1 00:00:00,000 --> 00:00:06,006 Okay, so in the last video we, we saw how the state of a qubit evolves through a 2 00:00:06,006 --> 00:00:14,002 rotation of the space. So now let's try to understand this a little better. So well, 3 00:00:14,002 --> 00:00:21,073 last time we, we talked about how you know for simplicity we thought of the qubit as 4 00:00:21,073 --> 00:00:28,008 having real amplitude, so we could represent the qubit on the two dimensional 5 00:00:28,008 --> 00:00:34,099 real space. And then we talked about how rotating the space, when you rotate the 6 00:00:34,099 --> 00:00:43,028 space through theta, you can write down a rotation matrix, r sub theta. Which says 7 00:00:43,028 --> 00:00:52,017 how the state of the system evolves. So, for example if the state was psi, then 8 00:00:52,017 --> 00:01:01,018 after the rotation it would become r sub theta times psi. Now, in actual fact, a 9 00:01:01,018 --> 00:01:07,080 qubits sits in a two dimensional complex vector space and so we've got to describe 10 00:01:07,080 --> 00:01:13,090 what, what the rotations of this complex vector space look like. Okay so, so 11 00:01:13,090 --> 00:01:23,084 rotations of this complex vector space are called unitary transformations. Okay and 12 00:01:23,084 --> 00:01:39,045 they are also specified by two by two matrices. So, so our complex rotation is 13 00:01:39,045 --> 00:01:51,017 going to be specified by a matrix like this a, b, c, d where a, b, c, and d are 14 00:01:51,017 --> 00:02:00,050 complex numbers. And what we're, what we're saying is if we start with a state 15 00:02:00,050 --> 00:02:08,058 psi, then under this unitary transformation, it evolves to the state u 16 00:02:08,058 --> 00:02:16,081 psi. Okay so, so now let's, let's try to understand what happens in different 17 00:02:16,081 --> 00:02:27,042 cases. What happens if psi = zero? What is u psi? Well, remember psi, psi = zero 18 00:02:27,042 --> 00:02:35,030 means, well, zero is also another way of writing it in standard vector notation 19 00:02:35,030 --> 00:02:44,044 would be as, as this vector10. And so if you multiply u times this vector, you 20 00:02:44,044 --> 00:02:56,027 would get, get the, get the vector ab which is a0+ b1. And similarly if you 21 00:02:56,027 --> 00:03:08,025 start in the state one, it gets mapped to c0+ d1, okay. Now, saying that u is a 22 00:03:08,025 --> 00:03:15,085 unitary transformation or it's a rotation of the two dimensional complex vector 23 00:03:15,085 --> 00:03:24,039 space, it's equivalent to saying this condition that u, u dagger = to u dagger u 24 00:03:24,039 --> 00:03:35,031 is the identity. Where u dagger, it's the conjugate transpose of u. Meaning first, 25 00:03:35,031 --> 00:03:42,053 we take complex conjugates of all the entries and then we transpose or we do it 26 00:03:42,053 --> 00:03:48,026 in the opposite order, it doesn't matter s o, meaning this one would become a 27 00:03:48,026 --> 00:03:54,081 conjugate, complex conjugate, dcomplex conjugate but these two entries are going 28 00:03:54,081 --> 00:04:02,090 to be exchanged. So this entry would become c conjugate, and this would become 29 00:04:02,090 --> 00:04:10,052 b conjugate. Okay, so now one of the conditions we want from this rigid body 30 00:04:10,052 --> 00:04:17,005 rotation is that angles are preserved. So for example if you do the rigid body 31 00:04:17,005 --> 00:04:24,015 rotation on the standard basis zero and one. What we want is that angles and 32 00:04:24,015 --> 00:04:33,021 lengths are preserved. So we want, we want that, since we started with the unit 33 00:04:33,021 --> 00:04:41,034 vector, this should still be a unit vector. Meaning that we want a^2 + b^2 = 34 00:04:41,034 --> 00:04:49,080 one = c^2 + d^2, right? We also want that these two should be orthogonal because a, 35 00:04:49,080 --> 00:04:56,091 zero, and one were orthogonal. So we want for example ac + bd = zero. So the 36 00:04:56,091 --> 00:05:05,027 inner product between these two is< /i>< /i> zero. Okay now, you can see that these 37 00:05:05,027 --> 00:05:18,003 conditions are expressed here in, in, in this because when you look at u dagger u 38 00:05:18,003 --> 00:05:38,062 well, this is. A b c d x a, sorry abcd which is what? Well, this is supposed to 39 00:05:38,062 --> 00:05:51,097 be the identity. But as you can see this entry is just the inner product of ab in 40 00:05:51,097 --> 00:06:01,004 itself. It's just magnitude of a^2 + magnitude of b^2. And similarly this entry 41 00:06:01,004 --> 00:06:12,075 is magnitude of c^2 + magnitude of d^2 where as this entry is the inner product 42 00:06:12,075 --> 00:06:22,042 of ab and cd. And so, so this is saying, this is saying that these two vectors are 43 00:06:22,042 --> 00:06:32,012 orthogonal and this says the same thing and these two say the length of, of the 44 00:06:32,012 --> 00:06:45,093 vectors are preserved. Another property of you know of unitary transformations these 45 00:06:45,093 --> 00:06:54,036 rotations is that they are linear, right? This is what gave us the right to, to 46 00:06:54,036 --> 00:07:04,041 specify the, the rotation by a matrix. So what this means is that if you say that u, 47 00:07:04,041 --> 00:07:11,092 you know u you know if, if I tell you what, what, what the unitary 48 00:07:11,092 --> 00:07:21,025 transformation does on zero, let's say that it maps it to phi zero, which people 49 00:07:21,025 --> 00:07:32,088 calling a0 + b1 and suppose it maps one to phi one which we were calling c0 + d1. 50 00:07:32,088 --> 00:07:47,044 Now, we could ask what does it do? What does u do to the, to the vector a0 + b1? 51 00:07:47,044 --> 00:08:00,010 And the answer is by linearity, it must map it to a phi zero + b phi one. Okay and 52 00:08:00,010 --> 00:08:07,091 you can, you can of course further simplify it and write what, what it looks 53 00:08:07,091 --> 00:08:11,091 like in terms of zero and one. You would just say it's a(a(0) + b(1)) + b ((c0) + 54 00:08:11,091 --> 00:08:33,010 (d1)) which is (aa + bc)(0) + (ab + bd)(1). Okay so, the main point is that we 55 00:08:33,010 --> 00:08:44,062 can specify this unitary transformation by saying what it does on each of the basis 56 00:08:44,062 --> 00:08:51,057 vectors, okay? Not just in two dimensions. Not, not just for qubit but this also 57 00:08:51,057 --> 00:08:58,001 holds, if you were specifying what it does for, you know if you had a, if you had a K 58 00:08:58,001 --> 00:09:03,065 level system then you could specify what the unitary transformation does on the K 59 00:09:03,065 --> 00:09:09,006 basis vectors and you'd have specified the, the unitary transformation 60 00:09:09,006 --> 00:09:15,001 completely. And of course, there's no mystery to this because as we said, the 61 00:09:15,001 --> 00:09:21,007 columns of u are exactly where you map each of these k basis factors.