1 00:00:00,000 --> 00:00:05,006 Okay, so in this video and the next video, we finally talk about quantum 2 00:00:05,006 --> 00:00:12,001 teleportation. I'm sure you've heard about quantum teleportation. You've seen, seen 3 00:00:12,001 --> 00:00:18,009 it in Star Trek. How to teleport an object from one place to another. This is, and 4 00:00:18,009 --> 00:00:25,082 quantum teleportation is a rough analogy with that so you will see that there are 5 00:00:25,082 --> 00:00:31,057 similarities. There are, there are very big differences but what, what is the 6 00:00:31,057 --> 00:00:37,054 basic idea in quantum teleportation? So, you know that it's impossible to clone a 7 00:00:37,054 --> 00:00:43,013 qubit but in quantum teleportation you want to, you want to transport a qubit 8 00:00:43,013 --> 00:00:48,097 from one location to another. And well, one way to transport it is if you have a 9 00:00:48,097 --> 00:00:54,013 quantum channel. If you could carry the quantum bit from one place to the other 10 00:00:54,013 --> 00:00:59,007 but it turns out that there is other way of transporting it from one place to 11 00:00:59,007 --> 00:01:05,052 another which is called teleportation. So lets see what that means. Well first, 12 00:01:05,052 --> 00:01:13,053 here, here is the set up. Okay, so you have two, let's say there are two 13 00:01:13,053 --> 00:01:19,009 physicists. Alice and Bob, experimentalists and Alice has created a 14 00:01:19,009 --> 00:01:26,009 state which it took her a long time to create it. It's a, it's a very interesting 15 00:01:26,009 --> 00:01:32,081 state. It, it's some superposition, a(0) + b(1). She doesn't quite know what, what 16 00:01:32,081 --> 00:01:38,071 alpha and beta are but it's just the output of some process that she designed. 17 00:01:38,071 --> 00:01:44,051 And now she wants to, she wants to run this through this quantum state through 18 00:01:44,051 --> 00:01:49,039 some, some other apparatus which she doesn't have but Bob has in his lab. But 19 00:01:49,039 --> 00:01:55,060 Bob is, Bob's lab is across campus at the other end of campus. How should she get a 20 00:01:55,060 --> 00:02:02,065 qubit from her lab to his lab? Maybe it's, it's all the way across the city. How, how 21 00:02:02,065 --> 00:02:08,091 should she transport this qubit from one location to another? Okay so, one thing 22 00:02:08,091 --> 00:02:15,069 she could do is she could try to send it from her lab to his lab but let's say that 23 00:02:15,069 --> 00:02:21,081 these two labs are very distant and it's very hard to transport it from one place 24 00:02:21,081 --> 00:02:28,034 to the other. So what else could she do? Well of course she, she, she cannot really 25 00:02:28,034 --> 00:02:35,074 make a copy of her qubit but what she might try to do is try to figure out. 26 00:02:35,074 --> 00:02:47,036 Alpha and beta. And then give that inform ation to both but of course alpha and beta 27 00:02:47,036 --> 00:02:53,085 are complex numbers. Not only do they require, you know she, what, what Alice 28 00:02:53,085 --> 00:02:59,065 has to do is figure out to sufficient precision, what alpha and beta are. But 29 00:02:59,065 --> 00:03:06,039 what you can do is measure the state and she doesn't, you know if she measures the 30 00:03:06,039 --> 00:03:13,096 state, she does not really get alpha and beta. What she sees is zero with 31 00:03:13,096 --> 00:03:25,039 probability a^2 and one with probability b^2. Moreover, as soon as she does the 32 00:03:25,039 --> 00:03:32,079 measurement the new state is whatever she saw, and so she can't repeat the 33 00:03:32,079 --> 00:03:40,037 experiment. So if she has only this one copy of the state, she really cannot 34 00:03:40,037 --> 00:03:47,067 figure out what alpha and beta are. So what is she to do? Okay, so this is where 35 00:03:47,067 --> 00:03:53,083 quantum teleportation comes in. So let's say that Alice and Bob were ready for this 36 00:03:53,083 --> 00:04:00,093 eventuality. So once upon a time, they set up, you know they shared a Bell pair with 37 00:04:00,093 --> 00:04:07,047 each other. So meaning that they have these two qubits. Alice has a qubit and 38 00:04:07,047 --> 00:04:14,021 Bob has a qubit. And they happen to be in the state (00) + (eleven) in the Bell 39 00:04:14,021 --> 00:04:20,060 State. Of course this, this Bell State has nothing to do with, with the qubit that 40 00:04:20,060 --> 00:04:28,084 Alice wants to transport both. So, quantum teleportation is this, remarkable protocol 41 00:04:28,084 --> 00:04:35,056 by which Alice can do something with her, with her two qubits. The one that she 42 00:04:35,056 --> 00:04:43,003 wants to transport and the qubit out of the Bell pair. She does something to these 43 00:04:43,003 --> 00:04:51,018 two, she measures these two. So what, what Alice does is she takes these two qubits 44 00:04:51,018 --> 00:04:58,084 and she performs a measurement. Okay, what's the result of this measurement? 45 00:04:58,084 --> 00:05:07,031 Well, you know she is measuring two qubits, the result is she gets two bits of 46 00:05:07,031 --> 00:05:17,068 information. She gets two bits, Lets call them b1 and b2. These are classical bits. 47 00:05:17,068 --> 00:05:26,039 Right its 00, 01, ten or eleven. So now she picks up the phone and calls up Bob 48 00:05:26,039 --> 00:05:39,034 and she transmits these two bits across to Bob by phone. At this point you might 49 00:05:39,034 --> 00:05:47,043 think, Alice what did you do? You've destroyed your qubit. Well, now what Bob 50 00:05:47,043 --> 00:05:56,038 does is based on what, what Alice told him about b1 and b2. He preforms some single 51 00:05:56,038 --> 00:06:05,086 qubit gates or he actually performs two gates on his qubit. Which gate? Really 52 00:06:05,086 --> 00:06:15,031 depends o n what b1 and b2 are. And lo and behold, what happens is, after performing, 53 00:06:15,031 --> 00:06:27,055 so, so he performs some unitary rotation, it's going to be a simple unitary rotation 54 00:06:27,055 --> 00:06:36,084 depending upon b1 and b2 and what happens? Well, what happens is magically this qubit 55 00:06:36,084 --> 00:06:45,044 gets transformed to being in the state a(0) + b(1). So this quantum state which 56 00:06:45,044 --> 00:06:53,097 got destroyed at Alice's end suddenly reappears on Bob's end. Okay? So, in what 57 00:06:53,097 --> 00:07:01,006 sense is it like teleportation? Well, it's like teleportation in the sense, you now 58 00:07:01,006 --> 00:07:07,020 on Star Trek, in the sense that well, you had this entity here on Alice's end and 59 00:07:07,020 --> 00:07:13,000 you perform a certain protocol. This entity disappear and then you do something 60 00:07:13,000 --> 00:07:18,047 on Bob's end and it reappears on that endso it's like teleportation. In what 61 00:07:18,047 --> 00:07:23,091 sense is it not like teleportation on Star Trek? Well it's not like that 62 00:07:23,091 --> 00:07:29,065 teleportation because you have, it's not instantaneous. You have to, Alice does her 63 00:07:29,065 --> 00:07:34,087 measurement at hand, she destroys her qubit, and she is left with just this 64 00:07:34,087 --> 00:07:40,096 classical information. And now, she has to pick up the phone. She has to send this 65 00:07:40,096 --> 00:07:46,071 information across to Bob and it's only after he gets this classical information 66 00:07:46,071 --> 00:07:52,049 that he can reconstruct the quantum information that Alice started with. Okay, 67 00:07:52,049 --> 00:07:59,091 so in this and the next video we are going to understand how this protocol works. 68 00:07:59,091 --> 00:08:07,049 Okay, and to make this easy, I am going to break this protocol down into two parts. 69 00:08:07,049 --> 00:08:13,097 First, we are going to start by making an unrealistic assumption. I am going to 70 00:08:13,097 --> 00:08:20,099 assume, okay so, on top here is Alice's lab so she has a qubit in the state a(0) + 71 00:08:20,099 --> 00:08:27,056 b(1). On bottom here is Bob's initial qubit which is initially in the state 72 00:08:27,056 --> 00:08:34,011 zero. I Am going to assume that Alice was able to perform a gate between her qubit 73 00:08:34,011 --> 00:08:40,040 and Bob's qubit. This is completely unrealistic but it'll help us understand 74 00:08:40,040 --> 00:08:48,055 how to proceed with teleportation. Okay, so she performs a CNOT, what happens? Well 75 00:08:48,055 --> 00:09:00,000 obviously the, the state of the two qubits after the CNOT becomes a(00) + b(11). Now 76 00:09:00,000 --> 00:09:11,006 what we'd like is, we'd like somehow to do something on Alice's end so that Bob gets 77 00:09:11,006 --> 00:09:19,054 left with a(0) + b(1). An d Alice gets left with just a measured qubit. Okay, so 78 00:09:19,054 --> 00:09:27,034 what should Alice do? Well, she could try measuring her qubit. I guess if she 79 00:09:27,034 --> 00:09:34,071 measures in the standard basis, what happens? Well, if she measures and the 80 00:09:34,071 --> 00:09:41,000 outcome is zero, then the new state of the qubits is just 00. And whats the state of 81 00:09:41,000 --> 00:09:46,071 Bob's qubit? Well its just, its just that, it's zero. On the other hand is, is she 82 00:09:46,071 --> 00:09:52,016 sees, sees a one, the state of his qubit, its just that, its one, So alpha and beta 83 00:09:52,016 --> 00:09:58,045 just disappear if you the do a measurement in the standard basis. So measuring the 84 00:09:58,045 --> 00:10:05,096 first qubit in the standard basis was a terrible idea. Okay, so here is an idea, 85 00:10:05,096 --> 00:10:12,030 why not use a different basis? So what basis should we use? So let's try the 86 00:10:12,030 --> 00:10:18,008 plus, minus basis. So now, what we are doing is, what Alice does is she measures 87 00:10:18,008 --> 00:10:25,005 the first qubit in the plus, minus basis and we want to know what's the result of 88 00:10:25,005 --> 00:10:30,093 this measurement. So let's write out her state, let's write out the state a(00) + 89 00:10:30,093 --> 00:10:41,089 b(11) and let's rewrite her part of the state in the plus, minus basis. So it's 90 00:10:41,089 --> 00:10:52,013 a(0) is one / square root two(+) + one / square root two (-) and then of course you 91 00:10:52,013 --> 00:10:58,057 have his qubit which is in the state (zero) + b(1), one is (one / square root 92 00:10:58,057 --> 00:11:26,047 two (+),, - one / square root two (-)) and then his qubit is in the state one and so 93 00:11:26,047 --> 00:11:35,028 now what we want to do is we want to collect them, so this is one / square root 94 00:11:35,028 --> 00:11:50,044 two (+) . So plus comes from here and< /i> then you have a(0) for his qubit. And 95 00:11:50,044 --> 00:12:02,007 then you have + and you have + b(1). I am just, I am just factoring things out and 96 00:12:02,007 --> 00:12:14,002 then you have - here, one / square root two (-) a(0). And for this you have -b(1). 97 00:12:14,002 --> 00:12:20,067 Okay, so that's the state of the two qubits. If you write the first qubit in 98 00:12:20,067 --> 00:12:27,033 the plus, minus basis in the second qubit you leave alone, as zero. So now what 99 00:12:27,033 --> 00:12:34,003 happens if you do a measurement? So, if Alice does a measurement well, how come 100 00:12:34,003 --> 00:12:41,056 either plus or minus if the outcome is, is plus? Then the new state, well her qubit 101 00:12:41,056 --> 00:12:50,013 is being in the state plus but also his qubit will be in the state a(0) + b(1). 102 00:12:50,013 --> 00:12:58,023 That's great because thats exactly the qubit that Alice was trying to send to Bob 103 00:12:58,023 --> 00:13:05,062 . And now he has got it, so thats great. But on the other hand if the outcome is 104 00:13:05,062 --> 00:13:14,008 minus. Then the state of his qubit is a(0) - b(1). So now what should he do? Well 105 00:13:14,008 --> 00:13:22,000 Alice can pick up the phone and she can call him and say, in this first case she 106 00:13:22,000 --> 00:13:28,004 says, the outcome is plus, you've got a bit, qubit. In the second case, she calls 107 00:13:28,004 --> 00:13:34,007 him up and says sorry, the outcome was minus but then Bob says you know that's 108 00:13:34,007 --> 00:13:41,001 not a problem because I have a phase flip gate sitting around. Phase flip gate is 109 00:13:41,001 --> 00:13:48,000 the z gate and what it does is it, it changes the freeze in front of the, of one 110 00:13:48,000 --> 00:13:55,001 to minus. And so that's going to take this qubit and restore it to look exactly like 111 00:13:55,001 --> 00:14:01,005 Alice's qubit. Okay so, so what did we, what did we discover? We discovered that 112 00:14:01,005 --> 00:14:08,049 if Alice was able to perform a CNOT from her qubit to Bob's qubit then she can 113 00:14:08,049 --> 00:14:16,017 create this special state a(00) + b(11). And then, if Alice does a measurement in 114 00:14:16,017 --> 00:14:22,056 the Hadamard basis, in the plus, minus basis, and the sign basis then, and she 115 00:14:22,056 --> 00:14:28,014 communicates that to Bob, he can apply a correction or not depending upon what the 116 00:14:28,014 --> 00:14:33,097 outcome was and he can recover Alice's cubit. So, the challenge is, how to create 117 00:14:33,097 --> 00:14:39,007 the state without the CNOT and that's what we'll see in the next video.