions? 891 01:08:35,380 --> 01:08:36,270 Yes. 892 01:08:36,270 --> 01:08:39,488 AUDIENCE: Does this also work if C is an entangled state? 893 01:08:39,488 --> 01:08:40,279 PROFESSOR: If what? 894 01:08:40,279 --> 01:08:43,570 AUDIENCE: If C say itself contains different parts which 895 01:08:43,570 --> 01:08:45,460 are entangled with each other? 896 01:08:45,460 --> 01:08:47,500 PROFESSOR: Well, it's a more complicated thing. 897 01:08:47,500 --> 01:08:50,399 I'm pretty sure it would work. 898 01:08:50,399 --> 01:08:54,560 Maybe you would need more than one entangled pair here. 899 01:08:54,560 --> 01:08:59,511 You would need a source that is more complicated. 900 01:08:59,511 --> 01:09:00,135 More questions. 901 01:09:00,135 --> 01:09:03,458 AUDIENCE: What do you mean about pushes the state 902 01:09:03,458 --> 01:09:05,930 into one of the [INAUDIBLE]? 903 01:09:05,930 --> 01:09:08,470 PROFESSOR: What do I mean by pushes it through one of them? 904 01:09:08,470 --> 01:09:10,380 Well you know, Hamiltonians. 905 01:09:10,380 --> 01:09:11,600 You get your state. 906 01:09:11,600 --> 01:09:14,350 You can put them in a magnetic field. 907 01:09:14,350 --> 01:09:16,080 Let them evolve a little bit. 908 01:09:16,080 --> 01:09:17,640 Those are machines. 909 01:09:17,640 --> 01:09:22,880 So any of these machines are some unitary time evolution. 910 01:09:22,880 --> 01:09:25,141 It does something to the state. 911 01:09:25,141 --> 01:09:28,417 AUDIENCE: But one [INAUDIBLE] 912 01:09:28,417 --> 01:09:29,125 PROFESSOR: Sorry. 913 01:09:29,125 --> 01:09:30,896 AUDIENCE: Are there Hamiltonians that 914 01:09:30,896 --> 01:09:33,670 are based off of what Alice measures? 915 01:09:33,670 --> 01:09:34,800 PROFESSOR: Yes. 916 01:09:34,800 --> 01:09:37,420 So they will be correlated as you will see. 917 01:09:37,420 --> 01:09:41,700 So if Alice measures that the light zero beeps, 918 01:09:41,700 --> 01:09:45,279 the instruction for Bob is to send the state through the zero 919 01:09:45,279 --> 01:09:49,979 Hamiltonian, and one, two, and three Hamiltonian. 920 01:09:49,979 --> 01:09:52,420 More questions? 921 01:09:52,420 --> 01:09:57,070 It's good to really have a good feeling of this 922 01:09:57,070 --> 01:09:59,850 or what we're trying to do and why it's nontrivial. 923 01:09:59,850 --> 01:10:02,270 Yes. 924 01:10:02,270 --> 01:10:04,240 AUDIENCE: This might be a little too intuitive, 925 01:10:04,240 --> 01:10:08,190 but in a state which-- Can a Hamiltonian which Bob needs 926 01:10:08,190 --> 01:10:14,050 to send B through in order to yield the same state that Alice 927 01:10:14,050 --> 01:10:15,920 had, can that also be transmitted quantumly 928 01:10:15,920 --> 01:10:16,460 through qubits? 929 01:10:16,460 --> 01:10:18,672 Or would you just get like an infinite line of qubits needing 930 01:10:18,672 --> 01:10:19,171 to-- 931 01:10:19,171 --> 01:10:20,300 PROFESSOR: No no. 932 01:10:20,300 --> 01:10:24,210 You know, this is a devise that they can build by themselves. 933 01:10:24,210 --> 01:10:27,340 As you will see once we do the calculation, 934 01:10:27,340 --> 01:10:31,500 Alice will construct a device that has these four lights 935 01:10:31,500 --> 01:10:33,360 and she knows what they mean. 936 01:10:33,360 --> 01:10:36,940 And Bob will construct a device that has these things, 937 01:10:36,940 --> 01:10:39,950 and they can use it to transport any state. 938 01:10:39,950 --> 01:10:43,710 So these machines are independent of the state 939 01:10:43,710 --> 01:10:44,990 you want to teleport. 940 01:10:44,990 --> 01:10:48,880 You teleported this, you want to teleport another state 941 01:10:48,880 --> 01:10:51,570 with alpha prime and beta prime? 942 01:10:51,570 --> 01:10:52,170 Sure. 943 01:10:52,170 --> 01:10:57,560 Use exactly the same machines, give me another entangled pair, 944 01:10:57,560 --> 01:10:58,470 and do it. 945 01:10:58,470 --> 01:11:00,355 AUDIENCE: Well, I think what I meant 946 01:11:00,355 --> 01:11:02,860 is that the information between the two machines, 947 01:11:02,860 --> 01:11:04,840 does that have to be transmitted classically, 948 01:11:04,840 --> 01:11:06,490 or is there some way to transmit-- 949 01:11:06,490 --> 01:11:08,490 PROFESSOR: There's no real information. 950 01:11:08,490 --> 01:11:13,810 The machines were built, say, in the same laboratory of IBM. 951 01:11:13,810 --> 01:11:18,180 And then they're built, and we will tell you 952 01:11:18,180 --> 01:11:20,540 how to build each of these machines. 953 01:11:20,540 --> 01:11:24,310 And then just put aside, taken away by these two people, 954 01:11:24,310 --> 01:11:28,320 and then we'll do it. 955 01:11:28,320 --> 01:11:33,690 There's no mystery of sending information about it. 956 01:11:33,690 --> 01:11:36,670 That probably will become clear with the computation, 957 01:11:36,670 --> 01:11:39,400 which I better start doing soon. 958 01:11:39,400 --> 01:11:41,016 Yes. 959 01:11:41,016 --> 01:11:42,450 AUDIENCE: The difference-- 960 01:11:42,450 --> 01:11:43,340 PROFESSOR: Louder. 961 01:11:43,340 --> 01:11:45,590 AUDIENCE: Just a question about the first part 962 01:11:45,590 --> 01:11:47,340 on the left side of the board. 963 01:11:47,340 --> 01:11:49,230 So, when we first do a measurement, 964 01:11:49,230 --> 01:11:50,730 does that mean it's something that's 965 01:11:50,730 --> 01:11:55,077 like a microscopic quantity, like an energy or something? 966 01:11:55,077 --> 01:11:57,285 Or does it just refer to any? 967 01:11:57,285 --> 01:12:00,070 PROFESSOR: When we refer to measurements and quantum 968 01:12:00,070 --> 01:12:02,690 mechanics, we talk-- Let me give you 969 01:12:02,690 --> 01:12:04,660 just a little bit of intuition here. 970 01:12:04,660 --> 01:12:08,389 We typically talk about measuring permission operators, 971 01:12:08,389 --> 01:12:09,930 because they have [INAUDIBLE] values. 972 01:12:09,930 --> 01:12:14,490 So we don't have to say what they are-- energy, momentum. 973 01:12:14,490 --> 01:12:17,080 It's a permission operator you measure. 974 01:12:17,080 --> 01:12:20,570 And projector operators into basic states 975 01:12:20,570 --> 01:12:22,160 of permission operators. 976 01:12:22,160 --> 01:12:25,560 So you could imagine that's one way 977 01:12:25,560 --> 01:12:28,790 of thinking about these measurements. 978 01:12:28,790 --> 01:12:29,910 OK. 979 01:12:29,910 --> 01:12:32,430 So let's do this. 980 01:12:32,430 --> 01:12:33,110 All right. 981 01:12:33,110 --> 01:12:37,860 The state to be teleported is this one, 982 01:12:37,860 --> 01:12:59,190 and the A B pair is an entangled state. 983 01:12:59,190 --> 01:13:03,115 So it will be one of the bell states, 984 01:13:03,115 --> 01:13:12,260 psi zero AB 1 over square root of 2 plus A plus 985 01:13:12,260 --> 01:13:23,750 b plus minus A plus minus B. So this is the state they share. 986 01:13:23,750 --> 01:13:29,610 Of course, Alic only has a handle on particle A, 987 01:13:29,610 --> 01:13:35,050 and Bob only has a handle on particle B. Nevertheless 988 01:13:35,050 --> 01:13:37,420 the state is entangled even though this 989 01:13:37,420 --> 01:13:42,010 could be 200 kilometers apart. 990 01:13:42,010 --> 01:13:51,880 So the total state-- well, we've been tensoring two things. 991 01:13:51,880 --> 01:13:54,990 Well, tensoring three is three particles. 992 01:13:54,990 --> 01:14:00,170 So I don't think you will be too unhappy to just tensor 993 01:14:00,170 --> 01:14:01,050 the whole thing. 994 01:14:01,050 --> 01:14:20,710 So psi zero AB tensor alpha plus C plus beta minus C. 995 01:14:20,710 --> 01:14:25,100 So here comes the interesting point. 996 01:14:25,100 --> 01:14:37,110 Alice has available the state A. The particle A is not the state 997 01:14:37,110 --> 01:14:40,115 A because A is in a funny thing. 998 01:14:40,115 --> 01:14:40,740 It's entangled. 999 01:14:40,740 --> 01:14:43,925 But it has a particle A available, 1000 01:14:43,925 --> 01:14:47,210 and it has a particle C available. 1001 01:14:47,210 --> 01:14:51,510 So Alice is going to do a measurement, 1002 01:14:51,510 --> 01:14:53,310 and it's going to be a sneaky measurement. 1003 01:14:53,310 --> 01:14:55,270 It's going to use a bases. 1004 01:14:55,270 --> 01:14:57,550 Since she has two particles, she can 1005 01:14:57,550 --> 01:15:01,210 choose a basis of two particle states. 1006 01:15:01,210 --> 01:15:04,502 Any orthonormal basis will do well by the idea 1007 01:15:04,502 --> 01:15:08,000 that we can measure with any orthonormal basis. 1008 01:15:08,000 --> 01:15:10,640 So what she's going to try to do is use 1009 01:15:10,640 --> 01:15:17,510 the bell basis for A and C. 1010 01:15:17,510 --> 01:15:21,080 So let's try to think of what that means. 1011 01:15:21,080 --> 01:15:24,550 That requires a small calculation here. 1012 01:15:24,550 --> 01:15:31,530 So this is equal to 1 over square root of 2 1013 01:15:31,530 --> 01:15:38,040 plus-- so I anticipate that this will become clear 1014 01:15:38,040 --> 01:15:42,860 in a second, what that measurement means-- plus minus 1015 01:15:42,860 --> 01:15:55,950 A minus b Tensor alpha plus plus beta minus C. So I just wrote 1016 01:15:55,950 --> 01:15:56,740 what this is. 1017 01:16:00,330 --> 01:16:00,830 OK. 1018 01:16:04,040 --> 01:16:05,350 Some algebra. 1019 01:16:05,350 --> 01:16:08,040 This is the total state, psi total. 1020 01:16:12,460 --> 01:16:18,550 Let's multiply these things out, and I will keep the labels 1021 01:16:18,550 --> 01:16:20,390 all the time because I don't want 1022 01:16:20,390 --> 01:16:23,160 there to be any confusion about what's happening. 1023 01:16:23,160 --> 01:16:25,420 So what do we get first? 1024 01:16:25,420 --> 01:16:32,310 Alpha multiplying plus of A. I should write in plus of B, 1025 01:16:32,310 --> 01:16:35,960 but the order doesn't really matter if I keep the labels. 1026 01:16:35,960 --> 01:16:46,450 So I'll put plus of C times plus of B. 1027 01:16:46,450 --> 01:16:48,390 Then keep multiplying. 1028 01:16:48,390 --> 01:16:53,050 So we have plus beta, from this with that. 1029 01:16:53,050 --> 01:17:05,570 So I'll have plus of A minus of C and plus of B. Maybe 1030 01:17:05,570 --> 01:17:08,300 it's easier to read if I use another line. 1031 01:17:08,300 --> 01:17:13,010 So I now must multiply the second state times this. 1032 01:17:13,010 --> 01:17:30,730 So I get plus alpha minus of A with plus of C and minus of B. 1033 01:17:30,730 --> 01:17:40,120 So this is this times that, minus of A plus of C minus of B 1034 01:17:40,120 --> 01:17:55,850 plus beta minus of A minus of C minus of B. 1035 01:17:55,850 --> 01:17:57,520 OK. 1036 01:17:57,520 --> 01:17:58,775 So there here my state. 1037 01:18:01,860 --> 01:18:08,696 But now I have written it in a way that I have here A and C A 1038 01:18:08,696 --> 01:18:14,700 and C A and C and A and C. So I could 1039 01:18:14,700 --> 01:18:17,330 decide to measure in this basis. 1040 01:18:17,330 --> 01:18:23,740 This is an orthonormal basis for A and C. 1041 01:18:23,740 --> 01:18:28,710 But it's not a very smart basis because it's not entangled. 1042 01:18:28,710 --> 01:18:32,030 So let's go to the entangled base. 1043 01:18:32,030 --> 01:18:35,780 So let's rewrite this state, this total state. 1044 01:18:35,780 --> 01:18:38,390 Nothing has been done yet to the state. 1045 01:18:38,390 --> 01:18:43,720 We're just mathematically rewriting it, nothing else. 1046 01:18:43,720 --> 01:18:46,420 We have this, this, this, and that. 1047 01:18:46,420 --> 01:18:53,380 And I want you now to use these formulas to do this. 1048 01:18:53,380 --> 01:18:56,106 So I'll do this on this blackboard. 1049 01:18:58,790 --> 01:19:02,185 We'll have to erase those important names. 1050 01:19:11,110 --> 01:19:14,060 So what do we get? 1051 01:19:14,060 --> 01:19:16,200 Well a little of algebra. 1052 01:19:16,200 --> 01:19:19,910 Let's do it. 1053 01:19:19,910 --> 01:19:24,470 A with C plus plus would be that. 1054 01:19:24,470 --> 01:19:27,830 So I'll write it with one over square root of 2 1055 01:19:27,830 --> 01:19:30,530 becomes one half. 1056 01:19:30,530 --> 01:19:44,450 A with C would be psi zero AC plus psi three AC multiplying 1057 01:19:44,450 --> 01:19:50,060 alpha plus on B. So I took care of the first term. 1058 01:19:50,060 --> 01:19:51,800 The alpha is there. 1059 01:19:51,800 --> 01:19:53,020 The B is there. 1060 01:19:53,020 --> 01:19:58,620 And AC is there, in which, you know, 1061 01:19:58,620 --> 01:20:00,930 you can put any labels you want to here. 1062 01:20:00,930 --> 01:20:04,950 AB, this is the AB state. 1063 01:20:04,950 --> 01:20:07,900 The entangled AB state. 1064 01:20:07,900 --> 01:20:08,660 We used AC. 1065 01:20:11,890 --> 01:20:17,270 Second term plus one half. 1066 01:20:17,270 --> 01:20:22,690 Now we have plus A minus C. So it's the second line in there. 1067 01:20:22,690 --> 01:20:39,490 So it would be psi one AC minus I psi 2 AC beta plus B. Next 1068 01:20:39,490 --> 01:20:43,100 line, I'll just copy it, one half. 1069 01:20:43,100 --> 01:20:44,160 Well not. 1070 01:20:44,160 --> 01:20:52,110 Alpha minus B and here you'll have the minus plus 1071 01:20:52,110 --> 01:21:02,280 which is the same thing, psi 1 AC plus I psi 2 AC. 1072 01:21:02,280 --> 01:21:16,154 And the last term is plus one half psi 0 AC minus psi 3 AC. 1073 01:21:16,154 --> 01:21:25,810 And we get beta minus B. 1074 01:21:25,810 --> 01:21:29,120 OK, almost was there. 1075 01:21:29,120 --> 01:21:36,790 Let's rewrite this as-- let's collect the psi zeroes, psi 0 1076 01:21:36,790 --> 01:21:37,830 and psi 0. 1077 01:21:37,830 --> 01:21:41,380 You see we're do nothing yet. 1078 01:21:41,380 --> 01:21:43,870 We're just mathematically rewriting 1079 01:21:43,870 --> 01:21:48,000 the states in a different basis, the total states. 1080 01:21:48,000 --> 01:21:56,950 So it is equal to one half psi 0 AC. 1081 01:21:56,950 --> 01:21:59,240 and look what you get here, very curiously. 1082 01:21:59,240 --> 01:22:11,410 You get alpha plus B plus beta minus B. Very curious, that 1083 01:22:11,410 --> 01:22:15,400 was precisely the state we wanted to teleport. 1084 01:22:15,400 --> 01:22:18,840 Alpha plus plus beta minus. 1085 01:22:18,840 --> 01:22:19,630 All right. 1086 01:22:19,630 --> 01:22:22,090 Let's see what else happens. 1087 01:22:22,090 --> 01:22:26,680 Here we get plus one half psi-- which 1088 01:22:26,680 --> 01:22:28,040 other one do I want to copy? 1089 01:22:28,040 --> 01:22:30,560 Psi 1 AC. 1090 01:22:39,080 --> 01:22:42,640 You see this is the state we wanted to teleport. 1091 01:22:42,640 --> 01:22:43,310 It's here. 1092 01:22:46,470 --> 01:22:50,840 And it sort of has appeared in the B space. 1093 01:22:50,840 --> 01:22:56,670 Psi 1 AC, well this time I have this term and this term. 1094 01:22:56,670 --> 01:22:58,960 So actually it seems a little different. 1095 01:22:58,960 --> 01:23:10,720 Now we get beta plus B plus alpha minus B. 1096 01:23:10,720 --> 01:23:12,520 Then we go to the next. 1097 01:23:12,520 --> 01:23:18,005 One half of psi 2 AC. 1098 01:23:21,880 --> 01:23:23,260 So psi 2 is here. 1099 01:23:25,970 --> 01:23:46,980 So you get I alpha minus B minus I beta plus B. OK. 1100 01:23:46,980 --> 01:23:48,810 Finally linear combinations. 1101 01:23:48,810 --> 01:23:50,740 And finally psi 3. 1102 01:23:50,740 --> 01:23:52,370 What is psi 3? 1103 01:23:56,430 --> 01:23:59,370 Well two terms also for psi 3. 1104 01:23:59,370 --> 01:24:00,820 This one and this one. 1105 01:24:00,820 --> 01:24:09,270 So you get alpha plus B minus beta 1106 01:24:09,270 --> 01:24:16,970 minus B. Kind of the end of math by now. 1107 01:24:16,970 --> 01:24:21,650 You've proven a funny identity actually in doing this. 1108 01:24:21,650 --> 01:24:25,880 And maybe this blackboard should-- 1109 01:24:25,880 --> 01:24:27,520 to make sure you understand. 1110 01:24:27,520 --> 01:24:30,923 This is the calculation of total state. 1111 01:24:36,520 --> 01:24:37,470 And here we go. 1112 01:24:37,470 --> 01:24:40,110 So let me show you one thing. 1113 01:24:40,110 --> 01:24:45,300 This is actually the state we wanted. 1114 01:24:45,300 --> 01:24:52,100 So this will be called psi in the B basis, in the B space. 1115 01:24:52,100 --> 01:24:54,730 The state that you wanted to teleport 1116 01:24:54,730 --> 01:25:00,920 that was psi in the C basis, now it's psi in the B basis. 1117 01:25:00,920 --> 01:25:06,250 Those ones look a little funny, but this one actually 1118 01:25:06,250 --> 01:25:15,700 looks like this thing, looks like sigma 3 times psi. 1119 01:25:15,700 --> 01:25:18,520 Because if you have sigma 3 on this state, 1120 01:25:18,520 --> 01:25:23,490 it gives you a plus 1 here and a minus [INAUDIBLE] value. 1121 01:25:23,490 --> 01:25:26,300 So that's sigma 3 psi. 1122 01:25:26,300 --> 01:25:29,400 This actually has flipped the plus and the minus. 1123 01:25:29,400 --> 01:25:36,230 So that actually is sigma 1 psi. 1124 01:25:36,230 --> 01:25:42,880 And this state is actually sigma 2 psi. 1125 01:25:42,880 --> 01:25:46,830 OK everything is in place now. 1126 01:25:46,830 --> 01:25:51,580 We've just done math, but now comes the physics. 1127 01:25:51,580 --> 01:25:59,580 Alice is going to measure in the bell space of A and C. 1128 01:25:59,580 --> 01:26:05,500 So these are the four bases states. 1129 01:26:05,500 --> 01:26:09,690 So she's going to measure in one of these bases states. 1130 01:26:09,690 --> 01:26:14,690 And as see measures, she falls and the wave function 1131 01:26:14,690 --> 01:26:17,690 of her collapses into one of them. 1132 01:26:17,690 --> 01:26:24,890 So when she gets the zero basis state, this light blanks. 1133 01:26:24,890 --> 01:26:28,780 If doing the measurement on AC, because she 1134 01:26:28,780 --> 01:26:31,880 has both particles A and C, she gets 1135 01:26:31,880 --> 01:26:36,850 this basis state-- recall the postulate of measurement-- 1136 01:26:36,850 --> 01:26:39,210 light one blinks. 1137 01:26:39,210 --> 01:26:46,460 If she gets the third like 2 and the fourth here. 1138 01:26:46,460 --> 01:26:52,260 Suppose the state light zero shines. 1139 01:26:52,260 --> 01:26:55,260 Well the state collapsed into this. 1140 01:26:55,260 --> 01:26:58,980 She is now sitting with psi 0 AC that 1141 01:26:58,980 --> 01:27:05,380 has no memory whatsoever of the original state C, 1142 01:27:05,380 --> 01:27:08,120 but B is sitting with this state, 1143 01:27:08,120 --> 01:27:10,770 the state we wanted to teleport. 1144 01:27:10,770 --> 01:27:15,070 So if light zero shines, she tells 1145 01:27:15,070 --> 01:27:18,220 Bob, let it go to machine zero where 1146 01:27:18,220 --> 01:27:20,910 there's no magnetic field, nothing. 1147 01:27:20,910 --> 01:27:25,500 So actually the same state goes out. 1148 01:27:25,500 --> 01:27:32,510 If she gets psi 1 as the measured state, 1149 01:27:32,510 --> 01:27:37,670 again no memory in this state about alpha and beta. 1150 01:27:37,670 --> 01:27:41,886 But Bob gets sigma 1 psi 1. 1151 01:27:41,886 --> 01:27:46,770 So he puts it into the first Hamiltonian for a picosecond, 1152 01:27:46,770 --> 01:27:48,430 produces a sigma 1. 1153 01:27:48,430 --> 01:27:53,060 This Hamiltonian, this box I takes a state 1154 01:27:53,060 --> 01:27:54,880 into sigma I state. 1155 01:27:54,880 --> 01:27:56,440 It's a unitary operation. 1156 01:27:56,440 --> 01:28:00,320 So puts a sigma 1 and gets psi. 1157 01:28:00,320 --> 01:28:05,910 If light two shines, goes to the machine two, 1158 01:28:05,910 --> 01:28:09,570 which produces a sigma 2, and so he gets the state. 1159 01:28:09,570 --> 01:28:13,530 Light four shines, the third Hamiltonian, he gets the state. 1160 01:28:13,530 --> 01:28:16,925 Any of the four options, he gets the precise state. 1161 01:28:16,925 --> 01:28:19,350 The state has been teleported. 1162 01:28:19,350 --> 01:28:23,060 You needed to send only the information of which light 1163 01:28:23,060 --> 01:28:26,770 shone, and the state is on the other side of the ocean. 1164 01:28:26,770 --> 01:28:27,420 All right. 1165 01:28:27,420 --> 01:28:29,610 That's it for today.