1 00:00:00,080 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,030 under a Creative Commons license. 3 00:00:04,030 --> 00:00:06,880 Your support will help MIT OpenCourseWare continue 4 00:00:06,880 --> 00:00:10,740 to offer high-quality educational resources for free. 5 00:00:10,740 --> 00:00:13,360 To make a donation or view additional materials 6 00:00:13,360 --> 00:00:17,260 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,260 --> 00:00:17,885 at ocw.mit.edu. 8 00:00:20,532 --> 00:00:21,615 PROFESSOR: Good afternoon. 9 00:00:24,180 --> 00:00:30,930 So, we continue our discussion of quantum states of light, 10 00:00:30,930 --> 00:00:36,360 and we focus on single photons as qubits today. 11 00:00:36,360 --> 00:00:40,600 But before I get into this, I want 12 00:00:40,600 --> 00:00:44,520 to ask you if you have any questions about the last unit 13 00:00:44,520 --> 00:00:49,190 we be completed on Wednesday-- namely, the discussion or non 14 00:00:49,190 --> 00:00:52,950 classical light-- in particular, squeezing light-- 15 00:00:52,950 --> 00:00:58,090 and we also discussed a lot of exciting quantum mechanical 16 00:00:58,090 --> 00:01:01,520 effects related to beam splitters. 17 00:01:01,520 --> 00:01:03,350 So, any questions? 18 00:01:03,350 --> 00:01:06,290 Anything you would like to see discussed? 19 00:01:14,050 --> 00:01:20,160 So, as I pointed out, we want to now talk 20 00:01:20,160 --> 00:01:25,240 about realization of quantum logic and quantum gates 21 00:01:25,240 --> 00:01:28,170 with single photons, but I also said that we are actually 22 00:01:28,170 --> 00:01:31,155 using the language of quantum communications-- 23 00:01:31,155 --> 00:01:34,060 a kind of information processing to describe 24 00:01:34,060 --> 00:01:37,970 general physics in a very nice way. 25 00:01:37,970 --> 00:01:41,360 So, first-- when we talk about photons, 26 00:01:41,360 --> 00:01:50,500 we should talk about our qubits, and right now, you 27 00:01:50,500 --> 00:01:53,410 may think that having a single photon 28 00:01:53,410 --> 00:01:56,910 and having no photon in a mode are two possibilities. 29 00:01:56,910 --> 00:01:59,430 And you could use them as qubits. 30 00:01:59,430 --> 00:02:03,160 However, I will tell you today that it is better, actually, 31 00:02:03,160 --> 00:02:07,340 to use always one photon, but in two different wave guides-- two 32 00:02:07,340 --> 00:02:10,050 different modes. 33 00:02:10,050 --> 00:02:13,220 But, we'll get there in a moment. 34 00:02:13,220 --> 00:02:20,410 So, the first task at hand is we want to manipulate photons. 35 00:02:20,410 --> 00:02:24,390 And for that, I want to introduce phase shifters 36 00:02:24,390 --> 00:02:26,170 and beam splitters. 37 00:02:26,170 --> 00:02:28,990 And just to give you the punchline right away, 38 00:02:28,990 --> 00:02:32,750 what I want to show you is that those two simple devices -- 39 00:02:32,750 --> 00:02:36,796 beam splitter, a half-silvered mirror, a phase shifter, 40 00:02:36,796 --> 00:02:39,170 which is just a piece of glass which you can put into one 41 00:02:39,170 --> 00:02:44,340 of the laser beams-- that those two elements and those two 42 00:02:44,340 --> 00:02:50,420 basic operations, if they are now put together-- 43 00:02:50,420 --> 00:02:52,760 if you do a beam splitter, a phase shifter, 44 00:02:52,760 --> 00:02:56,490 another beam splitter and such-- you can realize any single 45 00:02:56,490 --> 00:02:58,300 qubit operation. 46 00:02:58,300 --> 00:03:02,250 Therefore, if you have our qubit cubic state 47 00:03:02,250 --> 00:03:08,150 with single photons, any possible state of those qubit 48 00:03:08,150 --> 00:03:11,140 can be realized using these optical elements. 49 00:03:11,140 --> 00:03:14,670 Therefore, when I say we want to realize how we can manipulate 50 00:03:14,670 --> 00:03:16,570 photons with optical elements and such, 51 00:03:16,570 --> 00:03:19,390 I'm not going for a zoo of optical elements. 52 00:03:19,390 --> 00:03:23,430 Those two will do it all. 53 00:03:23,430 --> 00:03:27,520 So, with the phase shifter, that's 54 00:03:27,520 --> 00:03:30,020 what we discussed at the end of last class. 55 00:03:30,020 --> 00:03:36,210 If you have two modes, and we shift the phase in one 56 00:03:36,210 --> 00:03:39,930 of the modes, just simply put a piece of glass into it, 57 00:03:39,930 --> 00:03:41,630 then there is a phase shift. 58 00:03:41,630 --> 00:03:44,220 And why do we need two modes? 59 00:03:44,220 --> 00:03:47,070 Well, phases are usually-- at least 60 00:03:47,070 --> 00:03:49,010 in experiments-- not absolutely defined. 61 00:03:49,010 --> 00:03:52,020 You need a phase reference, and here, 62 00:03:52,020 --> 00:03:56,520 the mode a acts as a phase reference. 63 00:03:56,520 --> 00:03:59,830 Let me address one question which a student asked me 64 00:03:59,830 --> 00:04:02,950 after class, and this was related 65 00:04:02,950 --> 00:04:09,680 to what phases appear in more than one place. 66 00:04:09,680 --> 00:04:12,060 The student's question was motivated 67 00:04:12,060 --> 00:04:15,290 that if you have one photon, it's a flux state. 68 00:04:15,290 --> 00:04:18,360 A flux state is a circle in the causal probability, 69 00:04:18,360 --> 00:04:21,450 and therefore, the electric field has no phase. 70 00:04:21,450 --> 00:04:24,560 Well, this is the phase of the electric field, 71 00:04:24,560 --> 00:04:28,780 but here, we're talking about the phase of the wave function. 72 00:04:28,780 --> 00:04:32,050 And a flux state within photons is an eigenstate 73 00:04:32,050 --> 00:04:33,630 of the harmonic oscillator. 74 00:04:33,630 --> 00:04:36,780 It has a time dependence, e to the i omega t, 75 00:04:36,780 --> 00:04:38,660 and we can change its phase. 76 00:04:38,660 --> 00:04:41,950 So, this is a kind of phase shift I've introduced here. 77 00:04:49,110 --> 00:04:55,810 So, the second element we need to realize-- 78 00:04:55,810 --> 00:04:59,490 arbitrary single qubit operation is the beam splitter, 79 00:04:59,490 --> 00:05:03,600 and I introduced a beam splitter by just saying, hey look, 80 00:05:03,600 --> 00:05:05,460 I think that's a good Hamiltonian. 81 00:05:05,460 --> 00:05:08,300 Let's see what this Hamiltonian does to the two modes, 82 00:05:08,300 --> 00:05:11,550 and then we realize-- yes, it takes those two modes, 83 00:05:11,550 --> 00:05:15,710 and it mixes the modes with cosine theta sine theta reading 84 00:05:15,710 --> 00:05:16,500 factors. 85 00:05:16,500 --> 00:05:20,390 That's exactly what you expect a beam splitter to do. 86 00:05:20,390 --> 00:05:22,570 I'm not sure we need here now because it's 87 00:05:22,570 --> 00:05:24,380 part of your homework assignment. 88 00:05:24,380 --> 00:05:28,310 You will show that if you have a coherent state, 89 00:05:28,310 --> 00:05:32,430 the coherent state is split by a ratio which 90 00:05:32,430 --> 00:05:35,280 is cosine square theta sine square theta. 91 00:05:35,280 --> 00:05:38,270 So, it's exactly what you'd expect from a beam splitter. 92 00:05:38,270 --> 00:05:40,510 So from that, you realize now what 93 00:05:40,510 --> 00:05:44,860 this angle theta is, which I just put into the Hamiltonian. 94 00:05:44,860 --> 00:05:47,260 The cosine square and sine square of it 95 00:05:47,260 --> 00:05:50,715 is the reflection and the transmission 96 00:05:50,715 --> 00:05:51,590 of the beam splitter. 97 00:06:00,520 --> 00:06:01,900 Any questions at that point? 98 00:06:09,430 --> 00:06:15,870 Let me now introduce matrix representation, 99 00:06:15,870 --> 00:06:18,040 which will come in handy. 100 00:06:18,040 --> 00:06:23,370 You remember that you transform an operator-- the mode 101 00:06:23,370 --> 00:06:26,360 operator, the annihilation operator a and b. 102 00:06:26,360 --> 00:06:33,300 We multiply with the beam splitter operator on the left 103 00:06:33,300 --> 00:06:38,150 and on the right, b dega-- b ab dega. 104 00:06:38,150 --> 00:06:42,630 But there is often a simpler way how the right inside 105 00:06:42,630 --> 00:06:46,850 can be written, and this is by using a matrix representation, 106 00:06:46,850 --> 00:06:47,780 which goes as follows. 107 00:06:54,000 --> 00:07:02,770 We can say that the two operators are transformed 108 00:07:02,770 --> 00:07:16,650 by the following matrix, and this matrix 109 00:07:16,650 --> 00:07:18,460 represents the beam splitter. 110 00:07:21,460 --> 00:07:29,790 So that's what the beam splitter does to the mode operators. 111 00:07:29,790 --> 00:07:32,170 Now, we want to talk about what does the beam splitter do 112 00:07:32,170 --> 00:07:36,220 to quantum states, to single photon states. 113 00:07:36,220 --> 00:07:38,630 So now, we want to transform the single photon states. 114 00:07:41,990 --> 00:07:43,472 It's a little bit like, well-- we 115 00:07:43,472 --> 00:07:45,180 know what to do in the Heisenberg picture 116 00:07:45,180 --> 00:07:46,500 to the operator, but now, we want 117 00:07:46,500 --> 00:07:48,249 to see what happens to the wave functions. 118 00:07:53,060 --> 00:08:00,510 So, now-- I want to say it very slowly, 119 00:08:00,510 --> 00:08:07,470 because it shouldn't lead to any confusion later. 120 00:08:07,470 --> 00:08:12,720 We have single photons-- we can have the single photon in mode 121 00:08:12,720 --> 00:08:15,260 b, or we can have it in mode a. 122 00:08:15,260 --> 00:08:17,970 These are the two possibilities. 123 00:08:17,970 --> 00:08:23,025 And right now, I label it like this-- 124 00:08:23,025 --> 00:08:27,060 where the first number is your occupation number of mode a, 125 00:08:27,060 --> 00:08:34,689 and the second number is your occupation in mode b. 126 00:08:34,689 --> 00:08:36,230 You have to be a little bit careful-- 127 00:08:36,230 --> 00:08:39,740 I'm just saying that keep your alert level high. 128 00:08:39,740 --> 00:08:46,310 Later, I will use qubit representation, 129 00:08:46,310 --> 00:08:50,130 where the qubit one means the photon is in mode a, 130 00:08:50,130 --> 00:08:53,640 and the qubit zero means the photon is in mode b. 131 00:08:53,640 --> 00:08:57,200 So when we talked about logical states of our two levels used, 132 00:08:57,200 --> 00:08:59,640 and the photon can be here or the photon can be there, 133 00:08:59,640 --> 00:09:01,740 zero means the photon is there. 134 00:09:01,740 --> 00:09:04,480 It doesn't mean that we have zero photons. 135 00:09:04,480 --> 00:09:06,979 But sometimes, I have to talk about the photons-- 136 00:09:06,979 --> 00:09:08,520 and that's what I'm doing right now-- 137 00:09:08,520 --> 00:09:12,410 and zero means no photon in this mode. 138 00:09:12,410 --> 00:09:16,280 But I will remind you of that as we go along. 139 00:09:16,280 --> 00:09:20,720 So we have the two quantum states-- one zero, 140 00:09:20,720 --> 00:09:24,410 and zero one. 141 00:09:24,410 --> 00:09:27,490 And let's just see what the beam splitter is 142 00:09:27,490 --> 00:09:35,450 doing to one photon in mode b. 143 00:09:40,570 --> 00:09:44,740 I'm changing notation from one line to the next in my notes. 144 00:09:50,810 --> 00:09:55,300 Just give me a split second. 145 00:09:58,606 --> 00:09:59,106 Yes. 146 00:10:05,351 --> 00:10:06,600 So, let's use this convention. 147 00:10:11,850 --> 00:10:14,560 So, just to elaborate what I just said 148 00:10:14,560 --> 00:10:19,660 is-- this convention one zero is the direct product 149 00:10:19,660 --> 00:10:22,480 of the Hilbert space for mode b, with one 150 00:10:22,480 --> 00:10:26,800 photon with the Hilbert space of mode a, 151 00:10:26,800 --> 00:10:30,710 and we have zero photons there. 152 00:10:30,710 --> 00:10:33,150 And sometimes, you want to denote the state 153 00:10:33,150 --> 00:10:35,040 by putting a comma in between. 154 00:10:35,040 --> 00:10:36,060 That's all the same. 155 00:10:39,350 --> 00:10:46,340 So, how do we transform the quantum state 156 00:10:46,340 --> 00:10:48,310 with the beam splitter operator? 157 00:10:48,310 --> 00:10:51,150 Well, we could now apply to the quantum stage, 158 00:10:51,150 --> 00:10:54,910 but we just learned how to apply to operators, so let's 159 00:10:54,910 --> 00:10:58,530 just use what we already know, because we 160 00:10:58,530 --> 00:11:04,070 can write the state like this. 161 00:11:16,530 --> 00:11:23,120 And now, we want to insert the unit. 162 00:11:28,260 --> 00:11:43,420 We want to insert the unit operator b dega b, 163 00:11:43,420 --> 00:11:49,950 and now, we can use our knowledge of the operator. 164 00:11:49,950 --> 00:11:51,520 We know how this transforms. 165 00:11:51,520 --> 00:11:53,960 We had this above the transformation of the operators 166 00:11:53,960 --> 00:12:04,530 b and b dega, and when we apply b to nothing-- to no photons, 167 00:12:04,530 --> 00:12:05,300 we get no photons. 168 00:12:08,450 --> 00:12:12,310 So now, by taking from the page above the transformation 169 00:12:12,310 --> 00:12:16,180 or the operator b dega, it's a linear combination 170 00:12:16,180 --> 00:12:19,920 of a dega and b dega. 171 00:12:19,920 --> 00:12:24,130 We find what we expect-- the photon can now 172 00:12:24,130 --> 00:12:29,870 be in the same mode, or it can appear in the other mode. 173 00:12:33,170 --> 00:12:37,740 And the coefficients from the transformation of cosine theta 174 00:12:37,740 --> 00:12:39,310 minus sine theta. 175 00:12:43,680 --> 00:12:49,830 And if you would send the other state through the beam 176 00:12:49,830 --> 00:12:59,930 splitter, we find the cosine theta sine theta component. 177 00:13:10,810 --> 00:13:13,670 So, since the total probability to have a photon 178 00:13:13,670 --> 00:13:16,800 is cosine square plus sine square is unity, 179 00:13:16,800 --> 00:13:20,000 we find what we expected, but it's nice to see it. 180 00:13:20,000 --> 00:13:31,940 Namely, that b conserves the photon number-- of course, 181 00:13:31,940 --> 00:13:34,270 this should have been obvious from the outset, 182 00:13:34,270 --> 00:13:37,860 because we used a Hermitian operator, a unitary time 183 00:13:37,860 --> 00:13:40,190 evolution, and that is energy- conserving. 184 00:13:44,390 --> 00:13:49,440 What happens if we use a state which 185 00:13:49,440 --> 00:13:55,180 has one photon in each mode, and we 186 00:13:55,180 --> 00:13:58,570 act on it with the beam splitter? 187 00:14:04,740 --> 00:14:11,740 Well, it gives a superposition of the two photons 188 00:14:11,740 --> 00:14:15,770 can now be in either of the two modes, 189 00:14:15,770 --> 00:14:21,150 or they can be distributed one in each mode. 190 00:14:21,150 --> 00:14:27,800 And the coefficients are because we have two transformations, 191 00:14:27,800 --> 00:14:30,575 products of sine and cosine. 192 00:14:33,360 --> 00:14:39,630 Here is cosine square plus sine square theta. 193 00:14:39,630 --> 00:14:44,390 Here is a plus sine, square root two plus square root two. 194 00:14:48,760 --> 00:14:56,040 So that means if we allow one photon to be in each mode, 195 00:14:56,040 --> 00:14:59,740 it leads us actually out of the Hilbert space of single photon 196 00:14:59,740 --> 00:15:02,190 states, because we have a certain probability 197 00:15:02,190 --> 00:15:06,780 now that we have two photons in each mode. 198 00:15:06,780 --> 00:15:07,940 So, we don't want that. 199 00:15:07,940 --> 00:15:12,740 So if you want to deal with only one photon, 200 00:15:12,740 --> 00:15:20,990 we should restrict our attention, our formalism 201 00:15:20,990 --> 00:15:34,370 to the states which have nor more than one photon. 202 00:15:34,370 --> 00:15:37,820 And now, if we act on those states with beam splitters, 203 00:15:37,820 --> 00:15:43,740 we don't get out of this subspace or fill that space. 204 00:15:43,740 --> 00:15:59,150 However, we also want to omit this state because it's 205 00:15:59,150 --> 00:16:02,350 more elegant to do it, but also, if you have a single photon 206 00:16:02,350 --> 00:16:06,830 state and we have a detector which has not 100% efficiency 207 00:16:06,830 --> 00:16:08,770 and we do a measurement-- well, we 208 00:16:08,770 --> 00:16:10,900 don't know if we have the state zero one, 209 00:16:10,900 --> 00:16:14,550 but we just didn't detect it, or we had the state zero zero. 210 00:16:14,550 --> 00:16:17,410 Whereas if you deal with two states 211 00:16:17,410 --> 00:16:25,280 where you always have a photon-- exactly one photon-- 212 00:16:25,280 --> 00:16:28,610 and you read out your system and you detect nothing, 213 00:16:28,610 --> 00:16:31,100 well you discard the measurement. 214 00:16:31,100 --> 00:16:35,390 And then, you only take the measurements 215 00:16:35,390 --> 00:16:36,940 which have detected one photon. 216 00:16:36,940 --> 00:16:41,870 So you have the ability to deal with finite efficiencies 217 00:16:41,870 --> 00:16:43,890 and losses in a very straightforward way. 218 00:16:49,790 --> 00:16:51,084 Yes? 219 00:16:51,084 --> 00:16:53,461 AUDIENCE: The operation of dega 1, 1 220 00:16:53,461 --> 00:16:54,710 doesn't look so unitary to me. 221 00:17:00,730 --> 00:17:01,780 PROFESSOR: Yes. 222 00:17:01,780 --> 00:17:03,220 And it is in my lecture notes. 223 00:17:03,220 --> 00:17:04,822 Minus. 224 00:17:04,822 --> 00:17:05,764 AUDIENCE: [INAUDIBLE]. 225 00:17:13,119 --> 00:17:14,102 PROFESSOR: Yes. 226 00:17:14,102 --> 00:17:15,560 And it's right in my lecture notes. 227 00:17:15,560 --> 00:17:16,210 Sorry. 228 00:17:16,210 --> 00:17:19,731 Sometimes, when you're talking, explain it, build it up-- yes. 229 00:17:19,731 --> 00:17:20,230 Thank you. 230 00:17:24,839 --> 00:17:27,790 Other question? 231 00:17:27,790 --> 00:17:29,070 Good. 232 00:17:29,070 --> 00:17:31,700 So what I just said, that we want to restrict the Hilbert 233 00:17:31,700 --> 00:17:38,290 space to exactly one photon, this 234 00:17:38,290 --> 00:17:43,040 is so important for implementations and discussions 235 00:17:43,040 --> 00:17:47,620 of quantum information protocols that it has its own name. 236 00:17:47,620 --> 00:17:50,900 It's called the dual-rail photon state. 237 00:17:58,480 --> 00:18:01,860 The dual-rail photon state space. 238 00:18:01,860 --> 00:18:06,200 Exactly one photon, but the photon can either be in mode a 239 00:18:06,200 --> 00:18:09,242 or in mode b. 240 00:18:09,242 --> 00:18:13,690 So this part-- this dual-rail photon state space-- 241 00:18:13,690 --> 00:18:30,070 is a Hilbert space which has a basis which is zero one and one 242 00:18:30,070 --> 00:18:34,440 zero, and this is of course a two-level system. 243 00:18:38,970 --> 00:18:44,410 In this so-defined Hilbert space, 244 00:18:44,410 --> 00:18:52,930 we can have an arbitrary state, [? psi ?] which 245 00:18:52,930 --> 00:18:56,680 is-- as in any two-dimensional Hilbert space-- 246 00:18:56,680 --> 00:19:00,320 it's a linear superposition of your two base 247 00:19:00,320 --> 00:19:03,140 states with coefficients alpha and beta. 248 00:19:10,950 --> 00:19:18,660 And the few of them for which we have prepared right now 249 00:19:18,660 --> 00:19:28,050 is that any possible state of this Hilbert space 250 00:19:28,050 --> 00:19:36,575 can be created from any of the base state, that 251 00:19:36,575 --> 00:19:48,520 is from zero one simply by beam splitters and phase shifters. 252 00:19:59,740 --> 00:20:02,810 When I taught this unit the last time, I went through the proof, 253 00:20:02,810 --> 00:20:07,650 but I want to make room for more in-class discussion like we had 254 00:20:07,650 --> 00:20:09,590 on Wednesday with the clicker question, 255 00:20:09,590 --> 00:20:13,140 so what I decided is I give you the idea behind the proof, 256 00:20:13,140 --> 00:20:17,160 and the formal parts you can simply fill in 257 00:20:17,160 --> 00:20:20,220 by reading about it on the Wiki page. 258 00:20:20,220 --> 00:20:23,250 So, let me just focus on the idea. 259 00:20:23,250 --> 00:20:29,130 So the proof is that if I use some column 260 00:20:29,130 --> 00:20:36,670 vector for alpha beta-- that we can identify with the beam 261 00:20:36,670 --> 00:20:38,630 splitter and the phase shifter-- we 262 00:20:38,630 --> 00:20:42,780 can identify them as rotation operators. 263 00:20:42,780 --> 00:20:50,050 This is immediate obvious if I use the transformation on psi 264 00:20:50,050 --> 00:21:00,670 by beam splitter, we learned just a few minutes ago 265 00:21:00,670 --> 00:21:03,810 how the beam splitter operator acts on the base state. 266 00:21:03,810 --> 00:21:06,790 So now, we know how it acts on an arbitrary 267 00:21:06,790 --> 00:21:08,780 linear combination. 268 00:21:08,780 --> 00:21:13,030 And the matrix here is nothing else 269 00:21:13,030 --> 00:21:18,800 than the rotation matrix with the minus 270 00:21:18,800 --> 00:21:22,490 sign around the y-axis. 271 00:21:36,340 --> 00:21:40,220 So that's a beam splitter. 272 00:21:43,040 --> 00:21:51,110 The phase shifter is not changing any state. 273 00:21:51,110 --> 00:22:02,130 It shifts one of the states by e to the minus i phi, 274 00:22:02,130 --> 00:22:06,740 but I can sort of summarize it by taking half of it 275 00:22:06,740 --> 00:22:12,110 out as a prefector, and then adding here 276 00:22:12,110 --> 00:22:14,650 e to the i phi over two. 277 00:22:17,200 --> 00:22:23,780 And this here is immediately recognized 278 00:22:23,780 --> 00:22:26,970 as the rotation matrix around the z-axis. 279 00:22:30,480 --> 00:22:33,680 Now, you may ask what about this? 280 00:22:33,680 --> 00:22:37,840 Well, this is an irrelevant global phase factor. 281 00:22:42,340 --> 00:22:44,406 If you have any state in Hilbert space 282 00:22:44,406 --> 00:22:46,650 and you change the global phase, it doesn't matter. 283 00:22:51,880 --> 00:22:59,260 Therefore, we are realizing that the beam splitter 284 00:22:59,260 --> 00:23:04,910 is a rotation around the y-axis. 285 00:23:04,910 --> 00:23:07,720 It's now a definitional thing, but we put a factor of two 286 00:23:07,720 --> 00:23:16,360 here by an angle two theta, and the phase shifter 287 00:23:16,360 --> 00:23:23,131 is a rotation around the z-axis by an angle phi. 288 00:23:27,280 --> 00:23:37,370 Now, with that, we can formulate what 289 00:23:37,370 --> 00:23:42,390 is called Bloch's theorem, that any unitary operation 290 00:23:42,390 --> 00:23:44,310 of a two-dimensional Hilbert space 291 00:23:44,310 --> 00:23:47,340 can be written as a product of rotations. 292 00:23:47,340 --> 00:23:51,130 Now, let's first count an arbitrary unitary operation 293 00:23:51,130 --> 00:23:53,660 while a two by two matrix in complex space 294 00:23:53,660 --> 00:23:59,280 has eight numbers-- four real parts, four imaginary parts. 295 00:23:59,280 --> 00:24:01,320 But if the matrix is unitary, they 296 00:24:01,320 --> 00:24:03,350 are only four independent elements-- 297 00:24:03,350 --> 00:24:05,970 alpha, beta, gamma, delta. 298 00:24:05,970 --> 00:24:09,960 And if you parametrize the unitary matrix with alpha, 299 00:24:09,960 --> 00:24:13,950 beta, gamma, delta which is shown in the weekly notes, 300 00:24:13,950 --> 00:24:20,340 then we can write this arbitrary unitary matrix 301 00:24:20,340 --> 00:24:24,620 by a combination of an overall phase factor. 302 00:24:27,182 --> 00:24:29,140 Which is irrelevant because it's a global phase 303 00:24:29,140 --> 00:24:35,470 factor-- and then a rotation around z, a rotation around y, 304 00:24:35,470 --> 00:24:37,665 and a rotation around the z-axis. 305 00:24:47,640 --> 00:24:59,380 So, if I rephrase this theorem in modern language, 306 00:24:59,380 --> 00:25:05,930 I would say we have a qubit which 307 00:25:05,930 --> 00:25:08,790 is a two-level system characterized 308 00:25:08,790 --> 00:25:12,620 by two numbers alpha beta. 309 00:25:12,620 --> 00:25:18,950 And we can do an arbitrary operation in the Hilbert space 310 00:25:18,950 --> 00:25:30,780 of the qubit, which is called an arbitrary single qubit 311 00:25:30,780 --> 00:25:31,280 operation. 312 00:25:37,910 --> 00:25:50,250 So, can be performed by phase shifters and beam splitters. 313 00:25:57,260 --> 00:26:01,650 Just to make sure that this doesn't lead to any confusion, 314 00:26:01,650 --> 00:26:05,430 the phase shift which is relevant for this Hilbert space 315 00:26:05,430 --> 00:26:08,310 is a phase shift which is relative between the two 316 00:26:08,310 --> 00:26:12,590 states, and can be detected by an interference experiment. 317 00:26:12,590 --> 00:26:15,790 Those global phase shifts cannot be detected. 318 00:26:15,790 --> 00:26:17,420 There is no observable, no procedure 319 00:26:17,420 --> 00:26:23,510 to observe those-- unless you would have a a third mode. 320 00:26:23,510 --> 00:26:26,680 But then, we would expand the Hilbert space 321 00:26:26,680 --> 00:26:32,240 to three levels and then, of course, 322 00:26:32,240 --> 00:26:35,060 it's the global phase shift in the two-dimension Hilbert 323 00:26:35,060 --> 00:26:36,622 space becomes a relative phase shift 324 00:26:36,622 --> 00:26:38,330 within the three-dimension Hilbert space. 325 00:26:42,640 --> 00:26:47,685 And in modern language, those single qubit operations 326 00:26:47,685 --> 00:26:50,660 are called quantum gates. 327 00:26:50,660 --> 00:26:54,980 So, all gates which act on a single qubit 328 00:26:54,980 --> 00:27:00,120 can be realized with phase shifters and beam splitters. 329 00:27:00,120 --> 00:27:03,870 Let me give you one example for that. 330 00:27:03,870 --> 00:27:08,980 In quantum computation, quantum information science, 331 00:27:08,980 --> 00:27:14,710 there is a very important gate-- the Hadamard gate, 332 00:27:14,710 --> 00:27:19,130 which is described by a transformation which 333 00:27:19,130 --> 00:27:23,430 has 1 minus 111. 334 00:27:23,430 --> 00:27:28,790 And I said it can be realized with a beam splitter and phase 335 00:27:28,790 --> 00:27:29,930 shifter. 336 00:27:29,930 --> 00:27:37,930 So let me just show it to you in using our symbolic language. 337 00:27:37,930 --> 00:27:42,430 So if you look at the matrix we had derived for the beam 338 00:27:42,430 --> 00:27:48,480 splitter, you realize that we get the Hadamard gate 339 00:27:48,480 --> 00:27:50,730 by aiding the phase shifter by pi. 340 00:28:02,350 --> 00:28:07,220 So, in that sense, single qubit operations are checked off. 341 00:28:07,220 --> 00:28:09,850 We know how to deal with that. 342 00:28:09,850 --> 00:28:19,780 We want to now move towards two qubit operations-- 343 00:28:19,780 --> 00:28:22,490 and then, it's getting really interesting. 344 00:28:22,490 --> 00:28:26,230 As you may know, you can do any arbitrary operation 345 00:28:26,230 --> 00:28:28,650 in many qubits by just having single 346 00:28:28,650 --> 00:28:30,700 and two qubit operations. 347 00:28:30,700 --> 00:28:33,940 Therefore, once we know how two qubits interact-- 348 00:28:33,940 --> 00:28:38,970 how photons in two different qubits interact, 349 00:28:38,970 --> 00:28:43,500 we are done-- we have universal quantum gates. 350 00:28:43,500 --> 00:28:46,632 The element I need for two qubit operation 351 00:28:46,632 --> 00:28:48,090 is the Mach-Zehnder interferometer. 352 00:29:05,770 --> 00:29:11,200 Just a side remark, I introduce a Mach-Zehnder interferometer 353 00:29:11,200 --> 00:29:13,760 here as a way to manipulate qubits. 354 00:29:13,760 --> 00:29:15,850 On the other hand, most of you know 355 00:29:15,850 --> 00:29:19,900 that interferometers for light and interferometers for atoms 356 00:29:19,900 --> 00:29:24,120 are used for some of the most precise measurements ever done. 357 00:29:24,120 --> 00:29:27,110 So, in that sense, this shows the dual use 358 00:29:27,110 --> 00:29:29,340 of the formalism I'm introducing here. 359 00:29:29,340 --> 00:29:32,110 We really use some of the formalism and the language 360 00:29:32,110 --> 00:29:34,630 we develop here for qubit operation, 361 00:29:34,630 --> 00:29:39,350 and discuss next week the ultimate accuracy-- 362 00:29:39,350 --> 00:29:41,200 the fundamental limits to the accuracy 363 00:29:41,200 --> 00:29:43,100 you can get in precision measurements 364 00:29:43,100 --> 00:29:44,100 based on interferometry. 365 00:29:53,995 --> 00:29:55,520 Let me just write down the sentence 366 00:29:55,520 --> 00:29:58,080 because it has a lot of key words in it. 367 00:29:58,080 --> 00:30:04,390 So the dual-rail photon presentation of a qubit-- this 368 00:30:04,390 --> 00:30:05,800 is what we have achieved so far. 369 00:30:15,940 --> 00:30:20,340 So this allows us now to discuss interferometers simply 370 00:30:20,340 --> 00:30:21,268 as [INAUDIBLE] gates. 371 00:30:41,210 --> 00:30:48,746 And what isn't interferometer-- an interferometer is, 372 00:30:48,746 --> 00:30:51,870 in its basic [INAUDIBLE], you have a beam splitter, 373 00:30:51,870 --> 00:30:54,700 you have two paths, and then you use another beam splitter 374 00:30:54,700 --> 00:30:57,167 which recombines the two beams. 375 00:30:57,167 --> 00:30:58,625 So, let me draw two beam splitters. 376 00:31:09,020 --> 00:31:12,300 These are two beam splitters. 377 00:31:12,300 --> 00:31:20,770 We have an input [INAUDIBLE] b, an input mode a. 378 00:31:20,770 --> 00:31:22,030 They are sort of mixed. 379 00:31:22,030 --> 00:31:25,090 Here are the two arms of the interferometer. 380 00:31:25,090 --> 00:31:29,230 And now, we create output modes b prime and a prime. 381 00:31:33,070 --> 00:31:38,810 What we have introduced here is the beam splitter b, 382 00:31:38,810 --> 00:31:43,880 and by putting the dot on the other side, this is b dega. 383 00:31:43,880 --> 00:31:50,320 Since b dega times b equals one equals the identity, 384 00:31:50,320 --> 00:31:53,400 this interferometer is doing nothing. 385 00:31:53,400 --> 00:31:55,410 If the photon starts in mode b, it 386 00:31:55,410 --> 00:31:57,680 comes out in the upper mode b. 387 00:31:57,680 --> 00:32:01,580 If it starts in a, it comes out in the mode a. 388 00:32:01,580 --> 00:32:05,290 But of course, this is just the beginning, 389 00:32:05,290 --> 00:32:15,410 because we can now introduce here an arbitrary phase shift 390 00:32:15,410 --> 00:32:16,600 phi. 391 00:32:16,600 --> 00:32:18,470 And now, it is an interferometer. 392 00:32:18,470 --> 00:32:21,090 Now, we can read out with the output 393 00:32:21,090 --> 00:32:26,450 the phase shift, and ultimately, with very high precision. 394 00:32:26,450 --> 00:32:28,570 I what you to appreciate that. 395 00:32:28,570 --> 00:32:34,280 If we had used single qubits in one mode-- no photon, 396 00:32:34,280 --> 00:32:36,390 one photon. 397 00:32:36,390 --> 00:32:38,680 I gave you many reasons why we shouldn't do that, 398 00:32:38,680 --> 00:32:40,700 but then, we would be dealing with one mode, 399 00:32:40,700 --> 00:32:42,200 and this would actually be something 400 00:32:42,200 --> 00:32:44,570 which involves two qubits. 401 00:32:44,570 --> 00:32:48,420 But since our single qubit system 402 00:32:48,420 --> 00:32:52,580 is a two-level system where the photon is here and there, 403 00:32:52,580 --> 00:32:56,800 we can now send single photons with the interferometer-- 404 00:32:56,800 --> 00:33:02,140 describe the interferometer as a single qubit operation. 405 00:33:02,140 --> 00:33:04,810 At least it's nice. 406 00:33:04,810 --> 00:33:07,142 It allows us to keep things simple, 407 00:33:07,142 --> 00:33:09,100 and discuss things at a very fundamental level. 408 00:33:14,460 --> 00:33:18,470 So, what we want to figure out-- what 409 00:33:18,470 --> 00:33:21,080 is this interferometer doing? 410 00:33:21,080 --> 00:33:27,390 So, we want to know if we have an input state-- what 411 00:33:27,390 --> 00:33:28,950 is the output state? 412 00:33:28,950 --> 00:33:33,720 And the input state can now be any arbitrary wave function 413 00:33:33,720 --> 00:33:35,660 in our dual-rail Hilbert space. 414 00:33:35,660 --> 00:33:38,420 So, it's always one photon, but it 415 00:33:38,420 --> 00:33:43,730 can be in a random superposition in an arbitrary 416 00:33:43,730 --> 00:33:47,580 superposition of states a and b. 417 00:33:47,580 --> 00:33:50,950 Well, with the formalism we have developed now, 418 00:33:50,950 --> 00:33:52,720 it becomes very simple. 419 00:33:52,720 --> 00:33:57,220 All we have to do is we have to act with the first beam 420 00:33:57,220 --> 00:34:01,430 splitter with a phase shifter, and with a second beam splitter 421 00:34:01,430 --> 00:34:03,740 onto the state. 422 00:34:03,740 --> 00:34:14,070 And this is nothing else than taking two rotation matrices 423 00:34:14,070 --> 00:34:15,600 and multiplying them. 424 00:34:20,960 --> 00:34:23,659 The beam splitter we have chosen, 425 00:34:23,659 --> 00:34:26,170 I said there was sort of a phase convention 426 00:34:26,170 --> 00:34:29,445 when I peeked the beam splitter. 427 00:34:29,445 --> 00:34:33,590 It is a rotation by pi over 2. 428 00:34:33,590 --> 00:34:38,449 This is minus pi over 2, and the z-rotation is by minus phi. 429 00:34:46,010 --> 00:34:51,000 I'm not sure if I manage to do that, but if this is the y-axis 430 00:34:51,000 --> 00:34:54,980 and we start with the qubit, the first beam splitter 431 00:34:54,980 --> 00:34:57,340 rotates it down. 432 00:34:57,340 --> 00:35:02,620 Then, the rotation by z-- thus, that-- and the second beam 433 00:35:02,620 --> 00:35:05,700 splitter rotates by y. 434 00:35:05,700 --> 00:35:09,080 Therefore, the qubit is like that. 435 00:35:09,080 --> 00:35:15,000 Therefore, if you count all x's with the things 436 00:35:15,000 --> 00:35:17,530 I did is-- I started with a qubit here. 437 00:35:17,530 --> 00:35:20,170 It went down like this, like this. 438 00:35:20,170 --> 00:35:22,000 So what I did is-- in the end, I simply 439 00:35:22,000 --> 00:35:24,970 rotated it around the x-axis. 440 00:35:24,970 --> 00:35:27,630 If you want, you can multiply the matrices, 441 00:35:27,630 --> 00:35:33,440 or you can draw block sphere visualization. 442 00:35:33,440 --> 00:35:41,170 That you say we have an x-axis, y-axis, and z-axis. 443 00:35:41,170 --> 00:35:42,330 So the z is vertical. 444 00:35:46,160 --> 00:35:55,300 So, we started out with the qubit along the z-axis. 445 00:35:55,300 --> 00:35:58,110 The first one was rotation around the y-axis. 446 00:36:02,300 --> 00:36:03,980 So then, we were here. 447 00:36:03,980 --> 00:36:15,880 The second operation was rotation around the z-axis. 448 00:36:15,880 --> 00:36:26,340 And then, we have to rotate back around the y-axis, which 449 00:36:26,340 --> 00:36:30,920 eventually gives this as the final state. 450 00:36:30,920 --> 00:36:33,610 So ultimately, the product of all this 451 00:36:33,610 --> 00:36:37,225 is a rotation around the x-axis. 452 00:36:41,650 --> 00:36:44,354 Our interferometer-- I'm not sure 453 00:36:44,354 --> 00:36:45,770 if you've ever heard the language, 454 00:36:45,770 --> 00:36:47,840 but it's really elegant and powerful. 455 00:36:47,840 --> 00:36:52,330 An interferometer with an arbitrary phase 456 00:36:52,330 --> 00:36:55,350 in this two-level Hilbert space with this sort 457 00:36:55,350 --> 00:36:58,430 of geometric interpretation we have given, 458 00:36:58,430 --> 00:37:01,765 is nothing else than a rotation around the x-axis. 459 00:37:08,450 --> 00:37:15,250 Therefore, we know immediately the two limiting cases when 460 00:37:15,250 --> 00:37:21,100 we rotate by zero, it's sort of balanced. 461 00:37:21,100 --> 00:37:23,450 Output and input are the same, because we're not 462 00:37:23,450 --> 00:37:26,520 rotating at all. 463 00:37:26,520 --> 00:37:30,460 Whereas if we have a phase shift of 180 degrees, 464 00:37:30,460 --> 00:37:33,590 we swap the two qubits. 465 00:37:33,590 --> 00:37:40,840 So, we swap the two modes, and swapping modes 466 00:37:40,840 --> 00:37:43,550 is an inversion of the qubit. 467 00:37:53,430 --> 00:38:00,390 So that's almost trivial, but we need this definition-- 468 00:38:00,390 --> 00:38:03,490 we need this knowledge to take it to the next step. 469 00:38:03,490 --> 00:38:05,160 Remember, we want to do something 470 00:38:05,160 --> 00:38:07,870 more interesting than rotating two level systems. 471 00:38:07,870 --> 00:38:12,420 We want to get two qubits, combine them, entangle them, 472 00:38:12,420 --> 00:38:13,770 and all that. 473 00:38:13,770 --> 00:38:19,200 So what we need for that is-- we have 474 00:38:19,200 --> 00:38:21,660 to expand the Hilbert space. 475 00:38:21,660 --> 00:38:25,150 I introduced the Mach-Zehnder interferometer as the device 476 00:38:25,150 --> 00:38:26,400 to do it. 477 00:38:26,400 --> 00:38:30,000 But now, we need a way how a second photon 478 00:38:30,000 --> 00:38:33,410 from another qubit can interact with the photon 479 00:38:33,410 --> 00:38:35,420 of our first qubit. 480 00:38:35,420 --> 00:38:39,020 And we want to do that by introducing first 481 00:38:39,020 --> 00:38:41,440 the non-linear Mach-Zehnder interferometer. 482 00:38:41,440 --> 00:38:44,810 I want to throw in a non-linear element, 483 00:38:44,810 --> 00:38:48,640 and then we are ready to allow the second qubit 484 00:38:48,640 --> 00:38:53,430 through the non-linear element to manipulate the first qubit. 485 00:38:53,430 --> 00:38:55,960 And then, we have interactions between qubit. 486 00:38:55,960 --> 00:38:57,580 We have two qubit gates. 487 00:39:17,450 --> 00:39:21,202 Let me maybe just deviate for my notes. 488 00:39:21,202 --> 00:39:22,910 I really want to show you what I'm doing, 489 00:39:22,910 --> 00:39:26,510 and that you have it clearly in mind where I'm aiming it. 490 00:39:26,510 --> 00:39:31,160 So what we want to accomplish is-- instead of having a phase 491 00:39:31,160 --> 00:39:34,080 shifter with an externally controlled phase, 492 00:39:34,080 --> 00:39:38,480 I want to put in some non-linear crystal-- which 493 00:39:38,480 --> 00:39:41,820 has an index of refraction which has a phase shift. 494 00:39:41,820 --> 00:39:45,070 But now, the face that the non-linear crystal 495 00:39:45,070 --> 00:39:47,780 has the property, that its index of refraction 496 00:39:47,780 --> 00:39:54,740 changes when I take another laser beam and send it through. 497 00:39:54,740 --> 00:39:57,530 So, the laser beam through-- because the index of refraction 498 00:39:57,530 --> 00:40:01,400 is intensity-dependent, the green laser beam 499 00:40:01,400 --> 00:40:06,420 changes index of refraction of the non-linear crystal-- also 500 00:40:06,420 --> 00:40:08,340 called Kerr medium. 501 00:40:08,340 --> 00:40:11,240 And that means now, because the blue beam 502 00:40:11,240 --> 00:40:15,060 goes through the same crystal, that the green beam controls 503 00:40:15,060 --> 00:40:18,520 now the phase shift of the blue beam. 504 00:40:18,520 --> 00:40:23,302 And now, we have interacting photons-- one photon interacts 505 00:40:23,302 --> 00:40:24,260 with the second photon. 506 00:40:28,340 --> 00:40:29,120 Let's work it out. 507 00:40:32,160 --> 00:40:37,940 So our goal is to develop a description 508 00:40:37,940 --> 00:40:40,620 for non-linear Mach-Zehnder interferometer. 509 00:40:50,330 --> 00:40:53,080 What we need is a non-linear medium. 510 00:40:53,080 --> 00:40:56,140 So, let me just introduce for two or three minutes 511 00:40:56,140 --> 00:40:58,289 non-linear medium, how we describe it, 512 00:40:58,289 --> 00:41:00,080 and then we put in into our interferometer. 513 00:41:03,840 --> 00:41:08,060 So, in linear optics-- just to remind you, 514 00:41:08,060 --> 00:41:12,940 we have a polarization on electric dipole moment, which 515 00:41:12,940 --> 00:41:17,420 is proportional to the electric field of light, 516 00:41:17,420 --> 00:41:20,840 and x i is the polarizability. 517 00:41:20,840 --> 00:41:24,030 The most complicated case, it can be polarizability tensor. 518 00:41:32,930 --> 00:41:35,110 Now, this is linear. 519 00:41:35,110 --> 00:41:42,720 If you want two non-linear, then we have the linear relationship 520 00:41:42,720 --> 00:41:46,890 simply as you can say the first terminal [INAUDIBLE] expansion. 521 00:41:46,890 --> 00:41:50,900 So we expand the response of the medium in powers 522 00:41:50,900 --> 00:41:57,480 of the electric field, and we have the susceptibilities 523 00:41:57,480 --> 00:42:02,400 chi 1, chi 2, chi 3, and so on. 524 00:42:07,670 --> 00:42:13,230 We have already encountered a non-linear element which 525 00:42:13,230 --> 00:42:19,610 had a chi 2 susceptibility, and these where the crystals 526 00:42:19,610 --> 00:42:29,400 we put in our OPOs into our optical parametric oscillators. 527 00:42:29,400 --> 00:42:32,290 Remember, the optical parametric oscillator 528 00:42:32,290 --> 00:42:36,770 involved-- how many modes? 529 00:42:36,770 --> 00:42:38,730 Three. 530 00:42:38,730 --> 00:42:43,890 We had one photon, which was broken down into two photons. 531 00:42:43,890 --> 00:42:46,350 But that also means that the reverse process 532 00:42:46,350 --> 00:42:51,460 is you drive the beam-- you drive the OPO with omega, 533 00:42:51,460 --> 00:42:56,730 but the e square term creates an electric polarization 534 00:42:56,730 --> 00:42:57,800 as 2 omega. 535 00:42:57,800 --> 00:43:00,680 And if you have an oscillating polarization as 2 omega, 536 00:43:00,680 --> 00:43:02,720 you create light at 2 omega. 537 00:43:02,720 --> 00:43:06,020 So, I just want to remind you that the chi square term is 538 00:43:06,020 --> 00:43:08,779 what we have already encountered. 539 00:43:08,779 --> 00:43:11,070 That's not what we need for the non-linear Mach-Zehnder 540 00:43:11,070 --> 00:43:12,620 interferometer. 541 00:43:12,620 --> 00:43:17,100 What we need now is a chi 3 interaction, 542 00:43:17,100 --> 00:43:18,485 which is the Kerr medium. 543 00:43:22,790 --> 00:43:26,820 Because we want to realize the following Hamiltonian. 544 00:43:31,000 --> 00:43:34,410 The Hamiltonian, the Kerr medium is also called cross-- 545 00:43:34,410 --> 00:43:37,460 that's cx -- cross-phase modulation. 546 00:43:37,460 --> 00:43:40,795 One beam can affect the phase of another beam, 547 00:43:40,795 --> 00:43:42,545 and this is called cross-phase modulation. 548 00:43:45,190 --> 00:43:47,300 There's also self-phase modulation-- 549 00:43:47,300 --> 00:43:49,540 that one laser beam changes the index of refraction 550 00:43:49,540 --> 00:43:52,655 for itself-- but here, we want the cross-phase modulation. 551 00:43:56,710 --> 00:44:07,130 So, we know that if we have a mode, and we shift its phase, 552 00:44:07,130 --> 00:44:09,850 we're not changing any photons. 553 00:44:09,850 --> 00:44:12,420 So if we're in an eigenstate of mode a 554 00:44:12,420 --> 00:44:17,260 with one, two, or three photons and we simply shift its phase, 555 00:44:17,260 --> 00:44:21,870 we are diagonal in a dega a, and the phase shift 556 00:44:21,870 --> 00:44:24,170 is proportional to prefector here. 557 00:44:24,170 --> 00:44:28,115 So this Hamiltonian is simply a phase shift for photons in mode 558 00:44:28,115 --> 00:44:35,170 a, but if you now multiply that with b dega b, 559 00:44:35,170 --> 00:44:38,680 the phase shift is now proportional to the number 560 00:44:38,680 --> 00:44:40,900 of photons in mode b. 561 00:44:40,900 --> 00:44:43,830 Now, we have a situation that the phase shift from mode 562 00:44:43,830 --> 00:44:48,440 a proportional to the intensity in beam b, and vice versa. 563 00:44:48,440 --> 00:44:51,140 So, I think it's self-evident that this Hamiltonian is 564 00:44:51,140 --> 00:44:53,930 sequential description of the process we 565 00:44:53,930 --> 00:44:55,080 want to introduce know. 566 00:45:02,760 --> 00:45:05,300 This is the cross-phase modulation Hamiltonian. 567 00:45:11,010 --> 00:45:14,400 Isn't it much more elegant to describe non-linear optics 568 00:45:14,400 --> 00:45:18,150 with Hamiltonians than with parametrized susceptibilities 569 00:45:18,150 --> 00:45:18,994 and all that? 570 00:45:18,994 --> 00:45:20,910 It's just amazing-- you write in a few letters 571 00:45:20,910 --> 00:45:24,390 and it has all the power of the classical description-- 572 00:45:24,390 --> 00:45:27,060 plus extra, because it includes quantum fluctuations 573 00:45:27,060 --> 00:45:29,380 and everything. 574 00:45:29,380 --> 00:45:31,780 I hope you appreciate the power of the language 575 00:45:31,780 --> 00:45:33,460 we are developing. 576 00:45:33,460 --> 00:45:35,420 This is the cross-phase modulation Hamiltonian. 577 00:45:39,350 --> 00:45:45,320 And now, we want to use this Hamiltonian 578 00:45:45,320 --> 00:45:55,365 in a crystal of lengths L Well, you 579 00:45:55,365 --> 00:46:02,230 know if you have a time-propagation operator, 580 00:46:02,230 --> 00:46:06,410 you put the Hamiltonian in the exponent and multiply with t, 581 00:46:06,410 --> 00:46:11,300 but for propagating laser beam, t and L are related, 582 00:46:11,300 --> 00:46:14,240 so this is now the unitary transformation 583 00:46:14,240 --> 00:46:17,980 if you propagate through that crystal. 584 00:46:24,820 --> 00:46:33,780 We are now defining that by saying 585 00:46:33,780 --> 00:46:36,650 we have this non-linear susceptibility, 586 00:46:36,650 --> 00:46:39,540 things will be proportional to the links L, 587 00:46:39,540 --> 00:46:42,910 and then we have the operator a dega a, b dega b. 588 00:46:50,620 --> 00:46:58,680 And let me choose for the following discussion 589 00:46:58,680 --> 00:47:06,300 that the links are chosen, such that-- when 590 00:47:06,300 --> 00:47:08,360 multiplied with a non-linear susceptibility, 591 00:47:08,360 --> 00:47:09,880 we get a phase shift of pi. 592 00:47:13,920 --> 00:47:16,900 Now, we are ready to describe what 593 00:47:16,900 --> 00:47:21,730 happens when we have a Kerr medium, 594 00:47:21,730 --> 00:47:31,235 and we have two input modes. 595 00:47:35,310 --> 00:47:41,295 So, let's write down the table of possible combinations 596 00:47:41,295 --> 00:47:41,920 and operations. 597 00:47:44,990 --> 00:47:47,565 Well, when we act on the vacuum, we get nothing. 598 00:47:51,680 --> 00:47:54,884 What happens when we have one photon in one mode? 599 00:47:54,884 --> 00:47:56,050 What happens to that photon? 600 00:48:08,290 --> 00:48:14,500 Well, the extra phase shift coming 601 00:48:14,500 --> 00:48:17,130 from cross-phase modulation is zero, 602 00:48:17,130 --> 00:48:22,760 because we have no photon in one of the modes, and a dega a 603 00:48:22,760 --> 00:48:25,410 or b dega b acting on this mode gives zero. 604 00:48:30,040 --> 00:48:34,020 Therefore, as long as we have only one photon, 605 00:48:34,020 --> 00:48:35,810 we have no phase shift. 606 00:48:35,810 --> 00:48:38,600 The exponential factor is one, and we simply 607 00:48:38,600 --> 00:48:44,300 reproduce the original state. 608 00:48:44,300 --> 00:48:47,980 So, the only non-trivial situation by construction 609 00:48:47,980 --> 00:48:52,090 is when we have one photon in each mode, 610 00:48:52,090 --> 00:48:58,320 and then we get the matrix element e to the ikl. 611 00:49:01,380 --> 00:49:03,660 And we adjusted the lengths of the crystal 612 00:49:03,660 --> 00:49:07,030 that this is just minus one. 613 00:49:07,030 --> 00:49:10,540 In other words, in this Hilbert space, 614 00:49:10,540 --> 00:49:14,360 we get a phase shift of pi-- we get a minus sign when 615 00:49:14,360 --> 00:49:17,130 we have one photon in each mode. 616 00:49:17,130 --> 00:49:18,800 That's all what this medium does. 617 00:49:25,470 --> 00:49:28,400 And if you ever thought about it-- 618 00:49:28,400 --> 00:49:32,380 how to deal with this situation-- that when 619 00:49:32,380 --> 00:49:36,200 I said there is one photon which provides a phase 620 00:49:36,200 --> 00:49:38,200 shift to the other photon. 621 00:49:38,200 --> 00:49:40,590 But the other photon is also providing a phase shift 622 00:49:40,590 --> 00:49:41,990 to the first photon. 623 00:49:41,990 --> 00:49:43,340 So you may wonder. 624 00:49:43,340 --> 00:49:45,470 When you have a combined state of the two, 625 00:49:45,470 --> 00:49:47,970 how do you deal that one photon shifts 626 00:49:47,970 --> 00:49:50,500 the other photon, and vice versa? 627 00:49:50,500 --> 00:49:52,000 But when you apply the operator, you 628 00:49:52,000 --> 00:49:53,620 don't even have to think about it. 629 00:49:53,620 --> 00:49:56,540 You see that the operator for the combined Hilbert 630 00:49:56,540 --> 00:49:59,350 space of the two photons gives you a phase shift of pi. 631 00:49:59,350 --> 00:50:02,440 It's not pi for one photon times pi for the other one-- 632 00:50:02,440 --> 00:50:04,180 it's pi for the whole quantum state. 633 00:50:06,766 --> 00:50:07,550 Any questions? 634 00:50:11,530 --> 00:50:17,730 We have a Kerr medium, which means 635 00:50:17,730 --> 00:50:22,360 we are ready now to put it into our interferometer. 636 00:50:22,360 --> 00:50:24,135 I'm doing at this time-- insert picture. 637 00:50:55,290 --> 00:50:57,660 This is what you want to do now. 638 00:50:57,660 --> 00:51:00,990 We have now three modes of light. 639 00:51:00,990 --> 00:51:04,090 You would say, hey-- a qubit, each qubit is two. 640 00:51:04,090 --> 00:51:06,010 What is it-- is it one and a half qubit? 641 00:51:06,010 --> 00:51:10,490 Yes, it is-- but we extended two cubits in a moment. 642 00:51:10,490 --> 00:51:12,830 So what we simply have right now is 643 00:51:12,830 --> 00:51:18,300 we have the mode c-- which controls the non-linear medium, 644 00:51:18,300 --> 00:51:20,750 and we have what we have already discussed. 645 00:51:20,750 --> 00:51:23,250 Our interferometer with a phase shifter, 646 00:51:23,250 --> 00:51:24,520 which is now the Kerr medium. 647 00:51:29,560 --> 00:51:33,120 So I think even without doing any math, 648 00:51:33,120 --> 00:51:43,870 we know that by construction, when we have no photons in c, 649 00:51:43,870 --> 00:51:52,390 that the modes a prime b prime transform-- become a and b, 650 00:51:52,390 --> 00:51:54,610 because we have a balance interferometer which 651 00:51:54,610 --> 00:51:57,710 is the identity transformation, unless we have a phase shift. 652 00:52:00,980 --> 00:52:12,530 But when we have one photon in state c, then we actually swap. 653 00:52:12,530 --> 00:52:14,150 So, that's pretty cool. 654 00:52:14,150 --> 00:52:19,750 A single photon in mode c will redirect the photon from ab 655 00:52:19,750 --> 00:52:21,490 to ba. 656 00:52:21,490 --> 00:52:26,320 So that's how one photon in mode c 657 00:52:26,320 --> 00:52:29,410 controls what happens to the photon in ab. 658 00:52:35,130 --> 00:52:39,540 If you want a general description of that, 659 00:52:39,540 --> 00:52:46,060 we obtain the output state from the input state 660 00:52:46,060 --> 00:52:50,520 by using the matrix b. 661 00:52:50,520 --> 00:52:55,960 This was our interferometer based on two beam splitters. 662 00:52:55,960 --> 00:53:00,410 It involved mode a and b. 663 00:53:00,410 --> 00:53:03,340 But now, we throw in the phase shifter, 664 00:53:03,340 --> 00:53:07,000 which is our Kerr operator-- our non-linear phase 665 00:53:07,000 --> 00:53:08,840 shifter with b and c. 666 00:53:14,160 --> 00:53:18,540 This new addition to the interferometer 667 00:53:18,540 --> 00:53:25,130 is described by e to the i, non-linear susceptibility L, 668 00:53:25,130 --> 00:53:27,700 b dega b, c dega c. 669 00:53:38,520 --> 00:53:41,080 Let me give you one intermediate result. 670 00:53:41,080 --> 00:53:43,295 We have now beam splitters on either side. 671 00:53:49,210 --> 00:53:54,540 And that leads us to an expression. 672 00:53:54,540 --> 00:54:03,810 While the beam splitters do nothing to the mode c, 673 00:54:03,810 --> 00:54:07,390 but the beam splitters transform the modes 674 00:54:07,390 --> 00:54:09,026 in linear combinations. 675 00:54:17,810 --> 00:54:28,910 And if you do some tedious rearrangement of terms-- 676 00:54:28,910 --> 00:54:31,480 more details are given in the Wiki-- 677 00:54:31,480 --> 00:54:36,200 you find that the essential result is now 678 00:54:36,200 --> 00:54:54,460 the operator, which we get is a dega b minus b dega a over 2. 679 00:55:04,190 --> 00:55:08,390 I'm not giving you the non-essential terms, 680 00:55:08,390 --> 00:55:12,095 which are just phase shifts-- global phase shifts. 681 00:55:17,850 --> 00:55:20,680 So, let me write it down in blue here. 682 00:55:26,120 --> 00:55:29,420 This is just the result of the operator manipulation. 683 00:55:32,920 --> 00:55:36,320 If I would cover that-- what is that? 684 00:55:36,320 --> 00:55:40,710 Do you remember a dega b minus b dega a in the exponent? 685 00:55:40,710 --> 00:55:42,680 It's a beam splitter. 686 00:55:42,680 --> 00:55:45,120 And the beam splitter-- the beam splitter 687 00:55:45,120 --> 00:55:47,890 matrix was sine theta cosine theta. 688 00:55:47,890 --> 00:55:50,280 So we had a theta here. 689 00:55:50,280 --> 00:55:53,320 Now, you realize that this non-linear Mach-Zehnder 690 00:55:53,320 --> 00:55:56,100 interferometer, the three optical elements 691 00:55:56,100 --> 00:56:00,020 can be replaced by simply a single-beam splitter for mode 692 00:56:00,020 --> 00:56:04,520 a and b, but the angle theta of the beam splitter 693 00:56:04,520 --> 00:56:09,140 is now controlled by the photon field in state c. 694 00:56:11,770 --> 00:56:26,810 In other words, we have now a beam splitter. 695 00:56:37,762 --> 00:56:39,970 I told you that a beam splitter is 696 00:56:39,970 --> 00:56:43,235 nothing else than a rotation around the y-axis. 697 00:56:46,720 --> 00:56:51,170 So now, this non-linear Mach-Zehnder interferometer 698 00:56:51,170 --> 00:56:53,840 is simply a beam splitter with a rotation 699 00:56:53,840 --> 00:56:57,630 angle given by this term. 700 00:57:09,007 --> 00:57:09,590 Any questions? 701 00:57:32,020 --> 00:57:40,870 This is a situation we just had, and we can now 702 00:57:40,870 --> 00:57:51,010 get to two qubits by simply adding one more rail. 703 00:57:51,010 --> 00:57:59,120 Remember, two rails with exactly one photon in it is a qubit. 704 00:57:59,120 --> 00:58:05,570 So now, we have a qubit here, and we have a qubit there. 705 00:58:05,570 --> 00:58:08,980 And you see immediately one way how the upper qubit 706 00:58:08,980 --> 00:58:11,705 x on the lower qubit-- if the lower qubit has 707 00:58:11,705 --> 00:58:20,880 a photon in state c, it flips the other qubit. 708 00:58:20,880 --> 00:58:23,790 But if the upper qubit has a photon in d, 709 00:58:23,790 --> 00:58:26,100 it doesn't do anything to the lower qubit. 710 00:58:26,100 --> 00:58:27,825 Now, we have a two-qubit operation. 711 00:58:38,960 --> 00:58:41,870 This is shown here. 712 00:58:41,870 --> 00:58:49,490 But what I want to discuss now is the situation 713 00:58:49,490 --> 00:58:52,800 when we throw in one more beam splitter, 714 00:58:52,800 --> 00:58:55,130 and this beam splitter acts on the upper qubit. 715 00:59:04,611 --> 00:59:08,370 Well-- isn't it great how quickly 716 00:59:08,370 --> 00:59:10,460 we go from simple elements to something 717 00:59:10,460 --> 00:59:13,930 which looks quite sophisticated? 718 00:59:13,930 --> 00:59:21,520 So, this device now is a single optical device 719 00:59:21,520 --> 00:59:26,560 which can entangle qubits, which leads to entanglement. 720 00:59:26,560 --> 00:59:29,020 And actually, entanglement is our next big topic 721 00:59:29,020 --> 00:59:32,260 we want to talk about-- entanglement between particles, 722 00:59:32,260 --> 00:59:34,410 entanglement between photons. 723 00:59:34,410 --> 00:59:39,130 And by introducing the Mach-Zehnder interferometer, 724 00:59:39,130 --> 00:59:41,600 I was actually building up the situation, 725 00:59:41,600 --> 00:59:45,845 which is extremely simple elements which 726 00:59:45,845 --> 00:59:46,720 lead to entanglement. 727 00:59:49,570 --> 00:59:52,350 We have everything for the description-- 728 00:59:52,350 --> 00:59:54,700 each of those elements is described 729 00:59:54,700 --> 00:59:58,170 by a matrix which we have developed. 730 00:59:58,170 --> 01:00:01,130 Therefore, by just multiplying those matrices, 731 01:00:01,130 --> 01:00:05,990 you find immediately how this whole device is now 732 01:00:05,990 --> 01:00:08,490 manipulated into qubits. 733 01:00:08,490 --> 01:00:10,910 So, if you want, you can just take the crank, 734 01:00:10,910 --> 01:00:13,140 use what we have already done, and crank out 735 01:00:13,140 --> 01:00:14,426 the result for it. 736 01:00:27,570 --> 01:00:29,220 Now I wish I had a bigger screen, 737 01:00:29,220 --> 01:00:32,470 because I want to describe what's going on for you. 738 01:00:44,580 --> 01:00:46,190 What I want to discuss with you now 739 01:00:46,190 --> 01:00:49,560 is what happens when we take this device, 740 01:00:49,560 --> 01:00:54,245 and we use exactly the states 0,1 0,1 at the input 741 01:00:54,245 --> 01:00:54,870 as I indicated. 742 01:01:00,690 --> 01:01:04,960 In terms of qubit language, I say 743 01:01:04,960 --> 01:01:08,560 that the first qubit is in spin-up, 744 01:01:08,560 --> 01:01:10,550 and the lower qubit is also in spin-up. 745 01:01:13,400 --> 01:01:17,830 Spin-down would mean that the single photon 746 01:01:17,830 --> 01:01:19,820 is in the other state. 747 01:01:19,820 --> 01:01:23,140 Our two-level system-- our dual-rail single qubit 748 01:01:23,140 --> 01:01:26,790 two-level system is the photon can be either 749 01:01:26,790 --> 01:01:28,090 in one of those states. 750 01:01:28,090 --> 01:01:33,140 This one, we call spin-up-- this one, we call spin-down. 751 01:01:33,140 --> 01:01:38,690 What I want to discuss with you now, 752 01:01:38,690 --> 01:01:41,950 what happens when we start with this symmetric input 753 01:01:41,950 --> 01:01:46,240 up-up, which means 0,1 0,1. 754 01:01:46,240 --> 01:01:49,620 And since I don't want to scroll up and down so often, 755 01:01:49,620 --> 01:01:52,760 everything is trivial beam splitters, 756 01:01:52,760 --> 01:01:57,330 except for the situation when we have one photon each here, 757 01:01:57,330 --> 01:02:00,587 and then the Kerr medium gives us a minus sign. 758 01:02:00,587 --> 01:02:01,920 So that's what I want to put in. 759 01:02:13,670 --> 01:02:16,790 Remember, I want to develop the physics 760 01:02:16,790 --> 01:02:20,960 in one, two, three temporal steps. 761 01:02:20,960 --> 01:02:24,490 The first temporal step is simply two beam splitters, 762 01:02:24,490 --> 01:02:25,570 and this is shown here. 763 01:02:29,190 --> 01:02:32,650 We start out-- and now we have beam splitter 764 01:02:32,650 --> 01:02:37,270 for the upper qubit, beam splitter for the lower qubit. 765 01:02:37,270 --> 01:02:40,410 And well, by just multiplying it out, 766 01:02:40,410 --> 01:02:48,730 we are now obtaining this summation state 767 01:02:48,730 --> 01:02:53,290 for the four different modes. 768 01:02:53,290 --> 01:02:55,660 And I'm dropping here factors of square root-- 769 01:02:55,660 --> 01:02:58,420 or you can collect them at the end, if you want. 770 01:02:58,420 --> 01:03:01,370 And I said the only interesting situation 771 01:03:01,370 --> 01:03:06,800 is where we have two photons-- one each in the two 772 01:03:06,800 --> 01:03:11,320 middle modes, because now, the Kerr medium kicks in and gives 773 01:03:11,320 --> 01:03:14,570 us a minus sign. 774 01:03:14,570 --> 01:03:20,480 Now, you can show by inspection that if you now apply the beam 775 01:03:20,480 --> 01:03:24,190 splitter to modes a and b, you take 776 01:03:24,190 --> 01:03:28,740 1, 0 into a linear combination of 1 and 1, 777 01:03:28,740 --> 01:03:32,680 but the beam splitter is only acting on the last two numbers 778 01:03:32,680 --> 01:03:33,600 here. 779 01:03:33,600 --> 01:03:37,870 Then you obtain a state which looks very simple. 780 01:03:37,870 --> 01:03:40,070 So this is the output state of the device. 781 01:03:43,287 --> 01:03:43,870 Any questions? 782 01:03:47,870 --> 01:04:06,700 And this output state is-- if I use the spin language, 783 01:04:06,700 --> 01:04:08,020 it's an entangled state. 784 01:04:19,780 --> 01:04:21,860 So now, you have one simple example 785 01:04:21,860 --> 01:04:25,130 how just using linear optics, very simple elements 786 01:04:25,130 --> 01:04:28,540 you can start with one photon here, one photon here, 787 01:04:28,540 --> 01:04:30,980 a simple product state, and what comes out 788 01:04:30,980 --> 01:04:31,940 is an entangled state. 789 01:04:36,285 --> 01:04:36,785 Questions? 790 01:04:43,280 --> 01:04:47,000 Let me tell you what I'm not telling you-- namely, 791 01:04:47,000 --> 01:04:49,810 the Wiki developed by a professor [INAUDIBLE] 792 01:04:49,810 --> 01:04:54,030 has now a wonderful section on a famous quantum algorithm-- 793 01:04:54,030 --> 01:04:57,100 the Deutsch-Jozsa algorithm. 794 01:04:57,100 --> 01:04:59,750 Pretty much using the elements we discuss, 795 01:04:59,750 --> 01:05:03,220 you can now realize the Deutsch-Jozsa algorithm, 796 01:05:03,220 --> 01:05:06,150 which is one of the famous algorithms where 797 01:05:06,150 --> 01:05:09,620 quantum logic-- quantum computation is 798 01:05:09,620 --> 01:05:13,430 faster and more efficient than classical computation. 799 01:05:13,430 --> 01:05:16,530 I decided not to present it to you, because it just 800 01:05:16,530 --> 01:05:19,200 leads to an even more complicated diagram and more 801 01:05:19,200 --> 01:05:20,090 formulae. 802 01:05:20,090 --> 01:05:22,930 You should just sit back and slowly read it by yourself. 803 01:05:22,930 --> 01:05:25,290 There is no new idea introduced-- 804 01:05:25,290 --> 01:05:28,820 it's just the concepts we have discussed 805 01:05:28,820 --> 01:05:32,990 can lead to very powerful algorithms. 806 01:05:32,990 --> 01:05:37,050 What I want to rather do is-- I want to continue our discussion 807 01:05:37,050 --> 01:05:41,440 and now talk about the object we have encountered here-- 808 01:05:41,440 --> 01:05:42,625 namely, entangled states. 809 01:05:45,140 --> 01:05:48,650 But if you have any questions, if you 810 01:05:48,650 --> 01:05:52,100 want me to explain anything more about the single qubit 811 01:05:52,100 --> 01:05:55,150 manipulation, the dual-rail photon state, 812 01:05:55,150 --> 01:05:56,784 I would be happy to do that. 813 01:06:18,280 --> 01:06:51,630 Our next chapter is on entangled photons, 814 01:06:51,630 --> 01:06:57,050 and what we can accomplish in the next 10 or 15 minutes 815 01:06:57,050 --> 01:07:00,585 is mainly a definition and some of the properties. 816 01:07:04,970 --> 01:07:07,830 What we will do next week is we will 817 01:07:07,830 --> 01:07:11,617 talk about how we measure entanglement-- you already 818 01:07:11,617 --> 01:07:14,460 had one homework assignment where you discussed one 819 01:07:14,460 --> 01:07:16,690 way to measure entanglement, but there 820 01:07:16,690 --> 01:07:18,890 are others which we want to discuss next week. 821 01:07:21,810 --> 01:07:27,260 I also want to show you next week how entanglement leads 822 01:07:27,260 --> 01:07:30,830 to the Einstein-Podolsky-Rosen paradox and Bell's 823 01:07:30,830 --> 01:07:32,470 inequalities. 824 01:07:32,470 --> 01:07:34,310 And eventually, I want to show you 825 01:07:34,310 --> 01:07:39,520 how you can entangle not only photons, 826 01:07:39,520 --> 01:07:42,040 but how you can entangle atoms. 827 01:07:42,040 --> 01:07:43,610 But that's an outlook. 828 01:07:43,610 --> 01:07:46,340 I think, first of all, we want to understand 829 01:07:46,340 --> 01:07:47,260 what is entanglement. 830 01:07:50,780 --> 01:07:59,740 Entanglement is-- let me first [? motivate ?] it. 831 01:07:59,740 --> 01:08:05,850 Many people regard entanglement as the most quantum 832 01:08:05,850 --> 01:08:07,700 essence you can find. 833 01:08:12,080 --> 01:08:13,620 Well, there's always a discussion, 834 01:08:13,620 --> 01:08:16,510 but it's really quantum. 835 01:08:16,510 --> 01:08:19,760 Some people would say waves are quantum. 836 01:08:19,760 --> 01:08:23,774 But often, we find quantum systems-- 837 01:08:23,774 --> 01:08:26,720 very famously, Bose-Einstein condensates, which really 838 01:08:26,720 --> 01:08:29,380 behave like electromagnetic waves. 839 01:08:29,380 --> 01:08:30,410 They can be split. 840 01:08:30,410 --> 01:08:31,880 They can interfere. 841 01:08:31,880 --> 01:08:35,960 But Bose-Einstein condensates are so big, have so many atoms 842 01:08:35,960 --> 01:08:39,859 that you hardly ever encounter quantum fluctuations. 843 01:08:39,859 --> 01:08:42,790 You're really in the classical limit of matter waves, 844 01:08:42,790 --> 01:08:47,370 and I would actually say this is maybe not 845 01:08:47,370 --> 01:08:48,899 at the heart of quantum mechanics. 846 01:08:48,899 --> 01:08:50,520 While it's nice, it's powerful, it's 847 01:08:50,520 --> 01:08:54,540 important-- but this is not really the new feature which 848 01:08:54,540 --> 01:08:58,540 quantum mechanics has shown us in nature. 849 01:08:58,540 --> 01:09:01,010 You would say-- well, what else is quantum? 850 01:09:01,010 --> 01:09:02,810 Maybe certain quantum fluctuations 851 01:09:02,810 --> 01:09:05,040 going down to single photons, and I 852 01:09:05,040 --> 01:09:06,952 would agree that's a quantization 853 01:09:06,952 --> 01:09:08,160 of the electromagnetic field. 854 01:09:08,160 --> 01:09:11,290 That's much more quantum than having millions of condensates, 855 01:09:11,290 --> 01:09:15,330 millions of condensed atoms in one single wave. 856 01:09:15,330 --> 01:09:17,899 But what I would say is even more quantum 857 01:09:17,899 --> 01:09:20,950 than the single photon is the entanglement. 858 01:09:20,950 --> 01:09:25,420 Entanglement is sort of pure quantum mechanics. 859 01:09:25,420 --> 01:09:28,300 With single photons, you can still use intuition. 860 01:09:28,300 --> 01:09:32,689 But with entanglement, that is really bizarre. 861 01:09:32,689 --> 01:09:35,985 This is really- I hate the word, but let 862 01:09:35,985 --> 01:09:39,569 me use it-- this is sort of pure quantum weirdness. 863 01:09:39,569 --> 01:09:43,359 And there are people who have spent most of their career 864 01:09:43,359 --> 01:09:47,510 until now to just show the weirdness of quantum mechanics, 865 01:09:47,510 --> 01:09:52,779 and showing to what peculiar phenomena entanglement leads. 866 01:09:52,779 --> 01:09:57,709 So entanglement has really lead to a lot of-- I 867 01:09:57,709 --> 01:10:00,000 should be careful with the word surprising-- surprising 868 01:10:00,000 --> 01:10:02,060 always means you didn't have enough imagination 869 01:10:02,060 --> 01:10:05,370 to think about it, but yes, I would say-- really truly 870 01:10:05,370 --> 01:10:09,330 unanticipated, and in that sense, surprising developments. 871 01:10:15,030 --> 01:10:18,610 Entanglement is, therefore, for me, the most quantum 872 01:10:18,610 --> 01:10:20,085 aspect of quantum mechanics. 873 01:10:23,540 --> 01:10:32,590 It was through the Einstein-Podolsky-Rosen 874 01:10:32,590 --> 01:10:38,800 argument-- it was actually Einstein-Podolsky-Rosen 875 01:10:38,800 --> 01:10:42,890 in the '30s who argued-- Einstein-Podolsky-Rosen were 876 01:10:42,890 --> 01:10:45,237 really the first-- this was almost 5 877 01:10:45,237 --> 01:10:47,070 or 10 years after the development of quantum 878 01:10:47,070 --> 01:10:47,890 mechanics. 879 01:10:47,890 --> 01:10:50,600 But it was Einstein-Podolsky-Rosen 880 01:10:50,600 --> 01:10:56,185 who looked at the properties of entangled states. 881 01:11:01,420 --> 01:11:04,220 This famous experiment where you have something 882 01:11:04,220 --> 01:11:06,170 entangled in position and momentum space. 883 01:11:06,170 --> 01:11:08,040 And then, Einstein-Podolsky-Rosen 884 01:11:08,040 --> 01:11:11,760 found a paradox, and their conclusion 885 01:11:11,760 --> 01:11:16,250 was that the properties of entangled states 886 01:11:16,250 --> 01:11:19,560 implies that quantum mechanic is not 887 01:11:19,560 --> 01:11:23,390 compete-- is not a complete description of the word. 888 01:11:26,140 --> 01:11:32,260 Well, we don't any longer share this opinion. 889 01:11:32,260 --> 01:11:40,820 It was John Bell and others who said that entanglement really 890 01:11:40,820 --> 01:11:54,460 exemplifies that quantum mechanical correlations go 891 01:11:54,460 --> 01:12:02,180 beyond classical mechanics-- go beyond the classic picture. 892 01:12:04,780 --> 01:12:07,370 Therefore, you can say if you have a quantum 893 01:12:07,370 --> 01:12:10,600 system like a Bose-Einstein condensate, 894 01:12:10,600 --> 01:12:14,130 where you have end particles doing 895 01:12:14,130 --> 01:12:16,470 in unison what one particle does. 896 01:12:16,470 --> 01:12:19,100 This is sort of like the laser where many photons do 897 01:12:19,100 --> 01:12:21,640 what one photon does-- but it fits very, very 898 01:12:21,640 --> 01:12:25,310 well into the concept of classical probabilities 899 01:12:25,310 --> 01:12:28,800 and an intuitive understanding of why many particles do 900 01:12:28,800 --> 01:12:30,680 what one particle does. 901 01:12:30,680 --> 01:12:32,980 But if the particles are entangled, 902 01:12:32,980 --> 01:12:36,280 that's almost like you turbo-charge your system. 903 01:12:36,280 --> 01:12:39,960 Your system has now more oomph, more power in it. 904 01:12:39,960 --> 01:12:43,380 And this is shown that it has extra correlation-- there 905 01:12:43,380 --> 01:12:45,785 is something extra in it, which you would never 906 01:12:45,785 --> 01:12:48,090 get from any classical limit. 907 01:12:48,090 --> 01:12:52,340 And this extra oomph, this extra power 908 01:12:52,340 --> 01:12:55,600 turns out to be a real resource. 909 01:12:55,600 --> 01:13:02,490 So, entanglement-- this extra correlation-- is a resource. 910 01:13:02,490 --> 01:13:06,270 And resource means it's good for something. 911 01:13:06,270 --> 01:13:07,453 It enables teleportation. 912 01:13:14,340 --> 01:13:19,470 Remember when we talked about the teleportation scheme-- 913 01:13:19,470 --> 01:13:22,920 Alice and Bob could only teleport a quantum state 914 01:13:22,920 --> 01:13:27,240 because they shared entangled photons. 915 01:13:27,240 --> 01:13:32,020 It was this entangled state which Alice manipulated 916 01:13:32,020 --> 01:13:34,620 with a measurement, and Bob used this half 917 01:13:34,620 --> 01:13:37,740 of the entangled state to recreate the original state. 918 01:13:37,740 --> 01:13:44,320 So, it's the engine [INAUDIBLE] up 919 01:13:44,320 --> 01:13:51,380 behind teleportation in the world of quantum computation. 920 01:13:51,380 --> 01:13:55,860 The exponential speedup of quantum computers 921 01:13:55,860 --> 01:14:05,400 versus classical computers in quantum algorithms 922 01:14:05,400 --> 01:14:07,840 is due to the entanglement, which 923 01:14:07,840 --> 01:14:11,430 we can put into a quantum system. 924 01:14:11,430 --> 01:14:13,840 And eventually, next week, you will 925 01:14:13,840 --> 01:14:17,540 see that if you have an atom interferometer 926 01:14:17,540 --> 01:14:28,520 with entangled states, we can operate it 927 01:14:28,520 --> 01:14:32,900 at a precision which is better than short noise. 928 01:14:36,320 --> 01:14:41,480 So in other words, you use laser light to measure something, 929 01:14:41,480 --> 01:14:43,990 and you're limited by the fundamental noise 930 01:14:43,990 --> 01:14:45,650 limit of classical physics. 931 01:14:45,650 --> 01:14:48,630 Now, you entangle your laser beam 932 01:14:48,630 --> 01:14:50,380 and you get higher precision out of it. 933 01:14:50,380 --> 01:14:53,790 So this shows that entanglement is a very special resource. 934 01:14:53,790 --> 01:14:56,070 It's very precious, very powerful 935 01:14:56,070 --> 01:14:58,240 and can extend what physics can do. 936 01:15:08,630 --> 01:15:09,770 So, let's define it. 937 01:15:25,960 --> 01:15:27,880 I'm not describing to you entanglement 938 01:15:27,880 --> 01:15:29,860 in the most generous situation. 939 01:15:29,860 --> 01:15:32,880 If you talk about many modes-- if you 940 01:15:32,880 --> 01:15:36,570 talk about not just pure states, but statistical operators-- 941 01:15:36,570 --> 01:15:39,580 it can become quite involved. 942 01:15:39,580 --> 01:15:43,060 I'd rather start with the simple situation, which 943 01:15:43,060 --> 01:15:47,370 for conceptual discussions is also the most important one. 944 01:15:47,370 --> 01:15:52,840 I restrict our focus now on two modes. 945 01:16:05,530 --> 01:16:13,260 I want to ask if you have two modes a-- actually, two modes 946 01:16:13,260 --> 01:16:15,350 means two subsystems here. 947 01:16:15,350 --> 01:16:20,630 If you have two subsystems a and b-- 948 01:16:20,630 --> 01:16:23,530 let me tell you what entanglement is not, 949 01:16:23,530 --> 01:16:26,220 and then I make it the definition. 950 01:16:26,220 --> 01:16:28,100 There wouldn't be anything special 951 01:16:28,100 --> 01:16:34,200 if you have a system which has two parts a and b. 952 01:16:34,200 --> 01:16:37,100 The global system is psi ab. 953 01:16:37,100 --> 01:16:40,610 But if your system would simply factorize 954 01:16:40,610 --> 01:16:44,750 into a wave function of subsystem a, 955 01:16:44,750 --> 01:16:50,270 direct product with subsystem b, and this here 956 01:16:50,270 --> 01:16:51,305 is the tensor product. 957 01:16:54,720 --> 01:16:56,780 Then, there would be nothing special going on 958 01:16:56,780 --> 01:16:59,697 between subsystem a and subsystem b-- 959 01:16:59,697 --> 01:17:00,780 and this is non-entangled. 960 01:17:03,660 --> 01:17:05,455 Everything else is entangled. 961 01:17:05,455 --> 01:17:06,580 So, this is our definition. 962 01:17:10,800 --> 01:17:16,740 When we have two separate systems a and b, 963 01:17:16,740 --> 01:17:21,840 we call this situation the total system bipartite. 964 01:17:21,840 --> 01:17:27,220 So, a bipartite state which has sort of half of it in system a, 965 01:17:27,220 --> 01:17:28,700 half of it in system b. 966 01:17:39,720 --> 01:17:42,320 We need those two different systems-- 967 01:17:42,320 --> 01:17:49,230 it's a composite system a plus b. 968 01:17:49,230 --> 01:18:13,140 And this state is entangled-- if and only if you cannot find two 969 01:18:13,140 --> 01:18:25,496 states psi a and psi b, such that the state can be 970 01:18:25,496 --> 01:18:25,995 factorized. 971 01:18:34,770 --> 01:18:37,950 We need a few examples to fully comprehend 972 01:18:37,950 --> 01:18:43,030 this discussion-- what systems a and b qualify. 973 01:18:43,030 --> 01:18:46,940 What does it mean if there does not exist any combination? 974 01:18:46,940 --> 01:18:48,820 So, let me give you some examples now, 975 01:18:48,820 --> 01:18:50,060 and then I think we stop. 976 01:18:55,170 --> 01:19:02,750 If you have the 0, 0 state, photons in two mode, 977 01:19:02,750 --> 01:19:09,360 we can factorize it-- and this is not entangled. 978 01:19:13,950 --> 01:19:23,600 If you have a state which is 0, 0 plus 1, 1-- well, 979 01:19:23,600 --> 01:19:27,020 you cannot write it as a product of one state times another 980 01:19:27,020 --> 01:19:27,570 state. 981 01:19:27,570 --> 01:19:28,319 This is entangled. 982 01:19:34,530 --> 01:19:51,290 Well, if you take the following state-- Is that entangled 983 01:19:51,290 --> 01:19:53,330 or not? 984 01:19:53,330 --> 01:19:57,970 I've written it as a sum of four states, 985 01:19:57,970 --> 01:20:01,520 but if you just stare at it for a split second, 986 01:20:01,520 --> 01:20:05,590 you realize you can just write it as a product of two states. 987 01:20:05,590 --> 01:20:11,430 It's just a product of 0, 1 with 0, 1. 988 01:20:11,430 --> 01:20:14,160 So, a state is not entangled if you can write it 989 01:20:14,160 --> 01:20:17,470 as a linear superposition of those states. 990 01:20:17,470 --> 01:20:20,140 The definition of entanglement is-- 991 01:20:20,140 --> 01:20:22,220 if you try hard and hard and hard, 992 01:20:22,220 --> 01:20:25,726 and there doesn't exist any product state decomposition, 993 01:20:25,726 --> 01:20:26,600 then it is entangled. 994 01:20:30,200 --> 01:20:31,045 So, that was easy. 995 01:20:34,000 --> 01:20:38,330 If you use a state like-- let me just leave the 1, 1 out. 996 01:20:46,550 --> 01:20:47,530 What about this one? 997 01:20:55,520 --> 01:21:01,700 Well, you can try as hard as you want-- 998 01:21:01,700 --> 01:21:04,560 you will not find the decomposition. 999 01:21:04,560 --> 01:21:07,350 So eventually, when I say you have 1000 01:21:07,350 --> 01:21:09,230 to try hard to find decomposition, 1001 01:21:09,230 --> 01:21:11,440 maybe what we want is we want to apply 1002 01:21:11,440 --> 01:21:13,970 some operator or some procedure to this state, 1003 01:21:13,970 --> 01:21:15,870 and get the answer, yes-no. 1004 01:21:15,870 --> 01:21:18,795 This naturally asks for question-- how in general can 1005 01:21:18,795 --> 01:21:21,000 we measure if a state is entangled or not? 1006 01:21:21,000 --> 01:21:23,920 But this is something we'll discuss next week. 1007 01:21:23,920 --> 01:21:27,700 So let me just conclude with the following. 1008 01:21:27,700 --> 01:21:33,060 I'm focusing here and for pretty much all of this cause when 1009 01:21:33,060 --> 01:21:35,195 we talk about entanglement, about pure states. 1010 01:21:39,180 --> 01:21:45,900 If you have decoherence-- if you have the transition 1011 01:21:45,900 --> 01:21:48,890 from pure states to density matrices-- 1012 01:21:48,890 --> 01:21:52,040 it gets much more involved to talk about entanglement 1013 01:21:52,040 --> 01:21:53,740 and measure entanglement. 1014 01:21:53,740 --> 01:21:56,800 But at least the basic definition 1015 01:21:56,800 --> 01:22:00,040 can still be maintained for statistical operators. 1016 01:22:00,040 --> 01:22:02,920 If you describe the system-- the total system by one 1017 01:22:02,920 --> 01:22:07,730 statistical operator-- the system is not entangled. 1018 01:22:07,730 --> 01:22:10,600 If the statistical operator can be broken up 1019 01:22:10,600 --> 01:22:14,160 into a direct tensor product of statistical operator 1020 01:22:14,160 --> 01:22:16,260 for system a and system b. 1021 01:22:16,260 --> 01:22:18,690 Otherwise, the system is entangled. 1022 01:22:18,690 --> 01:22:21,960 So the basic definition can be generalized 1023 01:22:21,960 --> 01:22:25,470 to density matrices, but a lot of things 1024 01:22:25,470 --> 01:22:28,410 become much, much more messy if you don't have pure states. 1025 01:22:32,420 --> 01:22:35,460 Any last-minute questions? 1026 01:22:35,460 --> 01:22:37,900 See you on Monday.