1 00:00:00,000 --> 00:00:06,005 Okay. So, in this video we'll finally get to the, to Bell's experiment. Okay. So, 2 00:00:06,005 --> 00:00:12,004 let's, let me review where, what we're trying to do in Bell's experiment. 3 00:00:12,004 --> 00:00:18,009 Remember that in 1964, Bell devised this remarkable experiment, which had one of 4 00:00:18,009 --> 00:00:25,002 two outcomes. If outcome one is true, the nature is inconsistent with quantum 5 00:00:25,002 --> 00:00:30,085 mechanics, and in fact, can be explained by some local hidden variable theory. But 6 00:00:30,085 --> 00:00:37,003 if outcome two is true, then, nature is inconsistent with any local hidden 7 00:00:37,003 --> 00:00:42,008 variable theory. It rules out every local hidden variable theory, and in fact, 8 00:00:42,008 --> 00:00:48,000 nature is consistent with quantum mechanics. And so, Einstein's dream of 9 00:00:48,000 --> 00:00:53,007 finding this more complete picture for nature, you know, more complete theory, is 10 00:00:53,007 --> 00:00:58,079 doomed. Okay. So, and remember this was the abstract setup that we have. Apparatus 11 00:00:58,079 --> 00:01:04,079 was divided into two pieces which were at distant ends of the lab, so, that during 12 00:01:04,079 --> 00:01:11,067 the course of the experiment, light did not have enough time to travel from one to 13 00:01:11,067 --> 00:01:17,029 the other. And then, we give to each of these, so, each of these pieces of the 14 00:01:17,029 --> 00:01:24,001 apparatus we called boxes, and we gave each one, we input a bit 01 and each of 15 00:01:24,001 --> 00:01:30,002 these spit out, or output a bit. And what we wanted was, if both of the input bits 16 00:01:30,002 --> 00:01:36,006 are one, then we want the output bits from the boxes to be different. But in all the 17 00:01:36,006 --> 00:01:43,003 other three cases, we want the outputs to be the same. We also show in the first 18 00:01:43,003 --> 00:01:49,006 video, that if the boxes are described by local hidden variables, then they cannot 19 00:01:49,006 --> 00:01:55,003 succeed with probability greater than three quarters in this task. We also 20 00:01:55,003 --> 00:02:01,003 claimed that Bell showed that there's a way to use, in quantum mechanics, using 21 00:02:01,003 --> 00:02:06,085 entanglement to actually succeed at this task with probability as high as .85. So, 22 00:02:06,085 --> 00:02:13,049 in this video, we are going to see how exactly this is achieved quantumly. Okay. 23 00:02:13,049 --> 00:02:20,009 So, the way it's achieved is by using entanglement. So, the two boxes, each of 24 00:02:20,009 --> 00:02:28,001 the two boxes is going to contain a cubit and these two, two cubits are going to be 25 00:02:28,001 --> 00:02:36,007 entangled with each other, in this bell state, 00 plus eleven. Now, when the input 26 00:02:36,007 --> 00:02:44,007 is given, what we are going to do, i s we are going to measure the first cubit in 27 00:02:44,007 --> 00:02:52,047 one or another basis. So, we are going to measure it in, in some basis which we'll 28 00:02:52,047 --> 00:02:58,009 pick. Okay. So, we'll measure it in, in some particular basis, you, you perp, 29 00:02:58,009 --> 00:03:05,002 which depends upon, which basis depends upon whether the input was zero or one. 30 00:03:05,002 --> 00:03:12,003 Similarly, we are going to measure the second cubit in the second box, in some 31 00:03:12,003 --> 00:03:21,058 bases V, V perp, depending again, on whether the input Y is zero or one, which 32 00:03:21,058 --> 00:03:26,040 pieces we choose depends upon the input. And then, if the, if the outcome of the 33 00:03:26,040 --> 00:03:32,049 first measurement is, U, will, the box will output zero. And if it's U perp, it 34 00:03:32,049 --> 00:03:37,055 will output one. And similarly, if the outcome of the second measurement is V, 35 00:03:37,055 --> 00:03:43,005 the second box will output zero. And otherwise, it will output one. Now, in the 36 00:03:43,005 --> 00:03:50,054 last video we talked about what's the chance that the two outputs match? And we 37 00:03:50,054 --> 00:03:58,065 said, the chance of that is exactly given by this. So, you look at the angle between 38 00:03:58,065 --> 00:04:06,051 U and V and if that angle is theta, the chance that you have matching outputs, 39 00:04:06,051 --> 00:04:15,021 that your output zero in both cases are one and both cases is exactly. So, the 40 00:04:15,021 --> 00:04:24,026 probability of matching outputs is exactly cosine squared theta. Okay, so what we 41 00:04:24,026 --> 00:04:30,014 want to do, is we want to set things up cleverly, so that, in the three cases 42 00:04:30,014 --> 00:04:37,026 where we want the outputs to match, cosine square theta is close to one, and in the 43 00:04:37,026 --> 00:04:43,036 cases that the two matches, two outputs are supposed to not match, they are 44 00:04:43,036 --> 00:04:50,028 supposed to be different, we want to make sure that cosine square theta is almost 45 00:04:50,028 --> 00:04:56,023 zero. Right? Or sine square theta is close to one. Okay, so how do we achieve this? 46 00:04:56,023 --> 00:05:00,071 So, we have to pick these angles carefully. And in fact, it's a, once you 47 00:05:00,071 --> 00:05:06,081 think about it a little bit, there's sort of a unique way of picking these angles to 48 00:05:06,081 --> 00:05:13,029 maximize these probabilities. That some of these probabilities and that's, that's 49 00:05:13,029 --> 00:05:20,036 done like this. So, let me color code it, because I have to, I have to use several 50 00:05:20,036 --> 00:05:28,015 possibilities. So, okay. So, lets, lets draw the axis, the measurement basis for 51 00:05:28,015 --> 00:05:41,057 the first box first. So, in the case that, that X=0, let's say that w e'll do the 52 00:05:41,057 --> 00:05:48,085 measurement in the U knot, U knot perp basis, which is just the standard basis. 53 00:05:48,085 --> 00:06:00,098 And in the case X=1, we just rotate this basis by 45 degrees or Pi by four. And 54 00:06:00,098 --> 00:06:15,062 let's call this space as U1, and U1 perp. Okay. So, now what about the second box? 55 00:06:15,062 --> 00:06:24,080 How do the measurement? So, if Y=1, we do the measurement at an angle of Pi by 56 00:06:24,080 --> 00:06:37,084 eight. So let me, let me draw these dotted lines for the coordinate axis and then, we 57 00:06:37,084 --> 00:06:52,027 have an angle of Pi by eight. And let's call this V knot, V knot perp. On the 58 00:06:52,027 --> 00:07:05,091 other hand, if the, if Y=1, then we do measurement at minus Pi by eight, right? 59 00:07:05,091 --> 00:07:19,005 So, sorry. That minus Pi by eight, and that's our basis. P1, P1 perp. Okay. So, 60 00:07:19,005 --> 00:07:26,005 now what do we want from all these? Well, what we want is in the three cases, we 61 00:07:26,005 --> 00:07:32,006 want the angle between the first basis and the second basis to be small, so that 62 00:07:32,006 --> 00:07:38,002 cosine square theta is close to one. And in the last case, where X=1 and Y=1, we 63 00:07:38,002 --> 00:07:44,003 want the angle to be close to Pi by two, so, that cosine square theta is close to 64 00:07:44,003 --> 00:07:49,056 zero. So, there. So, we got different results in this, in this fourth case. So, 65 00:07:49,056 --> 00:07:56,094 let's just list out the cases, and see how well we do. Well, we can look at X equal 66 00:07:56,094 --> 00:08:08,046 to, or let's, let's look at the case where X, Y equal to, and what's the probability 67 00:08:08,046 --> 00:08:16,048 of match? Okay. So, if X=0, Y=0, well, then, yeah, we are measuring in the black 68 00:08:16,048 --> 00:08:22,057 pieces on both sides, and the angle between them is exactly Pi by eight. So, 69 00:08:22,057 --> 00:08:29,008 the probability of match is cosine square Pi by eight. If X=0, Y=1, then we are 70 00:08:29,008 --> 00:08:34,063 measuring black here and blue there. But the angle between these is also Pi by 71 00:08:34,063 --> 00:08:41,000 eight. So, the probability of matches cosine squared Pi by eight. What if X=1, 72 00:08:41,000 --> 00:08:46,095 Y=0? Then, we have blue here and black here. But look, the angle is still Pi by 73 00:08:46,095 --> 00:08:52,088 eight. So, the probability of match is cosine squared Pi by eight. Okay. That's a 74 00:08:52,088 --> 00:08:59,098 very nice choice of, choice of basis, because we don't have to, we, we just have 75 00:08:59,098 --> 00:09:07,026 this one number to calculate. What about the fourth case? Well now, we have X=1 and 76 00:09:07,026 --> 00:09:15,049 Y=1. In this case, we are measuring in the blue basis. So, the angle between them is 77 00:09:15,049 --> 00:09:22,044 Pi by four plus Pi by eight. So, it's three Pi by eight. So, the chance of match 78 00:09:22,044 --> 00:09:29,075 is cosine squared three Pi by eight, which is the same as, sine squared Pi over two 79 00:09:29,075 --> 00:09:36,074 minus three Pi by eight. This is the same as, sine squared, with Pi over two minus 80 00:09:36,074 --> 00:09:44,069 three Pi by eight is also, Pi over eight. But in this case, we succeed not when they 81 00:09:44,069 --> 00:09:53,081 match, but when they don't match, So, so in this case, whats the probability of not 82 00:09:53,081 --> 00:10:03,064 match? And the probability of not match in this case is, one minus sine squared Pi by 83 00:10:03,064 --> 00:10:10,020 eight which is, cosine squared Pi by eight. So, this was a very clever choice 84 00:10:10,020 --> 00:10:17,009 of basis because in each of these four cases, we succeed with probability exactly 85 00:10:17,009 --> 00:10:23,055 cosine squared Pi by eight. So, the total success probability of this particular 86 00:10:23,055 --> 00:10:30,003 strategy, this, this way of carrying out the experiment, so, probability of 87 00:10:30,003 --> 00:10:39,046 success, is exactly cosine squared Pi by eight, which if you work out, is 88 00:10:39,046 --> 00:10:50,007 approximately .85. And this is the content of Bell's theorem. Right? Okay. Now, let 89 00:10:50,007 --> 00:10:56,081 me just say one more thing which might make things a little more clearer, which 90 00:10:56,081 --> 00:11:07,007 is, what did we do here? Well, what we did here was, it says though we, we put four 91 00:11:07,007 --> 00:11:18,004 numbers on the, on the number line. So, we put the four X, wee put the, so, this was, 92 00:11:18,004 --> 00:11:31,000 this was X0. If, if X=0. And so, let me write this in blue. If this was X=1. And 93 00:11:31,000 --> 00:11:43,027 then we put a point down for Y=0, and this was, and then we put down a point for Y=1. 94 00:11:43,027 --> 00:11:50,098 And we let the distances between each of these, we let each of these distances be 95 00:11:50,098 --> 00:11:59,013 Pi over eight. So, think of these as the angles that we had. So, these were Pi over 96 00:11:59,013 --> 00:12:07,087 eight. And now, what we're saying is there are four cases. Either X=Y=0, the distance 97 00:12:07,087 --> 00:12:14,040 is Pi over eight, or X is zero, Y=1 Pi over eight, or Y is zero, X is one Pi over 98 00:12:14,040 --> 00:12:20,028 eight, or X is one, Y is one, three Pi over eight. Okay. So, this is, this is all 99 00:12:20,028 --> 00:12:26,056 we did in the, in setting up those angles. Okay. For some of you, this might help, 100 00:12:26,056 --> 00:12:32,080 make things clearer. If it doesn't, just ignore this. So, what did we actually see 101 00:12:32,080 --> 00:12:39,073 in all these? Well, to recap, John Bell devised this remarkable experime nt which 102 00:12:39,073 --> 00:12:47,011 has one of two outcomes. Either, you do this experiment, you set it up exactly the 103 00:12:47,011 --> 00:12:53,001 way, you know, quantum mechanics predicts. If the success probability ends up being 104 00:12:53,001 --> 00:12:58,085 less than or equal to three quarters, then you have to conclude that nature is 105 00:12:58,085 --> 00:13:04,077 inconsistent with quantum mechanics. But it's consistent with some local hidden 106 00:13:04,077 --> 00:13:10,048 variable theory, or you do the experiment, and the success probability is strictly 107 00:13:10,048 --> 00:13:15,038 greater than three quarters, to within your area parameters, and then you have to 108 00:13:15,038 --> 00:13:20,035 conclude that nature is inconsistent with any local hidden variable theory. 109 00:13:20,035 --> 00:13:26,040 Although, it might be consistent with quantum mechanics. The Bell experiment has 110 00:13:26,040 --> 00:13:32,030 been preformed numerous times. The results have always been consistent with quantum 111 00:13:32,030 --> 00:13:38,036 mechanics. There are still a few skeptics who say, well, you know, there are error 112 00:13:38,036 --> 00:13:44,036 bias, and if you were to assume, you know, because the detectors are not perfectly 113 00:13:44,036 --> 00:13:49,051 efficient, you don't have sources of single photons that are perfectly 114 00:13:49,051 --> 00:13:55,077 reliable, and if you assume that all these factors conspire against you in the worst 115 00:13:55,077 --> 00:14:01,022 possible way, then it's, it's still possible that quantum mechanics might be 116 00:14:01,022 --> 00:14:07,063 incorrect. That, that it might still be consistent with some sort of other theory. 117 00:14:07,063 --> 00:14:13,062 But although, dealing with these, with, with these kinds of loopholes is an 118 00:14:13,062 --> 00:14:20,016 interesting questions, at this point, one would have to say that, for all practical 119 00:14:20,016 --> 00:14:26,060 purposes, you know, these loopholes are so contrive, that we may as well believe that 120 00:14:26,060 --> 00:14:32,073 local hidden variable theories have not been ruled out by experiment. Here's one 121 00:14:32,073 --> 00:14:38,071 last word, which is Einstein spent the last twenty years of his life, trying to 122 00:14:38,071 --> 00:14:45,015 show that there is a local hidden variable theory that is consistent with everything 123 00:14:45,015 --> 00:14:51,055 we know in quantum mechanics, which is, which is consistent with everything we 124 00:14:51,055 --> 00:14:58,005 know about the nature. It wasn't until about ten years after his passing that, 125 00:14:58,005 --> 00:15:04,009 John Bell came up with this, with this experiment, but the remarkable thing to 126 00:15:04,009 --> 00:15:10,000 realize is that we're already in the fourth lecture, in the second week of 127 00:15:10,000 --> 00:15:15,060 your, for many of yo u, first encounter with quantum mechanics. You can already 128 00:15:15,060 --> 00:15:21,030 understand something which is simple enough and yet it is something that eluded 129 00:15:21,030 --> 00:15:27,011 Einstein himself and which if he had understood would have saved him twenty 130 00:15:27,011 --> 00:15:32,002 years of work. Isn't that a remarkable thing to think about?