1 00:02:32,091 --> 00:02:40,070 And then the rules of the, of the game are going to be completely analogous. So, the 2 00:02:40,070 --> 00:02:49,041 outcome is going to be you with probability. Let's call this angle theta 3 00:02:49,041 --> 00:03:02,031 prime. So, so, the outcome may be u with probability. Cosine squared theta prime 4 00:03:02,031 --> 00:03:10,031 and [inaudible] with probability. Sine squared theta prime. And of course the 5 00:03:10,031 --> 00:03:17,038 result of the measurement is that, if the outcome is U, then the new state is in 6 00:03:17,038 --> 00:03:24,063 fact this state U and in the other case the outcome is the new state is really U. 7 00:03:24,090 --> 00:03:33,008 So, here's another way of saying, of saying this whole thing. So, let's say 8 00:03:33,008 --> 00:03:47,084 that u was. And beta zero, zero + beta one, one. So [inaudible] might be - beta 9 00:03:47,084 --> 00:04:02,028 ten + beta 01. So these are orthogonal states. And now. The probability. You have 10 00:04:02,028 --> 00:04:11,040 this outcome cosine squared theta prime. This is exactly. The inner product between 11 00:04:11,040 --> 00:04:20,046 the state psi and u square. So if we were to write then a product this way, it's 12 00:04:20,046 --> 00:04:35,006 inner product of u. And sine. Squared. And this is in the product that we [inaudible] 13 00:04:35,006 --> 00:04:42,020 and sine squared. Okay, so let's see some examples and let's see how to work with 14 00:04:42,020 --> 00:04:50,049 all this. So. Let's first talk about a very special basis called the Hadamard 15 00:04:50,049 --> 00:04:56,092 Basis. This is this is a very important basis. It consist of, of these two states, 16 00:04:56,092 --> 00:05:02,088 the plus state which is this super position of zero and one with, with equal 17 00:05:02,088 --> 00:05:08,092 magnitudes one over square root of two, zero + one over square root of one. And 18 00:05:08,092 --> 00:05:14,015 the minus state which is. One / square root of zero - one / square root of one. 19 00:05:14,015 --> 00:05:19,024 So, what distinguished is that these two states is the, is the sign, the phase here 20 00:05:19,024 --> 00:05:25,056 in front of the, in front of the one. So now you could ask. Can we really, can we 21 00:05:25,056 --> 00:05:31,051 distinguish this plus state from the minus state through a measurement. Also, let's 22 00:05:31,051 --> 00:05:37,076 do a measurement in the 0,1 basis, in the standard basis so if you do a measurement 23 00:05:37,076 --> 00:05:43,049 in the standard basis, we end up saying, and if he, if he stop in the state, state 24 00:05:43,049 --> 00:05:53,047 plus. Then you see zero with probability one half and one with probability one 25 00:05:53,047 --> 00:06:00,013 half. In this case, the new state is zero. In this case the new state is one. What 26 00:06:00,013 --> 00:06:06,085 about if we start with the mindless state. Well once again, we see zero with 27 00:06:06,085 --> 00:06:12,078 probability. Square of this which is one-half and one with probability square 28 00:06:12,078 --> 00:06:18,051 of this of the phase goes away with probability one-half. And again in each of 29 00:06:18,051 --> 00:06:24,031 these cases the new status, zero, one. So, so the results are completely identical 30 00:06:24,031 --> 00:06:29,075 when we do a measurement. And the reason is clear, it's because, because the 31 00:06:29,075 --> 00:06:35,034 probability is determined by the square of the amplitude. And so whether it's 32 00:06:35,034 --> 00:06:40,031 positive or negative, it makes no difference to the outcome. So, standing 33 00:06:40,031 --> 00:06:45,092 there we can actually distinguish plus. The plus state from the minus state. Well, 34 00:06:45,092 --> 00:06:54,070 so. Here's an idea, why not measure in a different basis, so measure. In a 35 00:06:54,070 --> 00:07:01,027 different basis and some [inaudible] basis but which basis would you pick? Well, it's 36 00:07:01,027 --> 00:07:07,045 clear, right? If you want to distinguish plus or minus, why not measure it in the 37 00:07:07,045 --> 00:07:15,061 plus, minus basis. That's a perfectly good basis. And so, if you are mentioning the 38 00:07:15,061 --> 00:07:21,095 +,,,, - basis and you start with the plus state, what do you see? Well, the outcome 39 00:07:21,095 --> 00:07:28,076 is plus with probability one minus the probability zero. And on this case a new 40 00:07:28,076 --> 00:07:34,055 status plus, well this never happens but if it were to happen the new state would 41 00:07:34,055 --> 00:07:39,077 be minus. If you start with the minus state, you see plus with probability. 42 00:07:40,054 --> 00:07:47,054 Well, zero. And minus the probability of one. So, you end up distinguishing these 43 00:07:47,054 --> 00:07:52,053 two completely, it's probability one. You know for sure if you are measuring the 44 00:07:52,053 --> 00:07:57,047 plus minus basis and you start with the plus state, you always get plus and if 45 00:07:57,047 --> 00:08:02,045 it's other than minus state, you always get [inaudible] of minus. Okay. But now, 46 00:08:02,045 --> 00:08:08,025 suppose we start from a general state psi and you want to measure it in the plus, 47 00:08:08,025 --> 00:08:13,084 minus basis. How do you figure out what's the probability you, you see plus and 48 00:08:13,084 --> 00:08:19,093 what's probability you see minus? So this is this is an exercise in linear algebra 49 00:08:19,093 --> 00:08:25,067 and we are going to go through it just to make sure that, that you remember your 50 00:08:25,067 --> 00:08:30,078 linear algebra. And also, when we go through it. You know, we are actually 51 00:08:30,078 --> 00:08:36,065 going to be doing this in this [inaudible] notation. S o, so, not only the review of, 52 00:08:36,065 --> 00:08:42,089 of Linear Algebra for you but it will also be a way to familiarize yourself with this 53 00:08:42,089 --> 00:08:48,055 [inaudible] notation. Okay, so, so, here's a picture. There's, this is the standard 54 00:08:48,055 --> 00:08:54,008 basis, the 0,1 basis, has a +,,,, - basis to make my four angle. Okay? Here's, 55 00:08:54,008 --> 00:08:59,085 here's plus in terms of zero and one and so on. And now what we want to do is, we 56 00:08:59,085 --> 00:09:06,025 are given some state psi. Let's say it happens to be. You know, it, it happens to 57 00:09:06,025 --> 00:09:12,039 be a state psi which makes 60 degree angle. So this is what psi looks like. 58 00:09:12,039 --> 00:09:18,062 It's one-half zero + square root three over two, one. And now we want to measure 59 00:09:18,062 --> 00:09:24,020 the state in the plus, minus spaces. So, how do we figure out what's the 60 00:09:24,020 --> 00:09:30,042 probability of seeing the outcome plus? Okay. So, one way to compute is, is just 61 00:09:30,042 --> 00:09:35,086 by computing in a product, right? Remember, the probability that [inaudible] 62 00:09:35,086 --> 00:09:42,041 plus is the square in the product over that psi makes with plus. So we write it 63 00:09:42,041 --> 00:09:49,031 out, size this state pluses that state and we take the inner product. Take the square 64 00:09:49,031 --> 00:09:55,088 off of the magnitude so when you taking off products you get one-half times one 65 00:09:55,088 --> 00:10:02,046 over square root two + square root 3/2 times one over square root two. And so you 66 00:10:02,046 --> 00:10:08,088 add that, take the square and you get. Similarly, you can carry out and disperses 67 00:10:08,088 --> 00:10:14,072 also for probability of minus, it's the inner product that sine make at the minus 68 00:10:14,072 --> 00:10:20,056 state squared. But you can also, you know, you can work it out exactly like this so 69 00:10:20,056 --> 00:10:30,070 you can, you can also say that this is. The probability of plus which is just. 70 00:10:32,070 --> 00:10:41,082 Four - two square root three = eight. Okay. So, that's one way of computing 71 00:10:41,082 --> 00:10:49,052 these, these probabilities. Here's, here's another we, we can do with this. So, what 72 00:10:49,052 --> 00:11:01,047 we can do is, we can do a change of bases. So to figure out what the state si. What's 73 00:11:01,047 --> 00:11:08,009 the probability of plus. Minus when we measure psi to +,,,, - basis, what we'll 74 00:11:08,009 --> 00:11:13,073 do is first, we'll rewrite psi in the +,,,, - basis. So, we want to express it 75 00:11:13,073 --> 00:11:19,078 as beta one+ but beat 1- and then, if you are doing the measurement in the +,,,, - 76 00:11:19,078 --> 00:11:26,080 basis, we can just [inaudible] it off, the probability of + is beta bl magnitude 77 00:11:26,080 --> 00:11:33,025 squared and probability of - is beta one [inaudible] t squared. Okay, so how do we 78 00:11:33,025 --> 00:11:39,035 do this, this change of basis. Well, so, what we want to do is, we want to rewrite 79 00:11:39,035 --> 00:11:45,057 the zero state in terms of plus and minus and the one state in terms of plus and 80 00:11:45,057 --> 00:11:51,039 minus. So you can do that either by inspection, by seeing either zero must be. 81 00:11:52,020 --> 00:11:59,058 Equals to position of + and - or you can just, just solve, right. If you add these 82 00:11:59,058 --> 00:12:06,056 two together you will get the +,,,, +,,, -. Well, these two vectors will cancel. = 83 00:12:06,056 --> 00:12:16,058 two over square root two zero. Or zero is. Early you're just transposing. You get 84 00:12:16,058 --> 00:12:21,091 one / square root of two+ + one / square root of 2- and similarly, if you subtract 85 00:12:21,091 --> 00:12:28,093 these two. You get square root two x one and so one as square of two x plus, minus, 86 00:12:28,093 --> 00:12:35,064 minus. Okay, so now, we'll just go back and substitute this in, in here. So, psi 87 00:12:35,064 --> 00:12:42,062 which is, which is this super position, we just substitute for zero as being on 88 00:12:42,062 --> 00:12:50,076 number square of two+ + one / two squared- and similarly for one. And then we 89 00:12:50,076 --> 00:12:59,005 simplify so we gather the + together and the - together. And so this is beta 90 00:12:59,005 --> 00:13:05,027 [inaudible] that's beta one and now the probability that we see a plus is the 91 00:13:05,027 --> 00:13:12,013 outcome of the measurement is plus is just beta [inaudible] squared magnitude squared 92 00:13:12,013 --> 00:00:02,084 which again gives us the same values before. Okay. So now let's extend our 93 00:00:02,084 --> 00:00:10,090 understanding of the measurement principle to doing a measurement in an arbitrary 94 00:00:10,090 --> 00:00:18,067 basis. So, so let's, let's go back and review what we had. So. Again we are 95 00:00:18,067 --> 00:00:26,044 thinking about a two level system which is you know, the state of an electron of 96 00:00:26,044 --> 00:00:34,012 bonded energy so it's either in this ground state or excited state. Represented 97 00:00:34,012 --> 00:00:41,064 by zero and one so, okay so, so in general, the state of the system psi. Is 98 00:00:41,064 --> 00:00:48,006 some super position, alpha zero, zero + alpha one, one. And we already saw that we 99 00:00:48,006 --> 00:00:54,015 can view this as, as a vector and a two-dimensional vector space. It's a 100 00:00:54,015 --> 00:00:59,090 complex vector space but I'll, for convenience and drawing it out, I'll 101 00:00:59,090 --> 00:01:13,074 pretend it's all real. So this is zero and one and now I'll state psi. Is, is just 102 00:01:13,074 --> 00:01:27,099 some vector like t his. Okay. And we also saw what it meant to do a measurement. So 103 00:01:27,099 --> 00:01:34,018 when we do a measurement, this state collapses into either the zero state or 104 00:01:34,018 --> 00:01:40,071 the one state, either see the ground state or the excited state with probability 105 00:01:40,071 --> 00:01:47,015 cosine square theta. Theta is the angle that psi makes with, with the zero state. 106 00:01:47,081 --> 00:01:54,056 Okay and the new state is either the ground state or the excited state. But now 107 00:01:54,056 --> 00:02:01,021 in general, what we can do is we can measure not just to see that the state is 108 00:02:01,021 --> 00:02:07,096 a ground state or excited state. We can pick any orthogonal pieces for this, for 109 00:02:07,096 --> 00:02:16,025 this space so in other words what we can do is we can pick. Some basis consisting 110 00:02:16,025 --> 00:02:30,043 of, of the state u. [inaudible] And, and we can do a measurement in this arbitrary 111 00:02:30,043 --> 00:02:32,037 basis [inaudible].