1 00:00:00,000 --> 00:00:05,008 Okay. So in this video, we are going to look at some remarkable properties of the 2 00:00:05,008 --> 00:00:11,006 Bell state. So, the property that we'll be interested in is what's called the 3 00:00:11,006 --> 00:00:17,005 rotational invariance of the Bell State. So remember what the, what the Bell State 4 00:00:17,005 --> 00:00:23,004 is. It's a, it's a state of two qubits where they are in a equal superposition of 5 00:00:23,004 --> 00:00:30,000 being in the state 00 or eleven. So now you remember that if we measure the first 6 00:00:30,000 --> 00:00:35,009 qubit, if we see a zero, then if we measure the second qubit, its also going 7 00:00:35,009 --> 00:00:42,006 to be, we, we are going to see it in the state zero with probability one. Now what 8 00:00:42,006 --> 00:00:48,008 the rotational invariance of the Bell State says is the, is the following day 9 00:00:48,008 --> 00:00:55,001 remarkable property. What it says is, suppose that we were to measure this Bell 10 00:00:55,001 --> 00:01:13,084 State so that's a, base is 01. Instead of measuring it in this 01 basis, let's 11 00:01:13,084 --> 00:01:23,096 measure it in some basis uu perp. The u and u perp are just arbitrary rotations. 12 00:01:23,096 --> 00:01:31,013 They're, this is just some arbitrary angle here with the zero state in the real 13 00:01:31,013 --> 00:01:38,006 plane. And we do the same thing for the other Bell State. We measure this one the 14 00:01:38,006 --> 00:01:48,018 second, sorry, the other qubit in the Bell State. We measured also In you. You built. 15 00:01:48,018 --> 00:02:01,094 Okay. The remarkable fact is that if we, if you are to carry out this measurement, 16 00:02:01,094 --> 00:02:11,001 well first of all, we'll see u or u perp with equal probability with 50, 50 17 00:02:11,001 --> 00:02:17,095 probability. Okay, so the probability that the outcome is u is one-half. Probability 18 00:02:17,095 --> 00:02:24,083 of u perp is also one-half but if the probability we see the first outcome is u 19 00:02:24,083 --> 00:02:31,089 if that's what happens, then when we make a measurement on the second qubit, we'll 20 00:02:31,089 --> 00:02:39,024 also see u with probability one. So in other words, we'll always see matching 21 00:02:39,024 --> 00:02:45,061 outcomes in the two cases even though if you look at any one case it's 50, 50 22 00:02:45,061 --> 00:02:51,088 whether we get u or u perp. So this is exactly the same situation we had, when we 23 00:02:51,088 --> 00:02:57,073 measured in the 01 basis except it holds no matter what the basis is, okay. This 24 00:02:57,073 --> 00:03:03,080 remarkable property is called the Rotational Invariance of the Bell State. 25 00:03:03,080 --> 00:03:11,049 So now, let's try to understand this a little more. You know let's, let's see how 26 00:03:11,049 --> 00:03:21,074 well we understand this so let's say that we make a measu rement on this first qubit 27 00:03:21,074 --> 00:03:32,066 in some arbitrary basis uu perp and now they pick a different basis. So we pick, 28 00:03:32,066 --> 00:03:44,038 pick some, some different bases v, v perp for the second qubit and we measure it in 29 00:03:44,038 --> 00:03:53,070 this basis. So let's say, this was u and this angle is theta. So what we want to 30 00:03:53,070 --> 00:04:00,037 know is, suppose we do these measurements, we measure the first qubit of the Bell 31 00:04:00,037 --> 00:04:06,020 State in this uu perp basis. We measure the second cubit in the vv perp where 32 00:04:06,020 --> 00:04:13,059 these two bases make an angle of theta with each other. Now, what we want to know 33 00:04:13,059 --> 00:04:22,096 is. What is the probability of matching outcomes? Where we will see, we see that 34 00:04:22,096 --> 00:04:30,020 the outcomes are matching if we get this outcome on the first qubit and that 35 00:04:30,020 --> 00:04:37,041 outcome on the second qubit or if we get this outcome on the first cubit and that 36 00:04:37,041 --> 00:04:44,059 one on the second cubit right? So we see this matches with that and u perp matches 37 00:04:44,059 --> 00:04:51,008 with v perp. So, whats the chance that we will get matching outcomes? Well I claim 38 00:04:51,008 --> 00:04:57,008 that we already, from what we saw in the last slide, we already know how to answer 39 00:04:57,008 --> 00:05:03,009 this question. Why? Because when we measure the first qubit, we either get 40 00:05:03,009 --> 00:05:09,007 back in u or u perp with 50, 50 probability. So it doesn't matter which 41 00:05:09,007 --> 00:05:14,063 one we get. Let's say we get the outcome u. Now, what's the state of the second 42 00:05:14,063 --> 00:05:20,053 qubit? Well, I claim the second qubit is in the state u, right? That's what we saw 43 00:05:20,053 --> 00:05:26,033 in the previous slide but now, this is the state of the second qubit and we are 44 00:05:26,033 --> 00:05:33,065 measuring it in the v perp, vv perp basis. What's the chance we get outcome v? Well, 45 00:05:33,065 --> 00:05:42,005 the answers exactly cosine squared theta. Okay, so the same thing, the same 46 00:05:42,005 --> 00:05:48,003 reasoning holds if you happen to get u perp in the first as, as the outcome of 47 00:05:48,003 --> 00:05:54,055 our first measurement. Well, then the state of the second qubit must be u perp 48 00:05:54,055 --> 00:06:00,073 as well because they completely match. But now, if you measure the second qubit in 49 00:06:00,073 --> 00:06:07,014 the vv perp basis, what's the chance we get output v perp? Well, it's still cosine 50 00:06:07,014 --> 00:06:13,039 squared theta so the, so the, the answer is if you measure the two qubits in basis 51 00:06:13,039 --> 00:06:19,037 which make an angle of theta with each other, the chance that we get matching 52 00:06:19,037 --> 00:06:25,087 outcomes is exactly cosine squared theta, okay. So how do we prove this? Well the 53 00:06:25,087 --> 00:06:32,067 way you prove this is by saying that if you were to write your state psi it's, 54 00:06:32,067 --> 00:06:38,089 it's an equal superposition of 00 + eleven. In the last lecture we saw that we 55 00:06:38,089 --> 00:06:44,099 could also write it as an equal super position of plus, plus and minus, minus. 56 00:06:44,099 --> 00:06:54,052 What we are claiming is in fact if you have an arbitrary uu perp, then you can 57 00:06:54,052 --> 00:07:06,085 write 00 + eleven as uu perp + vv perp. Okay so, so just think of u as being some 58 00:07:06,085 --> 00:07:27,044 arbitrary vector is a(0) + b(1) where a and b are real numbers. And then u perp, 59 00:07:27,044 --> 00:07:33,081 you can write as b(0) - a(1). Actually, the way I've drawn it here, it's - b(0) + 60 00:07:33,081 --> 00:07:43,007 a(1). And now what you would do is you would substitute this into, into this 61 00:07:43,007 --> 00:07:56,099 formula and you'd say well, what's one / two^2 uu perp. It's just a(0) + b(1) a(0) 62 00:07:56,099 --> 00:08:13,032 + b(1) + one / two^2 x u perp u perp which is - b(0) + a(1) - b(0) + a(1). And if you 63 00:08:13,032 --> 00:08:25,042 expand this out and you simplify you'll get one / two^2 x 00 + one / two^2 x 64 00:08:25,042 --> 00:08:37,036 eleven. Let's see, should I leave this as an exercise? Well, here's the other 65 00:08:37,036 --> 00:08:44,045 condition we have. Of course u and u perp unit vectors, they have a normalization 66 00:08:44,045 --> 00:08:50,069 condition which isa^2 + b^2 = one. Okay, so let's just, let's just eyeball it and 67 00:08:50,069 --> 00:08:55,056 lets see, what's the amplitude of 00? Well, there are two ways of achieving 00. 68 00:08:55,056 --> 00:09:01,023 The, the amplitude here is a squared. The amplitude from here is b^2 x one-half^2. 69 00:09:01,023 --> 00:09:10,003 So it's one / two^2 x a^2 + b^2 but a^2 + b^2 = one so you get total amplitude is 70 00:09:10,003 --> 00:09:17,014 one / two^2. The same thing holds for the amplitude of eleven. It's a^2 + b^2 and so 71 00:09:17,014 --> 00:09:25,074 that's, that's one / two^2. What about the amplitude of the 01? Will you achieve 01 72 00:09:25,074 --> 00:09:32,064 from here with amplitude ab one / two^2? And from here with amplitude -b so b 73 00:09:32,064 --> 00:09:38,068 cancel, and so you get amplitude zero, okay. The same is true for ten and so, so 74 00:09:38,068 --> 00:09:46,034 you get, that the Bell State can be written in any basis as uu perp. Uu + u 75 00:09:46,034 --> 00:09:55,043 perp u perp. Now, there's one other thing that I should say that here I use the 76 00:09:55,043 --> 00:10:03,039 restriction that, that this uu perp basis has to be a real, valued basis, okay. So 77 00:10:03,039 --> 00:10:10,019 the, the a and b, the, the coefficients here are not allowed to be complex value. 78 00:10:10,019 --> 00:10:17,012 If you do want to use complex coefficients, then you can use a different 79 00:10:17,012 --> 00:10:26,061 kind of Bell State. This, this Bell State is called psi minus. And it's of the form 80 00:10:26,061 --> 00:10:37,048 one / two^2 01 - one / two^2 ten. And this Bell State is actually invariant on the, 81 00:10:37,048 --> 00:10:46,076 any change of base. You know if you, if you measure it in uu perp. Where u is now 82 00:10:46,076 --> 00:10:59,082 of the form a(0) + b(1) where the alpha and beta are complex numbers. And u perp 83 00:10:59,082 --> 00:11:10,060 is -b0 + a1. These are< /i>< /i> complex conjugates. You know of course 84 00:11:10,060 --> 00:11:17,087 they satisfy this condition a^2 + b^2 = one. And now you can, you can actually 85 00:11:17,087 --> 00:11:24,085 verify if you like that, you know the same sort of rotational invariance holds except 86 00:11:24,085 --> 00:11:30,089 now its not just real rotations but also complex rotations on the complex plane but 87 00:11:30,089 --> 00:11:36,038 this is not, you know if you don't follow this, its okay because we are not going to 88 00:11:36,038 --> 00:11:42,038 be dealing with this. For the purposes of this lecture we are only going to be 89 00:11:42,038 --> 00:11:48,001 dealing with real rotations and what I said so far is all that you need to know.