1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,238 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,238 --> 00:00:17,863 at ocw.mit.edu. 8 00:00:25,224 --> 00:00:26,890 PROFESSOR: So good afternoon, everybody. 9 00:00:29,940 --> 00:00:33,230 You see the topic for today's class 10 00:00:33,230 --> 00:00:36,730 on the screen-- entangled photons. 11 00:00:36,730 --> 00:00:39,220 Just to remind you, when we talked 12 00:00:39,220 --> 00:00:43,930 about single photons and Mach-Zender interferometers, 13 00:00:43,930 --> 00:00:49,690 we realized that when we have a nonlinear interferometer where 14 00:00:49,690 --> 00:00:56,710 one mode, you can say one beam, affects the phase shift 15 00:00:56,710 --> 00:00:59,850 in the interferometer for the other photon. 16 00:00:59,850 --> 00:01:02,780 Then we get a nonlinear situation, 17 00:01:02,780 --> 00:01:08,130 and we can create photon states which no longer factorize. 18 00:01:08,130 --> 00:01:10,570 So we have a system a, a system b, 19 00:01:10,570 --> 00:01:14,180 and we can no longer write down the total wave function 20 00:01:14,180 --> 00:01:17,650 in a wave function for system a times system b. 21 00:01:17,650 --> 00:01:19,660 And this is something very interesting, 22 00:01:19,660 --> 00:01:24,520 and this is what we feature in this section. 23 00:01:24,520 --> 00:01:36,450 So we finished the last class by defining entanglement. 24 00:01:36,450 --> 00:01:41,500 And just to remind you, we said something is entangled 25 00:01:41,500 --> 00:01:47,240 if it is impossible to write it in a product of two wave 26 00:01:47,240 --> 00:01:48,890 functions. 27 00:01:48,890 --> 00:01:54,750 So therefore, if you have some correlation between the two 28 00:01:54,750 --> 00:01:58,840 systems, then we call the two systems 29 00:01:58,840 --> 00:02:02,270 and the state of the system entangled. 30 00:02:02,270 --> 00:02:06,100 Now this is a definition which needs explanation, 31 00:02:06,100 --> 00:02:09,430 so we went through some examples. 32 00:02:09,430 --> 00:02:18,510 And we showed that certain states which on first sight 33 00:02:18,510 --> 00:02:20,760 look entangled are not entangled, 34 00:02:20,760 --> 00:02:22,780 because if you try harder, you find 35 00:02:22,780 --> 00:02:25,950 the way to factorize the state. 36 00:02:25,950 --> 00:02:28,960 So I want to continue now in explaining 37 00:02:28,960 --> 00:02:31,670 different aspects of the definition, 38 00:02:31,670 --> 00:02:36,140 in particular what it means to have a system which 39 00:02:36,140 --> 00:02:39,440 has two subsystems, a and b. 40 00:02:39,440 --> 00:02:42,870 Before I do that, do you have any questions up to that point? 41 00:03:01,230 --> 00:03:07,345 So we want to talk about some standard entangled state. 42 00:03:16,040 --> 00:03:24,990 And the most basic state is a single state. 43 00:03:37,910 --> 00:03:49,450 So if you have a state which is 0 1 minus 1 0 normalized, 44 00:03:49,450 --> 00:03:52,040 this is an Einstein-Podolsky-Rosen state. 45 00:03:56,480 --> 00:03:59,920 This needs some explanation. 46 00:03:59,920 --> 00:04:02,410 We have often encountered, in physics, 47 00:04:02,410 --> 00:04:06,230 states which are simply a superposition of if you 48 00:04:06,230 --> 00:04:14,820 interpret 0 1 as spin up or spin down, 49 00:04:14,820 --> 00:04:16,430 you encounter that quite often. 50 00:04:16,430 --> 00:04:18,649 And I want to explain to you now what 51 00:04:18,649 --> 00:04:25,730 is not an entangled state for the reasons of using 52 00:04:25,730 --> 00:04:28,920 entanglement as a resource. 53 00:04:28,920 --> 00:04:33,370 So first of all, I want to point out 54 00:04:33,370 --> 00:04:43,430 this state is not a single photon, 55 00:04:43,430 --> 00:04:47,420 because we have entangled here two qubits. 56 00:04:47,420 --> 00:04:48,990 Let me just contrast it. 57 00:04:48,990 --> 00:05:01,360 If you have a single photon after beam splitter, 58 00:05:01,360 --> 00:05:03,890 the single photon after beam splitter 59 00:05:03,890 --> 00:05:13,430 is in a superposition of mode a, mode b, 0 1 minus 1 0 60 00:05:13,430 --> 00:05:16,270 divided by square root 2. 61 00:05:16,270 --> 00:05:22,040 But now 0 is the vacuum. 62 00:05:25,720 --> 00:05:29,470 So we have not a system which can 63 00:05:29,470 --> 00:05:34,230 be decomposed into two partial systems a and b. 64 00:05:34,230 --> 00:05:38,310 You may even separate the system and then manipulate 65 00:05:38,310 --> 00:05:42,240 your single qubits individually by putting phase shifts on 66 00:05:42,240 --> 00:05:44,790 and doing other operations. 67 00:05:44,790 --> 00:05:48,240 If you have this state which is a superposition between having 68 00:05:48,240 --> 00:05:51,930 a photon, not having a vacuum, and having a vacuum state, 69 00:05:51,930 --> 00:05:54,850 you cannot-- the vacuum state itself is not a separate 70 00:05:54,850 --> 00:05:55,350 system. 71 00:05:55,350 --> 00:05:58,260 You cannot take the vacuum and perform operations 72 00:05:58,260 --> 00:06:00,400 on the vacuum. 73 00:06:00,400 --> 00:06:05,060 So I know it has been confusing for me 74 00:06:05,060 --> 00:06:09,040 when I heard about it for the first time. 75 00:06:09,040 --> 00:06:12,490 We have, so to speak, here an entangled mode, 76 00:06:12,490 --> 00:06:13,940 but not an entangled state. 77 00:06:13,940 --> 00:06:15,370 It's a singlet state. 78 00:06:15,370 --> 00:06:17,810 It has some aspects of entanglement, 79 00:06:17,810 --> 00:06:21,355 but it is not the entanglement we have defined is a resource. 80 00:06:27,030 --> 00:06:32,480 So let me just repeat-- this may be an entangled mode, 81 00:06:32,480 --> 00:06:36,340 but to be in an entangled state requires systems 82 00:06:36,340 --> 00:06:38,425 where you have two parts which you can separate. 83 00:06:47,480 --> 00:06:50,730 The second comment is we have situations 84 00:06:50,730 --> 00:06:58,390 where we have two different parts, 85 00:06:58,390 --> 00:07:01,050 but they can't be physically separated. 86 00:07:01,050 --> 00:07:04,170 So part of our definition of entanglement as a resource 87 00:07:04,170 --> 00:07:22,660 is that the state must be of two physically distinct systems. 88 00:07:28,020 --> 00:07:31,010 And we want to have parts which can 89 00:07:31,010 --> 00:07:46,880 be separately addressed, manipulated, and measured. 90 00:07:54,170 --> 00:07:58,435 In other words, if you read out one part of an EPR pair, 91 00:07:58,435 --> 00:08:01,410 the other part still exists. 92 00:08:01,410 --> 00:08:04,640 So that's sort of what we want. 93 00:08:04,640 --> 00:08:09,040 Let me illustrate that with examples. 94 00:08:12,500 --> 00:08:20,140 For instance, in the helium atom, you have two electrons. 95 00:08:20,140 --> 00:08:36,559 And the ground state-- if I use spin notation, is a singlet. 96 00:08:43,120 --> 00:08:45,570 So that's this. 97 00:08:45,570 --> 00:08:47,300 But does the ground state of helium 98 00:08:47,300 --> 00:08:49,740 fulfill our definition of an entangled state? 99 00:08:54,520 --> 00:08:58,690 Well, if you find a way to switch off the Coulomb 100 00:08:58,690 --> 00:09:01,990 potential, and then the electrons separate 101 00:09:01,990 --> 00:09:06,080 from each other, they still maintain their spin singlet 102 00:09:06,080 --> 00:09:10,640 character, then you can take them, measure them, 103 00:09:10,640 --> 00:09:12,920 take measurements, manipulate them. 104 00:09:12,920 --> 00:09:15,420 But since nobody has come up with a good idea 105 00:09:15,420 --> 00:09:19,010 how to switch off the Coulomb potential in an atom, 106 00:09:19,010 --> 00:09:21,550 you can never separate the states. 107 00:09:21,550 --> 00:09:27,920 And it is physically impossible to address this 108 00:09:27,920 --> 00:09:30,860 so the two electrons see them separately and such. 109 00:09:34,750 --> 00:09:46,640 So this state is not entangled, because it's not-- 110 00:09:46,640 --> 00:09:48,910 the two systems cannot be separated. 111 00:09:52,970 --> 00:09:55,850 What is usually a good choice for entanglement-- 112 00:09:55,850 --> 00:09:58,460 and this is why we discuss it here with photons-- 113 00:09:58,460 --> 00:10:00,010 is you have photon states. 114 00:10:00,010 --> 00:10:02,140 Photons always fly away. 115 00:10:02,140 --> 00:10:04,290 Photons are in a certain superposition state. 116 00:10:04,290 --> 00:10:09,130 You can always separate them, and take them individually. 117 00:10:09,130 --> 00:10:13,250 So let's assume we have two photons, two modes. 118 00:10:13,250 --> 00:10:21,020 We have two photons-- actually, the two photons can 119 00:10:21,020 --> 00:10:22,920 be in one spatial mode, but we are now 120 00:10:22,920 --> 00:10:25,850 playing with the polarization-- horizontal, vertical. 121 00:10:30,710 --> 00:10:35,810 So if we have a state with horizontal- vertical 122 00:10:35,810 --> 00:10:40,400 polarization, this is a nice entangled state of two photons. 123 00:10:40,400 --> 00:10:44,510 So what did means is that we have photon each. 124 00:10:44,510 --> 00:10:46,060 One is vertical. 125 00:10:46,060 --> 00:10:47,970 And one is horizontal. 126 00:10:47,970 --> 00:10:50,270 But you don't know which one. 127 00:10:50,270 --> 00:10:52,200 If you would just look at one photon, 128 00:10:52,200 --> 00:10:55,030 it would be 50% horizontal, 50% vertical. 129 00:10:55,030 --> 00:10:58,040 It would be completely unpolarized, 130 00:10:58,040 --> 00:11:01,220 which would be a random state, which would require 131 00:11:01,220 --> 00:11:03,630 and density matrix for its description. 132 00:11:03,630 --> 00:11:06,950 But if one photon is horizontal, the other one 133 00:11:06,950 --> 00:11:08,840 is vertical and vice versa. 134 00:11:08,840 --> 00:11:12,350 So it's a pure state, but all the pureness of the state 135 00:11:12,350 --> 00:11:16,760 comes from the entanglement and not 136 00:11:16,760 --> 00:11:18,790 from what one photon does by itself. 137 00:11:21,710 --> 00:11:24,120 So these are two photons in one mode, 138 00:11:24,120 --> 00:11:27,170 and they are polarization entangled. 139 00:11:27,170 --> 00:11:38,140 Or this brings us back to our dual [? Rail ?] single photon 140 00:11:38,140 --> 00:11:40,200 states. 141 00:11:40,200 --> 00:11:47,330 We have two photons, two qubits and each is in two modes. 142 00:12:22,650 --> 00:12:26,620 So our 0 state-- and just to make sure 143 00:12:26,620 --> 00:12:32,710 that you do not confuse it with no photon, our logic 0 state, 144 00:12:32,710 --> 00:12:39,550 so L means logic here-- is that the photon 145 00:12:39,550 --> 00:12:45,900 is in the second mode, and the 1 state of our logic state 146 00:12:45,900 --> 00:12:48,760 means that the single photon is in the other mode. 147 00:13:00,310 --> 00:13:11,490 So we can now have an entangled state, 148 00:13:11,490 --> 00:13:17,445 which is now this 0 1 1 0 state. 149 00:13:20,810 --> 00:13:31,850 But these are now logic states, which 150 00:13:31,850 --> 00:13:34,930 means that the 0 has one photon. 151 00:13:34,930 --> 00:13:37,180 It has a photon in one of the modes, 152 00:13:37,180 --> 00:13:39,560 and the one has a photon in the other mode. 153 00:13:39,560 --> 00:13:42,440 So each state here has two photons, 154 00:13:42,440 --> 00:13:48,370 but then the two photons have-- 0 1 155 00:13:48,370 --> 00:13:52,845 and 1 0 are switched in the two parts of the wave function. 156 00:13:55,920 --> 00:14:07,690 And we actually saw in the last unit 157 00:14:07,690 --> 00:14:13,500 how Kerr medium and an interferometer 158 00:14:13,500 --> 00:14:14,770 can generate this state. 159 00:14:31,790 --> 00:14:33,695 So yes? 160 00:14:33,695 --> 00:14:34,570 AUDIENCE: [INAUDIBLE] 161 00:14:37,299 --> 00:14:38,840 PROFESSOR: There are two such states. 162 00:14:38,840 --> 00:14:43,310 I mean, I will later tell you what the four famous Bell 163 00:14:43,310 --> 00:14:44,100 states are. 164 00:14:44,100 --> 00:14:45,850 There is one which has a plus sign and one 165 00:14:45,850 --> 00:14:47,780 which has a minus sign. 166 00:14:47,780 --> 00:14:51,470 So when we talk about spins in singlet state, 167 00:14:51,470 --> 00:14:53,610 it's often more natural the minus sign. 168 00:14:53,610 --> 00:14:57,100 Here what naturally emerged was the plus sign, 169 00:14:57,100 --> 00:14:59,320 but they are both sort of Bell states 170 00:14:59,320 --> 00:15:04,760 and therefore, they have what Einstein-Podolsky-Rosen 171 00:15:04,760 --> 00:15:06,230 introduced into it. 172 00:15:06,230 --> 00:15:08,360 So tolerate both signs. 173 00:15:08,360 --> 00:15:11,370 They are two different states, but for the purpose 174 00:15:11,370 --> 00:15:13,759 of the current discussion, they have the same property-- 175 00:15:13,759 --> 00:15:14,925 they're maximally entangled. 176 00:15:17,674 --> 00:15:18,340 Other questions? 177 00:15:22,030 --> 00:15:31,930 OK, so let me point out that the properties of entangled states 178 00:15:31,930 --> 00:15:34,090 always involve two qualities. 179 00:15:34,090 --> 00:15:40,380 One is the non-local character, because we have correlation 180 00:15:40,380 --> 00:15:44,700 between two subsystems which may be together when they interact, 181 00:15:44,700 --> 00:15:46,610 but then they can be separated. 182 00:15:46,610 --> 00:15:50,010 So we have correlations which happen between the two 183 00:15:50,010 --> 00:15:53,410 systems which are a distance apart. 184 00:15:53,410 --> 00:15:57,300 And we later come back when we talk about Gauss inequalities 185 00:15:57,300 --> 00:16:02,290 and that we know that physics has non-local aspects. 186 00:16:02,290 --> 00:16:06,540 And secondly, if you can separate two parts 187 00:16:06,540 --> 00:16:08,530 and they interact with the environment, 188 00:16:08,530 --> 00:16:11,350 the environment may interact with them 189 00:16:11,350 --> 00:16:13,670 differently in those two different parts. 190 00:16:13,670 --> 00:16:19,687 And therefore, entangled states are always regarded as fragile 191 00:16:19,687 --> 00:16:20,520 against decoherence. 192 00:16:26,890 --> 00:16:31,010 And it's a technical challenge how do you find states? 193 00:16:31,010 --> 00:16:35,550 How do you implement entangled states which are robust? 194 00:16:35,550 --> 00:16:37,110 Just to give you one example, if you 195 00:16:37,110 --> 00:16:40,280 have an entangled state which is based on electron spin, 196 00:16:40,280 --> 00:16:43,320 you may be more than 1,000 times more sensitive 197 00:16:43,320 --> 00:16:46,550 to magnetic field fluctuations in your laboratory 198 00:16:46,550 --> 00:16:50,460 than if you have qubits which are entangled states which 199 00:16:50,460 --> 00:16:52,870 are based on nuclear spin. 200 00:16:52,870 --> 00:16:57,150 So that's a big research area to find states which 201 00:16:57,150 --> 00:17:00,880 are less sensitive, or even you immune against decoherence. 202 00:17:08,530 --> 00:17:19,089 OK, so I've mentioned to you that entangled states are 203 00:17:19,089 --> 00:17:22,300 states which you cannot factorize. 204 00:17:22,300 --> 00:17:26,410 But now we can sort of start playing with that definition. 205 00:17:26,410 --> 00:17:29,480 And we say, OK, if you have an entangled state which 206 00:17:29,480 --> 00:17:35,630 is up-down plus down-up, but now the contribution of down-up 207 00:17:35,630 --> 00:17:38,180 has only a tiny, little amplitude. 208 00:17:38,180 --> 00:17:40,780 So it's almost a pure state which 209 00:17:40,780 --> 00:17:44,680 can be factorized with a little bit of an extra configuration 210 00:17:44,680 --> 00:17:47,039 which prevents us from factorizing that. 211 00:17:47,039 --> 00:17:49,080 I mean, that doesn't look like good entanglement. 212 00:17:49,080 --> 00:17:50,660 It looks at the whole entanglement 213 00:17:50,660 --> 00:17:55,200 of the state depends on very small admixture 214 00:17:55,200 --> 00:17:56,990 to the wave function. 215 00:17:56,990 --> 00:18:00,010 And so what we want to address now 216 00:18:00,010 --> 00:18:02,650 is how can we quantify that? 217 00:18:02,650 --> 00:18:05,110 How can we look at this data and say, hey, 218 00:18:05,110 --> 00:18:08,210 this is sort of not a strongly entangled state. 219 00:18:08,210 --> 00:18:12,230 It has only a weak amount of entanglement. 220 00:18:12,230 --> 00:18:15,160 So let's not forget, entanglement is a resource. 221 00:18:15,160 --> 00:18:17,650 Entanglement allows you to do teleportation. 222 00:18:17,650 --> 00:18:20,660 Entanglement allows you to do more precise measurements. 223 00:18:20,660 --> 00:18:25,050 And what I want to convey you, if a state is only 224 00:18:25,050 --> 00:18:27,630 weakly entangled, it doesn't help 225 00:18:27,630 --> 00:18:34,610 you much to achieve precision beyond the standard limit, 226 00:18:34,610 --> 00:18:36,510 effective teleportation, and such things. 227 00:18:39,270 --> 00:18:45,180 Before I introduce several measures for entanglement, 228 00:18:45,180 --> 00:18:47,863 let me talk about entanglement purification. 229 00:18:52,030 --> 00:18:56,940 It's a very nice subject which tells you 230 00:18:56,940 --> 00:18:59,570 that if you have weakly entangled state, 231 00:18:59,570 --> 00:19:01,790 you can make them more entangled. 232 00:19:01,790 --> 00:19:03,520 And actually, the effort you have 233 00:19:03,520 --> 00:19:06,870 to spend to make a weakly entangled state more entangled 234 00:19:06,870 --> 00:19:09,850 can actually act as a measurement how entangled 235 00:19:09,850 --> 00:19:11,650 was your state in the first place. 236 00:19:11,650 --> 00:19:14,320 So the purification is introducing us 237 00:19:14,320 --> 00:19:16,260 with one measurement for entanglement, 238 00:19:16,260 --> 00:19:21,700 as I just said, but it also gives you an idea how 239 00:19:21,700 --> 00:19:23,250 quantum state can be manipulated. 240 00:19:28,110 --> 00:19:30,720 Purification is also the first example 241 00:19:30,720 --> 00:19:36,590 we encounter in this course for new insight 242 00:19:36,590 --> 00:19:38,830 into quantum physics. 243 00:19:38,830 --> 00:19:41,970 A lot of people thought quantum physics, at least 244 00:19:41,970 --> 00:19:44,040 non-relativistic quantum physics that 245 00:19:44,040 --> 00:19:48,050 was invented in the '20s in the last century, 246 00:19:48,050 --> 00:19:50,490 and by now we've understood it all. 247 00:19:50,490 --> 00:19:54,410 But there was an aspect of quantum physics, 248 00:19:54,410 --> 00:19:56,200 which I think nobody understood. 249 00:19:56,200 --> 00:20:01,080 And this is when you have a quantum state, which 250 00:20:01,080 --> 00:20:04,920 may decohere, a quantum state which may no longer be pure, 251 00:20:04,920 --> 00:20:10,140 that you have ways to error correction. 252 00:20:10,140 --> 00:20:13,980 You have ways to get the pure quantum state back. 253 00:20:13,980 --> 00:20:16,730 And ultimately, I mean, in the early days 254 00:20:16,730 --> 00:20:18,170 when I learned quantum physics, I 255 00:20:18,170 --> 00:20:20,640 thought if you have a quantum state, that's nice. 256 00:20:20,640 --> 00:20:23,070 But if the quantum-ness has decayed away, 257 00:20:23,070 --> 00:20:25,030 you can't get it back. 258 00:20:25,030 --> 00:20:30,270 But this is something we learn now from the new methods 259 00:20:30,270 --> 00:20:32,910 and new approach in quantum information science 260 00:20:32,910 --> 00:20:35,100 that you can do quantum error correction. 261 00:20:35,100 --> 00:20:37,330 You can have a stage which has decohered 262 00:20:37,330 --> 00:20:39,320 and you can get back to it. 263 00:20:39,320 --> 00:20:43,170 And there is something which-- it's not yet 264 00:20:43,170 --> 00:20:46,450 that which we're discussing today-- but in purification, 265 00:20:46,450 --> 00:20:49,250 you have states which have inferior entanglement, 266 00:20:49,250 --> 00:20:52,790 but you don't need to get stuck with that. 267 00:20:52,790 --> 00:20:56,630 You can take several of the purely entangled states 268 00:20:56,630 --> 00:21:00,354 and create the maximally entangled state out of it. 269 00:21:00,354 --> 00:21:01,770 So that's what I want to show you. 270 00:21:01,770 --> 00:21:04,180 So you should look at it as one example 271 00:21:04,180 --> 00:21:07,360 that wow, it's really cool how we 272 00:21:07,360 --> 00:21:12,150 can have quantum states with inferior quality. 273 00:21:12,150 --> 00:21:15,350 And by doing quantum operation on those states, 274 00:21:15,350 --> 00:21:18,450 we get something which is more entangled, 275 00:21:18,450 --> 00:21:21,405 and therefore, if that's our purpose, has a higher quality. 276 00:21:29,330 --> 00:21:39,100 So to introduce purification, I'm 277 00:21:39,100 --> 00:21:48,920 simply mentioning what are-- for two qubits-- the standard, 278 00:21:48,920 --> 00:21:50,620 maximally entangled states. 279 00:21:55,670 --> 00:21:58,240 In 10 minutes or so, we talk about how do we 280 00:21:58,240 --> 00:22:00,470 measure entanglement, and indeed, those states 281 00:22:00,470 --> 00:22:03,600 will come out as being maximally entangled. 282 00:22:03,600 --> 00:22:06,480 But you already get the idea. 283 00:22:06,480 --> 00:22:09,860 Maximally entangled means they're not 284 00:22:09,860 --> 00:22:13,490 factorizable and not just by a small margin. 285 00:22:13,490 --> 00:22:16,930 They may be two equal parts of-- you always 286 00:22:16,930 --> 00:22:20,420 need two equal parts of 0 1 and 1 0, 287 00:22:20,420 --> 00:22:22,670 and this state has maximally entangled. 288 00:22:22,670 --> 00:22:26,220 It's maximally non-factorizable. 289 00:22:26,220 --> 00:22:30,710 So the four states which are actually the Bell 290 00:22:30,710 --> 00:22:37,440 states-- the famous Bell basis-- are 0 1 plus minus 1 0 291 00:22:37,440 --> 00:22:43,440 and 0 0 plus minus 1 1. 292 00:22:43,440 --> 00:22:48,140 Just to remind you if if you have two systems, each of them 293 00:22:48,140 --> 00:22:49,610 has two states. 294 00:22:49,610 --> 00:22:52,650 You can think of spin-- spin up, down for particle one, 295 00:22:52,650 --> 00:22:54,610 spin up down for particle two. 296 00:22:54,610 --> 00:22:57,040 That Hilbert space is four-dimensional. 297 00:22:57,040 --> 00:22:59,090 So you need a four-dimensional basis. 298 00:22:59,090 --> 00:23:03,430 And the trivial basis is up up, down down, up down, down up. 299 00:23:03,430 --> 00:23:10,780 But what is relevant for entangled states, 300 00:23:10,780 --> 00:23:16,580 we often use the basis of Bell states. 301 00:23:16,580 --> 00:23:19,980 And this is a new basis which spans 302 00:23:19,980 --> 00:23:21,910 the four-dimensional Hilbert space, 303 00:23:21,910 --> 00:23:26,610 but each of those bases function is maximally entangled. 304 00:23:26,610 --> 00:23:31,500 OK we should correctly normalize them. 305 00:23:31,500 --> 00:23:36,610 And that state is often called psi plus minus. 306 00:23:36,610 --> 00:23:39,625 And this here is phi plus minus. 307 00:23:44,010 --> 00:23:51,030 OK, so now let's get a purely entangled state. 308 00:23:51,030 --> 00:23:57,040 And let this state 0 0 plus 1 1. 309 00:23:57,040 --> 00:24:00,110 But we will have an example where 310 00:24:00,110 --> 00:24:01,990 b may be very, very small. 311 00:24:01,990 --> 00:24:11,020 So those states, this state is entangled for all choices 312 00:24:11,020 --> 00:24:13,410 of a and b. 313 00:24:13,410 --> 00:24:18,550 So the question is now how can we take such an arbitrary state 314 00:24:18,550 --> 00:24:22,820 where a and b-- one of them may be small, and create 315 00:24:22,820 --> 00:24:24,070 a standard Bell state? 316 00:24:44,570 --> 00:24:46,270 So what we have to assume for that 317 00:24:46,270 --> 00:24:49,960 is that we have a large supply of such states. 318 00:24:52,960 --> 00:24:59,945 And so let's assume we have large supply, identical copies. 319 00:25:05,590 --> 00:25:07,650 And now we want to take two such copies. 320 00:25:17,849 --> 00:25:19,890 And what I want to outline you is the following-- 321 00:25:19,890 --> 00:25:22,510 you take sort of two of your copies, 322 00:25:22,510 --> 00:25:24,940 and you do a measurement. 323 00:25:24,940 --> 00:25:31,400 And I will tell you what kind of qubit operation we need, 324 00:25:31,400 --> 00:25:33,730 what kind of measurement we perform. 325 00:25:33,730 --> 00:25:38,120 And then when the outcome of the measurement is such and such, 326 00:25:38,120 --> 00:25:42,900 you say-- OK let me be specific. 327 00:25:42,900 --> 00:25:45,910 So if you take two copies, we have a total 328 00:25:45,910 --> 00:25:50,080 of two states with two photons each. 329 00:25:50,080 --> 00:25:53,330 And now we perform a quantum operation 330 00:25:53,330 --> 00:25:56,790 on two of the photons, and the other two photons we leave 331 00:25:56,790 --> 00:25:58,470 untouched. 332 00:25:58,470 --> 00:26:02,580 So now it depends when the measurement of the two photons 333 00:26:02,580 --> 00:26:06,320 has a good outcome, we know those other two 334 00:26:06,320 --> 00:26:07,930 photons are maximally entangled. 335 00:26:07,930 --> 00:26:09,360 They're in a Bell state. 336 00:26:09,360 --> 00:26:12,310 If the outcome of the measurement is bad, 337 00:26:12,310 --> 00:26:15,150 it tells us the two photons are not entangled, 338 00:26:15,150 --> 00:26:17,210 and we throw them away. 339 00:26:17,210 --> 00:26:19,850 So therefore, we have a finite probability 340 00:26:19,850 --> 00:26:27,070 by performing measurements that a pair of our sample 341 00:26:27,070 --> 00:26:29,560 state with the coefficient a and b 342 00:26:29,560 --> 00:26:32,920 will result into a maximally entangled state-- in a Bell 343 00:26:32,920 --> 00:26:33,700 state. 344 00:26:33,700 --> 00:26:36,446 And I want to describe now what is the protocol, what 345 00:26:36,446 --> 00:26:39,720 is the procedure to implement that. 346 00:26:39,720 --> 00:26:42,620 And what we will find-- and this is 347 00:26:42,620 --> 00:26:47,120 what I think you should expect here-- if the initial state has 348 00:26:47,120 --> 00:26:51,600 very bad entanglement in the sense that b is very small, 349 00:26:51,600 --> 00:26:55,760 we will need many, many attempts doing many measurements 350 00:26:55,760 --> 00:27:00,450 on our pair of states before we produce a Bell state. 351 00:27:00,450 --> 00:27:03,420 It's probabilistic, but the probability 352 00:27:03,420 --> 00:27:08,390 to succeed in preparing a Bell state will depend on-- we 353 00:27:08,390 --> 00:27:11,447 will see-- the product of a and b. 354 00:27:11,447 --> 00:27:12,030 Any questions? 355 00:27:12,030 --> 00:27:12,700 Yes. 356 00:27:12,700 --> 00:27:17,420 AUDIENCE: Does the assumption of large supply of the state 357 00:27:17,420 --> 00:27:19,592 violate no-cloning in any sense? 358 00:27:24,240 --> 00:27:25,860 PROFESSOR: The question is, does it 359 00:27:25,860 --> 00:27:27,840 violate the no-cloning theorem? 360 00:27:27,840 --> 00:27:29,470 No, it doesn't, otherwise I would not 361 00:27:29,470 --> 00:27:31,886 say we have a large supply, because the no-cloning theorem 362 00:27:31,886 --> 00:27:32,690 is absolute. 363 00:27:32,690 --> 00:27:36,000 But it simply means we cannot have one state, 364 00:27:36,000 --> 00:27:40,280 and then clone and clone more and more copies. 365 00:27:40,280 --> 00:27:41,700 Let me be specific. 366 00:27:41,700 --> 00:27:43,850 If you have an experiment which produces 367 00:27:43,850 --> 00:27:46,190 a certain superposition state, you 368 00:27:46,190 --> 00:27:49,510 can just push the button on your experiment many times, 369 00:27:49,510 --> 00:27:52,340 and produce, in the identical create, 370 00:27:52,340 --> 00:27:53,990 as many states as you want. 371 00:27:53,990 --> 00:27:55,890 If you have spin-up state-- well, 372 00:27:55,890 --> 00:28:01,310 this is now for two photons, but if you have a spin-up state, 373 00:28:01,310 --> 00:28:03,760 you can make as many copies as you 374 00:28:03,760 --> 00:28:09,690 want of the state which has been rotated by a certain angle. 375 00:28:09,690 --> 00:28:12,510 So therefore, in state preparation, 376 00:28:12,510 --> 00:28:14,670 by going through the exact procedure, 377 00:28:14,670 --> 00:28:18,810 we can just create as many copies of a state we want. 378 00:28:18,810 --> 00:28:21,410 The no-cloning [INAUDIBLE] meant the following-- 379 00:28:21,410 --> 00:28:24,960 I give you one state which may be a spin which 380 00:28:24,960 --> 00:28:26,990 has been rotated at a certain angle. 381 00:28:26,990 --> 00:28:29,290 You know nothing about the state. 382 00:28:29,290 --> 00:28:32,540 And now you should try to make a copy out of it. 383 00:28:32,540 --> 00:28:37,340 And the answer is you can't, because any measurement you do 384 00:28:37,340 --> 00:28:40,390 is-- if the particle were spin up 385 00:28:40,390 --> 00:28:42,570 and you would measure spin up or speed down, 386 00:28:42,570 --> 00:28:44,070 you could say I got spin up. 387 00:28:44,070 --> 00:28:47,560 Now I produce 10 spin up particles. 388 00:28:47,560 --> 00:28:49,570 But you don't know along which axis 389 00:28:49,570 --> 00:28:51,400 the spin has been prepared. 390 00:28:51,400 --> 00:28:57,280 So unless you know which axis the spin has been oriented, 391 00:28:57,280 --> 00:29:05,457 and if you choose another axis, you have irreversibly lost 392 00:29:05,457 --> 00:29:07,040 information which cannot be retrieved. 393 00:29:10,990 --> 00:29:12,890 I don't know if it helps you, but if you 394 00:29:12,890 --> 00:29:16,400 have a certain state, and you're going to measure it 395 00:29:16,400 --> 00:29:19,890 without destroying it, you need a quantum non-demolition 396 00:29:19,890 --> 00:29:20,980 measurement. 397 00:29:20,980 --> 00:29:23,430 If you're in energy eigenstate, you can measure energy. 398 00:29:23,430 --> 00:29:26,170 If you're in a spin eigenstate which points along Z, 399 00:29:26,170 --> 00:29:31,110 you can measure the direction of the spin in the Z direction 400 00:29:31,110 --> 00:29:32,690 without destroying it. 401 00:29:32,690 --> 00:29:35,690 So if you can do a quantum non-demolition measurement 402 00:29:35,690 --> 00:29:38,070 on a state, you could clone it. 403 00:29:38,070 --> 00:29:43,040 But that violates the assumption that if I give you an arbitrary 404 00:29:43,040 --> 00:29:46,890 state, you do not know by definition what kind 405 00:29:46,890 --> 00:29:48,950 of measurement is a non-demolition measurement. 406 00:29:48,950 --> 00:29:51,690 You just take your chance, you try 407 00:29:51,690 --> 00:29:55,440 to take a Stern-Gerlach experiment, separate the spins 408 00:29:55,440 --> 00:29:57,980 in the Z component, and then it turns out 409 00:29:57,980 --> 00:30:00,680 I gave you a state which is polarized along x. 410 00:30:00,680 --> 00:30:07,224 So that's a subtle, but important difference. 411 00:30:07,224 --> 00:30:07,890 Other questions? 412 00:30:10,490 --> 00:30:21,680 So we take two copies and let's bring in Alice and Bob. 413 00:30:21,680 --> 00:30:28,050 So the first photon we associate with Alice. 414 00:30:28,050 --> 00:30:32,640 And the second photon is associated with Bob. 415 00:30:38,250 --> 00:30:44,120 So if you take two copies-- we have now 416 00:30:44,120 --> 00:30:47,290 is an Hilbert space, a four-dimensional Hilbert 417 00:30:47,290 --> 00:30:50,180 space, a direct product. 418 00:30:50,180 --> 00:30:55,180 And if you just take the state and calculate 419 00:30:55,180 --> 00:31:10,940 the direct product, you get four terms-- 0 0 1 1, 0 0 1 1, 1 1 420 00:31:10,940 --> 00:31:15,820 0 0, and 1 1 1 1. 421 00:31:15,820 --> 00:31:22,980 And the coefficients are given here. 422 00:31:26,580 --> 00:31:29,510 Let me just underline that it if you 423 00:31:29,510 --> 00:31:33,170 think we have some state with some entanglement, 424 00:31:33,170 --> 00:31:37,180 and we separate the system, one goes to Alice 425 00:31:37,180 --> 00:31:39,910 and one goes to Bob. 426 00:31:39,910 --> 00:31:45,969 So the one which Alice has is the first part of it 427 00:31:45,969 --> 00:31:46,510 those states. 428 00:31:53,110 --> 00:31:56,380 And Bob has the other part. 429 00:31:56,380 --> 00:32:09,060 So the protocol is now that Alice and Bob first, they're 430 00:32:09,060 --> 00:32:11,040 not doing a measurement. 431 00:32:11,040 --> 00:32:13,080 They're not reading out the system. 432 00:32:13,080 --> 00:32:15,780 They perform a unitary operation. 433 00:32:15,780 --> 00:32:17,690 And what Alice and Bob have formed 434 00:32:17,690 --> 00:32:22,940 is the controlled NOT operation on the two qubits. 435 00:32:25,800 --> 00:32:27,950 Let me just write it down and then explain it. 436 00:32:27,950 --> 00:32:37,105 Perform what is called the controlled NOT or CNOT gate. 437 00:32:48,820 --> 00:32:51,480 And I've explained before in the last section how 438 00:32:51,480 --> 00:32:54,070 the controlled NOT can be performed 439 00:32:54,070 --> 00:32:57,100 using a non-linear Mach-Zehnder interferometer. 440 00:32:57,100 --> 00:32:59,000 So Bob and Alice both run their two photons 441 00:32:59,000 --> 00:33:01,000 with the non-linear Mach-Zehnder interferometer. 442 00:33:07,480 --> 00:33:09,280 So what is a controlled NOT? 443 00:33:09,280 --> 00:33:11,470 Let me remind you. 444 00:33:11,470 --> 00:33:17,260 If you have two qubits, the first one is the control. 445 00:33:17,260 --> 00:33:20,840 If the control is 1, you flip the second one. 446 00:33:20,840 --> 00:33:25,020 If the control is 0, you do nothing to the second one. 447 00:33:25,020 --> 00:33:33,993 So the controlled CNOT transforms. 448 00:33:41,280 --> 00:33:46,650 So since the first qubit is the control qubit, out 449 00:33:46,650 --> 00:33:50,410 of those four combinations, the controlled NOT only 450 00:33:50,410 --> 00:33:58,190 does something if Alice's controls and Bob's controls 451 00:33:58,190 --> 00:34:02,800 is 1, and then the second bit is flipped. 452 00:34:02,800 --> 00:34:06,310 If Alice's and Bob's controls are 0, 453 00:34:06,310 --> 00:34:08,870 they do nothing to the second bit. 454 00:34:08,870 --> 00:34:14,699 So therefore, the 1 1 0 is transformed into 1 1 1, 455 00:34:14,699 --> 00:34:20,329 and the 1 1 1 1 is transformed into 1 1 0 0. 456 00:34:30,889 --> 00:34:35,820 So that's the first operation. 457 00:34:35,820 --> 00:34:37,130 Let me just indicate it. 458 00:34:42,650 --> 00:34:47,270 So what has happened here, the 0 0 has been flipped into 1 1 459 00:34:47,270 --> 00:34:50,650 and the 1 1 has been flipped into 0 0. 460 00:34:50,650 --> 00:34:51,949 That's the first step. 461 00:34:51,949 --> 00:34:55,150 The second step is that now Alice and Bob 462 00:34:55,150 --> 00:34:56,925 measure their target qubit. 463 00:35:09,410 --> 00:35:10,920 What does that mean? 464 00:35:10,920 --> 00:35:13,050 We have a controlled NOT operation. 465 00:35:13,050 --> 00:35:15,470 In the controlled NOT operation, we 466 00:35:15,470 --> 00:35:18,470 have the first one is the control qubit. 467 00:35:18,470 --> 00:35:22,200 The second one is the target of the operation. 468 00:35:22,200 --> 00:35:27,190 So in our, the way how we write it, 469 00:35:27,190 --> 00:35:31,710 Alice's target is the third in line, 470 00:35:31,710 --> 00:35:34,430 and Bob is the fourth in line. 471 00:35:34,430 --> 00:35:39,980 So now Alice and Bob measure the target qubits. 472 00:35:39,980 --> 00:35:41,130 Let me just be specific. 473 00:35:41,130 --> 00:35:43,920 So Alice has number one and number three. 474 00:35:43,920 --> 00:35:47,200 Bob has number two and four, and the target qubits 475 00:35:47,200 --> 00:35:48,460 are the third and fourth. 476 00:36:02,260 --> 00:36:08,470 So what is the probability that Alice and Bob measure both 1 1, 477 00:36:08,470 --> 00:36:13,810 that they both find the target qubit of 1? 478 00:36:13,810 --> 00:36:18,050 Well, it is this one and it is this one 479 00:36:18,050 --> 00:36:20,980 where the controlled NOT has flipped it. 480 00:36:20,980 --> 00:36:23,940 So one has the probability a squared b squared. 481 00:36:23,940 --> 00:36:26,650 The other one has probability a squared b squared. 482 00:36:26,650 --> 00:36:38,270 So with probability 2 a squared b squared-- well, 483 00:36:38,270 --> 00:36:41,425 let's allow a and b to be complex. 484 00:36:45,590 --> 00:36:51,900 They obtain 1 and 1. 485 00:36:51,900 --> 00:37:02,600 So in this case, what is left is here 0 0. 486 00:37:02,600 --> 00:37:05,810 What is here left is 1 1. 487 00:37:05,810 --> 00:37:07,590 You may just need an intermediate line 488 00:37:07,590 --> 00:37:11,890 to write down what the state is after the measurement. 489 00:37:11,890 --> 00:37:15,850 But if you read it off here, you find 490 00:37:15,850 --> 00:37:20,610 that when they obtain 1 1 1, that 491 00:37:20,610 --> 00:37:43,170 implies the post measurement state is then 0 0 plus 1 1 492 00:37:43,170 --> 00:37:48,275 divided by square of 2, and this is one of our Bell states. 493 00:37:58,450 --> 00:38:02,070 So what we had is we had a system of four photons 494 00:38:02,070 --> 00:38:04,850 after a qubit operation, the controlled NOT. 495 00:38:04,850 --> 00:38:08,930 Alice and Bob do a measurement together on two of the photons. 496 00:38:08,930 --> 00:38:09,860 Collins. 497 00:38:09,860 --> 00:38:13,500 And if the outcome of this measurement is 1 1, 498 00:38:13,500 --> 00:38:15,780 the rest of the system is in the Bell state. 499 00:38:26,280 --> 00:38:30,000 So we assume that Alice and Bob have 500 00:38:30,000 --> 00:38:33,970 a large supply of those copies. 501 00:38:33,970 --> 00:38:41,820 So let's assume that they start with n copies of psi. 502 00:38:41,820 --> 00:38:47,490 And then, because the probability is m over n, 503 00:38:47,490 --> 00:38:54,320 they successfully obtain m copies of the Bell state. 504 00:38:54,320 --> 00:39:01,090 And the question is, what is the probability 505 00:39:01,090 --> 00:39:03,320 in the limit of a large ensemble. 506 00:39:05,980 --> 00:39:10,680 And we will see in a few moments that this is actually 507 00:39:10,680 --> 00:39:13,770 a measurement of how entangled the original space is. 508 00:39:20,255 --> 00:39:21,630 Any questions about purification? 509 00:39:29,250 --> 00:39:32,900 Well, then let's measure entanglement. 510 00:40:04,650 --> 00:40:09,420 The basic idea here is that if you 511 00:40:09,420 --> 00:40:14,430 have a state up/down plus down/up, it's a pure state. 512 00:40:14,430 --> 00:40:18,850 But this pure state has a correlation 513 00:40:18,850 --> 00:40:21,410 between the two subsystems. 514 00:40:21,410 --> 00:40:25,080 And the idea is now entanglement is 515 00:40:25,080 --> 00:40:27,890 that there is a correlation between the two subsystems. 516 00:40:27,890 --> 00:40:30,290 And you can say well, it would be a good way 517 00:40:30,290 --> 00:40:34,070 to characterize entanglement if I only look at one subsystem. 518 00:40:34,070 --> 00:40:36,400 In this case, you look at one subsystem, 519 00:40:36,400 --> 00:40:40,320 and you would just randomly see spin up, spin down. 520 00:40:40,320 --> 00:40:44,660 Spin up spin down is described by the unity density matrix, 521 00:40:44,660 --> 00:40:48,490 which is, therefore, the most random state on earth. 522 00:40:48,490 --> 00:40:58,200 So if you have a pure state and you only look at one subsystem, 523 00:40:58,200 --> 00:41:00,570 the more random the subsystem is, 524 00:41:00,570 --> 00:41:04,970 the more the pureness of the initial state 525 00:41:04,970 --> 00:41:08,340 comes from correlations, comes what is entanglement. 526 00:41:08,340 --> 00:41:10,520 So therefore, what I want to introduce now 527 00:41:10,520 --> 00:41:14,230 as a measure of entanglement is that we take the total system 528 00:41:14,230 --> 00:41:16,660 and then we perform the partial trace. 529 00:41:16,660 --> 00:41:18,930 We only look at one subsystem. 530 00:41:18,930 --> 00:41:23,620 And the purer the subsystem is, the less entangled 531 00:41:23,620 --> 00:41:26,200 it is, because in the ultimate limit 532 00:41:26,200 --> 00:41:29,790 that our system factorizes into two pure states, when we look 533 00:41:29,790 --> 00:41:32,430 at the subsystem, we still have a pure state. 534 00:41:32,430 --> 00:41:36,550 So the purity of the subsystem in terms of pure state 535 00:41:36,550 --> 00:41:39,010 is now a measure of entanglement. 536 00:41:39,010 --> 00:41:43,260 The purer the subsystem is, the less entangled it was. 537 00:41:51,070 --> 00:42:05,820 So the basic idea here is entanglement is 538 00:42:05,820 --> 00:42:09,420 related to correlations. 539 00:42:17,010 --> 00:42:25,760 And if you take half of a Bell state-- 540 00:42:25,760 --> 00:42:28,920 so if two particles which are an EPR pair and we 541 00:42:28,920 --> 00:42:36,330 take half of it-- then half of it is completely random. 542 00:42:36,330 --> 00:42:42,150 So let me illustrate that. 543 00:42:42,150 --> 00:42:47,520 So if you take one half of-- let's just 544 00:42:47,520 --> 00:42:50,400 take one of the Bell states, phi plus, 545 00:42:50,400 --> 00:42:56,410 which was the superposition of 0 0 plus 1 1. 546 00:43:00,160 --> 00:43:06,870 So let me be specific, because we need it for the definition. 547 00:43:06,870 --> 00:43:14,310 We describe this ensemble of this system in a pure EPR state 548 00:43:14,310 --> 00:43:16,660 by a density matrix. 549 00:43:16,660 --> 00:43:18,960 The density matrix is nothing else 550 00:43:18,960 --> 00:43:24,060 than you take your total system. 551 00:43:28,839 --> 00:43:30,255 So this is now the density matrix. 552 00:43:33,020 --> 00:43:42,750 So in our case, psi a b is just the state. 553 00:43:42,750 --> 00:43:51,720 And now we describe the subsystem 554 00:43:51,720 --> 00:43:56,210 by performing a partial trace on rho. 555 00:43:56,210 --> 00:44:00,900 The partial trace is over the system b. 556 00:44:00,900 --> 00:44:06,460 And that means we take all eigenfunctions k b of state b, 557 00:44:06,460 --> 00:44:11,480 sum over all k's, and this is our partial trace. 558 00:44:11,480 --> 00:44:16,440 So therefore, we would take our statistical operator 559 00:44:16,440 --> 00:44:25,970 from the line above, and perform the partial trace 560 00:44:25,970 --> 00:44:33,640 where those states are the state 0 and 1 of b. 561 00:44:38,470 --> 00:44:39,970 So these are the states of b. 562 00:44:42,620 --> 00:44:46,220 So when you do that, you just insert that. 563 00:44:46,220 --> 00:44:50,110 You find that what you get is 1/2. 564 00:44:50,110 --> 00:44:54,700 a has been traced out, so b has been traced out. 565 00:44:54,700 --> 00:45:02,180 And what we obtain for a is just from the two terms above. 566 00:45:02,180 --> 00:45:04,710 This gives that, and this gives that. 567 00:45:04,710 --> 00:45:14,800 And you immediately realize that this is 1/2 times the identity 568 00:45:14,800 --> 00:45:16,520 matrix. 569 00:45:16,520 --> 00:45:22,450 So therefore, we have shown that this 570 00:45:22,450 --> 00:45:24,000 is a completely random state. 571 00:45:31,880 --> 00:45:40,270 So now we can characterize the randomness 572 00:45:40,270 --> 00:45:46,570 of the partial trace of the density operator obtained 573 00:45:46,570 --> 00:45:49,560 by performing the partial trace with the standard von Neumann 574 00:45:49,560 --> 00:45:50,060 entropy. 575 00:46:00,010 --> 00:46:02,420 As a reminder, the von Neumann entropy 576 00:46:02,420 --> 00:46:08,470 for statistical operator rho is defined as the expectation 577 00:46:08,470 --> 00:46:15,490 value of rho log rho, where we take the logarithm with respect 578 00:46:15,490 --> 00:46:17,560 to the base 2. 579 00:46:17,560 --> 00:46:22,200 So this is the trace of rho log rho. 580 00:46:25,870 --> 00:46:31,300 Or if we use the eigenvalues of rho, 581 00:46:31,300 --> 00:46:35,700 we multiply eigenvalues with the logarithm of the eigenvalues. 582 00:46:48,560 --> 00:46:59,870 So for a pure state, the entropy is 0, 583 00:46:59,870 --> 00:47:04,000 because a pure state has one eigenvalue, which is 1, 584 00:47:04,000 --> 00:47:07,480 and the log of 1 is 0, so we get 0 for pure state. 585 00:47:16,370 --> 00:47:21,510 For completely mixed state, we're 586 00:47:21,510 --> 00:47:24,350 talking about a state which has two dimensions, so it 587 00:47:24,350 --> 00:47:25,580 can be up and down. 588 00:47:25,580 --> 00:47:31,780 A completely unique state has probabilities of 1/2 each. 589 00:47:31,780 --> 00:47:36,150 And then we say the entropy of this state is one, 590 00:47:36,150 --> 00:47:38,225 or we call it one bit. 591 00:47:38,225 --> 00:47:40,205 Yes. 592 00:47:40,205 --> 00:47:42,680 AUDIENCE: About the volume [INAUDIBLE] 593 00:47:42,680 --> 00:47:45,401 expectation of rho log rho is called 594 00:47:45,401 --> 00:47:50,600 a trace [INAUDIBLE] expectation of log rho, expectation 595 00:47:50,600 --> 00:47:54,560 of [INAUDIBLE] trace of the operator times density matrix. 596 00:48:09,922 --> 00:48:11,630 PROFESSOR: OK, so this is just a reminder 597 00:48:11,630 --> 00:48:14,110 of how we measure entropy of density matrix. 598 00:48:14,110 --> 00:48:15,720 And now we apply it to entanglement. 599 00:48:18,780 --> 00:48:27,730 We define now the entanglement for the entanglement 600 00:48:27,730 --> 00:48:41,620 e over state psi a b to be the entropy of the density 601 00:48:41,620 --> 00:48:50,160 matrix for system a after tracing out system b. 602 00:48:50,160 --> 00:49:04,690 And for pure state, this is-- it doesn't matter whether we trace 603 00:49:04,690 --> 00:49:08,930 out a or b if you start with a pure state. 604 00:49:08,930 --> 00:49:13,530 The entanglement, the entropy of the statistical operator 605 00:49:13,530 --> 00:49:17,490 rho a and rho b are the same. 606 00:49:17,490 --> 00:49:19,210 I tried for a moment to prove it. 607 00:49:19,210 --> 00:49:20,410 I saw it quoted somewhere. 608 00:49:20,410 --> 00:49:22,820 I didn't succeed in a split second, 609 00:49:22,820 --> 00:49:24,871 so either I overlooked something, 610 00:49:24,871 --> 00:49:26,870 or it's a little bit more involved to show that. 611 00:49:29,790 --> 00:49:33,100 So therefore, to use the inverts, 612 00:49:33,100 --> 00:49:40,680 our definition says that the entropy, so the entanglement, 613 00:49:40,680 --> 00:49:46,415 is nothing else than the entropy of the reduced density matrix. 614 00:49:56,860 --> 00:50:04,700 And we immediately see if we have any of the four Bell 615 00:50:04,700 --> 00:50:09,640 states, by performing the partial trace over one qubit, 616 00:50:09,640 --> 00:50:11,820 we obtain the identity matrix. 617 00:50:11,820 --> 00:50:15,000 So therefore, the entropy of all the Bell states is 1. 618 00:50:26,840 --> 00:50:30,123 Let me state without proof-- when we come back 619 00:50:30,123 --> 00:50:38,350 to the purification scheme the result 620 00:50:38,350 --> 00:50:45,440 is that the probability or the optimum probability-- 621 00:50:45,440 --> 00:50:47,620 if you do stupid measurements on your states, 622 00:50:47,620 --> 00:50:49,110 of course you get nothing. 623 00:50:49,110 --> 00:50:54,540 But the optimum strategy to create pure Bell states out 624 00:50:54,540 --> 00:50:59,520 of your reservoir of poorly entangled states-- 625 00:50:59,520 --> 00:51:06,180 so for an optimum strategy, the success probability m over n 626 00:51:06,180 --> 00:51:09,285 actually turns out to be not a different measure 627 00:51:09,285 --> 00:51:10,580 of entanglement. 628 00:51:10,580 --> 00:51:13,490 It is the entanglement which we have just 629 00:51:13,490 --> 00:51:16,210 defined through the entropy of the partial trace. 630 00:51:21,090 --> 00:51:23,630 So therefore-- and I think this is nicely 631 00:51:23,630 --> 00:51:25,930 illustrated with the purification scheme-- 632 00:51:25,930 --> 00:51:28,170 entanglement is a real resource. 633 00:51:28,170 --> 00:51:32,170 When you have better entanglement to start with, 634 00:51:32,170 --> 00:51:37,590 then you can get more copies. 635 00:51:37,590 --> 00:51:40,590 You can get more pure Bell states 636 00:51:40,590 --> 00:51:44,660 out of your supply of poorly entangled states. 637 00:51:44,660 --> 00:51:48,800 So therefore, you lose more if your states are not 638 00:51:48,800 --> 00:51:49,490 fully entangled. 639 00:51:49,490 --> 00:51:52,035 You lose more of them, and therefore, the success 640 00:51:52,035 --> 00:51:54,980 of the purification scheme makes it clear 641 00:51:54,980 --> 00:51:56,690 how entanglement is a resource. 642 00:51:56,690 --> 00:51:58,850 If you have entanglement, it's precious. 643 00:51:58,850 --> 00:52:00,340 You had to do something to get it. 644 00:52:08,080 --> 00:52:13,020 I didn't point out entanglement is not 645 00:52:13,020 --> 00:52:18,580 something which one number characterize it, it's all. 646 00:52:18,580 --> 00:52:21,250 We introduced to you already another measurement 647 00:52:21,250 --> 00:52:43,300 of entanglement through the Schmidt number, 648 00:52:43,300 --> 00:52:46,320 which was in homework number two. 649 00:52:46,320 --> 00:52:49,490 And usually when you have different measures 650 00:52:49,490 --> 00:52:52,310 for entanglement, they're not one to one related. 651 00:52:52,310 --> 00:52:56,120 It seems that, similarly when we measured non-classic light, 652 00:52:56,120 --> 00:52:57,320 we had a G2 function. 653 00:52:57,320 --> 00:52:59,090 We had [INAUDIBLE] bunching, antibunching. 654 00:52:59,090 --> 00:53:01,740 We have negative quasi probabilities. 655 00:53:01,740 --> 00:53:05,370 And it's often clear that one system which 656 00:53:05,370 --> 00:53:08,910 is truly non-classical, fulfills all the criteria, 657 00:53:08,910 --> 00:53:12,320 but how the quantitative measurements are related 658 00:53:12,320 --> 00:53:15,010 to each other is really subtle. 659 00:53:15,010 --> 00:53:18,350 In the case of entanglement, for a long time it 660 00:53:18,350 --> 00:53:20,490 has even been a big question in research-- 661 00:53:20,490 --> 00:53:22,490 if you have an arbitrary density matrix 662 00:53:22,490 --> 00:53:24,830 with a complicated many-body system, 663 00:53:24,830 --> 00:53:28,310 how can you even characterize the entanglement? 664 00:53:28,310 --> 00:53:30,470 We are focusing here on pure states 665 00:53:30,470 --> 00:53:32,390 where things are fairly simple. 666 00:53:32,390 --> 00:53:36,400 But in the general situation of a many-body system, 667 00:53:36,400 --> 00:53:40,502 it can be quite challenging just define and measure 668 00:53:40,502 --> 00:53:41,043 entanglement. 669 00:53:44,080 --> 00:53:44,880 Any questions? 670 00:53:44,880 --> 00:53:45,902 Yes. 671 00:53:45,902 --> 00:53:47,985 AUDIENCE: If you have a general [INAUDIBLE] that's 672 00:53:47,985 --> 00:53:50,895 not necessarily a pure state, can you 673 00:53:50,895 --> 00:53:52,835 show that it's always the reduced 674 00:53:52,835 --> 00:53:55,260 trace of the bigger [INAUDIBLE]? 675 00:54:05,020 --> 00:54:08,872 PROFESSOR: Say again, if I have a general state which is-- 676 00:54:08,872 --> 00:54:13,291 AUDIENCE: You have a subsystem-- let's say you have two spins, 677 00:54:13,291 --> 00:54:14,764 and then you have a density matrix 678 00:54:14,764 --> 00:54:19,680 for the first-- or actually just to say that you have one spin 679 00:54:19,680 --> 00:54:21,804 and you have a density matrix for that single spin, 680 00:54:21,804 --> 00:54:26,232 it's a matrix, not necessarily a pure state. 681 00:54:26,232 --> 00:54:27,708 Let's say it's a mixed state. 682 00:54:27,708 --> 00:54:31,152 Then you can introduce a fake second spin 683 00:54:31,152 --> 00:54:36,072 and show that this matrix is the trace of a matrix [INAUDIBLE] 684 00:54:36,072 --> 00:54:39,516 and maybe just put that one in a pure state and do some stuff, 685 00:54:39,516 --> 00:54:42,665 do some calculations more easily. 686 00:54:42,665 --> 00:54:43,290 PROFESSOR: Yes. 687 00:54:48,266 --> 00:54:50,450 If you have a density matrix, you 688 00:54:50,450 --> 00:54:53,570 can always regard it as a partial trace of a bigger 689 00:54:53,570 --> 00:54:54,820 system. 690 00:54:54,820 --> 00:55:01,050 That means you always represent your state as a pure state, 691 00:55:01,050 --> 00:55:03,570 but it is entangled with a bigger system. 692 00:55:03,570 --> 00:55:09,500 However, what the big system is, is by no means unique. 693 00:55:09,500 --> 00:55:15,170 I'm missing the technical word-- it's 694 00:55:15,170 --> 00:55:17,410 called unraveling the density matrix. 695 00:55:17,410 --> 00:55:21,170 You can always represent the density matrix written down 696 00:55:21,170 --> 00:55:23,300 in forms of pure state. 697 00:55:23,300 --> 00:55:26,055 But this unraveling of the density matrix-- no, 698 00:55:26,055 --> 00:55:27,550 actually, its related. 699 00:55:33,830 --> 00:55:36,090 When I said yes, I thought about the unraveling 700 00:55:36,090 --> 00:55:37,430 of the density matrix. 701 00:55:37,430 --> 00:55:39,760 You can always write down a density matrix 702 00:55:39,760 --> 00:55:42,950 as a mixture of pure states, but which are the pure states 703 00:55:42,950 --> 00:55:44,130 is not unique. 704 00:55:44,130 --> 00:55:46,270 So when you say my density matrix is 705 00:55:46,270 --> 00:55:48,460 half of the atoms are spin up and half of the atoms 706 00:55:48,460 --> 00:55:51,040 are spin down, somebody else would say, no, that's not true. 707 00:55:51,040 --> 00:55:53,630 Half of the atoms are spin [INAUDIBLE] and spin x 708 00:55:53,630 --> 00:55:57,665 and some are in spin minus x, and those representations 709 00:55:57,665 --> 00:55:59,330 are equivalent. 710 00:55:59,330 --> 00:56:02,300 So I was just thinking of that as to write down the density 711 00:56:02,300 --> 00:56:04,720 matrix in the pure state basis. 712 00:56:04,720 --> 00:56:07,006 But you are asking about-- 713 00:56:07,006 --> 00:56:11,510 AUDIENCE: [INAUDIBLE] what he's asking about [INAUDIBLE] 714 00:56:11,510 --> 00:56:13,490 PROFESSOR: Somehow, but I was thinking of that, 715 00:56:13,490 --> 00:56:15,448 but I think the answer to your question is yes. 716 00:56:15,448 --> 00:56:17,094 So you said, you confirmed that. 717 00:56:17,094 --> 00:56:18,510 AUDIENCE: Yes, but I am forgetting 718 00:56:18,510 --> 00:56:19,966 the name of the theorem here. 719 00:56:19,966 --> 00:56:20,800 AUDIENCE: Isn't it just purification again? 720 00:56:20,800 --> 00:56:23,000 AUDIENCE: Yeah, it's related to purification. 721 00:56:23,000 --> 00:56:26,175 You can prove that you're using purification. 722 00:56:26,175 --> 00:56:27,050 AUDIENCE: [INAUDIBLE] 723 00:56:27,050 --> 00:56:28,716 AUDIENCE: Using purification [INAUDIBLE] 724 00:56:28,716 --> 00:56:32,717 but I'm completely forgetting the names of [INAUDIBLE] 725 00:56:32,717 --> 00:56:33,550 PROFESSOR: But wait. 726 00:56:33,550 --> 00:56:36,060 We've talked here about-- just to be clear, 727 00:56:36,060 --> 00:56:39,030 we've talked here about purification of a pure state. 728 00:56:39,030 --> 00:56:44,070 We started with a pure state and we 729 00:56:44,070 --> 00:56:48,555 purified it to be a Bell state by doing certain measurements. 730 00:56:52,020 --> 00:56:54,170 So we've not talked here about density matrices, 731 00:56:54,170 --> 00:56:58,250 but it sounds very plausible that you can always 732 00:56:58,250 --> 00:56:59,630 construct a bigger system. 733 00:56:59,630 --> 00:57:06,030 But I should look it up and see if there is an exact proof. 734 00:57:06,030 --> 00:57:09,839 It's-- intuitively it sounds correct. 735 00:57:09,839 --> 00:57:10,505 Other questions? 736 00:57:16,210 --> 00:57:22,310 OK, so we have discussed the definition of entangled state. 737 00:57:22,310 --> 00:57:25,650 We've talked about purification of entangled states 738 00:57:25,650 --> 00:57:28,700 and how to measure it. 739 00:57:28,700 --> 00:57:38,060 I want to talk now about how we can 740 00:57:38,060 --> 00:57:39,960 create entangled states for atoms. 741 00:57:49,670 --> 00:57:54,390 Maybe let me say the following-- so by now we are convinced. 742 00:57:54,390 --> 00:57:57,090 Entangled states are great, and we want to create them. 743 00:57:57,090 --> 00:58:02,670 And for photons, I showed you that some simple element-- 744 00:58:02,670 --> 00:58:08,660 beam splitters, Kerr medium-- can create entanglement. 745 00:58:08,660 --> 00:58:12,390 It's much harder to do that with atoms. 746 00:58:12,390 --> 00:58:15,140 Now, we want to do it with atoms, 747 00:58:15,140 --> 00:58:17,950 because atoms-- in contrast to light-- they're 748 00:58:17,950 --> 00:58:20,730 pretty much staying still, whereas photons always 749 00:58:20,730 --> 00:58:22,610 move at the speed of light. 750 00:58:22,610 --> 00:58:25,690 And the only way to make photons stand still 751 00:58:25,690 --> 00:58:27,650 is you put them in a cavity and then they 752 00:58:27,650 --> 00:58:28,860 bounce back and forth. 753 00:58:28,860 --> 00:58:31,940 But even in super cavities with the highest reflectivity 754 00:58:31,940 --> 00:58:35,160 mirror, you get-- what are the longest ring-down times 755 00:58:35,160 --> 00:58:37,970 you get-- fraction of a second, milliseconds, 756 00:58:37,970 --> 00:58:39,570 depending kind of in which domain 757 00:58:39,570 --> 00:58:42,010 you work-- microwave domain, optical domain. 758 00:58:42,010 --> 00:58:45,960 Whereas atoms, you can hold onto your qubits for a long time. 759 00:58:45,960 --> 00:58:50,460 So therefore, if you want to use entanglement 760 00:58:50,460 --> 00:58:52,520 as a resource for certain protocols, 761 00:58:52,520 --> 00:58:56,420 you want to have entangled atoms where the entanglement would 762 00:58:56,420 --> 00:58:58,290 like for a long time. 763 00:58:58,290 --> 00:59:00,700 So the question is now for atoms, 764 00:59:00,700 --> 00:59:03,610 we do not have perfect beam splitters and perfect Kerr 765 00:59:03,610 --> 00:59:05,340 mediums. 766 00:59:05,340 --> 00:59:08,273 Also, we can control interactions between atoms 767 00:59:08,273 --> 00:59:10,180 for [INAUDIBLE] using VSEPR resonances, 768 00:59:10,180 --> 00:59:11,720 but that's another story. 769 00:59:11,720 --> 00:59:14,510 But now let me ask a question, how do we entangle atoms? 770 00:59:30,630 --> 00:59:35,090 And I want to first show you that if you 771 00:59:35,090 --> 00:59:40,100 had the right system, things can be fairly simple. 772 00:59:40,100 --> 00:59:46,370 This is a suggestion which was made almost 20 years ago, 773 00:59:46,370 --> 00:59:49,580 and it goes like follows-- if you 774 00:59:49,580 --> 00:59:54,420 have a diatomic molecule of two identical atoms, 775 00:59:54,420 --> 00:59:58,520 in this case mercury, and mercury 776 00:59:58,520 --> 01:00:03,270 doesn't have-- this molecule doesn't have any electron spin, 777 01:00:03,270 --> 01:00:06,730 but mercury has a nuclear spin. 778 01:00:06,730 --> 01:00:11,680 And if you now photo-dissociate mercury and two mercury atoms 779 01:00:11,680 --> 01:00:17,070 fly away, then you have separated a spin singlet 780 01:00:17,070 --> 01:00:18,860 into two parts. 781 01:00:18,860 --> 01:00:21,480 And now you have created the Bell state 782 01:00:21,480 --> 01:00:22,840 up/down minus down/up. 783 01:00:25,880 --> 01:00:33,680 So you could say this is sort of the example-- this realizes 784 01:00:33,680 --> 01:00:37,420 very closely the example I gave you with the helium atom, 785 01:00:37,420 --> 01:00:40,210 where I said you have an electron in spin up and spin 786 01:00:40,210 --> 01:00:44,220 down, but there was no way to separate the electrons. 787 01:00:44,220 --> 01:00:47,730 Therefore, you can say it is a state which has entanglement, 788 01:00:47,730 --> 01:00:50,970 but it's not entanglement as a resource. 789 01:00:50,970 --> 01:00:54,620 But right now, here it becomes a resource 790 01:00:54,620 --> 01:00:58,610 once you have found a method to separate the two 791 01:00:58,610 --> 01:01:02,700 parts of the wave function that you can give one to Alice, give 792 01:01:02,700 --> 01:01:07,570 one to Bob, and they can perform the operations on it. 793 01:01:07,570 --> 01:01:09,470 Well, if it looks so simple, why don't we 794 01:01:09,470 --> 01:01:12,810 have entangled atoms everywhere? 795 01:01:12,810 --> 01:01:15,650 This experiment has been suggested 20 years ago, 796 01:01:15,650 --> 01:01:18,900 but nobody has done it, or some several groups 797 01:01:18,900 --> 01:01:21,280 have worked on it. 798 01:01:21,280 --> 01:01:26,750 You need a molecule with suitable states. 799 01:01:26,750 --> 01:01:30,660 You don't want any electron spin which interferes with that. 800 01:01:30,660 --> 01:01:33,970 All you want to have is two nuclear spins. 801 01:01:33,970 --> 01:01:35,650 You want a singlet state here. 802 01:01:35,650 --> 01:01:38,830 So those requirements are not easily fulfilled 803 01:01:38,830 --> 01:01:41,760 with real atoms. 804 01:01:41,760 --> 01:01:45,400 Of course we liked-- and also who wants to work with mercury? 805 01:01:45,400 --> 01:01:49,170 Mercury has transitions in the ultraviolet. 806 01:01:49,170 --> 01:01:51,215 I think there's only one group in the world who 807 01:01:51,215 --> 01:01:53,710 has operated a magneto optic trap with mercury. 808 01:01:53,710 --> 01:01:56,440 So it's not your tabletop atom. 809 01:01:56,440 --> 01:02:00,550 So therefore, let's now talk about a method 810 01:02:00,550 --> 01:02:06,120 how we can entangle atoms by using light. 811 01:02:06,120 --> 01:02:08,540 So if you take atoms, maybe we can 812 01:02:08,540 --> 01:02:10,630 use the light which has been emitted 813 01:02:10,630 --> 01:02:13,770 from the atoms, performing measurement on the light, 814 01:02:13,770 --> 01:02:17,730 and then depending on the outcome of the measurement, 815 01:02:17,730 --> 01:02:20,795 we know the atoms are entangled. 816 01:02:25,710 --> 01:02:35,770 So Professor [? Swann ?] calls this the poor man's entangler. 817 01:02:38,510 --> 01:02:39,830 I don't know why poor man's. 818 01:02:39,830 --> 01:02:43,160 You still need quite a bit of equipment to do it. 819 01:02:43,160 --> 01:02:46,250 But at least it seems the poor man's solution 820 01:02:46,250 --> 01:02:49,680 if you can't make the above experiment work. 821 01:02:49,680 --> 01:02:55,280 So this addresses a question that, 822 01:02:55,280 --> 01:03:05,330 just for technical reasons matter 823 01:03:05,330 --> 01:03:07,900 is more difficult to entangle. 824 01:03:11,300 --> 01:03:12,830 Whereas photons are easy. 825 01:03:28,640 --> 01:03:34,360 Yeah so you can say the idea is related 826 01:03:34,360 --> 01:03:36,290 to the purification scheme. 827 01:03:36,290 --> 01:03:45,680 We can't take a system of two atoms, atom one and atom two 828 01:03:45,680 --> 01:03:51,810 which are unentangled, and we shine some laser light on them, 829 01:03:51,810 --> 01:03:52,890 excite them. 830 01:03:52,890 --> 01:03:54,780 Then they emit photons. 831 01:03:54,780 --> 01:03:57,520 So all we can do is-- we can only 832 01:03:57,520 --> 01:03:59,260 talk to the atoms with the photons. 833 01:03:59,260 --> 01:04:00,790 So the only thing we can do now is 834 01:04:00,790 --> 01:04:02,830 we can measure the two photons. 835 01:04:02,830 --> 01:04:06,380 And then the situation will be similar as in the purification 836 01:04:06,380 --> 01:04:09,170 scheme, where Alice and Bob did a measurement. 837 01:04:09,170 --> 01:04:15,140 If Alice and Bob said both of our target qubits are one, 838 01:04:15,140 --> 01:04:21,940 then what was left behind was in a pure, entangled Bell state. 839 01:04:21,940 --> 01:04:25,560 And similarly, what you want to do here is we have two atoms. 840 01:04:25,560 --> 01:04:28,940 There was nothing special about them but they scatter light. 841 01:04:28,940 --> 01:04:32,480 And if you now perform a measurement on the photons 842 01:04:32,480 --> 01:04:36,030 and the outcome of the measurement is positive, 843 01:04:36,030 --> 01:04:39,800 then we know for sure what has been left behind is entangled. 844 01:04:39,800 --> 01:04:42,420 That's the idea. 845 01:04:42,420 --> 01:04:46,560 So it shares with the purification scheme 846 01:04:46,560 --> 01:04:48,610 that it is a probabilistic entanglement. 847 01:04:48,610 --> 01:04:51,090 You run your experiment many times. 848 01:04:51,090 --> 01:04:52,062 You do a measurement. 849 01:04:52,062 --> 01:04:53,520 If the measurement is positive, you 850 01:04:53,520 --> 01:04:55,340 say now I have an entangled state. 851 01:04:55,340 --> 01:04:58,220 And maybe then you can move on to measure the entanglement. 852 01:04:58,220 --> 01:05:01,815 You can move on to do teleportation, other things you 853 01:05:01,815 --> 01:05:03,190 want to do with entangled states. 854 01:05:03,190 --> 01:05:07,035 But if your measurements says no, bad luck. 855 01:05:07,035 --> 01:05:09,770 The probability hasn't worked out this time. 856 01:05:09,770 --> 01:05:12,150 You just press a button again, scatter light 857 01:05:12,150 --> 01:05:14,209 again of your two atoms and/or your two ions 858 01:05:14,209 --> 01:05:16,000 and hope that the next outcome is positive. 859 01:05:20,990 --> 01:05:26,780 So the idea is we want to introduce now 860 01:05:26,780 --> 01:05:31,550 a probabilistic method. 861 01:05:39,390 --> 01:05:44,355 It's based on two atoms emitting light. 862 01:05:48,070 --> 01:05:52,170 And the result is with a certain probability 863 01:05:52,170 --> 01:05:53,690 that we get entangled atoms. 864 01:06:30,320 --> 01:06:36,036 Let me just scroll down and show you one thing I want to-- 865 01:06:46,790 --> 01:06:49,170 When I prepare the notes and everything is clear to me, 866 01:06:49,170 --> 01:06:51,260 but then I want to explain it to you and say hey, 867 01:06:51,260 --> 01:06:52,480 I have to motivate you. 868 01:06:52,480 --> 01:06:54,110 If I just go through a few lines, 869 01:06:54,110 --> 01:06:55,550 you wonder what it leads to. 870 01:06:55,550 --> 01:06:57,600 So let me maybe first give you the explanation 871 01:06:57,600 --> 01:06:59,350 I would have given you a little bit later. 872 01:07:03,750 --> 01:07:07,160 What is an entangled state is if the atoms are 873 01:07:07,160 --> 01:07:10,870 in maybe one of two ground states-- one atom is in one 874 01:07:10,870 --> 01:07:13,990 ground state, the other one is in the other one, 875 01:07:13,990 --> 01:07:15,490 or it is flipped. 876 01:07:15,490 --> 01:07:18,430 So we know one atom is in the ground state one. 877 01:07:18,430 --> 01:07:20,640 One atom is in the ground state two, 878 01:07:20,640 --> 01:07:22,350 but we don't know which one is in which. 879 01:07:22,350 --> 01:07:25,320 It is in the superposition state. 880 01:07:25,320 --> 01:07:28,100 So what you need now in this scheme 881 01:07:28,100 --> 01:07:32,270 is the following-- if the atoms scatter light, 882 01:07:32,270 --> 01:07:35,840 they can go to two different ground states. 883 01:07:40,200 --> 01:07:43,600 And we know to which ground state 884 01:07:43,600 --> 01:07:47,320 they have gone, because due to selection rules, 885 01:07:47,320 --> 01:07:50,050 they reach one ground state with polarization one. 886 01:07:50,050 --> 01:07:53,310 They reach one ground state with polarization two. 887 01:07:53,310 --> 01:07:56,840 So therefore, when we had two atoms, they emit light 888 01:07:56,840 --> 01:07:59,480 and we would measure the polarization of the light, 889 01:07:59,480 --> 01:08:01,340 we would know in which state they are. 890 01:08:01,340 --> 01:08:03,570 But if you know atom 1 is in state 1, 891 01:08:03,570 --> 01:08:06,940 and atom 2 is in state 2, this is not entangled. 892 01:08:06,940 --> 01:08:10,010 So what we have to do is when the two atoms have emitted 893 01:08:10,010 --> 01:08:14,590 photons, we have to mix the photon at the beam splitter. 894 01:08:14,590 --> 01:08:16,500 And after the beam splitter, when 895 01:08:16,500 --> 01:08:19,420 we measure that the photon is polarized, 896 01:08:19,420 --> 01:08:23,710 we know that one of the atoms is in one of the ground states, 897 01:08:23,710 --> 01:08:25,470 but we don't know which. 898 01:08:25,470 --> 01:08:31,069 So therefore, we can now use-- the photon carries 899 01:08:31,069 --> 01:08:35,510 through the polarization the information in which ground 900 01:08:35,510 --> 01:08:37,270 state the atom is. 901 01:08:37,270 --> 01:08:40,790 But now we have to perform operations to the photons 902 01:08:40,790 --> 01:08:44,680 that for fundamental reasons, fundamental quantum 903 01:08:44,680 --> 01:08:48,310 measurement, we know there is one photon in one state, 904 01:08:48,310 --> 01:08:50,910 but we have no way to ever figure out 905 01:08:50,910 --> 01:08:54,220 which atom has emitted the photon. 906 01:08:54,220 --> 01:08:56,170 So that's the idea, and the protocol 907 01:08:56,170 --> 01:08:59,130 I want to show you now is what do we have to do to the two 908 01:08:59,130 --> 01:09:02,390 photons to make sure that we never know, 909 01:09:02,390 --> 01:09:06,130 that we can't find out which atom has emitted the photon. 910 01:09:06,130 --> 01:09:08,210 And we have to do little bit more tricks also 911 01:09:08,210 --> 01:09:10,930 to make sure that when we do a measurement on the photon, 912 01:09:10,930 --> 01:09:14,870 we know the atoms are in the Bell state. 913 01:09:14,870 --> 01:09:15,830 Questions about that? 914 01:09:37,899 --> 01:09:41,254 So therefore, we need a beam splitter. 915 01:09:45,689 --> 01:09:48,870 So we can come back to what we have already 916 01:09:48,870 --> 01:09:50,274 introduced in the last section. 917 01:09:58,540 --> 01:10:02,990 So each atom will emit a photon. 918 01:10:02,990 --> 01:10:07,020 And I will actually show you at the end of the class 919 01:10:07,020 --> 01:10:09,350 that people have entangled with that scheme 920 01:10:09,350 --> 01:10:12,000 two ions which were in two different ion traps. 921 01:10:12,000 --> 01:10:14,290 So you have two distance atoms. 922 01:10:14,290 --> 01:10:16,960 They emit light, and after the measurement process, 923 01:10:16,960 --> 01:10:18,190 you know they are entangled. 924 01:10:18,190 --> 01:10:19,990 And that's pretty cool. 925 01:10:19,990 --> 01:10:24,190 So the situation how we do it is we have two photons which come. 926 01:10:24,190 --> 01:10:26,130 But the first thing we have to make sure 927 01:10:26,130 --> 01:10:28,250 is we have to scramble the photons. 928 01:10:28,250 --> 01:10:30,190 We have to make sure that we can't find out 929 01:10:30,190 --> 01:10:32,470 from which atom the photon has come, 930 01:10:32,470 --> 01:10:35,320 and this is done with the beam splitter. 931 01:10:35,320 --> 01:10:38,330 So there is one aspect of beam splitters 932 01:10:38,330 --> 01:10:41,710 and two photons which have to explain to you now. 933 01:10:41,710 --> 01:10:49,620 And this is this famous HUM, Hong-Ou-Mandel. 934 01:10:52,540 --> 01:11:00,430 This is the Hong-Ou-Mandel interference. 935 01:11:00,430 --> 01:11:02,480 It's a very famous effect, and it's actually 936 01:11:02,480 --> 01:11:04,140 very, very special. 937 01:11:04,140 --> 01:11:07,280 So let me explain what happens when 938 01:11:07,280 --> 01:11:13,160 we have two photons in the same state-- 939 01:11:13,160 --> 01:11:18,327 two identical photons, same frequency, same polarization-- 940 01:11:18,327 --> 01:11:19,410 coming to a beam splitter. 941 01:11:28,820 --> 01:11:32,010 So we have already all the tools. 942 01:11:32,010 --> 01:11:35,050 So we have a beam splitter characterized 943 01:11:35,050 --> 01:11:37,560 by this angle theta, which through cosine theta, 944 01:11:37,560 --> 01:11:40,930 sine theta determines what the beam splitter is doing. 945 01:11:40,930 --> 01:11:46,730 And all we want to do is we apply it now to the state 1 1. 946 01:11:49,790 --> 01:11:54,190 So what we will find is that there is a probability 947 01:11:54,190 --> 01:11:58,050 to get one photon each. 948 01:11:58,050 --> 01:12:03,140 Then there is a probability to get two photons in one output. 949 01:12:03,140 --> 01:12:06,010 And it will actually be the same probability 950 01:12:06,010 --> 01:12:10,370 with a minus sign in the amplitude to get 0 2. 951 01:12:10,370 --> 01:12:14,120 And the matrix which acts on each of the photons 952 01:12:14,120 --> 01:12:17,040 has cosines and sines. 953 01:12:17,040 --> 01:12:25,690 So what we get is products of trigonometric functions. 954 01:12:25,690 --> 01:12:31,370 And here for getting two photons in one arm, 955 01:12:31,370 --> 01:12:41,390 it's square root 2, cosine theta, sine theta. 956 01:12:43,940 --> 01:12:48,680 And the spectacular thing here is 957 01:12:48,680 --> 01:12:52,090 what happens when we have a balanced beam splitter. 958 01:13:06,770 --> 01:13:11,510 If you have the 50-50 beam splitter-- 959 01:13:11,510 --> 01:13:15,040 and this is called the Hong-Ou-Mandel interference, 960 01:13:15,040 --> 01:13:23,630 you have the situation where you have a beam splitter, 961 01:13:23,630 --> 01:13:28,840 you have one photon, you have two photons. 962 01:13:28,840 --> 01:13:34,560 And it is absolutely impossible that afterwards, you 963 01:13:34,560 --> 01:13:36,590 have one photon in each arm. 964 01:13:36,590 --> 01:13:40,210 So you send two photons on a beam splitter, 965 01:13:40,210 --> 01:13:42,890 and after the beam splitter is, you 966 01:13:42,890 --> 01:13:46,190 have either two photons coming out here, 967 01:13:46,190 --> 01:13:48,470 or two photons coming out there. 968 01:13:52,470 --> 01:13:56,630 You can say it's Poissonic stimulation photons and bosons. 969 01:13:56,630 --> 01:13:58,010 That all plays a role. 970 01:13:58,010 --> 01:14:02,500 If one photon goes one path, the other photons follow suit. 971 01:14:06,910 --> 01:14:10,970 This is now very powerful, because it already 972 01:14:10,970 --> 01:14:14,620 happens, of course, when the two photons are identical. 973 01:14:14,620 --> 01:14:17,840 We had to use in this formalism the photon is 974 01:14:17,840 --> 01:14:20,030 in the same mode on each part. 975 01:14:20,030 --> 01:14:22,560 But that means now the following-- 976 01:14:22,560 --> 01:14:25,340 if you set up photo detectors here, 977 01:14:25,340 --> 01:14:28,260 and each photon detector makes click, 978 01:14:28,260 --> 01:14:30,850 you know you had one photon each. 979 01:14:30,850 --> 01:14:36,310 And that tells you that the two photons were not identical-- 980 01:14:36,310 --> 01:14:40,380 for instance, because they have different polarizations. 981 01:14:40,380 --> 01:14:41,980 So if you want to now, with the atoms, 982 01:14:41,980 --> 01:14:44,840 get a Bell state, where atoms decay-- one atom 983 01:14:44,840 --> 01:14:47,110 decays to state one, one atom decays 984 01:14:47,110 --> 01:14:51,160 to state two-- the signature of that 985 01:14:51,160 --> 01:14:55,050 would be that we have one atom each. 986 01:14:55,050 --> 01:15:04,660 And if you do it right, one atom each is an ingredient for 1 2 987 01:15:04,660 --> 01:15:06,780 plus 2 1 for Bell state. 988 01:15:06,780 --> 01:15:10,010 We can detect that we have one photon each because at such 989 01:15:10,010 --> 01:15:13,260 a beam splitter, it's only in this situation 990 01:15:13,260 --> 01:15:16,500 that we can get one photon after each beam splitter 991 01:15:16,500 --> 01:15:21,020 if we start with two non-identical photons. 992 01:15:21,020 --> 01:15:24,990 Of course, by the way, experimentally there are quite 993 01:15:24,990 --> 01:15:28,130 some challenges, even if you have identical photons 994 01:15:28,130 --> 01:15:31,870 of the same polarization, if they arrive as a nanosecond 995 01:15:31,870 --> 01:15:35,710 pulse, and they don't arrive exactly at the same time 996 01:15:35,710 --> 01:15:39,390 at the beam splitter, then you first split one photon, 997 01:15:39,390 --> 01:15:41,800 and then the next photon, and the two photons cannot 998 01:15:41,800 --> 01:15:43,330 influence each other. 999 01:15:43,330 --> 01:15:46,030 So there are a lot of experimental requirements 1000 01:15:46,030 --> 01:15:48,065 to realize this ideal experiment. 1001 01:15:57,340 --> 01:16:02,590 So as I was preparing for today's class, 1002 01:16:02,590 --> 01:16:12,317 I actually saw a paper which just came out in "Science." 1003 01:16:32,390 --> 01:16:35,750 So in this month's "Science" they 1004 01:16:35,750 --> 01:16:39,215 discussed the Hong-Ou-Mandel interference experiment 1005 01:16:39,215 --> 01:16:41,440 with fermions. 1006 01:16:41,440 --> 01:16:44,150 So let me just explain what happens. 1007 01:17:11,930 --> 01:17:12,783 OK, now it fits. 1008 01:17:21,330 --> 01:17:24,720 So if you have two identical photons, 1009 01:17:24,720 --> 01:17:29,530 if you have two particles which impinge on the beam splitter-- 1010 01:17:29,530 --> 01:17:33,380 they are the two input beams for the beam splitter-- 1011 01:17:33,380 --> 01:17:36,620 you would say you have four different outcomes. 1012 01:17:36,620 --> 01:17:40,630 One is both come out here, both come out here. 1013 01:17:40,630 --> 01:17:43,730 Or they are both reflected, or they are both 1014 01:17:43,730 --> 01:17:45,500 transmitted through. 1015 01:17:45,500 --> 01:17:49,080 And what happens is bosons-- as I just pointed out-- 1016 01:17:49,080 --> 01:17:52,690 bosons characterized by photons can only do that. 1017 01:17:52,690 --> 01:17:55,430 Identical photons want two bunch up. 1018 01:17:55,430 --> 01:18:02,390 They appear in pairs-- 50% left output, 50% right output. 1019 01:18:02,390 --> 01:18:05,260 Well, for fermions, they just do the opposite. 1020 01:18:05,260 --> 01:18:08,680 For fermions, you will always get one particle each. 1021 01:18:08,680 --> 01:18:10,637 You may immediately, of course, explain it 1022 01:18:10,637 --> 01:18:12,720 with the Pauli Exclusion Principle, which does not 1023 01:18:12,720 --> 01:18:14,650 allow two particles to be in the same state 1024 01:18:14,650 --> 01:18:16,240 after the beam splitter. 1025 01:18:16,240 --> 01:18:19,850 And you should contrast it with classical particles. 1026 01:18:19,850 --> 01:18:22,620 When you have two classical particles, 1027 01:18:22,620 --> 01:18:26,950 you will actually find that all of those four possibilities, 1028 01:18:26,950 --> 01:18:28,865 each of them has a 25% probability. 1029 01:18:31,560 --> 01:18:37,830 So it was a major experiment which 1030 01:18:37,830 --> 01:18:40,070 was featured in "Science" when people realized 1031 01:18:40,070 --> 01:18:42,720 that with electrons, they created electrons 1032 01:18:42,720 --> 01:18:48,950 in a semiconductor structure, and showed 1033 01:18:48,950 --> 01:18:53,210 through some statistical measurement 1034 01:18:53,210 --> 01:18:56,920 that this was the physics which happened. 1035 01:18:56,920 --> 01:18:59,090 The measurement they did is, I forgot details-- 1036 01:18:59,090 --> 01:19:02,820 if you have bosons and you get either two here or two there, 1037 01:19:02,820 --> 01:19:05,720 you have more fluctuations in your system than for fermions. 1038 01:19:05,720 --> 01:19:07,810 And they conclusively showed that they 1039 01:19:07,810 --> 01:19:11,040 had realized the Hong-Ou-Mandel interference for fermions. 1040 01:19:37,390 --> 01:19:39,190 I think we have to stop. 1041 01:19:39,190 --> 01:19:46,780 Let me just add one more thing and then we are done. 1042 01:19:50,070 --> 01:19:54,760 So this Hong-Ou-Mandel interference 1043 01:19:54,760 --> 01:19:59,640 is at the heart of how perform the measurement which 1044 01:19:59,640 --> 01:20:03,200 is ultimately entangling the atoms. 1045 01:20:03,200 --> 01:20:05,180 But we need one more element. 1046 01:20:05,180 --> 01:20:09,130 We need sort of to scramble the photons in one more way, 1047 01:20:09,130 --> 01:20:17,860 and this is by adding circular polarizers at the input. 1048 01:20:17,860 --> 01:20:21,650 So we assume for now that we start out 1049 01:20:21,650 --> 01:20:26,860 with linear polarization in those states. 1050 01:20:26,860 --> 01:20:30,140 And here is our beam splitter. 1051 01:20:30,140 --> 01:20:35,710 Here is mode a and mode be, which we detect. 1052 01:20:35,710 --> 01:20:42,990 And before we measure this Hong interference, 1053 01:20:42,990 --> 01:20:47,930 we put in quarter-wave plates which 1054 01:20:47,930 --> 01:20:49,900 provide circular polarization. 1055 01:21:10,550 --> 01:21:13,120 If you start with linearly polarized light-- we 1056 01:21:13,120 --> 01:21:19,610 put in a polarizer-- after the circular polarizer, 1057 01:21:19,610 --> 01:21:21,640 we have this linear superposition 1058 01:21:21,640 --> 01:21:27,330 of horizontal and vertical at mode one, 1059 01:21:27,330 --> 01:21:30,130 and we have linear polarization-- superposition 1060 01:21:30,130 --> 01:21:34,340 of linear polarization in mode two. 1061 01:21:34,340 --> 01:21:42,100 So this is a situation at the input of the beam splitter. 1062 01:21:42,100 --> 01:21:47,440 And if we expand it, we have probabilities 1063 01:21:47,440 --> 01:21:51,860 where the polarization is different, 1064 01:21:51,860 --> 01:21:56,040 and where the polarization is the same. 1065 01:21:56,040 --> 01:21:59,280 So we have two detectors here. 1066 01:21:59,280 --> 01:22:10,310 And if both detectors click, then we 1067 01:22:10,310 --> 01:22:16,050 know that the input to the interferometer 1068 01:22:16,050 --> 01:22:20,840 was not H H nor V V, because in that case, 1069 01:22:20,840 --> 01:22:23,820 the Hong-Ou-Mandel interference would have directed 1070 01:22:23,820 --> 01:22:27,300 both photons to one output, and we would not 1071 01:22:27,300 --> 01:22:29,950 have obtained clicks from both. 1072 01:22:29,950 --> 01:22:34,840 So therefore, when both detectors click, 1073 01:22:34,840 --> 01:22:39,840 we know that the quantum state before the interferometer-- 1074 01:22:39,840 --> 01:22:48,620 before the beam splitter-- was H V or V H-- or, of course, 1075 01:22:48,620 --> 01:22:53,200 in another basis, one of those. 1076 01:22:53,200 --> 01:22:55,230 And this is sort of the ingredient, which 1077 01:22:55,230 --> 01:22:56,810 I will show you on Wednesday, which 1078 01:22:56,810 --> 01:23:00,230 can lead to probabilistic entanglement of atoms. 1079 01:23:00,230 --> 01:23:01,010 Any questions? 1080 01:23:04,250 --> 01:23:07,760 OK, one announcement-- we post this week's homework 1081 01:23:07,760 --> 01:23:09,340 assignment later today. 1082 01:23:09,340 --> 01:23:11,290 It will be due in a week. 1083 01:23:11,290 --> 01:23:13,300 See you on Wednesday.