1 00:00:00,000 --> 00:00:05,000 Good morning. In today's lecture, we'll talk about how to describe the quantum 2 00:00:05,000 --> 00:00:14,004 state of a system of two qubits. And in the process we'll, we'll talk about a very 3 00:00:14,004 --> 00:00:22,001 fundamental concept, that of entanglement, quantum entanglement. Quantum entanglement 4 00:00:22,001 --> 00:00:29,000 is, is, is probably the most important concept in, in quantum information, 5 00:00:29,000 --> 00:00:35,006 quantum computation. It's the key resource that makes exponential speedups possible 6 00:00:35,006 --> 00:00:41,009 in quantum computation. In fact, one way one can think, describe it is quantum 7 00:00:41,009 --> 00:00:46,007 computation, quantum information, an exploration of quantum entanglement. 8 00:00:46,007 --> 00:00:52,001 Something that was discovered in the early days of quantum mechanics and then, not 9 00:00:52,001 --> 00:00:59,001 really pursued too, too much, much depth. So, entanglement was discovered by. 10 00:00:59,001 --> 00:01:08,004 Einstein Podolsky Rosen, and Einstein later described it once derisively as spooky 11 00:01:08,004 --> 00:01:16,005 action at a distance. So, entanglement is a very counter intuitive concept and you 12 00:01:16,005 --> 00:01:21,009 know, in today's lecture and the next lecture we'll try to come to grips with 13 00:01:21,009 --> 00:01:28,003 this concept. Now, actually there's a part from Erwin Schrödinger which I found very 14 00:01:28,003 --> 00:01:32,008 interesting because it, you know, this is from the 1930's, where he already talked 15 00:01:32,008 --> 00:01:37,006 about, he says, I would not call entanglement one but rather the 16 00:01:37,006 --> 00:01:43,006 characteristic trait of quantum mechanics, the one that enforces its entire departure 17 00:01:43,006 --> 00:01:49,008 from classical lines of thought. So this is a remarkably modern viewpoint on 18 00:01:49,008 --> 00:01:56,009 entanglement. Okay, but before we do all this, I'd like to do a review of what 19 00:01:56,009 --> 00:02:04,002 we've seen so far. So, what we've studied so far is, what's a qubit. And a model of the 20 00:02:04,002 --> 00:02:11,001 qubit is, well, we've been describing it in terms of atomic qubits, in terms of the 21 00:02:11,001 --> 00:02:16,000 energy levels of an electron. And we are thinking of two distinct energy levels, 22 00:02:16,000 --> 00:02:20,005 the ground state and the first excited state which, if this was a classical 23 00:02:20,005 --> 00:02:26,005 system, we could use to store a bit of information and quoting zero as a ground 24 00:02:26,005 --> 00:02:33,004 state and one by the excited state. But of course, in quantum mechanics, the electron 25 00:02:33,004 --> 00:02:39,004 doesn't make up it's mind, whether it's in the ground or excited state, in general. 26 00:02:39,004 --> 00:02:44,002 And it can be in a superposition of the ground and excited state where it has some 27 00:02:44,002 --> 00:02:49,002 complex amplitude of being in the ground state and some other complex amplitude of 28 00:02:49,002 --> 00:02:53,004 being in the excited state. The interesting thing being of these, these, 29 00:02:53,004 --> 00:03:01,000 these amplitudes are complex numbers and they're, and they're normalized okay so, 30 00:03:01,000 --> 00:03:10,000 so in general, we understate, α0|0> + α1|1> where α0 and α1 are complex 31 00:03:10,000 --> 00:03:17,000 numbers. And the sum of the squares of their magnitude add up to one, so it's 32 00:03:17,000 --> 00:03:26,000 normalized. Okay, the normalization condition is, is interesting because, 33 00:03:26,000 --> 00:03:34,002 because when we actually go to look and see which state the electron is, is in, it 34 00:03:34,002 --> 00:03:39,005 quickly makes up it's mind and it ends up in the ground state with amplitude alpha 35 00:03:39,005 --> 00:03:45,000 zero, magnitude squared. And in the excited state with amplitude alpha one, 36 00:03:45,000 --> 00:03:50,000 magnitude squared. Moreover, as soon as we may look, the state of the system is, is 37 00:03:50,000 --> 00:03:56,007 disturbed and the new state is exactly consistent with what, what we measured. 38 00:03:56,007 --> 00:04:03,003 So, if we were to look at this example that we had earlier, the probability that 39 00:04:03,003 --> 00:04:09,000 we'd see zero would be one over square root squared which would be one-half and 40 00:04:09,000 --> 00:04:18,003 then the new state would be exactly the zero state. And the probability that we 41 00:04:18,003 --> 00:04:28,008 see one would also be half because that would be one over two squared plus one 42 00:04:28,008 --> 00:04:37,004 over two squared. And the new state would now be the one state. Okay, so all this 43 00:04:37,004 --> 00:04:44,003 we, we saw last time. We also saw that there's a geometric interpretation of, of 44 00:04:44,003 --> 00:04:48,008 what it means for, to have the state of the qubits. So for example, let me just 45 00:04:48,008 --> 00:05:02,007 use real numbers. So let's say that, that the state of the qubit was 46 00:05:02,007 --> 00:05:12,003 1/√2|0> - √3/2|1>, sorry, that doesn't add up, so, I should actually, let's say it's 47 00:05:12,003 --> 00:05:18,000 1/2|0> - √3/2|1>. So you can see it's normalized. Now, the geometric 48 00:05:18,000 --> 00:05:26,000 interpretation says, that the state of this qubit is a unit vector in a two 49 00:05:26,000 --> 00:05:34,006 dimensional vector space where we have the axis, the coordinating axis are labeled 50 00:05:34,006 --> 00:05:41,002 with zero and one, the ground and excited state. And, the state of the system sits 51 00:05:41,002 --> 00:05:49,001 on a, on a unit circle and we can plot it out here and it, it might look something 52 00:05:49,001 --> 00:05:54,000 like this. So this would be |Ψ>, okay. And now there's, there's also an 53 00:05:54,000 --> 00:06:03,000 interpretation of, of this in terms, when we do a measurement. What we're saying is 54 00:06:03,000 --> 00:06:10,000 that, that the state collapses to the zero state with, probability cos²θ 55 00:06:10,000 --> 00:06:18,000 and it collapses to the one state with probability, well, cosine square of 56 00:06:18,000 --> 00:06:22,000 the angle it makes with the one step which, which actually ends up being 57 00:06:22,000 --> 00:06:27,000 sin²Θ. So this is the probability of |0>, and sin²θ is the 58 00:06:27,000 --> 00:06:36,000 probability of |1>. Okay, now, okay, let, let me just give you a couple of other 59 00:06:36,000 --> 00:06:42,000 examples of qubits. So it turns out that photons, particles of light, have a qubit 60 00:06:42,000 --> 00:06:46,000 associated with them, which is called the polarization which is roughly the 61 00:06:46,000 --> 00:06:52,000 orientation of the electric field oscillations associated with, with this, 62 00:06:53,000 --> 00:06:57,000 you know, and we can think of these, these, oscillations as either being 63 00:06:57,000 --> 00:07:03,000 horizontal or vertical, and that corresponds to qubit. The spin of particle 64 00:07:03,000 --> 00:07:12,000 like an electron is a quantized version of its, its angular momentum and it, it also 65 00:07:12,000 --> 00:07:18,000 forms a qubit where the spin is either pointing up or down. So now, how would you 66 00:07:18,000 --> 00:07:24,000 measure these, these qubits? Well, in the case of photon polarization, this is 67 00:07:24,000 --> 00:07:29,000 particularly easy. It's you know, it's in terms you do this with the polarizing 68 00:07:29,000 --> 00:07:34,006 filter. Whether filter, depending upon it's orientation, it might allow photons 69 00:07:34,006 --> 00:07:41,000 that are polarized vertically but not horizontally. So think of the, you know, 70 00:07:41,000 --> 00:07:48,002 of the photon sort of, moving horizontally, the polarizing filter is, is 71 00:07:48,002 --> 00:07:53,005 orthogonal to it and it's aligned let's say, vertically. So then, it only allows 72 00:07:53,005 --> 00:07:59,000 photons through that are polarized vertically and blocks photons that are 73 00:07:59,000 --> 00:08:07,000 aligned horizontally. But of course, in general, the, the state of the photon, so, 74 00:08:07,000 --> 00:08:13,000 so let's say this is, this is horizontally polarized, that's vertically polarized. So 75 00:08:13,000 --> 00:08:18,000 in general, the state of the photon will be some superposition of vertical and 76 00:08:18,000 --> 00:08:26,002 horizontal. This is going to be α times horizontal plus, α0 times 77 00:08:26,002 --> 00:08:33,000 horizon, horizontal plus α1 times vertical. And so, if you, i f you were to 78 00:08:33,000 --> 00:08:42,004 pass such a photon through these filters then the net effect is it, it gets blocked 79 00:08:42,004 --> 00:08:55,004 with probability |α0|², and it's transmitted with probability |α1|². If 80 00:08:55,004 --> 00:09:05,002 it's transmitted, then it's new state is vertically polarized. Okay, so this is 81 00:09:05,002 --> 00:09:11,001 what you see when you, when you, hold up a polarizing filter. You see photons coming 82 00:09:11,001 --> 00:09:16,009 through, and they are polarized in various different directions. But of course, you 83 00:09:16,009 --> 00:09:21,002 have the ones that are horizontally polarized are blocked, ones the vertically 84 00:09:21,002 --> 00:09:26,006 are, are transmitted. The ones that are polarized diagonally, they are transmitted 85 00:09:26,006 --> 00:09:32,009 with some probability but then what you see is vertically polarized photons, only. 86 00:09:32,009 --> 00:09:39,002 Okay. Well, there's another question you could ask which is, what happens if you 87 00:09:39,002 --> 00:09:45,004 orient this filter so that it's not aligned either vertically or horizontally 88 00:09:45,004 --> 00:09:52,001 but rather diagonally at a 45 degree angle? Well, it makes sense that what it 89 00:09:52,001 --> 00:09:59,001 would do is allow through those photons which are at a 45 degree orientation, but 90 00:09:59,001 --> 00:10:06,001 block everything which is at a -45 degree orientation. So in terms of the qubit 91 00:10:06,001 --> 00:10:14,000 picture, here's what it corresponds to. What it corresponds to is that, we are now 92 00:10:14,000 --> 00:10:22,000 measuring in this diagonal basis, this is |+> and this is |->. So now if we 93 00:10:22,000 --> 00:10:29,007 started with a photon in this, in this particular state, the probability that it 94 00:10:29,007 --> 00:10:38,001 will go through is given by cos²θ, where θ is this angle. And the 95 00:10:38,001 --> 00:10:45,001 probability that it's blocked is sin²θ. Moreover, if it, if it goes 96 00:10:45,001 --> 00:10:52,004 through, then it must be orientated in a diagonal, at a 45 degree angle. Okay? So 97 00:10:52,004 --> 00:10:58,004 this is how measurement at an arbitrary in an arbitrary orthogonal basis works. So 98 00:10:58,004 --> 00:11:05,004 this is what it means to measure at an arbitrary angle. Now, what happens if you, 99 00:11:05,004 --> 00:11:10,000 if you take two of these polarizers and you align one horizontally and one 100 00:11:10,000 --> 00:11:16,000 vertically? Well, you know, if you look through it, well, you have lots of light 101 00:11:16,000 --> 00:11:21,000 coming through. Some of it, some of it passes through the horizontally polarized 102 00:11:21,000 --> 00:11:25,000 one, some of it passes through the vertically polarized one. But where the 103 00:11:25,000 --> 00:11:32,000 two intersect , you see a dark, dark patch. And the reason is, simple. The 104 00:11:32,000 --> 00:11:36,000 light that is coming through, it might be lets say, you know, lets say, we have only 105 00:11:36,000 --> 00:11:43,000 one photon coming through. Well, you know, it might be polarized diagonally 106 00:11:43,000 --> 00:11:49,005 α0|→> + α1|↑>. And lets say, this first polarized that 107 00:11:49,005 --> 00:11:55,003 goes through is horizontally, it's horizontally aligned. So then, with 108 00:11:55,003 --> 00:12:02,001 probability |α0|², it goes through and now the light is horizontally 109 00:12:02,001 --> 00:12:09,000 polarized. But now it's blocked with probability one by the second filter. And 110 00:12:09,000 --> 00:12:15,002 so, the net effect is no matter what it, what it's original orientation, it's 111 00:12:15,002 --> 00:12:20,004 definitely blocked by the two filters in conjunction. But now you can do, do a very 112 00:12:20,004 --> 00:12:27,001 interesting experiment. What you could do is, between these two filters, you could 113 00:12:27,001 --> 00:12:35,005 insert a third one which is, which is aligned at a 45 degree angle. Or π/4 114 00:12:35,005 --> 00:12:43,000 So this is our third filter, which is inserted between the two. And now what 115 00:12:43,000 --> 00:12:49,001 happens? Well, the interesting thing that happens is, that this middle patch is no 116 00:12:49,001 --> 00:12:55,004 longer dark. And the reason is, some of the light that comes in, it goes through 117 00:12:55,004 --> 00:13:01,000 the first filter. But now it's horizontally polarized. None of it would 118 00:13:01,000 --> 00:13:06,000 get through this, this second filter, but on the other hand, we have dispersed this 119 00:13:06,000 --> 00:13:10,004 third filter in the middle and that's at 45 degree angle. And so, these 120 00:13:10,004 --> 00:13:19,001 horizontally polarized photons get through this middle filter with probability half. 121 00:13:19,001 --> 00:13:28,007 Okay, so, so, so through the middle filter, they get through with probability, 122 00:13:28,007 --> 00:13:36,003 half. And then, the output, the, the photons that get through are polarized at 123 00:13:36,003 --> 00:13:44,004 a 45 degree angle. Now when they try to get through the, the second filter, which 124 00:13:44,004 --> 00:13:51,009 was vertically polarized, they are going to get through with probability half, 125 00:13:51,009 --> 00:13:57,004 because they, they are oriented at 45 degrees to the, to the orientation of this 126 00:13:57,004 --> 00:14:03,000 filter. And of course the new, the photons, that get through are now 127 00:14:03,000 --> 00:14:10,009 vertically polarized. Okay, so, so that's it as far as, as far as measurement goes. 128 00:14:10,009 --> 00:14:18,000 Now if you remember, we, we also have this notion of the uncertainty principle for 129 00:14:18,000 --> 00:14:24,003 qubits where we, we could measure the qubit either in the bit basis |0>,|1>, or in 130 00:14:24,003 --> 00:14:33,005 the sign basis. So remember this is, this is our bit basis, |0>, |1>. This is |+> and 131 00:14:33,005 --> 00:14:45,005 |->. |+> is where, where |+> is, and equals to superposition of |0> and |1>, 132 00:14:45,005 --> 00:14:54,001 and |-> is an opposite superposition of both |0> and |1>. What the uncertainty 133 00:14:54,001 --> 00:15:00,008 principle tells us is that, we cannot know both the bit value and the sign value 134 00:15:00,008 --> 00:15:06,003 simultaneously. In other words, suppose we were to measure in the bit basis, well 135 00:15:06,003 --> 00:15:11,001 then, we know that, that the qubit is either in the state |0> or in the state 136 00:15:11,001 --> 00:15:16,006 |1>. But now it's maximally uncertain with respect to the sign basis, and vice versa. 137 00:15:16,006 --> 00:15:22,006 If we were to measure in the sign basis, it's maximally uncertain with respect to 138 00:15:22,006 --> 00:15:24,000 the bit basis.