0 00:00:00,000 --> 00:00:04,730 1 00:00:04,730 --> 00:00:07,570 THOMAS VIDICK: So so far we've seen two different kinds 2 00:00:07,570 --> 00:00:11,750 of quantum states-- pure states and mixed states. 3 00:00:11,750 --> 00:00:16,149 And we've been thinking of mixed states as probability distributions. 4 00:00:16,149 --> 00:00:20,310 The mixed state is simply a mixture over pure states. 5 00:00:20,310 --> 00:00:24,040 So a natural question is what does this mixture represent? 6 00:00:24,040 --> 00:00:29,550 So we say that the mixed state is in state psi with probability p. 7 00:00:29,550 --> 00:00:31,490 Where does that probability come from? 8 00:00:31,490 --> 00:00:34,044 Where does this uncertainty arise from? 9 00:00:34,044 --> 00:00:35,960 And so what we're going to see in this lecture 10 00:00:35,960 --> 00:00:42,540 is that, in fact, every mixed state arises as the reduced density matrix 11 00:00:42,540 --> 00:00:46,220 of a pure state on a larger system. 12 00:00:46,220 --> 00:00:49,040 So the fact that we are in a mixed state, a state that is not 13 00:00:49,040 --> 00:00:52,400 pure-- so a mixed state that has rank larger than 1-- 14 00:00:52,400 --> 00:00:57,500 reflects the fact that this mixed state must be part of a larger system. 15 00:00:57,500 --> 00:01:00,120 That larger system is in a pure state, but we 16 00:01:00,120 --> 00:01:01,800 don't have access to the whole system. 17 00:01:01,800 --> 00:01:06,280 And this is why we end up with a density matrix that has rank larger than 1, 18 00:01:06,280 --> 00:01:08,580 so a mixed state that is not pure. 19 00:01:08,580 --> 00:01:11,570 So let's see how these purifications work. 20 00:01:11,570 --> 00:01:14,840 So we've already seen before that density matrices can 21 00:01:14,840 --> 00:01:17,310 arise by taking the reduced density. 22 00:01:17,310 --> 00:01:20,780 And the sort of prototypical example is-- 23 00:01:20,780 --> 00:01:23,870 if you start with an entangled state like this psi 24 00:01:23,870 --> 00:01:27,080 here, which is initialized in an EPR pair. 25 00:01:27,080 --> 00:01:30,650 So let's quickly compute the reduced density on A. 26 00:01:30,650 --> 00:01:31,640 We know how to do that. 27 00:01:31,640 --> 00:01:35,070 We have to choose a basis B. For the B system 28 00:01:35,070 --> 00:01:39,750 here, it's appropriate to choose the standard basis for B. 29 00:01:39,750 --> 00:01:41,080 So let's choose this. 30 00:01:41,080 --> 00:01:44,280 To this basis we can associate POVM elements 31 00:01:44,280 --> 00:01:47,550 that represent the partial measurement of the B system 32 00:01:47,550 --> 00:01:50,460 in the basis B. So this would be M0, which 33 00:01:50,460 --> 00:01:56,420 is the identity on A tensored rank 1 projection on 0 for B and M1, 34 00:01:56,420 --> 00:02:01,600 which is the identity on A tensored the rank 1 projection on 1 for B. 35 00:02:01,600 --> 00:02:04,330 We can choose as Kraus operators the same 36 00:02:04,330 --> 00:02:06,470 as the M's, because they're projectors. 37 00:02:06,470 --> 00:02:09,470 This lets us compute the reduced densities. 38 00:02:09,470 --> 00:02:11,560 If you do the computation, you'll see that you'll 39 00:02:11,560 --> 00:02:14,390 get the outcome 0 with probability 1/2. 40 00:02:14,390 --> 00:02:20,530 And in which case, you get the state rho 0 on A and B, which is simply 0 0 41 00:02:20,530 --> 00:02:22,890 or you could get a 1 with probability 1/2. 42 00:02:22,890 --> 00:02:24,450 The state will be rho 1. 43 00:02:24,450 --> 00:02:27,290 And so this tells us that the reduced density on A 44 00:02:27,290 --> 00:02:30,510 is the totally mixed state-- half of the identity 45 00:02:30,510 --> 00:02:35,350 matrix, half of the projection on 0 plus the projection on 1. 46 00:02:35,350 --> 00:02:37,540 So here we have a pure state, a global state. 47 00:02:37,540 --> 00:02:38,430 It's entangled. 48 00:02:38,430 --> 00:02:43,850 I take the reduced density on system A and I obtain a mixed state. 49 00:02:43,850 --> 00:02:47,540 And the question is whether this process is reversible or more generally, 50 00:02:47,540 --> 00:02:50,670 if whether given the density matrix, we can always 51 00:02:50,670 --> 00:02:56,120 write it as the reduced density on the system of a pure state of a larger 52 00:02:56,120 --> 00:02:57,510 system. 53 00:02:57,510 --> 00:03:00,140 And the answer is yes, and let's see how that works. 54 00:03:00,140 --> 00:03:02,751 So let's take an arbitrary density matrix rho 55 00:03:02,751 --> 00:03:07,190 A. We know that we can write it as a mixture 56 00:03:07,190 --> 00:03:10,900 with here some coefficients p_i that are non-negative and sum to 1. 57 00:03:10,900 --> 00:03:13,790 These are the eigenvalues of the reduced density matrix 58 00:03:13,790 --> 00:03:16,280 and of psi i, which are the eigenvectors. 59 00:03:16,280 --> 00:03:19,910 So you can always compute this decomposition using the singular value 60 00:03:19,910 --> 00:03:21,020 decomposition. 61 00:03:21,020 --> 00:03:24,530 So the way we interpreted this is by saying that, all right, 62 00:03:24,530 --> 00:03:25,730 this is a mixture. 63 00:03:25,730 --> 00:03:29,856 And what it means is that with probability p_i the state on A 64 00:03:29,856 --> 00:03:32,580 is in the pure state psi i. 65 00:03:32,580 --> 00:03:34,900 So where does this probability arise from? 66 00:03:34,900 --> 00:03:39,020 Who gets to decide what state we're actually on? 67 00:03:39,020 --> 00:03:42,040 So now what I claim is that I can introduce a new state 68 00:03:42,040 --> 00:03:46,540 on A and a new system B, where B is going to be the system that 69 00:03:46,540 --> 00:03:49,240 controls which i we get. 70 00:03:49,240 --> 00:03:52,920 So it's natural to introduce this state, psi A B, which 71 00:03:52,920 --> 00:03:55,580 is going to be the sum over all i. 72 00:03:55,580 --> 00:04:03,500 Let me put root p_i psi i on A tensored I on B. Where B is a new system, 73 00:04:03,500 --> 00:04:07,770 let's have this index A_i range up to the dimension of the system B. 74 00:04:07,770 --> 00:04:10,060 So B-- this is a new system. 75 00:04:10,060 --> 00:04:13,530 It has the same dimension as the A system. 76 00:04:13,530 --> 00:04:17,140 And it has a basis that I called i. 77 00:04:17,140 --> 00:04:18,920 So this is certainly a pure state. 78 00:04:18,920 --> 00:04:20,000 It's a valid state. 79 00:04:20,000 --> 00:04:22,180 It's normalized, because the norm of psi will 80 00:04:22,180 --> 00:04:26,180 be the sum of the p_i's, which is 1 because rho A was 81 00:04:26,180 --> 00:04:28,790 the normalized density matrix. 82 00:04:28,790 --> 00:04:31,830 And now let's check that it achieves what I claim it does. 83 00:04:31,830 --> 00:04:36,630 So we have to compute the reduced density of psi A B on A. 84 00:04:36,630 --> 00:04:37,840 How do we do that? 85 00:04:37,840 --> 00:04:39,480 We choose a basis for the B system. 86 00:04:39,480 --> 00:04:40,360 We measure. 87 00:04:40,360 --> 00:04:41,870 We see what happens. 88 00:04:41,870 --> 00:04:44,720 So, of course, what basis are we going to choose? 89 00:04:44,720 --> 00:04:48,790 The one we just introduced, so let's use that basis for B. 90 00:04:48,790 --> 00:04:53,050 Then if we compute the probability that we get a certain outcome i-- 91 00:04:53,050 --> 00:04:54,410 we're getting used to this now. 92 00:04:54,410 --> 00:04:59,486 It's just going to be the squared norm of this vector square root p_i psi 93 00:04:59,486 --> 00:05:03,190 i, which is p_i and the corresponding projected state rho 94 00:05:03,190 --> 00:05:08,430 i will be psi i on the A system tensored i on the B system. 95 00:05:08,430 --> 00:05:14,630 And so this tells us that the overall reduced density rho A on the A system, 96 00:05:14,630 --> 00:05:17,390 obtained by taking the partial trace operation, 97 00:05:17,390 --> 00:05:20,580 will be the sum over i of the probability that we get i. 98 00:05:20,580 --> 00:05:24,700 This is p_i tensored with the reduced density on A 99 00:05:24,700 --> 00:05:26,400 that's associated with rho i. 100 00:05:26,400 --> 00:05:29,780 So that's just a rank 1 projection on psi i. 101 00:05:29,780 --> 00:05:33,930 And you can verify that, as claimed, the results that I obtain here 102 00:05:33,930 --> 00:05:36,970 is exactly the state that I started from. 103 00:05:36,970 --> 00:05:39,060 And I didn't make any assumption about that state. 104 00:05:39,060 --> 00:05:43,290 This was just an arbitrary density matrix on A. So 105 00:05:43,290 --> 00:05:47,440 any such state that can be obtained the way that I've obtained it, 106 00:05:47,440 --> 00:05:52,990 or in any way that is such that it's reduced density on the system A-- 107 00:05:52,990 --> 00:05:57,010 this is called a purification of the reduced density.