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PROFESSOR: So today
I want to finish up

9
00:00:26,800 --> 00:00:28,660
a couple of loose
ends in the class.

10
00:00:28,660 --> 00:00:30,252
The first is by
the end of the day,

11
00:00:30,252 --> 00:00:31,960
we'll talk about EPR
that we've picked up

12
00:00:31,960 --> 00:00:33,487
on the last couple of lectures.

13
00:00:33,487 --> 00:00:35,070
We're going to close
up Bell's theorem

14
00:00:35,070 --> 00:00:36,109
from the very beginning.

15
00:00:36,109 --> 00:00:37,650
So we talked about
Bell's Inequality,

16
00:00:37,650 --> 00:00:39,270
and we'll re-prove
Bell's Inequality,

17
00:00:39,270 --> 00:00:41,945
and then we'll see what quantum
mechanics has to say about it.

18
00:00:41,945 --> 00:00:43,820
But first, I want to
show you something neat.

19
00:00:43,820 --> 00:00:46,069
So a couple of times, we've
talked about entanglement,

20
00:00:46,069 --> 00:00:47,190
and what it can do.

21
00:00:47,190 --> 00:00:49,731
And I want to spend the first
good chunk of the class talking

22
00:00:49,731 --> 00:00:53,344
about what entanglement can
do, and also what it can't.

23
00:00:53,344 --> 00:00:55,760
So before we get started, I
want to remind you of a couple

24
00:00:55,760 --> 00:00:57,410
things from the last lecture.

25
00:00:57,410 --> 00:01:01,940
So the first is
particles can have spin.

26
00:01:01,940 --> 00:01:03,540
Half-integer intrisic
angular momentum

27
00:01:03,540 --> 00:01:06,080
that has nothing to
do with being a rigid

28
00:01:06,080 --> 00:01:08,470
object that rotates, but is
just an intrinsic property

29
00:01:08,470 --> 00:01:09,440
of angular momentum.

30
00:01:09,440 --> 00:01:11,230
And this spin is
observable-- it's

31
00:01:11,230 --> 00:01:12,700
represented by an operator.

32
00:01:12,700 --> 00:01:14,960
In fact, a vector of
operators, Sx, Sy, and Sz,

33
00:01:14,960 --> 00:01:17,085
that satisfy the same
[? computational ?] relations

34
00:01:17,085 --> 00:01:19,100
as the orbital angular momentum.

35
00:01:19,100 --> 00:01:22,030
Now last time, one thing we
showed is that a state which is

36
00:01:22,030 --> 00:01:25,780
up-- or, for example,
down-- at an angle theta,

37
00:01:25,780 --> 00:01:27,970
so if I measure the
spin along this axis,

38
00:01:27,970 --> 00:01:30,460
I will always find it's
other up or down, spin-1/2,

39
00:01:30,460 --> 00:01:33,190
either plus h-bar upon
2 or minus h-bar upon 2.

40
00:01:33,190 --> 00:01:36,620
But I can express this
state in terms of the states

41
00:01:36,620 --> 00:01:39,850
with spin, which are up
or down along the z-axis.

42
00:01:39,850 --> 00:01:42,540
And we gave an
expression-- and I

43
00:01:42,540 --> 00:01:44,900
challenge you to
follow the logic given

44
00:01:44,900 --> 00:01:46,770
to derive this expression.

45
00:01:46,770 --> 00:01:48,310
An expression for
the state, which

46
00:01:48,310 --> 00:01:50,820
is, in this case, down
along the angle theta--

47
00:01:50,820 --> 00:01:54,610
it went down when measuring
spin along the angle theta--

48
00:01:54,610 --> 00:02:01,590
in terms of up and down of spin
along the z-axis, or long angle

49
00:02:01,590 --> 00:02:04,430
0, with theta here being the
declination from the vertical.

50
00:02:04,430 --> 00:02:05,520
OK.

51
00:02:05,520 --> 00:02:07,000
Yep.

52
00:02:07,000 --> 00:02:13,870
AUDIENCE: [INAUDIBLE] angle.

53
00:02:13,870 --> 00:02:14,590
PROFESSOR: Yeah.

54
00:02:14,590 --> 00:02:16,460
There's a general
thing for this,

55
00:02:16,460 --> 00:02:17,970
that says for a general angle.

56
00:02:17,970 --> 00:02:20,460
And it's actually
in the notes, which

57
00:02:20,460 --> 00:02:22,250
I guess weren't
posted from last time.

58
00:02:22,250 --> 00:02:23,880
But it's in the notes.

59
00:02:23,880 --> 00:02:26,189
And it's also easy
to derive right?

60
00:02:26,189 --> 00:02:27,230
So what's the point here?

61
00:02:27,230 --> 00:02:29,320
The point is you
have a spin vector,

62
00:02:29,320 --> 00:02:31,230
and it's a vector, Sx, Sy, Sz.

63
00:02:31,230 --> 00:02:33,170
And you can compute the
operator representing

64
00:02:33,170 --> 00:02:34,180
spin along a
particular direction

65
00:02:34,180 --> 00:02:35,800
by taking a unit vector
in that direction

66
00:02:35,800 --> 00:02:36,850
and taking the dot product.

67
00:02:36,850 --> 00:02:38,891
So that gives you a
particular linear combination

68
00:02:38,891 --> 00:02:40,190
of Sx, Sy, and Sz.

69
00:02:40,190 --> 00:02:42,930
But, from last time, you know
that's just some stupid matrix.

70
00:02:42,930 --> 00:02:44,460
And you could express
a matrix in any basis

71
00:02:44,460 --> 00:02:45,630
you want, for example,
the basis of up

72
00:02:45,630 --> 00:02:47,713
and down in the z-direction,
which is what we used

73
00:02:47,713 --> 00:02:48,339
last time.

74
00:02:48,339 --> 00:02:50,130
Then you can find the
eigenvectors of that,

75
00:02:50,130 --> 00:02:51,994
and that's how you
find these states.

76
00:02:51,994 --> 00:02:53,660
So there's a very
simple generalization.

77
00:02:53,660 --> 00:02:55,510
It just involves an
extra angle theta,

78
00:02:55,510 --> 00:02:58,664
which is the angle
around the equator.

79
00:02:58,664 --> 00:03:00,030
OK?

80
00:03:00,030 --> 00:03:01,570
AUDIENCE: [INAUDIBLE].

81
00:03:01,570 --> 00:03:05,800
PROFESSOR: Sorry,
phi, extra angle phi.

82
00:03:05,800 --> 00:03:08,089
So we then did the
following thing.

83
00:03:08,089 --> 00:03:09,880
I talked about the
Stern-Gerlach experiment

84
00:03:09,880 --> 00:03:12,004
and, here in an abstract
way from the Stern Gerlach

85
00:03:12,004 --> 00:03:14,270
experiment, the
core of the nugget.

86
00:03:14,270 --> 00:03:16,000
And the core of it is this.

87
00:03:16,000 --> 00:03:20,294
Suppose I have some spin, and
I put it in a particular state.

88
00:03:20,294 --> 00:03:22,710
Let's say, for simplicity, up
in the z-direction plus down

89
00:03:22,710 --> 00:03:24,670
in the z-direction
with equal amplitudes.

90
00:03:24,670 --> 00:03:27,660
So they're equally
likely to be measured.

91
00:03:27,660 --> 00:03:29,912
And then, these guys,
these two states,

92
00:03:29,912 --> 00:03:32,120
should be degenerate in
energy, because the system is

93
00:03:32,120 --> 00:03:32,990
rotationally invariant.

94
00:03:32,990 --> 00:03:34,323
It's just a spin, sitting there.

95
00:03:34,323 --> 00:03:36,030
There's rotational
invariance, so we

96
00:03:36,030 --> 00:03:40,320
know the energy can't depend on
the z-component of the angular

97
00:03:40,320 --> 00:03:40,870
momentum.

98
00:03:40,870 --> 00:03:43,161
But we can break that symmetry
and break the degeneracy

99
00:03:43,161 --> 00:03:44,460
by turning on a magnetic field.

100
00:03:44,460 --> 00:03:45,930
So for a Stern-Gerlach
experiment,

101
00:03:45,930 --> 00:03:47,560
we turn on a magnetic
field that had a gradient.

102
00:03:47,560 --> 00:03:49,430
But I just want to look at
a constant magnetic field

103
00:03:49,430 --> 00:03:50,990
for the moment and
see what it does.

104
00:03:50,990 --> 00:03:52,706
So we turn a constant
magnetic field.

105
00:03:52,706 --> 00:03:54,830
That's a contribution of
the energy, which is minus

106
00:03:54,830 --> 00:03:56,980
the magnetic field dot
the magnetic moment, which

107
00:03:56,980 --> 00:03:58,580
is a constant times the spin.

108
00:03:58,580 --> 00:04:00,580
But if the magnetic field
is in the z-direction,

109
00:04:00,580 --> 00:04:03,490
then this is just Bz Sz.

110
00:04:03,490 --> 00:04:06,990
So we then found yesterday
that this led to a splitting--

111
00:04:06,990 --> 00:04:09,160
that the up state had
energy plus h-bar omega,

112
00:04:09,160 --> 00:04:11,160
and the down state had
energy minus h-bar omega,

113
00:04:11,160 --> 00:04:14,271
where omega is given by
mu naught Bz upon two.

114
00:04:14,271 --> 00:04:15,770
Yesterday, we had
an additional term

115
00:04:15,770 --> 00:04:18,690
which involved the z
dependence, the beta z term,

116
00:04:18,690 --> 00:04:21,107
but here I'm just looking at
a constant magnetic field.

117
00:04:21,107 --> 00:04:22,690
But if we know the
energies are, these

118
00:04:22,690 --> 00:04:25,060
are still the eigenstates
of the energy operator.

119
00:04:25,060 --> 00:04:26,080
If we know the
energies are, we know

120
00:04:26,080 --> 00:04:27,480
how the system evolves in time.

121
00:04:27,480 --> 00:04:28,860
Here is the initial state.

122
00:04:28,860 --> 00:04:30,070
These are the eigenstates.

123
00:04:30,070 --> 00:04:31,447
The energies are these.

124
00:04:31,447 --> 00:04:33,280
So we can let the system
evolve in the time,

125
00:04:33,280 --> 00:04:34,850
and we find that
all we do, is we

126
00:04:34,850 --> 00:04:36,580
evolve the system with phases.

127
00:04:36,580 --> 00:04:38,420
This is as usual in
quantum evolution.

128
00:04:38,420 --> 00:04:41,550
If you have an expansion of the
state in energy eigenstates,

129
00:04:41,550 --> 00:04:45,600
time evolution is just
phases for each term.

130
00:04:45,600 --> 00:04:48,680
But note that there's
a simple thing.

131
00:04:48,680 --> 00:04:50,240
This omega, the
thing that determines

132
00:04:50,240 --> 00:04:51,990
the rate of evolution
of the phase,

133
00:04:51,990 --> 00:04:53,630
is controlled by
magnetic moment, which

134
00:04:53,630 --> 00:04:55,760
is something intrinsic to spin,
and the magnetic field, which

135
00:04:55,760 --> 00:04:57,660
is something I can
dial in an experiment.

136
00:04:57,660 --> 00:04:58,859
I can really tune it.

137
00:04:58,859 --> 00:05:01,400
Meanwhile, we decide how long
to leave the magnetic field on.

138
00:05:01,400 --> 00:05:02,790
I can turn the magnetic
field on for a while,

139
00:05:02,790 --> 00:05:04,760
and I could turn
it off while later.

140
00:05:04,760 --> 00:05:09,050
So suppose I leave the
field on for time t,

141
00:05:09,050 --> 00:05:13,821
such that omega t is
equal to pi upon 2.

142
00:05:13,821 --> 00:05:14,320
OK.

143
00:05:14,320 --> 00:05:16,900
So you turn on the magnetic
field for a time t,

144
00:05:16,900 --> 00:05:18,580
such that omega t
is pi upon 2, or t

145
00:05:18,580 --> 00:05:20,350
is equal to pi upon 2 omega.

146
00:05:20,350 --> 00:05:22,060
What is the state
then, afterwards?

147
00:05:22,060 --> 00:05:27,590
The state subsequently psi
after, is equal to this guy--

148
00:05:27,590 --> 00:05:30,510
omega t is equal to pi upon 2.

149
00:05:30,510 --> 00:05:34,080
But what's e to the i pi upon 2?

150
00:05:34,080 --> 00:05:35,190
i.

151
00:05:35,190 --> 00:05:35,690
Right?

152
00:05:35,690 --> 00:05:39,530
And e to the minus i
pi over 2, is minus i.

153
00:05:39,530 --> 00:05:40,780
So this is an i and a minus i.

154
00:05:40,780 --> 00:05:43,430
I can pull out the i so that
the phi after is i, a phase,

155
00:05:43,430 --> 00:05:50,490
upon root 2 times
up z, minus down z.

156
00:05:50,490 --> 00:05:52,340
Yeah.

157
00:05:52,340 --> 00:05:53,290
That's cool.

158
00:05:53,290 --> 00:05:55,304
And this is a state
that we saw last time.

159
00:05:55,304 --> 00:05:56,720
It's actually up
to normalization,

160
00:05:56,720 --> 00:06:00,670
which is an i, up or down
x in the x-direction.

161
00:06:03,299 --> 00:06:05,590
So what this lets us do is
something really quite cool.

162
00:06:05,590 --> 00:06:07,460
What this tells us is that all
these calculations are just

163
00:06:07,460 --> 00:06:09,040
the Stern-Gerlach
calculation, but even easier,

164
00:06:09,040 --> 00:06:10,415
because we don't
have a gradient.

165
00:06:10,415 --> 00:06:13,950
If we have the spin in
some superposition, which,

166
00:06:13,950 --> 00:06:17,380
incidentally, was initially up
and down in the z-direction,

167
00:06:17,380 --> 00:06:21,870
this is equal to up
in the x-direction.

168
00:06:21,870 --> 00:06:26,020
So in x, we have it up.

169
00:06:26,020 --> 00:06:28,820
We turn on a magnetic field
in z, and what happens

170
00:06:28,820 --> 00:06:31,490
is the spin precesses
around the z-axis.

171
00:06:31,490 --> 00:06:33,840
And we get down.

172
00:06:33,840 --> 00:06:35,070
We get precession.

173
00:06:35,070 --> 00:06:36,210
Here's the important point.

174
00:06:36,210 --> 00:06:37,550
The important point
for everything

175
00:06:37,550 --> 00:06:38,841
that's going to follow is this.

176
00:06:38,841 --> 00:06:40,970
If I have some spin
state, and I want

177
00:06:40,970 --> 00:06:44,450
to take it to some other
spin state, how do I do so?

178
00:06:44,450 --> 00:06:46,617
Well, I can build a
machine that does it,

179
00:06:46,617 --> 00:06:48,450
by turning on the
appropriate magnetic field

180
00:06:48,450 --> 00:06:51,141
and having the magnetic
field precess the spin.

181
00:06:51,141 --> 00:06:51,640
OK.

182
00:06:51,640 --> 00:06:53,500
Here I found exactly
what magnetic field

183
00:06:53,500 --> 00:06:56,530
I need to turn on, with what
amplitude, and for how long,

184
00:06:56,530 --> 00:06:59,850
such that I think the
state up x to down x,

185
00:06:59,850 --> 00:07:01,740
with a known phase i.

186
00:07:01,740 --> 00:07:03,890
Everyone cool with that?

187
00:07:03,890 --> 00:07:06,240
So any unit-- here's the
thing I want to convince you

188
00:07:06,240 --> 00:07:08,250
of-- any unitary
operation that takes you

189
00:07:08,250 --> 00:07:12,730
from from one spinner to
another spinner-- up z,

190
00:07:12,730 --> 00:07:15,970
down y, linear combination
of up and down z,

191
00:07:15,970 --> 00:07:19,640
some particular linear
combination of up and down x.

192
00:07:19,640 --> 00:07:24,260
Any such pair of
vectors can be related

193
00:07:24,260 --> 00:07:25,502
by unitary transformation.

194
00:07:25,502 --> 00:07:26,960
And any such unitary
transformation

195
00:07:26,960 --> 00:07:30,160
can be built by some
series of magnetic fields.

196
00:07:30,160 --> 00:07:31,980
That cool?

197
00:07:31,980 --> 00:07:34,150
You can just can prove this
yourself quite quickly,

198
00:07:34,150 --> 00:07:37,450
but let me just state it.

199
00:07:37,450 --> 00:07:40,020
So now this gives us the
ability to do the following.

200
00:07:40,020 --> 00:07:47,460
One, we have spins,
one, spins that can

201
00:07:47,460 --> 00:07:50,200
be put in a system
that can be put

202
00:07:50,200 --> 00:07:54,910
in states like up z and down z.

203
00:07:54,910 --> 00:07:58,450
And two, the ability
to rotate states.

204
00:07:58,450 --> 00:08:06,660
The ability to evolve
states from state to state.

205
00:08:06,660 --> 00:08:12,760
From spin state 1 to
psi 2 by turning on some

206
00:08:12,760 --> 00:08:18,500
magnetic fields, so by suitably
choosing our magnetic fields.

207
00:08:18,500 --> 00:08:20,470
Everyone cool with that?

208
00:08:20,470 --> 00:08:23,000
So here's a question I want to
ask, what do we do with this?

209
00:08:23,000 --> 00:08:25,470
Imagine we really had this
equipment in front of us.

210
00:08:25,470 --> 00:08:29,530
What power, what awesomeness
could we realize?

211
00:08:29,530 --> 00:08:35,100
And this is the entry
point to quantum computing.

212
00:08:35,100 --> 00:08:37,350
The answer to what can you
do with this cool machinery

213
00:08:37,350 --> 00:08:39,690
is you build a quantum computer.

214
00:08:39,690 --> 00:08:41,190
So let's talk about
what that means.

215
00:08:47,190 --> 00:08:50,395
So quick question, do
quantum computers exist?

216
00:08:50,395 --> 00:08:50,895
Yeah.

217
00:08:50,895 --> 00:08:52,190
Are they very big?

218
00:08:52,190 --> 00:08:52,990
No, OK.

219
00:08:52,990 --> 00:08:54,750
The biggest quantum computer
that's been built my knowledge

220
00:08:54,750 --> 00:08:56,440
is one that has factored
the number fifteen

221
00:08:56,440 --> 00:08:58,148
Or they might have
done 21 at this point,

222
00:08:58,148 --> 00:08:59,611
I'm not exactly sure.

223
00:08:59,611 --> 00:09:00,110
21?

224
00:09:00,110 --> 00:09:00,720
Done?

225
00:09:00,720 --> 00:09:01,220
Yeah, OK.

226
00:09:01,220 --> 00:09:01,910
21's done.

227
00:09:01,910 --> 00:09:05,440
So that's the upper end.

228
00:09:05,440 --> 00:09:06,961
Well, they're large, physically.

229
00:09:06,961 --> 00:09:07,460
It's true.

230
00:09:07,460 --> 00:09:09,270
They take up a room.

231
00:09:09,270 --> 00:09:12,040
Or at least a large table
top, an optics table.

232
00:09:12,040 --> 00:09:13,980
But they're not very
big in a useful sense.

233
00:09:13,980 --> 00:09:14,480
OK.

234
00:09:14,480 --> 00:09:18,110
And there's actually reasonable
people, not just reasonable,

235
00:09:18,110 --> 00:09:20,660
intelligent people--
and reasonable.

236
00:09:20,660 --> 00:09:22,530
That's a nontrivial-- Ok.

237
00:09:22,530 --> 00:09:24,370
The people who live
at the intersection

238
00:09:24,370 --> 00:09:28,199
of intelligent and
reasonable, which,

239
00:09:28,199 --> 00:09:29,990
admittedly as my
colleague is pointing out,

240
00:09:29,990 --> 00:09:32,940
is not a large overlap.

241
00:09:32,940 --> 00:09:35,640
There are people
in that overlap who

242
00:09:35,640 --> 00:09:38,865
suspect that, for some
fundamental reason,

243
00:09:38,865 --> 00:09:40,990
it's impossible to build
a useful quantum computer,

244
00:09:40,990 --> 00:09:42,394
in some deep, deep sense.

245
00:09:42,394 --> 00:09:44,060
I don't know whether
that's true or not.

246
00:09:44,060 --> 00:09:46,760
But we're about to engage in a
discussion of what happens when

247
00:09:46,760 --> 00:09:48,880
you build quantum
computers for n bits

248
00:09:48,880 --> 00:09:51,890
were n gets large, like
millions and billions

249
00:09:51,890 --> 00:09:55,412
as you need for codes
or for studying images.

250
00:09:55,412 --> 00:09:57,120
And, of course, this
is a little strange,

251
00:09:57,120 --> 00:09:59,070
because such a
compter doesn't exist,

252
00:09:59,070 --> 00:10:03,080
which leads to my colleague
and friend Scott Aaronson's

253
00:10:03,080 --> 00:10:06,010
lovely comment that his job,
as someone who studies quantum

254
00:10:06,010 --> 00:10:07,640
computers, is to
prove things that

255
00:10:07,640 --> 00:10:10,080
can't be done with
computers that don't exist.

256
00:10:10,080 --> 00:10:12,420
Which is what we're
about to do, so.

257
00:10:12,420 --> 00:10:14,450
So let's talk about
quantum computing.

258
00:10:14,450 --> 00:10:16,140
So what do I mean by
quantum computing?

259
00:10:16,140 --> 00:10:16,810
Here's the basic idea.

260
00:10:16,810 --> 00:10:18,190
Let's talk about
classical computing.

261
00:10:18,190 --> 00:10:20,064
Classical computing
says, I have these things

262
00:10:20,064 --> 00:10:22,460
called bits, which
are binary objects.

263
00:10:22,460 --> 00:10:23,790
They're either 0, or they're 1.

264
00:10:23,790 --> 00:10:25,665
And we realize them in
a very particular way.

265
00:10:25,665 --> 00:10:27,270
We realize them with
a little magnet,

266
00:10:27,270 --> 00:10:28,497
which points north or south.

267
00:10:28,497 --> 00:10:30,080
Now, it's not always
exactly a magnet.

268
00:10:30,080 --> 00:10:32,781
It isn't like an old-style
hard disk in your computer.

269
00:10:32,781 --> 00:10:35,030
It's something a little more
fancy than just a magnet.

270
00:10:35,030 --> 00:10:36,479
But it's still
basically a magnet,

271
00:10:36,479 --> 00:10:37,770
and you have some north, south.

272
00:10:37,770 --> 00:10:39,500
And whether it's pointing
north, or whether it's

273
00:10:39,500 --> 00:10:41,333
pointing south down
here, which I'll call 0.

274
00:10:41,333 --> 00:10:44,690
Or whether it's pointing north
down here, which I'll call a 1,

275
00:10:44,690 --> 00:10:46,570
determines whether
you call this 0 or 1.

276
00:10:46,570 --> 00:10:49,760
And classical
computation is your data

277
00:10:49,760 --> 00:10:53,070
are a set of numbers--
what's pointing down.

278
00:10:53,070 --> 00:10:57,520
And the process of doing
a classical computation

279
00:10:57,520 --> 00:11:00,740
is, build a machine
which is governed

280
00:11:00,740 --> 00:11:05,110
by classical mechanics,
OK, that takes

281
00:11:05,110 --> 00:11:07,440
this initial configuration
and replaces it

282
00:11:07,440 --> 00:11:10,070
with a new one, f of 0, 1.

283
00:11:10,070 --> 00:11:12,730
Which is also some number,
so maybe it's 0, 0.

284
00:11:12,730 --> 00:11:14,612
I don't know--
it's whatever f is.

285
00:11:14,612 --> 00:11:16,070
And what you want
to do is you want

286
00:11:16,070 --> 00:11:18,960
to perform some calculation,
which is some function, a known

287
00:11:18,960 --> 00:11:21,282
function, of your
input variables--

288
00:11:21,282 --> 00:11:23,490
a function, in this case,
of two bits, which produces

289
00:11:23,490 --> 00:11:24,960
the output that
you want, and you

290
00:11:24,960 --> 00:11:26,381
build a machine that does that.

291
00:11:26,381 --> 00:11:26,880
OK.

292
00:11:26,880 --> 00:11:28,129
So let me give you an example.

293
00:11:28,129 --> 00:11:30,800
I want to build a
classical computer that

294
00:11:30,800 --> 00:11:37,370
takes a string of binary
integers-- 0, 0, 1, 0, 0.

295
00:11:37,370 --> 00:11:40,684
And performs a logical [? NOT ?]
on each bit independently.

296
00:11:40,684 --> 00:11:43,350
So I need to build that computer
out of objects available to me.

297
00:11:43,350 --> 00:11:43,850
OK.

298
00:11:43,850 --> 00:11:45,900
And I need to use nothing
other than the laws

299
00:11:45,900 --> 00:11:47,550
of classical mechanics.

300
00:11:47,550 --> 00:11:50,030
That has to suffice to
describe the system.

301
00:11:50,030 --> 00:11:53,270
So let me give an example
of such a computer.

302
00:11:53,270 --> 00:11:55,650
Why, Allen, why don't
you do this calculation?

303
00:11:55,650 --> 00:11:56,250
OK.

304
00:11:56,250 --> 00:11:58,260
So, that's the
input, and I am now

305
00:11:58,260 --> 00:12:00,840
the classical computer 0, 1, 1.

306
00:12:00,840 --> 00:12:01,340
Right?

307
00:12:01,340 --> 00:12:03,590
Now that's not actually what
you do in your cell phone

308
00:12:03,590 --> 00:12:04,810
or on your laptop.

309
00:12:04,810 --> 00:12:07,260
That uses transistors, but
it's the same basic idea.

310
00:12:07,260 --> 00:12:08,510
You build a set of machines.

311
00:12:08,510 --> 00:12:11,880
You build a set of objects,
you know, magnetic fields,

312
00:12:11,880 --> 00:12:14,840
electric currents that
effect this calculation.

313
00:12:14,840 --> 00:12:17,410
And so there's some relationship
between what electric fields

314
00:12:17,410 --> 00:12:18,240
you turn on and
what currents you

315
00:12:18,240 --> 00:12:19,870
induce to flow and
the calculation

316
00:12:19,870 --> 00:12:22,852
you want to perform--
f of the input.

317
00:12:22,852 --> 00:12:24,893
OK, this is a basic idea
of a classical computer.

318
00:12:27,462 --> 00:12:28,420
Oh, and one last thing.

319
00:12:28,420 --> 00:12:30,740
Suppose we have n bits.

320
00:12:30,740 --> 00:12:34,460
Suppose we have
n classical bits.

321
00:12:34,460 --> 00:12:37,669
Then we have 0, 1, the end.

322
00:12:37,669 --> 00:12:39,460
And how many numbers
do you have to specify

323
00:12:39,460 --> 00:12:41,380
to specify the
configuration of n bits?

324
00:12:41,380 --> 00:12:43,720
This is sort of [INAUDIBLE].

325
00:12:43,720 --> 00:12:46,770
No, you just need to specify
the number of each bit, right?

326
00:12:46,770 --> 00:12:49,180
So we need n bits.

327
00:12:49,180 --> 00:12:50,440
So n binary numbers.

328
00:12:55,230 --> 00:12:56,590
Everyone cool with that?

329
00:12:56,590 --> 00:13:00,200
I specify what each
register is, and I'm done.

330
00:13:00,200 --> 00:13:02,570
Imagine this is
quantum mechanical.

331
00:13:02,570 --> 00:13:05,710
Instead of having bits, let's
take quantum mechanical bits.

332
00:13:05,710 --> 00:13:09,250
By which I'm going
to mean a system

333
00:13:09,250 --> 00:13:14,550
that has two possible
observable states, 0 and 1.

334
00:13:14,550 --> 00:13:15,400
OK.

335
00:13:15,400 --> 00:13:17,233
So these are the possible
observable states.

336
00:13:19,710 --> 00:13:23,890
And to specify a
general configuration,

337
00:13:23,890 --> 00:13:31,410
I see that psi is equal to
some linear combination, alpha

338
00:13:31,410 --> 00:13:34,700
0 plus beta 1.

339
00:13:34,700 --> 00:13:39,260
Now, if I measure, we know
that we'll find either 0 or 1.

340
00:13:39,260 --> 00:13:41,010
If we measure the spin
in the z-direction,

341
00:13:41,010 --> 00:13:43,820
we will measure
either up or down.

342
00:13:43,820 --> 00:13:45,762
However, at a given
moment in time,

343
00:13:45,762 --> 00:13:47,720
the system need not be
in a state corresponding

344
00:13:47,720 --> 00:13:48,616
to a definite value.

345
00:13:48,616 --> 00:13:50,240
It could be in a
general superposition.

346
00:13:50,240 --> 00:13:51,516
Agreed?

347
00:13:51,516 --> 00:13:52,640
So now here's the question.

348
00:13:52,640 --> 00:13:54,098
How do you how many
numbers does it

349
00:13:54,098 --> 00:13:59,410
take to specify the state
of a single quantum bit?

350
00:13:59,410 --> 00:14:01,125
Two complex numbers.

351
00:14:01,125 --> 00:14:01,625
Right?

352
00:14:01,625 --> 00:14:04,140
It takes two complex numbers
to specify the state of a bit.

353
00:14:04,140 --> 00:14:08,160
And if we have n qubits, and
there are n of these guys,

354
00:14:08,160 --> 00:14:10,100
well then, how many
numbers does it take?

355
00:14:10,100 --> 00:14:11,475
Well, I have to
specify the state

356
00:14:11,475 --> 00:14:15,140
of every possible
superposition for

357
00:14:15,140 --> 00:14:16,970
every possible
configuration the system.

358
00:14:16,970 --> 00:14:21,500
So, for example, it could be
all 0, 0, 0, dot dot dot 0,

359
00:14:21,500 --> 00:14:22,590
plus beta.

360
00:14:22,590 --> 00:14:25,635
The first one is 1, and the
rest are 0, dot dot dot plus.

361
00:14:25,635 --> 00:14:26,760
And how it terms are there?

362
00:14:26,760 --> 00:14:28,680
There are 2 to the n terms.

363
00:14:28,680 --> 00:14:29,180
Right.

364
00:14:29,180 --> 00:14:34,250
So how many numbers do I need to
specify this state of n qubits?

365
00:14:34,250 --> 00:14:36,870
I need 2 to the n
complex numbers.

366
00:14:42,370 --> 00:14:43,330
Yeah?

367
00:14:43,330 --> 00:14:44,940
Everyone see that?

368
00:14:44,940 --> 00:14:48,750
So this immediately leads to
a really frustrating reality

369
00:14:48,750 --> 00:14:49,750
we have to face.

370
00:14:49,750 --> 00:14:53,330
Suppose you want to simulate,
using a classical computer,

371
00:14:53,330 --> 00:14:56,850
such as sits on my desktop, I
wanted to simulate a quantum

372
00:14:56,850 --> 00:15:00,460
mechanical system of n
bits, or n spin-1/2 states,

373
00:15:00,460 --> 00:15:02,610
evolving according to
some energy function--

374
00:15:02,610 --> 00:15:06,170
evolving according to some
Hamiltonian [INAUDIBLE].

375
00:15:06,170 --> 00:15:11,926
How many variables, and how
much memory will it take?

376
00:15:11,926 --> 00:15:12,749
2 to the n, right?

377
00:15:12,749 --> 00:15:14,540
If I've got n bits that
I want to describe,

378
00:15:14,540 --> 00:15:17,091
it's going to take 2 to
the n complex numbers.

379
00:15:17,091 --> 00:15:18,590
So that's how many
variables I have.

380
00:15:18,590 --> 00:15:23,220
So if I have 10 bits, literally,
10 little spin-1/2 objects,

381
00:15:23,220 --> 00:15:27,240
to accurately specify the
quantum configuration system,

382
00:15:27,240 --> 00:15:30,550
I need 2 to the 10
complex numbers.

383
00:15:30,550 --> 00:15:32,330
And that's for 10 spins.

384
00:15:32,330 --> 00:15:35,240
How many things make
this piece of chalk?

385
00:15:35,240 --> 00:15:35,920
Right?

386
00:15:35,920 --> 00:15:39,200
So could you ever,
in an honest way,

387
00:15:39,200 --> 00:15:41,600
simulate on a
classical computer,

388
00:15:41,600 --> 00:15:43,790
like sits on my desktop, a
quantum mechanical system

389
00:15:43,790 --> 00:15:45,220
the size of this chalk.

390
00:15:45,220 --> 00:15:46,020
Absolutely not.

391
00:15:46,020 --> 00:15:47,740
It's completely and
utterly intractable.

392
00:15:47,740 --> 00:15:50,198
You have to come up with some
sort of approximation scheme.

393
00:15:50,198 --> 00:15:52,300
So simulating quantum
systems directly,

394
00:15:52,300 --> 00:15:55,520
by just taking the spins and
representing them with a wave

395
00:15:55,520 --> 00:15:57,880
function, is wildly inefficient.

396
00:15:57,880 --> 00:16:00,310
Incredibly difficult.

397
00:16:00,310 --> 00:16:03,730
Interestingly, the converse
is almost certainly not

398
00:16:03,730 --> 00:16:04,630
so difficult.

399
00:16:04,630 --> 00:16:07,570
It's almost certainly
possible for a quantum system

400
00:16:07,570 --> 00:16:10,360
to simulate classical
evolution quite easily.

401
00:16:10,360 --> 00:16:12,730
And how do you know that?

402
00:16:12,730 --> 00:16:13,620
Yeah, here we are.

403
00:16:13,620 --> 00:16:15,020
Exactly, right?

404
00:16:15,020 --> 00:16:18,100
Nature apparently has no trouble
simulating classical mechanics

405
00:16:18,100 --> 00:16:19,910
with underlying
quantum mechanics.

406
00:16:19,910 --> 00:16:21,850
Quantum mechanics is
the operating system,

407
00:16:21,850 --> 00:16:25,136
and classical mechanics is
the simulation it's running.

408
00:16:25,136 --> 00:16:28,970
Yeah, in a very useful
sense, a very real sense.

409
00:16:28,970 --> 00:16:33,930
So, at this point,
Feynman, I think,

410
00:16:33,930 --> 00:16:36,593
was the first person who
really pointed this out.

411
00:16:36,593 --> 00:16:38,460
I don't know the
details of the history,

412
00:16:38,460 --> 00:16:42,340
but the lore is that this was
really from his observation.

413
00:16:42,340 --> 00:16:43,890
Look, if that's
true, this opens up

414
00:16:43,890 --> 00:16:45,990
a real possibility
for computation.

415
00:16:45,990 --> 00:16:50,830
If things like spins-- hard
problems like calculating

416
00:16:50,830 --> 00:16:54,360
how spins evolve or even the
motion of chalk in the room--

417
00:16:54,360 --> 00:16:56,590
can be done pretty efficiently
by nature, a comput

418
00:16:56,590 --> 00:17:00,064
if we can build a computer that
used quantum mechanical bits,

419
00:17:00,064 --> 00:17:01,730
whose variables were
quantum mechanical,

420
00:17:01,730 --> 00:17:03,750
and involved all the
superpositions and interference

421
00:17:03,750 --> 00:17:05,470
effects of quantum
mechanics, perhaps we

422
00:17:05,470 --> 00:17:08,550
could build computers
that run exponentially

423
00:17:08,550 --> 00:17:10,740
faster and with
exponentially less memory

424
00:17:10,740 --> 00:17:12,832
and less resources than
classical computer would.

425
00:17:12,832 --> 00:17:14,290
Because apparently
it works, right?

426
00:17:14,290 --> 00:17:17,030
We're here, as was
previously said.

427
00:17:17,030 --> 00:17:19,000
So this is the question
of quantum computing.

428
00:17:19,000 --> 00:17:20,250
Can you use quantum mechanics?

429
00:17:20,250 --> 00:17:24,250
Can use, in particular, the
quantum evolution of the system

430
00:17:24,250 --> 00:17:27,030
to perform calculations
that you care about,

431
00:17:27,030 --> 00:17:31,000
rather than classical
computation?

432
00:17:31,000 --> 00:17:34,726
And if you do so, do
you gain anything?

433
00:17:34,726 --> 00:17:36,100
Do you get any
sort of speed ups?

434
00:17:36,100 --> 00:17:37,099
Is there an enhancement?

435
00:17:37,099 --> 00:17:39,030
Is anything better?

436
00:17:39,030 --> 00:17:41,172
So, here's the basic gig.

437
00:17:41,172 --> 00:17:43,540
The basic gig is that we're
going to take our system,

438
00:17:43,540 --> 00:17:45,970
considering the following
kinds of systems.

439
00:17:45,970 --> 00:17:49,600
Our systems are going to
be n qubits or n spins.

440
00:17:49,600 --> 00:17:52,000
But because I want to
emphasize that the substrate

441
00:17:52,000 --> 00:17:54,594
doesn't matter--- the underlying
material doesn't matter--

442
00:17:54,594 --> 00:17:56,010
I'm going to refer
to them purely,

443
00:17:56,010 --> 00:17:58,360
and this is totally typical
in the field, as qubits,

444
00:17:58,360 --> 00:17:59,530
rather than spin systems.

445
00:17:59,530 --> 00:18:02,820
And the reason is it might
work use little isolated

446
00:18:02,820 --> 00:18:04,927
spinning particles
with intrinsic spin.

447
00:18:04,927 --> 00:18:07,260
Or that might turn out to be
technologically infeasible.

448
00:18:07,260 --> 00:18:09,840
It shouldn't matter at the end
of the day, in the same way

449
00:18:09,840 --> 00:18:11,756
that if I ask you, how
does you computer work?

450
00:18:11,756 --> 00:18:13,440
Or if I ask you to
write a code, you

451
00:18:13,440 --> 00:18:15,885
write some code in C
or Python or whatever--

452
00:18:15,885 --> 00:18:17,510
what the hip kids
are using these days.

453
00:18:17,510 --> 00:18:22,340
So you write some little Scheme
code-- I still love Scheme.

454
00:18:22,340 --> 00:18:23,844
You write some
little Scheme code,

455
00:18:23,844 --> 00:18:25,760
and it performs some
calculation as to defined

456
00:18:25,760 --> 00:18:30,344
in just logic, in lambda
calculus, in abstract logic.

457
00:18:30,344 --> 00:18:31,760
Do you need to
know the substrate?

458
00:18:31,760 --> 00:18:34,176
So you need to know whether
your transistors are built out

459
00:18:34,176 --> 00:18:35,760
of silicon or germanium?

460
00:18:35,760 --> 00:18:39,110
Or indeed, whether it's
running on vacuum tubes?

461
00:18:39,110 --> 00:18:42,302
No, that's the whole point of
abstracting away computation.

462
00:18:42,302 --> 00:18:43,510
The substrate doesn't matter.

463
00:18:43,510 --> 00:18:44,477
So we can use spin-1/2.

464
00:18:44,477 --> 00:18:46,810
And that's going to be useful
for us at various moments.

465
00:18:46,810 --> 00:18:48,601
But I want to emphasize
the important thing

466
00:18:48,601 --> 00:18:50,860
is the actual logic
of the process,

467
00:18:50,860 --> 00:18:52,210
not the underlying substrate.

468
00:18:52,210 --> 00:18:53,001
Here's what I need.

469
00:18:53,001 --> 00:18:54,670
My systems are going
to be n copies,

470
00:18:54,670 --> 00:19:00,830
or n qubits, where each bit
is specified by either 1,

471
00:19:00,830 --> 00:19:04,220
represented by up, or
0, represented by down.

472
00:19:04,220 --> 00:19:06,010
I'll use these interchangeably.

473
00:19:06,010 --> 00:19:09,580
So that the full system, psi,
is a superposition of sum

474
00:19:09,580 --> 00:19:13,720
over all possible-- sorry,
I didn't mean to write that.

475
00:19:13,720 --> 00:19:15,390
So this is my system.

476
00:19:15,390 --> 00:19:19,490
It's going to evolve according
to some energy operation.

477
00:19:19,490 --> 00:19:25,070
And so my input is going to
be some wave function, psi n,

478
00:19:25,070 --> 00:19:27,900
for these n qubits.

479
00:19:27,900 --> 00:19:34,320
Computation is going to
be, evolve with some energy

480
00:19:34,320 --> 00:19:41,670
operator-- which I've chosen--
to implement an algorithm,

481
00:19:41,670 --> 00:19:45,044
in the same way that the precise
series of magnetic fields

482
00:19:45,044 --> 00:19:46,710
that we turn on in a
classical computer,

483
00:19:46,710 --> 00:19:49,126
or currents that we allow to
flow in a classical computer,

484
00:19:49,126 --> 00:19:51,730
implement the algorithm
that we want to effect.

485
00:19:51,730 --> 00:19:53,230
We evolve with some
energy operator,

486
00:19:53,230 --> 00:19:54,130
implementing our
linear algorithm,

487
00:19:54,130 --> 00:19:55,671
and we get some
output wave function,

488
00:19:55,671 --> 00:19:59,991
psi-- again-- for our n
bits, n quantum bits, out.

489
00:19:59,991 --> 00:20:02,240
So this is the basic gig
with the quantum computation.

490
00:20:02,240 --> 00:20:06,510
We're just replacing strings of
binary numbers and functions--

491
00:20:06,510 --> 00:20:09,600
evolution according to
classical mechanics in Maxwell--

492
00:20:09,600 --> 00:20:13,070
to other strings of numbers
with superpositions of states.

493
00:20:13,070 --> 00:20:15,300
Superpositions of
strings, as it were,

494
00:20:15,300 --> 00:20:18,310
evolving according to
the Schrodinger equation

495
00:20:18,310 --> 00:20:21,220
into again general
superposition.

496
00:20:21,220 --> 00:20:22,340
So how does this differ?

497
00:20:22,340 --> 00:20:24,892
What are the key differences
from a classical computer?

498
00:20:24,892 --> 00:20:27,100
So the key things are first
off, the input and output

499
00:20:27,100 --> 00:20:28,890
are not definite values.

500
00:20:28,890 --> 00:20:30,640
They could, in general,
be superpositions.

501
00:20:30,640 --> 00:20:34,850
They do not have to correspond
to a definite state in the 1, 0

502
00:20:34,850 --> 00:20:36,660
basis.

503
00:20:36,660 --> 00:20:46,340
These, in general, are
superpositions of 0, 0, 0, 1,

504
00:20:46,340 --> 00:20:49,280
0, 0, dot dot dot.

505
00:20:49,280 --> 00:20:57,180
So the input and output
are superpositions

506
00:20:57,180 --> 00:20:59,290
of the values we'll measure.

507
00:21:05,180 --> 00:21:06,520
OK.

508
00:21:06,520 --> 00:21:10,880
Second, when we do measure, at
the end of the day, the output,

509
00:21:10,880 --> 00:21:13,846
we get interference from
the various different terms

510
00:21:13,846 --> 00:21:14,720
in the superposition.

511
00:21:14,720 --> 00:21:15,920
The different terms
in the superposition

512
00:21:15,920 --> 00:21:17,336
will lead to
interference effects.

513
00:21:17,336 --> 00:21:19,940
And so our output will be
sensitive to that interference

514
00:21:19,940 --> 00:21:22,260
between the different
terms in the superposition.

515
00:21:22,260 --> 00:21:24,350
It may be that we're
in some pure state.

516
00:21:24,350 --> 00:21:26,080
But it may be, more
generally, we'll

517
00:21:26,080 --> 00:21:28,840
be in some superposition of
the states we want to measure.

518
00:21:28,840 --> 00:21:30,298
So we're going to
get interference.

519
00:21:32,670 --> 00:21:36,210
So naively, this is a
disaster because this

520
00:21:36,210 --> 00:21:38,185
means we get
probabilistic outcomes.

521
00:21:42,130 --> 00:21:44,470
That sounds bad.

522
00:21:44,470 --> 00:21:50,417
And so, this leads to the key
move in quantum computation.

523
00:21:50,417 --> 00:21:52,250
Given that we're going
to have interference,

524
00:21:52,250 --> 00:21:54,375
and given that those
interference effects are going

525
00:21:54,375 --> 00:21:59,620
to affect the probabilities of
various outcomes, what we're

526
00:21:59,620 --> 00:22:03,980
going to want is to
tune, or orchestrate,

527
00:22:03,980 --> 00:22:11,340
the interference to
get a definite outcome.

528
00:22:11,340 --> 00:22:14,870
We want to get the correct
answer out, rather than

529
00:22:14,870 --> 00:22:16,925
just probably the
correct answer out.

530
00:22:16,925 --> 00:22:18,550
Now there's a slight
variation of this.

531
00:22:18,550 --> 00:22:20,450
It's not obvious at the
beginning, how to do that.

532
00:22:20,450 --> 00:22:21,700
It's not even
clear that you can.

533
00:22:21,700 --> 00:22:22,970
So I'm going to show
you that you can.

534
00:22:22,970 --> 00:22:24,970
I'm going to show you an
explicit algorithm that

535
00:22:24,970 --> 00:22:26,340
realizes this.

536
00:22:26,340 --> 00:22:29,110
But, OK, that's obviously
going to be tricky.

537
00:22:29,110 --> 00:22:31,480
Here's something
else you can do.

538
00:22:31,480 --> 00:22:34,294
You can also focus
on special problems.

539
00:22:34,294 --> 00:22:35,460
Focus on checkable problems.

540
00:22:39,850 --> 00:22:42,370
And what I mean
by this is imagine

541
00:22:42,370 --> 00:22:45,600
we have some algorithm
that gives us an output,

542
00:22:45,600 --> 00:22:49,740
and that output is
probably the right answer.

543
00:22:49,740 --> 00:22:50,680
But we're not sure.

544
00:22:50,680 --> 00:22:52,160
There's some quantum
mechanical probability

545
00:22:52,160 --> 00:22:54,290
that this is the correct
answer to our computation--

546
00:22:54,290 --> 00:22:55,914
there's some probability
that it's not.

547
00:22:55,914 --> 00:22:58,170
So, if the calculation
was difficult,

548
00:22:58,170 --> 00:23:01,481
but it's easy to check whether
you have the right answer

549
00:23:01,481 --> 00:23:02,230
then that's great.

550
00:23:02,230 --> 00:23:04,188
Imagine it's 10% that
you get the right answer,

551
00:23:04,188 --> 00:23:06,220
but it's trivial to check.

552
00:23:06,220 --> 00:23:08,010
So, for example,
factoring numbers, right?

553
00:23:08,010 --> 00:23:09,750
If you multiply the
numbers, that's easy.

554
00:23:09,750 --> 00:23:11,791
You check to see whether
you got the right thing.

555
00:23:11,791 --> 00:23:14,980
So, for example, if you imagine
that I build a computer that

556
00:23:14,980 --> 00:23:16,987
is supposed to factor
numbers-- I almost

557
00:23:16,987 --> 00:23:18,070
said factor prime numbers.

558
00:23:18,070 --> 00:23:19,403
That would be a boring computer.

559
00:23:19,403 --> 00:23:21,750
So imagine that we build a
machine that factors numbers.

560
00:23:21,750 --> 00:23:22,249
Right?

561
00:23:22,249 --> 00:23:24,650
And so imagine its
output is probabilistic.

562
00:23:24,650 --> 00:23:26,860
So, you say 15, and
I say 3 times 5.

563
00:23:26,860 --> 00:23:28,360
You say 15, I say 5 times 3.

564
00:23:28,360 --> 00:23:29,692
You say 15, I say 7 tiimes 2.

565
00:23:29,692 --> 00:23:31,900
At that point, you're like,
well, which one is right?

566
00:23:31,900 --> 00:23:34,050
Well, you can check
by multiplying.

567
00:23:34,050 --> 00:23:36,120
So if you have a problem,
which is easy to check,

568
00:23:36,120 --> 00:23:39,460
but hard to do, then
probablistic is already OK.

569
00:23:39,460 --> 00:23:42,710
That's just as true classically
as it is quantum mechanically.

570
00:23:42,710 --> 00:23:44,890
Those are our basic moves.

571
00:23:44,890 --> 00:23:49,290
And so, the key thing
for us is that when

572
00:23:49,290 --> 00:23:53,140
we have n quantum bits, we have
these interference effects.

573
00:23:53,140 --> 00:23:56,100
And in particular, as we
started talking about last time,

574
00:23:56,100 --> 00:23:57,377
we get entanglement effects.

575
00:23:57,377 --> 00:23:59,210
What we're going to
discover is that the key

576
00:23:59,210 --> 00:24:00,580
to making a good
quantum algorithm

577
00:24:00,580 --> 00:24:02,820
is going to go to attune the
entanglement appropriately.

578
00:24:02,820 --> 00:24:04,520
So going to have to define
that for you in just a minute.

579
00:24:04,520 --> 00:24:05,396
Yeah.

580
00:24:05,396 --> 00:24:08,030
AUDIENCE: [INAUDIBLE].

581
00:24:08,030 --> 00:24:08,920
PROFESSOR: Sorry?

582
00:24:08,920 --> 00:24:10,170
AUDIENCE: [INAUDIBLE] process.

583
00:24:10,170 --> 00:24:12,010
PROFESSOR: Well, it
may or may not be.

584
00:24:12,010 --> 00:24:15,110
So the check, for example, ,
imagine the process I just gave

585
00:24:15,110 --> 00:24:15,610
you.

586
00:24:15,610 --> 00:24:16,980
So the question
is, what do you do

587
00:24:16,980 --> 00:24:18,688
if the checking process
is probabilistic?

588
00:24:18,688 --> 00:24:21,040
But you can use a classical
computer to check,

589
00:24:21,040 --> 00:24:23,560
if you have an easy checking
algorithm-- for example,

590
00:24:23,560 --> 00:24:24,940
multiplying numbers together.

591
00:24:24,940 --> 00:24:26,480
AUDIENCE: [INAUDIBLE].

592
00:24:26,480 --> 00:24:27,250
PROFESSOR: No.

593
00:24:27,250 --> 00:24:27,460
Good.

594
00:24:27,460 --> 00:24:28,670
And so the question
then becomes,

595
00:24:28,670 --> 00:24:30,150
doesn't that defeat the point
of using a quantum computer,

596
00:24:30,150 --> 00:24:32,066
if you're using a classical
computer to check?

597
00:24:32,066 --> 00:24:33,570
And so here's the thing.

598
00:24:33,570 --> 00:24:35,372
If I ask you to factor
a gigantic number,

599
00:24:35,372 --> 00:24:36,330
what do you have to do?

600
00:24:36,330 --> 00:24:38,470
Well, you have to take all
numbers up to its square root,

601
00:24:38,470 --> 00:24:40,095
and you have to
multiply them together,

602
00:24:40,095 --> 00:24:41,130
and that takes forever.

603
00:24:41,130 --> 00:24:41,630
Right?

604
00:24:41,630 --> 00:24:43,810
But if I tell you,
do these two numbers

605
00:24:43,810 --> 00:24:45,740
multiply to give you
the third number?

606
00:24:45,740 --> 00:24:46,864
That's easy, right?

607
00:24:46,864 --> 00:24:48,780
So I use a quantum
computer for the hard part,

608
00:24:48,780 --> 00:24:50,840
and the classical computer
for the forward part.

609
00:24:50,840 --> 00:24:51,750
For the checking.

610
00:24:51,750 --> 00:24:54,230
And that's a special class of
problems which are checkable.

611
00:24:54,230 --> 00:24:57,030
These all have fancy names
in information theory.

612
00:24:57,030 --> 00:24:59,372
I'm not going to
use the fancy names.

613
00:24:59,372 --> 00:25:02,029
AUDIENCE: [INAUDIBLE]
the wave function?

614
00:25:02,029 --> 00:25:03,070
PROFESSOR: Yeah, exactly.

615
00:25:03,070 --> 00:25:05,670
I mean, suppose I
give you some output,

616
00:25:05,670 --> 00:25:08,621
and let's say the output
is 0, 0 plus 1, 1.

617
00:25:08,621 --> 00:25:09,120
Right?

618
00:25:09,120 --> 00:25:11,510
What are you going to get,
if you measure the first bit?

619
00:25:11,510 --> 00:25:13,176
Well, you're either
going to get 0 or 1,

620
00:25:13,176 --> 00:25:15,980
with one probability
or another, right?

621
00:25:15,980 --> 00:25:19,905
When we compute
the probabilities

622
00:25:19,905 --> 00:25:22,280
in general, when we have many
terms in our superposition,

623
00:25:22,280 --> 00:25:23,905
we're going to get
interference effects

624
00:25:23,905 --> 00:25:26,080
from different terms
in the superposition.

625
00:25:26,080 --> 00:25:29,340
And those interference terms
will tune the probability,

626
00:25:29,340 --> 00:25:33,670
so they're not just the naive
probability of that one thing.

627
00:25:33,670 --> 00:25:35,422
But you get these
interference terms.

628
00:25:35,422 --> 00:25:37,130
Norm squared of one,
norm squared of two,

629
00:25:37,130 --> 00:25:39,840
and then the real part twice
real part of the cross term.

630
00:25:39,840 --> 00:25:41,080
And that twice the real
part of the cross term

631
00:25:41,080 --> 00:25:42,380
didn't exist classically.

632
00:25:42,380 --> 00:25:43,974
Quantum mechanically,
it's important,

633
00:25:43,974 --> 00:25:45,640
and it can change the
final probability.

634
00:25:45,640 --> 00:25:47,980
That's what I mean by
the interference effects.

635
00:25:47,980 --> 00:25:50,310
OK, so let's make all
this much more explicit.

636
00:25:50,310 --> 00:25:52,140
So far I've just
given you philosophy,

637
00:25:52,140 --> 00:25:54,490
and you should be deeply
suspicious of philosophy

638
00:25:54,490 --> 00:25:57,740
in a physics classroom.

639
00:25:57,740 --> 00:26:03,890
So, again, to be explicit,
a quantum bit, or qubit,

640
00:26:03,890 --> 00:26:06,450
is equal to a two state
system, and, again,

641
00:26:06,450 --> 00:26:09,410
the substrate doesn't matter.

642
00:26:09,410 --> 00:26:12,480
I could be talking
about spin-1/2 systems,

643
00:26:12,480 --> 00:26:17,800
or I could be talking about
ropes on a cylinder with

644
00:26:17,800 --> 00:26:18,550
winding mod 2.

645
00:26:18,550 --> 00:26:20,424
I could be talking about
all sorts of things.

646
00:26:20,424 --> 00:26:25,910
But it's some system with
two quantum states, 0 and 1.

647
00:26:25,910 --> 00:26:28,530
I want to emphasize this is
not going to be on the final.

648
00:26:28,530 --> 00:26:32,220
So this is just for
your moral well being.

649
00:26:32,220 --> 00:26:34,587
So a quantum bit is
a two state system.

650
00:26:34,587 --> 00:26:36,670
We have these two states,
which I'll call 0 and 1,

651
00:26:36,670 --> 00:26:39,360
and a general state, the
general wave function, psi,

652
00:26:39,360 --> 00:26:43,730
is equal to alpha 0 plus beta 1.

653
00:26:43,730 --> 00:26:44,440
OK.

654
00:26:44,440 --> 00:26:47,900
And what this means is the
probability that I measure 0

655
00:26:47,900 --> 00:26:50,320
is equal to norm alpha
squared, et cetera.

656
00:26:50,320 --> 00:26:52,680
OK.

657
00:26:52,680 --> 00:26:54,287
Now, again many
systems can be used--

658
00:26:54,287 --> 00:26:55,370
many different substrates.

659
00:26:55,370 --> 00:26:57,602
This is what I'm going
to mean by a qubit.

660
00:26:57,602 --> 00:26:59,340
So that's one qubit.

661
00:26:59,340 --> 00:27:01,767
The much more interesting
system is two qubits.

662
00:27:01,767 --> 00:27:02,850
So let's study two qubits.

663
00:27:06,670 --> 00:27:09,220
So in the case of two qubits,
what's a general state?

664
00:27:09,220 --> 00:27:10,862
Well a general state
is going to be,

665
00:27:10,862 --> 00:27:12,320
well, the first
particle could be 0

666
00:27:12,320 --> 00:27:14,985
and the second particle
could also be 0.

667
00:27:14,985 --> 00:27:16,790
Or we could have
the first particle 0

668
00:27:16,790 --> 00:27:19,460
and the second particle
1, with some coefficient,

669
00:27:19,460 --> 00:27:28,860
plus beta, plus gamma
1, 0, plus delta 0, 1.

670
00:27:28,860 --> 00:27:29,384
OK.

671
00:27:29,384 --> 00:27:31,050
So this is just a
general superposition.

672
00:27:31,050 --> 00:27:32,716
Now, you might worry
at this point look,

673
00:27:32,716 --> 00:27:35,180
are these identical, or
these non-identical spins?

674
00:27:35,180 --> 00:27:36,270
But here's the thing.

675
00:27:36,270 --> 00:27:38,000
I've got a spin
here-- it's in a box--

676
00:27:38,000 --> 00:27:39,020
and I've got another
spin here-- it's

677
00:27:39,020 --> 00:27:40,410
in a box-- I've got
another qubit over here.

678
00:27:40,410 --> 00:27:40,980
It's in a box.

679
00:27:40,980 --> 00:27:42,030
So they're distinguishable
because they're

680
00:27:42,030 --> 00:27:43,920
in different places
in my laboratory.

681
00:27:43,920 --> 00:27:46,400
So these are
distinguishable particles.

682
00:27:46,400 --> 00:27:48,900
The particle in this box
is either up or down.

683
00:27:48,900 --> 00:27:49,400
OK.

684
00:27:49,400 --> 00:27:50,858
So these are
distinguishable, and I

685
00:27:50,858 --> 00:27:53,176
don't have to worry about
symmetrizatoin, Bosonic

686
00:27:53,176 --> 00:27:54,800
or Fermionc statistics,
or any of that.

687
00:27:54,800 --> 00:27:56,170
They're distinguishable.

688
00:27:56,170 --> 00:27:58,040
And we need normalization,
so norm alpha

689
00:27:58,040 --> 00:28:01,010
squared plus dot dot
dot is equal to 1.

690
00:28:01,010 --> 00:28:02,010
And what does this mean?

691
00:28:02,010 --> 00:28:04,218
What this means is that,
for example, the probability

692
00:28:04,218 --> 00:28:07,220
of the first part, which
I'll call a is equal to 0,

693
00:28:07,220 --> 00:28:10,542
is equal to-- well, we sum
up all the possible ways

694
00:28:10,542 --> 00:28:11,250
we could do that.

695
00:28:11,250 --> 00:28:16,000
We have norm alpha squared
plus norm beta squared.

696
00:28:16,000 --> 00:28:18,910
Whoops-- this was 1, 1.

697
00:28:18,910 --> 00:28:20,340
Thank you.

698
00:28:20,340 --> 00:28:24,590
And if we're more specific, the
probability of the first qubit

699
00:28:24,590 --> 00:28:28,890
is 0, and the second qubit
is 1-- is equal to 1.

700
00:28:28,890 --> 00:28:29,650
I need 0, 1.

701
00:28:29,650 --> 00:28:32,740
That's this guy,
norm beta squared.

702
00:28:32,740 --> 00:28:33,830
OK.

703
00:28:33,830 --> 00:28:34,980
What I mean by two qubits.

704
00:28:34,980 --> 00:28:37,430
But this immediately
leads to a funny thing.

705
00:28:37,430 --> 00:28:40,670
There are two kinds of
states that a pair of qubits

706
00:28:40,670 --> 00:28:43,540
could be in, a very
special set of states,

707
00:28:43,540 --> 00:28:48,800
which are called
separable states.

708
00:28:48,800 --> 00:28:50,120
And these are special.

709
00:28:53,760 --> 00:28:58,400
And these states correspond
to the full system

710
00:28:58,400 --> 00:29:00,735
being in the state where
the first particles is

711
00:29:00,735 --> 00:29:02,380
in one state, and
the second particle

712
00:29:02,380 --> 00:29:04,370
is in the second state.

713
00:29:04,370 --> 00:29:04,870
OK.

714
00:29:04,870 --> 00:29:09,730
So, for example,
this could be eg

715
00:29:09,730 --> 00:29:18,690
the state 1 over
root 2-- sorry-- 0

716
00:29:18,690 --> 00:29:23,110
plus 1 for the first particle,
and the second particle

717
00:29:23,110 --> 00:29:29,078
to be in the state 1
over root 2, 0, minus 1.

718
00:29:33,020 --> 00:29:38,180
[INAUDIBLE] Let's just call
these general coefficients.

719
00:29:38,180 --> 00:29:45,010
a plus b and times c plus d.

720
00:29:45,010 --> 00:29:46,230
So what does this equal to?

721
00:29:46,230 --> 00:29:47,875
Well, this is of
the form-- there's

722
00:29:47,875 --> 00:29:53,140
going to be a term that's
0, 0, and that's ac.

723
00:29:53,140 --> 00:29:58,420
Plus a term that's 0, 1,
and that's ad with a minus.

724
00:29:58,420 --> 00:30:00,970
Minus ad.

725
00:30:00,970 --> 00:30:03,340
Plus a term that's 1, 0.

726
00:30:03,340 --> 00:30:04,500
That's bc.

727
00:30:04,500 --> 00:30:11,200
And minus a term that's
1, 1, and that's bd.

728
00:30:11,200 --> 00:30:11,890
OK.

729
00:30:11,890 --> 00:30:13,582
This is clearly not
generic, because it

730
00:30:13,582 --> 00:30:15,790
implies relationships amongst
the alpha, beta, delta,

731
00:30:15,790 --> 00:30:19,020
gamma, apart from
just normalizability.

732
00:30:19,020 --> 00:30:20,444
Everyone see that?

733
00:30:20,444 --> 00:30:21,860
So it's a pretty
non-trivial thing

734
00:30:21,860 --> 00:30:24,027
that you can write a thing
as, the first particle is

735
00:30:24,027 --> 00:30:26,485
in one state, and the second
particles in the second state.

736
00:30:26,485 --> 00:30:28,260
And here's physically
what that means.

737
00:30:28,260 --> 00:30:34,163
Physically, what that
means is that, if you're

738
00:30:34,163 --> 00:30:36,300
in a state like this,
and I ask you, what's

739
00:30:36,300 --> 00:30:40,260
the probability that I measure
the first particle to be 0,

740
00:30:40,260 --> 00:30:42,320
do I need to know anything
about the state of b?

741
00:30:45,200 --> 00:30:45,764
No.

742
00:30:45,764 --> 00:30:47,180
If I want to know
what probability

743
00:30:47,180 --> 00:30:49,100
of the first particle is 0,
I just take norm a squared.

744
00:30:49,100 --> 00:30:50,560
Because I'm going
to get plus norm

745
00:30:50,560 --> 00:30:51,810
c squared plus norm d squared.

746
00:30:56,515 --> 00:30:57,140
So that's fine.

747
00:30:57,140 --> 00:30:59,320
So, imagine the probability
of the first particle

748
00:30:59,320 --> 00:31:02,130
being up or down is
independent of any information

749
00:31:02,130 --> 00:31:03,540
about the second
particle, right?

750
00:31:03,540 --> 00:31:04,740
There is another thing
that's important.

751
00:31:04,740 --> 00:31:06,406
Suppose I tell you,
ah ha I've measured,

752
00:31:06,406 --> 00:31:08,490
and the first particle
is in the state 0.

753
00:31:08,490 --> 00:31:09,020
OK.

754
00:31:09,020 --> 00:31:09,450
Cool?

755
00:31:09,450 --> 00:31:11,491
What is the state subsequent
to that measurement?

756
00:31:13,950 --> 00:31:16,920
So if we measure
a is equal to 0,

757
00:31:16,920 --> 00:31:18,640
what is the state subsequent?

758
00:31:18,640 --> 00:31:21,670
Psi is equal to--
well, that's 0,

759
00:31:21,670 --> 00:31:24,645
and we know we've lost this,
so this particular subsystem,

760
00:31:24,645 --> 00:31:27,930
this particular qubit has
been collapsed to the state 0,

761
00:31:27,930 --> 00:31:37,544
so we have 0 times c, 0, minus
d, 1 for the second particle.

762
00:31:37,544 --> 00:31:39,210
Have we learned
anything about the state

763
00:31:39,210 --> 00:31:40,168
of the second particle?

764
00:31:43,644 --> 00:31:45,310
Have we learned
anything about the state

765
00:31:45,310 --> 00:31:46,850
of the second particle?

766
00:31:46,850 --> 00:31:47,640
Absolutely not.

767
00:31:47,640 --> 00:31:48,140
Right?

768
00:31:48,140 --> 00:31:49,640
Beforehand, what
was the probability

769
00:31:49,640 --> 00:31:52,360
that the second article is 0?

770
00:31:52,360 --> 00:31:53,180
Norm c squared.

771
00:31:53,180 --> 00:31:54,450
And the second particle 1?

772
00:31:54,450 --> 00:31:55,282
Norm d squared.

773
00:31:55,282 --> 00:31:56,240
Now what's the problem?

774
00:31:56,240 --> 00:31:56,739
Same.

775
00:31:56,739 --> 00:31:58,140
Norm c squared, norm d squared.

776
00:31:58,140 --> 00:31:59,640
So when you have a
seperable system,

777
00:31:59,640 --> 00:32:02,990
measuring one qubit tells you
nothing about the other qubit.

778
00:32:02,990 --> 00:32:03,930
Cool?

779
00:32:03,930 --> 00:32:08,330
On the other hand, consider a
state which is not separable.

780
00:32:08,330 --> 00:32:19,369
So the generic states
are not separable.

781
00:32:19,369 --> 00:32:21,410
And let me give you an
example of the state which

782
00:32:21,410 --> 00:32:22,750
is not seperable.

783
00:32:22,750 --> 00:32:23,980
Which one do I want to pick?

784
00:32:23,980 --> 00:32:24,930
Yeah, what the hell.

785
00:32:24,930 --> 00:32:34,430
Psi is 1 over root
2, 0, 0, plus 1, 1.

786
00:32:34,430 --> 00:32:35,960
Can this be written
as the product

787
00:32:35,960 --> 00:32:37,376
of the first
particle in one state

788
00:32:37,376 --> 00:32:38,918
and the second
particle in the other?

789
00:32:38,918 --> 00:32:41,042
No, because they were have
to be cross terms, which

790
00:32:41,042 --> 00:32:41,800
don't exist here.

791
00:32:41,800 --> 00:32:43,383
Right, just compare
this to that form.

792
00:32:43,383 --> 00:32:45,670
So, we have that the
coefficient of 0, 0 is ac.

793
00:32:45,670 --> 00:32:47,960
So a and c must
both be non-zero,

794
00:32:47,960 --> 00:32:49,880
and the coefficient
of 1, 1 is bd.

795
00:32:49,880 --> 00:32:52,810
So the coefficient bd must be--
both of those must be non-zero.

796
00:32:52,810 --> 00:32:54,554
So ac and bd are all non-zero.

797
00:32:54,554 --> 00:32:56,720
That means these terms have
to exist in order for it

798
00:32:56,720 --> 00:32:59,392
to be separable.

799
00:32:59,392 --> 00:33:00,257
Yeah?

800
00:33:00,257 --> 00:33:01,715
Because a and b
[? can't ?] vanish,

801
00:33:01,715 --> 00:33:04,131
and b and c [? can't ?] vanish
And these orthogonal states

802
00:33:04,131 --> 00:33:05,010
are independent.

803
00:33:05,010 --> 00:33:07,389
So this is not a
separable state.

804
00:33:07,389 --> 00:33:08,680
We call these states entangled.

805
00:33:11,720 --> 00:33:18,900
And it's funny,
because [INAUDIBLE] I

806
00:33:18,900 --> 00:33:20,825
think there's an e in there.

807
00:33:20,825 --> 00:33:21,700
AUDIENCE: [INAUDIBLE]

808
00:33:26,417 --> 00:33:27,000
PROFESSOR: OK.

809
00:33:27,000 --> 00:33:29,920
That's better.

810
00:33:29,920 --> 00:33:33,710
Look, I'm not a
professor of spelling.

811
00:33:33,710 --> 00:33:36,300
It's a little bit funny to
give these a special name

812
00:33:36,300 --> 00:33:41,220
and call them entangled, because
the generic state is entangled.

813
00:33:41,220 --> 00:33:45,080
It's sort of like
calling mice, mice,

814
00:33:45,080 --> 00:33:48,650
and calling all the
other mammals non-mice.

815
00:33:48,650 --> 00:33:50,660
Oh look, well, that
was bad example.

816
00:33:50,660 --> 00:33:52,605
Oh look at mice right
across from [INAUDIBLE].

817
00:33:52,605 --> 00:33:55,280
AUDIENCE: [LAUGHS].

818
00:33:55,280 --> 00:33:58,309
PROFESSOR: Hearkening back
to an earlier lecture.

819
00:33:58,309 --> 00:34:00,225
So, in any case, we give
these a special name,

820
00:34:00,225 --> 00:34:01,830
and the reason we give
them a special name

821
00:34:01,830 --> 00:34:03,940
is that these seperable states
do you more or less what

822
00:34:03,940 --> 00:34:04,760
you'd expect classically.

823
00:34:04,760 --> 00:34:05,990
There are no
strange correlations

824
00:34:05,990 --> 00:34:08,080
between the state of one
and the state of the other.

825
00:34:08,080 --> 00:34:08,980
They're independent quantities.

826
00:34:08,980 --> 00:34:11,409
But when you have a generic
state, something funny happens.

827
00:34:11,409 --> 00:34:12,030
They're entangled.

828
00:34:12,030 --> 00:34:13,696
And here's the funny
thing that happens.

829
00:34:13,696 --> 00:34:16,920
Suppose given the state, what's
the probability that I measure

830
00:34:16,920 --> 00:34:20,520
the first particle
be up, or to be 0?

831
00:34:20,520 --> 00:34:21,250
1/2.

832
00:34:21,250 --> 00:34:23,583
What's the probability that
I measure the first particle

833
00:34:23,583 --> 00:34:25,389
to be 1?

834
00:34:25,389 --> 00:34:26,770
1/2.

835
00:34:26,770 --> 00:34:29,510
So, before doing
any measurements,

836
00:34:29,510 --> 00:34:31,790
the first particle is
equally likely to be 0 or 1.

837
00:34:31,790 --> 00:34:35,719
So suppose I measure the
second particle to be up.

838
00:34:35,719 --> 00:34:37,000
OK.

839
00:34:37,000 --> 00:34:40,130
Then the probability that I
measure the first particle is 0

840
00:34:40,130 --> 00:34:43,090
is equal to 0, and
the probability

841
00:34:43,090 --> 00:34:46,462
that the first particle
is 1 is equal to 1,

842
00:34:46,462 --> 00:34:48,670
because I've this collapse
onto this term in the wave

843
00:34:48,670 --> 00:34:51,070
pack, the wave function.

844
00:34:51,070 --> 00:34:54,909
So measuring the second
qubit alters the probability

845
00:34:54,909 --> 00:34:56,900
distribution for
the first qubit.

846
00:34:56,900 --> 00:34:59,166
These guys aren't independent.

847
00:34:59,166 --> 00:35:00,790
They are correllated,
in a way that you

848
00:35:00,790 --> 00:35:01,960
studied on the problem set.

849
00:35:01,960 --> 00:35:02,793
They are correlated.

850
00:35:02,793 --> 00:35:05,570
They're in a correlated state.

851
00:35:05,570 --> 00:35:08,866
We call this correlation
entanglement.

852
00:35:08,866 --> 00:35:10,990
And here's the thing that's
most spooky about this,

853
00:35:10,990 --> 00:35:12,690
and we'll come back to
this in a few minutes,

854
00:35:12,690 --> 00:35:14,860
but I didn't tell you
anything about the geometry

855
00:35:14,860 --> 00:35:15,541
of the set up.

856
00:35:15,541 --> 00:35:17,790
But, in fact, when I was
thinking of that measurement,

857
00:35:17,790 --> 00:35:20,900
I built the two little qubits
in my lab, and I took one,

858
00:35:20,900 --> 00:35:24,180
and I sent it to France--
because France, you know.

859
00:35:24,180 --> 00:35:27,630
And I took the other one, and
I said it to the planet Zorg.

860
00:35:27,630 --> 00:35:30,070
And so off on the planet is
some poor experimentalist

861
00:35:30,070 --> 00:35:31,240
huddling in the cold.

862
00:35:31,240 --> 00:35:34,630
And our French experimentalist
makes a measurement,

863
00:35:34,630 --> 00:35:37,500
altering the probability
distribution instantaneously

864
00:35:37,500 --> 00:35:39,890
on Zorg.

865
00:35:39,890 --> 00:35:41,980
That should make you
a little worried.

866
00:35:41,980 --> 00:35:43,480
That sounds a little crazy.

867
00:35:43,480 --> 00:35:45,890
We'll come back to why
that is or isn't crazy,

868
00:35:45,890 --> 00:35:50,680
and the EPR analysis,
which puts flesh on it,

869
00:35:50,680 --> 00:35:52,892
sounds crazy, in a little bit.

870
00:35:52,892 --> 00:35:54,600
But for the moment,
let me just emphasize

871
00:35:54,600 --> 00:35:56,170
that, while the generic
state is entangled,

872
00:35:56,170 --> 00:35:57,810
the generic state is
also different than what

873
00:35:57,810 --> 00:35:59,590
your classical
intuition would expect.

874
00:35:59,590 --> 00:36:02,780
There are correlations
between the particles.

875
00:36:02,780 --> 00:36:05,457
So this is something
that happens

876
00:36:05,457 --> 00:36:07,040
with quantum mechanical
particles that

877
00:36:07,040 --> 00:36:08,744
doesn't happen with
classical particles.

878
00:36:08,744 --> 00:36:10,160
And that means
it's something that

879
00:36:10,160 --> 00:36:12,620
can be used in a
quantum computer that

880
00:36:12,620 --> 00:36:15,520
can't be used in a
classical computer.

881
00:36:15,520 --> 00:36:17,740
So let's see what
using entanglement

882
00:36:17,740 --> 00:36:21,230
gives us for computation.

883
00:36:21,230 --> 00:36:24,420
So let's go come back to the
quantum computing problem.

884
00:36:24,420 --> 00:36:27,430
And how do we
compute with qubits?

885
00:36:27,430 --> 00:36:28,449
So how to compute.

886
00:36:33,301 --> 00:36:33,800
OK.

887
00:36:33,800 --> 00:36:39,519
So again, as usual, the way we
take our input to our output

888
00:36:39,519 --> 00:36:41,060
is we build a machine
that implements

889
00:36:41,060 --> 00:36:42,940
an algorithm of our
choice by arranging

890
00:36:42,940 --> 00:36:44,940
the physical evolution
of the system under time.

891
00:36:44,940 --> 00:36:48,800
So that means picking
an electric field.

892
00:36:48,800 --> 00:36:52,250
Sorry, an energy operator--
an electric field, good lord.

893
00:36:52,250 --> 00:36:55,090
So it means picking
an energy operator.

894
00:36:55,090 --> 00:36:55,590
OK.

895
00:36:55,590 --> 00:37:02,180
So computing is Schroedinger
evolution with our chosen,

896
00:37:02,180 --> 00:37:06,780
our attuned, energy operator.

897
00:37:06,780 --> 00:37:08,850
So, for example, I
want to build for you

898
00:37:08,850 --> 00:37:10,910
now a couple of basic
circuit elements

899
00:37:10,910 --> 00:37:13,020
that you might want to
use in a quantum computer.

900
00:37:13,020 --> 00:37:14,806
So example one.

901
00:37:14,806 --> 00:37:17,820
The first example is, I
want to build a NOT gate

902
00:37:17,820 --> 00:37:20,880
And what NOT means is
that it takes the state 0

903
00:37:20,880 --> 00:37:25,581
and gives me 1, and it takes
the state 1, and it gives me 0.

904
00:37:25,581 --> 00:37:26,080
OK.

905
00:37:26,080 --> 00:37:27,742
This is what NOT does.

906
00:37:27,742 --> 00:37:28,950
So how do I build a NOT gate?

907
00:37:28,950 --> 00:37:34,050
Well, I can realize
this in a nice way.

908
00:37:34,050 --> 00:37:34,770
Doot do do.

909
00:37:34,770 --> 00:37:35,210
Do I want to say it that way?

910
00:37:35,210 --> 00:37:35,950
Yeah, good.

911
00:37:35,950 --> 00:37:39,970
So if I realize 0
as 1, 0, and 1 as 0,

912
00:37:39,970 --> 00:37:50,690
1 from last lecture and 0, 1 to
1, 0, so we need an operator.

913
00:37:50,690 --> 00:37:55,180
We need time evolution to effect
multiplication by an operator.

914
00:37:55,180 --> 00:37:56,960
That takes this
vector to this vector,

915
00:37:56,960 --> 00:37:58,340
and this vector to this vector.

916
00:37:58,340 --> 00:38:00,580
We know what that
unitary operation is.

917
00:38:00,580 --> 00:38:04,380
That unitary operation,
which I'll call NOT,

918
00:38:04,380 --> 00:38:07,000
must equal to 0, 1, 1, 0.

919
00:38:07,000 --> 00:38:09,570
And this operation takes
this guy to this guy,

920
00:38:09,570 --> 00:38:11,440
and it takes this
guy to this guy.

921
00:38:11,440 --> 00:38:13,630
Yeah?

922
00:38:13,630 --> 00:38:18,115
But I can write this
as-- I like this-- shoot,

923
00:38:18,115 --> 00:38:26,100
there's a phase-- minus i, e
to the i pi over 2, 0, 1, 1, 0.

924
00:38:28,715 --> 00:38:30,090
I mean, you can't
stop me, right?

925
00:38:30,090 --> 00:38:34,930
So this expanded out, as we've
done before, expanding this out

926
00:38:34,930 --> 00:38:37,260
gives me, with
the exponential, 1

927
00:38:37,260 --> 00:38:39,580
plus the thing and
all the other terms.

928
00:38:39,580 --> 00:38:45,240
This becomes minus i times
cosine of pi over 2 times

929
00:38:45,240 --> 00:38:47,710
the identity, the
2 by 2 identity,

930
00:38:47,710 --> 00:38:55,170
up plus i sine of pi
over 2 times 0, 1, 1, 0.

931
00:38:57,720 --> 00:38:59,620
But what's cosine of pi upon 2?

932
00:39:02,400 --> 00:39:03,890
Yeah, 0.

933
00:39:03,890 --> 00:39:04,390
Come on.

934
00:39:04,390 --> 00:39:05,360
Is everyone that tired?

935
00:39:05,360 --> 00:39:06,670
So cosine pi over 2 is 0.

936
00:39:06,670 --> 00:39:08,040
Up.

937
00:39:08,040 --> 00:39:10,970
Sine of pi upon 2 is 1,
because i times minus i,

938
00:39:10,970 --> 00:39:13,470
that's 1 times 0, 1, 1, 0.

939
00:39:13,470 --> 00:39:14,110
Pretty solid.

940
00:39:14,110 --> 00:39:14,820
But what is this?

941
00:39:14,820 --> 00:39:16,960
Well this is the form
Schroedinger evolution

942
00:39:16,960 --> 00:39:19,460
with a magnetic field,
and this is just

943
00:39:19,460 --> 00:39:24,549
the unitary transformation
unitary transformation

944
00:39:24,549 --> 00:39:28,010
for a magnetic field.

945
00:39:28,010 --> 00:39:28,510
Well, OK.

946
00:39:28,510 --> 00:39:29,330
We can say this.

947
00:39:29,330 --> 00:39:32,270
This is the x polymatrix
from last time.

948
00:39:32,270 --> 00:39:35,170
So this is like Sx,
unitary transformation

949
00:39:35,170 --> 00:39:39,770
for a magnetic field in the
x-direction for some time t.

950
00:39:39,770 --> 00:39:43,420
For time t times the frequency,
omega, which is given by mu B

951
00:39:43,420 --> 00:39:48,761
upon 2 is equal to pi upon 2.

952
00:39:48,761 --> 00:39:49,260
OK.

953
00:39:49,260 --> 00:39:51,385
Just like before, but for
a slightly different one.

954
00:39:51,385 --> 00:39:52,885
A slightly different
magnetic field.

955
00:39:52,885 --> 00:39:54,610
So my point here
is that we can pick

956
00:39:54,610 --> 00:39:56,800
a magnetic field that does this.

957
00:39:56,800 --> 00:39:58,910
We turn a magnetic field
with a known amplitude

958
00:39:58,910 --> 00:40:00,780
with a known amount
of time, details here

959
00:40:00,780 --> 00:40:01,687
don't matter so much.

960
00:40:01,687 --> 00:40:02,770
The point is we can do it.

961
00:40:02,770 --> 00:40:04,603
We turn a magnetic field
in the x-direction,

962
00:40:04,603 --> 00:40:06,060
and it takes 0 to 1 and 1 to 0.

963
00:40:06,060 --> 00:40:07,240
Everyone cool with that?

964
00:40:07,240 --> 00:40:09,860
So here is a substrate,
an actual physical system

965
00:40:09,860 --> 00:40:11,550
that effects this
particular evolution.

966
00:40:11,550 --> 00:40:12,840
I can build a NOT.

967
00:40:12,840 --> 00:40:15,020
The crucial thing is that
I can build a NOT gate.

968
00:40:15,020 --> 00:40:19,200
And I'll represent that not with
some unitary transformation U

969
00:40:19,200 --> 00:40:19,700
sub NOT.

970
00:40:22,290 --> 00:40:25,250
So that's a useful one, but
that's not the most useful gate

971
00:40:25,250 --> 00:40:28,620
because, if you only
ever impose logical NOTs,

972
00:40:28,620 --> 00:40:30,100
you just get everyone angry.

973
00:40:30,100 --> 00:40:32,400
But you don't actually
get anything done.

974
00:40:32,400 --> 00:40:36,170
Second example-- let that be
a lesson to you, Congress.

975
00:40:36,170 --> 00:40:39,020
So, the second
example, if we turn

976
00:40:39,020 --> 00:40:43,160
on the magnetic field
in the y-direction

977
00:40:43,160 --> 00:40:50,420
for a particular
time t, what we find

978
00:40:50,420 --> 00:40:59,430
is that 0 goes to 1
over root 2, 0 plus 1,

979
00:40:59,430 --> 00:41:08,794
and 1 goes to 1 over
upon root 2 0 minus 1.

980
00:41:08,794 --> 00:41:09,960
And this should be familiar.

981
00:41:09,960 --> 00:41:12,390
This is the up x state, and
this is the down x state.

982
00:41:12,390 --> 00:41:13,717
Just as we talked before.

983
00:41:13,717 --> 00:41:15,550
So we turn on some B
field, and we get this.

984
00:41:20,530 --> 00:41:23,100
So this operation has
a name because it's

985
00:41:23,100 --> 00:41:25,080
going to turn out to
be very useful for us.

986
00:41:25,080 --> 00:41:27,540
It's taking a system that's
definitely in the state 0,

987
00:41:27,540 --> 00:41:28,930
for sure, right?

988
00:41:28,930 --> 00:41:31,050
And it put us in a
superposition of 0, 1.

989
00:41:31,050 --> 00:41:32,800
It's a definite
superposition, so it's not

990
00:41:32,800 --> 00:41:34,174
like we don't know
what happened.

991
00:41:34,174 --> 00:41:36,755
But it's a superposition,
and you've lost certainty

992
00:41:36,755 --> 00:41:39,880
that you'll measure
up in the z-direction.

993
00:41:39,880 --> 00:41:42,620
You've gained certainty that you
measure up in the x-direction.

994
00:41:42,620 --> 00:41:44,330
But if we do all our
measurements in z,

995
00:41:44,330 --> 00:41:47,200
we just taking ourselves from
definite to superposition.

996
00:41:47,200 --> 00:41:47,700
Cool?

997
00:41:47,700 --> 00:41:50,158
So that's useful because we
know that's something a quantum

998
00:41:50,158 --> 00:41:52,460
computer can do, that a
classical computer can't do.

999
00:41:52,460 --> 00:41:54,440
Something a quantum
computer can take advantage

1000
00:41:54,440 --> 00:41:55,981
of that classical
computer can't take

1001
00:41:55,981 --> 00:41:58,170
advantage of is this
process of putting things

1002
00:41:58,170 --> 00:41:59,080
into superpositions.

1003
00:41:59,080 --> 00:42:00,140
So here we've got
an operation that

1004
00:42:00,140 --> 00:42:01,390
puts things in superpositions.

1005
00:42:01,390 --> 00:42:02,759
And I'll call this Hadamard.

1006
00:42:02,759 --> 00:42:05,050
I don't know the history of
why that's called Hadamard,

1007
00:42:05,050 --> 00:42:07,425
presumably there's some guy
with a last name of Hadamard.

1008
00:42:07,425 --> 00:42:08,980
Anyway, the U
Hadamard does this.

1009
00:42:08,980 --> 00:42:11,840
And as a matrix, it's
represented as 1 over root 2,

1010
00:42:11,840 --> 00:42:13,790
times 1, 1, 1, minus 1.

1011
00:42:19,230 --> 00:42:20,920
And there's a the
last one, which

1012
00:42:20,920 --> 00:42:30,350
is going to be useful for me,
another one is called C C-NOT.

1013
00:42:30,350 --> 00:42:32,154
Controlled-NOT.

1014
00:42:32,154 --> 00:42:33,820
Controlled-NOT does
a really cool thing.

1015
00:42:33,820 --> 00:42:37,830
It takes 0, 0, and 0,
1 and 1, 0, and 1, 1.

1016
00:42:37,830 --> 00:42:39,980
What does is it says,
I'm going to apply a NOT

1017
00:42:39,980 --> 00:42:45,180
to the second qubit if, and
only if, the first qubit is 1.

1018
00:42:45,180 --> 00:42:48,250
So this takes me to--
0, 0 goes to-- well,

1019
00:42:48,250 --> 00:42:49,710
do I perform an NOT on this bit?

1020
00:42:49,710 --> 00:42:51,440
No, so 0, 0.

1021
00:42:51,440 --> 00:42:52,950
Do we perform a NOT on this bit?

1022
00:42:52,950 --> 00:42:55,700
No, 0, 1.

1023
00:42:55,700 --> 00:42:58,340
Now I do perform a NOT
on 0, so I get 1, 1.

1024
00:42:58,340 --> 00:43:01,060
And 1, 1-- I perform a NOT on
this bit, which gives me 1, 0.

1025
00:43:04,420 --> 00:43:05,900
So this is called
controlled-NOT.

1026
00:43:05,900 --> 00:43:08,120
It's a very useful thing.

1027
00:43:08,120 --> 00:43:10,867
I'm going to represent
this in the following way.

1028
00:43:10,867 --> 00:43:12,200
I should represent all of these.

1029
00:43:12,200 --> 00:43:16,242
So this NOT gate, first,
I take some initial bit,

1030
00:43:16,242 --> 00:43:17,200
and it's in some state.

1031
00:43:17,200 --> 00:43:22,790
And then I impose U-NOT, and
it gives me an out state.

1032
00:43:22,790 --> 00:43:26,700
Similarly, with the Hadamard,
I take an initial state n,

1033
00:43:26,700 --> 00:43:29,590
and I've apply U Hadamard.

1034
00:43:29,590 --> 00:43:33,370
And I get u out.

1035
00:43:33,370 --> 00:43:36,830
And for controlled-NOT,
I now have two qubits.

1036
00:43:36,830 --> 00:43:40,780
And I take the two qubits,
and I do a controlled-NOT,

1037
00:43:40,780 --> 00:43:42,550
which is represented
in this form.

1038
00:43:42,550 --> 00:43:47,960
Which says, do a NOT
on this guy, controlled

1039
00:43:47,960 --> 00:43:49,780
by this first bit.

1040
00:43:49,780 --> 00:43:54,960
And so this acts as U C-NOT.

1041
00:43:54,960 --> 00:43:56,989
OK.

1042
00:43:56,989 --> 00:43:59,030
And the key thing here,
is that while it's always

1043
00:43:59,030 --> 00:44:01,580
possible to find a physical
real representation

1044
00:44:01,580 --> 00:44:03,384
of some particular
unitary transformation,

1045
00:44:03,384 --> 00:44:06,050
at the end of the day, all we're
going need is some truth table.

1046
00:44:06,050 --> 00:44:07,466
At the end of the
day, all we need

1047
00:44:07,466 --> 00:44:09,160
is the logic that's
being effected.

1048
00:44:09,160 --> 00:44:10,700
And so, the details
of the substrate

1049
00:44:10,700 --> 00:44:11,710
can be abstracted away.

1050
00:44:15,260 --> 00:44:17,290
So what we do with this?

1051
00:44:17,290 --> 00:44:19,580
First, so what can we do?

1052
00:44:19,580 --> 00:44:22,600
Before we actually talk about
what we can do with it, let's

1053
00:44:22,600 --> 00:44:25,330
briefly talk about what
you can't do with it.

1054
00:44:25,330 --> 00:44:29,260
So what are things you can't do
with these sorts of operations?

1055
00:44:29,260 --> 00:44:30,392
What can't you do?

1056
00:44:38,500 --> 00:44:41,960
And to me this is among
the more surprising things.

1057
00:44:41,960 --> 00:44:43,630
Remember that what
we're doing here

1058
00:44:43,630 --> 00:44:46,290
is going to be evolving a system
for a Schroedinger evolution,

1059
00:44:46,290 --> 00:44:50,076
and a Schroedinger
evolution is linear,

1060
00:44:50,076 --> 00:44:52,040
it respects superpositions,
it's unitary,

1061
00:44:52,040 --> 00:44:57,595
it preserves probability,
and let's just focus on that.

1062
00:44:57,595 --> 00:44:58,970
It's linear,
unitary, and it lots

1063
00:44:58,970 --> 00:45:01,630
of other properties,
[? temporal ?] invariance,

1064
00:45:01,630 --> 00:45:04,059
unless you turn on a
magnetic field, which you do.

1065
00:45:04,059 --> 00:45:05,850
But in particular, it's
linear and unitary.

1066
00:45:05,850 --> 00:45:08,520
And those facts are going
to constrain, powerfully,

1067
00:45:08,520 --> 00:45:13,100
the kinds of operations we can
effect on a quantum system.

1068
00:45:13,100 --> 00:45:15,230
So, in particular, when we
look at just two qubits,

1069
00:45:15,230 --> 00:45:16,813
there's a beautiful,
beautiful theorem

1070
00:45:16,813 --> 00:45:18,815
says there's no cloning.

1071
00:45:18,815 --> 00:45:19,940
And here's what that means.

1072
00:45:19,940 --> 00:45:22,784
The no cloning theorem,
pretty high flautin'

1073
00:45:22,784 --> 00:45:24,700
for what it really is,
which is the following.

1074
00:45:24,700 --> 00:45:27,160
Suppose we have a system,
which has input xy.

1075
00:45:29,454 --> 00:45:31,370
And I want to build a
machine that says, look,

1076
00:45:31,370 --> 00:45:33,710
I've got this first qubit
it in the state x, and what

1077
00:45:33,710 --> 00:45:35,140
I want to do is I
want to make a copy.

1078
00:45:35,140 --> 00:45:36,490
I want to make another
quantum system that's

1079
00:45:36,490 --> 00:45:38,580
in exactly the same
state as whatever x is.

1080
00:45:38,580 --> 00:45:41,640
So you hand me a system where
first bit's in state tax,

1081
00:45:41,640 --> 00:45:43,170
and the second
qubit's in state y.

1082
00:45:43,170 --> 00:45:44,830
And I want to make a copy of x.

1083
00:45:44,830 --> 00:45:48,310
And y is just, who
cares what's in it?

1084
00:45:48,310 --> 00:45:52,680
So I want this to go to x, x.

1085
00:45:52,680 --> 00:45:53,180
OK.

1086
00:45:53,180 --> 00:45:55,654
For all y.

1087
00:45:55,654 --> 00:45:57,470
So regardless of what
data was in here,

1088
00:45:57,470 --> 00:45:59,790
I want to overwrite
that data, and rewrite--

1089
00:45:59,790 --> 00:46:02,600
or that datem--
rewrite it with x.

1090
00:46:02,600 --> 00:46:04,490
Can you do this?

1091
00:46:04,490 --> 00:46:05,150
No, right.

1092
00:46:05,150 --> 00:46:05,650
Why?

1093
00:46:08,117 --> 00:46:08,992
AUDIENCE: [INAUDIBLE]

1094
00:46:11,425 --> 00:46:12,300
PROFESSOR: Excellent.

1095
00:46:12,300 --> 00:46:15,690
It would violate linearity
and also unitarity, indeed.

1096
00:46:15,690 --> 00:46:18,830
So to see that quickly, it's
easiest to see the unitarity,

1097
00:46:18,830 --> 00:46:19,356
I think.

1098
00:46:19,356 --> 00:46:21,480
Well, it violates them
both, but for unitarity, you

1099
00:46:21,480 --> 00:46:24,110
manage to take a linear
combination of these guys,

1100
00:46:24,110 --> 00:46:25,855
where the two states
y are orthogonal,

1101
00:46:25,855 --> 00:46:27,280
and you take the norm squared.

1102
00:46:27,280 --> 00:46:29,594
So you've normalized it to 1.

1103
00:46:29,594 --> 00:46:31,010
The linear combination
of each one

1104
00:46:31,010 --> 00:46:33,620
goes to Sx, where
the coefficient

1105
00:46:33,620 --> 00:46:35,240
is the sum of those two terms.

1106
00:46:35,240 --> 00:46:42,000
So we have, for example,
x z, alpha x z plus beta

1107
00:46:42,000 --> 00:46:47,835
x y goes to alpha plus beta x x.

1108
00:46:50,370 --> 00:46:53,480
And that's really bad, because
if x, z, and y are orthogonal,

1109
00:46:53,480 --> 00:46:56,640
then normalization is alpha
squared plus beta squared is 1.

1110
00:46:56,640 --> 00:46:58,400
But x x, the
normalization is alpha

1111
00:46:58,400 --> 00:47:00,170
plus beta quantity squared is 1.

1112
00:47:00,170 --> 00:47:02,340
And in general that's not true.

1113
00:47:02,340 --> 00:47:03,860
In fact, this could be 0.

1114
00:47:03,860 --> 00:47:06,815
So this violates linearity
and a unitarity rather badly.

1115
00:47:06,815 --> 00:47:07,777
So you can't clone.

1116
00:47:07,777 --> 00:47:08,860
This is really disturbing.

1117
00:47:08,860 --> 00:47:10,420
That means if you
have a quantum,

1118
00:47:10,420 --> 00:47:13,140
and you want make a
copy of it, you can't.

1119
00:47:13,140 --> 00:47:15,700
You can't ever make a copy
of your quantum system.

1120
00:47:15,700 --> 00:47:16,700
One copy.

1121
00:47:16,700 --> 00:47:18,000
One chance.

1122
00:47:18,000 --> 00:47:20,650
That's it.

1123
00:47:20,650 --> 00:47:22,370
No cut and paste.

1124
00:47:22,370 --> 00:47:24,470
So, as you can imagine
that pretty powerfully

1125
00:47:24,470 --> 00:47:26,300
constrains things
that you can do.

1126
00:47:26,300 --> 00:47:28,690
So, a related thing here is
that there's no forgetting.

1127
00:47:32,980 --> 00:47:38,959
Quantum evolution is, unlike an
elephant, it is highly-- well,

1128
00:47:38,959 --> 00:47:40,250
it's like an elephant, I guess.

1129
00:47:40,250 --> 00:47:41,330
It remembers very well.

1130
00:47:41,330 --> 00:47:43,524
It never forgets anything.

1131
00:47:43,524 --> 00:47:44,815
And you can see that from this.

1132
00:47:44,815 --> 00:47:46,300
This would be an
example of forgetting.

1133
00:47:46,300 --> 00:47:47,860
You forgot what
was in the state y.

1134
00:47:47,860 --> 00:47:49,101
You can't ever do that.

1135
00:47:49,101 --> 00:47:49,600
OK.

1136
00:47:49,600 --> 00:47:54,710
So I leave this as a challenge
you to prove this show.

1137
00:47:54,710 --> 00:47:58,510
It's a simple extension
of the same logic.

1138
00:47:58,510 --> 00:47:59,580
So what can you do?

1139
00:48:02,856 --> 00:48:05,501
What you can do, is you
can entangle two qubits.

1140
00:48:05,501 --> 00:48:07,250
And that's really the
juice of everything.

1141
00:48:07,250 --> 00:48:12,280
You can entangle.

1142
00:48:12,280 --> 00:48:15,550
So let me show you entanglement.

1143
00:48:15,550 --> 00:48:17,180
Good, no e.

1144
00:48:17,180 --> 00:48:18,804
Sorry, question?

1145
00:48:18,804 --> 00:48:20,720
So you can entangle, and
here's how you do it.

1146
00:48:20,720 --> 00:48:22,774
Let's take this state 0, 0.

1147
00:48:22,774 --> 00:48:23,690
So we have two qubits.

1148
00:48:23,690 --> 00:48:25,094
The first one's in the
state 0, and the second one

1149
00:48:25,094 --> 00:48:26,150
is in the state 0.

1150
00:48:26,150 --> 00:48:27,790
And now, I'm going to do the
following set of operations

1151
00:48:27,790 --> 00:48:28,260
to it.

1152
00:48:28,260 --> 00:48:29,760
I'm first going to
impose a Hadamard

1153
00:48:29,760 --> 00:48:33,610
operation on the first qubit,
and nothing on the second.

1154
00:48:33,610 --> 00:48:39,287
And then I'm going to
apply controlled-NOT,

1155
00:48:39,287 --> 00:48:40,870
and we're going to
see what I get out.

1156
00:48:40,870 --> 00:48:42,590
So the initial state is 0, 0.

1157
00:48:47,070 --> 00:48:53,090
After I Hadamard, well, the
first bit is no longer in 0.

1158
00:48:53,090 --> 00:48:56,120
Hadamard on 0 gives me 0 plus 1.

1159
00:48:56,120 --> 00:48:58,830
So this is now the
state 1 upon root 2.

1160
00:48:58,830 --> 00:49:12,230
0 plus 1 times 0, also known as
1 over root 2, 0, 0 plus 1, 0.

1161
00:49:12,230 --> 00:49:14,270
Now is this the separable state?

1162
00:49:14,270 --> 00:49:17,326
Yes, there is separated.

1163
00:49:17,326 --> 00:49:19,200
And now I'm going to
perform a controlled-NOT

1164
00:49:19,200 --> 00:49:20,575
and what the
controlled-NOT does,

1165
00:49:20,575 --> 00:49:23,520
is that it switches the second
bit, if the first bit is a 1.

1166
00:49:23,520 --> 00:49:25,520
So what is the state
after we've done this?

1167
00:49:25,520 --> 00:49:30,120
The state after we've done this
is, well, from the first term,

1168
00:49:30,120 --> 00:49:33,920
1 upon root 2,
from the first term

1169
00:49:33,920 --> 00:49:36,349
0, 0-- what happens to that
when we controlled-NOT?

1170
00:49:36,349 --> 00:49:37,640
Well, we NOT this if this is 1.

1171
00:49:37,640 --> 00:49:38,760
This is not 1, so
we don't NOT it.

1172
00:49:38,760 --> 00:49:39,770
We leave it alone.

1173
00:49:39,770 --> 00:49:44,600
And the second term--
1, 0 plus well, we

1174
00:49:44,600 --> 00:49:47,720
flip this, if this is 1, and
not if it's not, so this is 1.

1175
00:49:47,720 --> 00:49:49,090
We flip it, and we get 1, 1.

1176
00:49:49,090 --> 00:49:53,140
And now this is the
prototypical entangled state--

1177
00:49:53,140 --> 00:49:56,019
that I think I just erased.

1178
00:49:56,019 --> 00:49:57,310
But this is our entitled state.

1179
00:49:57,310 --> 00:49:58,516
It's not separable.

1180
00:49:58,516 --> 00:49:59,890
But if I measure
the first one, I

1181
00:49:59,890 --> 00:50:01,431
know what the state
of the second one

1182
00:50:01,431 --> 00:50:04,400
is, which is to
say it's entangled.

1183
00:50:04,400 --> 00:50:05,820
Cool?

1184
00:50:05,820 --> 00:50:07,999
So by performing this
series of operations, which

1185
00:50:07,999 --> 00:50:10,040
is nothing other than a
series of magnetic fields

1186
00:50:10,040 --> 00:50:12,170
which I'm going to
impose to the system,

1187
00:50:12,170 --> 00:50:14,220
I've taken a state
with initial conditions

1188
00:50:14,220 --> 00:50:18,361
0, 0, and put it into an
entangled state, 0, 0 plus 1,

1189
00:50:18,361 --> 00:50:18,860
1.

1190
00:50:21,560 --> 00:50:27,600
And that's all we need for
the first basic algorithm

1191
00:50:27,600 --> 00:50:28,680
of quantum computation.

1192
00:50:28,680 --> 00:50:30,230
So this idea the
quantum computers

1193
00:50:30,230 --> 00:50:31,940
might be able to
do things faster

1194
00:50:31,940 --> 00:50:35,450
than classical computers
floated around for a while.

1195
00:50:35,450 --> 00:50:38,230
It took a while for
people to make that sharp.

1196
00:50:38,230 --> 00:50:44,710
And David Deutsch, who is a
very entertaining and bombastic

1197
00:50:44,710 --> 00:50:49,840
speaker, and he wrote-- I
guess it's several now-- pretty

1198
00:50:49,840 --> 00:50:52,030
entertaining books on the topic.

1199
00:50:52,030 --> 00:50:53,590
And he sounds crazy.

1200
00:50:53,590 --> 00:50:56,280
You listen to the guy
talk, and he sounds nuts.

1201
00:50:56,280 --> 00:50:59,620
He sounds like he's
just way out there.

1202
00:50:59,620 --> 00:51:02,020
The thing he's just-- gah!

1203
00:51:02,020 --> 00:51:04,840
As a theorist, you listen to
him like, just slow down there,

1204
00:51:04,840 --> 00:51:05,520
buddy.

1205
00:51:05,520 --> 00:51:06,110
Right?

1206
00:51:06,110 --> 00:51:09,054
And so for a long time,
I thought the guy--

1207
00:51:09,054 --> 00:51:10,720
I only knew his sort
of public persona--

1208
00:51:10,720 --> 00:51:12,650
I thought, yeah, he's
a little bit crazy;

1209
00:51:12,650 --> 00:51:14,130
I'm not exactly
sure-- and this is

1210
00:51:14,130 --> 00:51:16,500
why everyone thinks
he's such a damn genius.

1211
00:51:16,500 --> 00:51:17,950
Because this is beautiful.

1212
00:51:17,950 --> 00:51:20,180
So here is-- I don't know
how he came up with this,

1213
00:51:20,180 --> 00:51:21,230
but he's clever.

1214
00:51:21,230 --> 00:51:23,850
So here is what's now
called the Deutsch--

1215
00:51:23,850 --> 00:51:26,280
and it's really
the one bit version

1216
00:51:26,280 --> 00:51:32,350
of the Deutsch-Josza algorithm.

1217
00:51:32,350 --> 00:51:34,150
So there is a first
algorithm by Deutsch

1218
00:51:34,150 --> 00:51:35,860
that didn't quite what
it was supposed to do,

1219
00:51:35,860 --> 00:51:37,500
then it was improved
together with Jozsa,

1220
00:51:37,500 --> 00:51:39,300
and they made an n particle
version and everything was

1221
00:51:39,300 --> 00:51:39,799
awesome.

1222
00:51:39,799 --> 00:51:43,187
But here's the
Deutsch-Jozsa algorithm.

1223
00:51:43,187 --> 00:51:44,770
And what it is, it's
a series of rules

1224
00:51:44,770 --> 00:51:47,340
for how to make a quantum
mechanical system evolve

1225
00:51:47,340 --> 00:51:52,570
so as to effect the calculation
you wanted to calculate.

1226
00:51:52,570 --> 00:51:55,600
So you have to
grant, to begin, you

1227
00:51:55,600 --> 00:51:58,860
have to let me pose
a problem to solve

1228
00:51:58,860 --> 00:52:00,790
that can be solved
in this fashion.

1229
00:52:00,790 --> 00:52:04,290
And this problem is going
to sound a little contrived.

1230
00:52:04,290 --> 00:52:05,795
And, in fact, it's
wildly contrived.

1231
00:52:05,795 --> 00:52:07,170
It was contrived
so that it could

1232
00:52:07,170 --> 00:52:08,253
be solved in this fashion.

1233
00:52:08,253 --> 00:52:13,880
But it's actually one that
preexists the algorithm itself,

1234
00:52:13,880 --> 00:52:16,280
so it's not quite as ridiculous

1235
00:52:16,280 --> 00:52:17,674
So here's the problem.

1236
00:52:17,674 --> 00:52:19,090
So the statement
of the problem is

1237
00:52:19,090 --> 00:52:25,020
that someone has
a function f of x.

1238
00:52:25,020 --> 00:52:32,310
So, let's say Matt
knows a function f of x.

1239
00:52:38,600 --> 00:52:42,070
Now the thing is, it's
extremely expensive

1240
00:52:42,070 --> 00:52:44,120
to evaluate this
function f of x.

1241
00:52:44,120 --> 00:52:47,960
So the way you evaluate
involves putting 20 kilomeres

1242
00:52:47,960 --> 00:52:50,022
in superposition
states with each other.

1243
00:52:50,022 --> 00:52:51,480
You have to run a
whole experiment.

1244
00:52:51,480 --> 00:53:02,150
And it costs a lot of money
to run, so he charges--

1245
00:53:02,150 --> 00:53:04,900
$1 million dollars order to--

1246
00:53:04,900 --> 00:53:06,220
AUDIENCE: [LAUGHTER]

1247
00:53:06,220 --> 00:53:08,220
PROFESSOR: Thank you, you
guys are not quite old

1248
00:53:08,220 --> 00:53:10,870
enough to-- so he knows
the function f of x and he

1249
00:53:10,870 --> 00:53:13,370
charges a million dollars in
order to evaluate the function.

1250
00:53:13,370 --> 00:53:16,010
You say, hey, Matt, look, I
know this is a function-- which

1251
00:53:16,010 --> 00:53:19,860
I should tell you f
is a function that

1252
00:53:19,860 --> 00:53:24,065
takes a single bit, 0 or 1,
to another single bit, 0 or 1.

1253
00:53:24,065 --> 00:53:26,190
So it sounds like, how hard
could this possibly be?

1254
00:53:26,190 --> 00:53:27,810
But in fact, it's a very
hard function to evaluate.

1255
00:53:27,810 --> 00:53:29,210
So you say, hey
Matt, what's f of 0?

1256
00:53:29,210 --> 00:53:30,380
And he's like, give
me a million bucks.

1257
00:53:30,380 --> 00:53:31,250
So you give him a million bucks.

1258
00:53:31,250 --> 00:53:32,042
And he's like, 1.

1259
00:53:32,042 --> 00:53:34,000
And you're like damn,
that cost a lot of money.

1260
00:53:36,640 --> 00:53:38,000
So now here's the question.

1261
00:53:38,000 --> 00:53:39,320
So this is not yet the problem.

1262
00:53:39,320 --> 00:53:41,560
The problem is this.

1263
00:53:41,560 --> 00:53:49,170
Is f of 0 equal
to f of 1 or not?

1264
00:53:49,170 --> 00:53:51,630
OK.

1265
00:53:51,630 --> 00:53:56,500
So f of 0 is either 0 or
1. f of 1 is either 0 or 1.

1266
00:53:56,500 --> 00:53:58,800
Are they equal to each other?

1267
00:53:58,800 --> 00:53:59,830
So this is easy, right?

1268
00:53:59,830 --> 00:54:00,996
Classically, this is stupid.

1269
00:54:00,996 --> 00:54:03,060
You calculate the
function f twice.

1270
00:54:03,060 --> 00:54:04,790
You evaluate f of
0, you get a number.

1271
00:54:04,790 --> 00:54:05,910
You evaluate f of 1,
and you get number.

1272
00:54:05,910 --> 00:54:07,576
You look at your piece
of paper, and you

1273
00:54:07,576 --> 00:54:09,250
say it's either the
same or different.

1274
00:54:09,250 --> 00:54:11,400
How much does that cost you?

1275
00:54:11,400 --> 00:54:12,580
Two million bucks.

1276
00:54:12,580 --> 00:54:13,820
Better have good funding.

1277
00:54:13,820 --> 00:54:15,310
So this is an expensive--

1278
00:54:15,310 --> 00:54:17,250
And here's what Deutsch
and Josza have to say.

1279
00:54:17,250 --> 00:54:20,250
This is really Deutsche
at the beginning.

1280
00:54:20,250 --> 00:54:22,620
It's really quite spectacular.

1281
00:54:22,620 --> 00:54:26,110
Deutsch says actually, I
tell you what, give me a

1282
00:54:26,110 --> 00:54:29,170
million and a half, and I'll
do the computation give you

1283
00:54:29,170 --> 00:54:30,430
the answer.

1284
00:54:30,430 --> 00:54:32,430
At which point you think,
like I did previously,

1285
00:54:32,430 --> 00:54:33,740
the guy's clearly raving.

1286
00:54:33,740 --> 00:54:36,782
But, in fact, he's
going to make a profit,

1287
00:54:36,782 --> 00:54:38,240
And here's how he's
going to do it.

1288
00:54:38,240 --> 00:54:40,400
He's going to build, not
a classical computer,

1289
00:54:40,400 --> 00:54:43,430
but a quantum computer
using quantum interference

1290
00:54:43,430 --> 00:54:46,970
and entanglement to
do this calculation.

1291
00:54:46,970 --> 00:54:48,524
One evaluation.

1292
00:54:48,524 --> 00:54:49,940
And here's how
it's going to work.

1293
00:54:49,940 --> 00:54:56,130
And the first thing we have
to do is a preview, or set up,

1294
00:54:56,130 --> 00:54:59,720
in order to do this calculation,
you need two things.

1295
00:54:59,720 --> 00:55:05,180
First off, you need Matt to be
able to evaluate his function

1296
00:55:05,180 --> 00:55:08,194
in a way that respects
quantum mechanics.

1297
00:55:08,194 --> 00:55:09,610
So, in particular,
Matt had better

1298
00:55:09,610 --> 00:55:13,000
be able to do his experiment,
if I give him an superpositon.

1299
00:55:13,000 --> 00:55:15,080
So we better be able to
effect the calculation

1300
00:55:15,080 --> 00:55:16,460
in a quantum mechanical way.

1301
00:55:16,460 --> 00:55:18,120
The same way that we
implemented a NOT quantum

1302
00:55:18,120 --> 00:55:19,850
mechanically, or the
controlled-NOT quantum

1303
00:55:19,850 --> 00:55:21,290
mechanically, or the
Hadamard, with some set

1304
00:55:21,290 --> 00:55:22,112
of magnetic fields.

1305
00:55:22,112 --> 00:55:24,320
He must be able to implement
it quantum mechanically.

1306
00:55:24,320 --> 00:55:26,153
Otherwise, it's not an
interesting function.

1307
00:55:26,153 --> 00:55:28,654
And let me just point out that
any function you can think of

1308
00:55:28,654 --> 00:55:30,319
can be implemented
quantum mechanically,

1309
00:55:30,319 --> 00:55:31,750
because you are
quantum mechanics.

1310
00:55:31,750 --> 00:55:32,130
OK?

1311
00:55:32,130 --> 00:55:33,921
You're just not an
elegant implementation--

1312
00:55:33,921 --> 00:55:40,210
and no offense-- of the
quantum mechanical computation.

1313
00:55:40,210 --> 00:55:45,220
So the set up is that Matt
needs to be able to give me--

1314
00:55:45,220 --> 00:55:49,565
Matt provides-- a
unitary transformation,

1315
00:55:49,565 --> 00:55:54,620
a unitary operation, use of
f that takes two qubits, x

1316
00:55:54,620 --> 00:56:03,550
and y to x and f of x plus y.

1317
00:56:03,550 --> 00:56:09,070
Where what this means, f of x
plus y, is addition mod two.

1318
00:56:11,850 --> 00:56:14,920
So what this says is,
if y is 0, then this

1319
00:56:14,920 --> 00:56:16,760
gives me f of x plus 0.

1320
00:56:16,760 --> 00:56:19,640
If f of x is 0, that's 0
plus 0, so that gives me 0.

1321
00:56:19,640 --> 00:56:22,890
If f of x is 1, is this
gives me 1 plus 0, that's 1.

1322
00:56:22,890 --> 00:56:28,450
On the other hand, if y is 1,
then this gives me-- if f of x

1323
00:56:28,450 --> 00:56:30,260
is 0, it gives me 1
plus 1, which is 0.

1324
00:56:30,260 --> 00:56:34,600
And if f of x is 0, it's going
to be 0 plus 1, which is 1.

1325
00:56:34,600 --> 00:56:37,250
Everyone cool with that?

1326
00:56:37,250 --> 00:56:38,238
Yeah.

1327
00:56:38,238 --> 00:56:39,226
AUDIENCE: [INAUDIBLE].

1328
00:56:39,226 --> 00:56:42,559
--question, but like actually,
how do you know that the matrix

1329
00:56:42,559 --> 00:56:46,520
actually [INAUDIBLE] I mean,
how can we know [INAUDIBLE]

1330
00:56:46,520 --> 00:56:51,420
if matrices actually prove that
quantum mechanics [INAUDIBLE]

1331
00:56:51,420 --> 00:56:54,322
What if the matrix is is
just an approximation--

1332
00:56:54,322 --> 00:56:56,530
PROFESSOR: You mean what if
quantum mechanics is only

1333
00:56:56,530 --> 00:56:57,415
an approximate description--?

1334
00:56:57,415 --> 00:56:57,790
Of the--

1335
00:56:57,790 --> 00:56:59,248
AUDIENCE: No I'm
sorry. [INAUDIBLE]

1336
00:56:59,248 --> 00:57:03,617
To what if quantum mechanics--
the inelegant reprensentation

1337
00:57:03,617 --> 00:57:05,700
of [INAUDIBLE] implementation
of quantum mechanics

1338
00:57:05,700 --> 00:57:06,540
PROFESSOR: Is inescapable--?

1339
00:57:06,540 --> 00:57:09,123
AUDIENCE: [INAUDIBLE] is just
an approximation of the problem,

1340
00:57:09,123 --> 00:57:12,430
or is a really, really
good approximation--

1341
00:57:12,430 --> 00:57:14,310
PROFESSOR: This is an
interesting question.

1342
00:57:14,310 --> 00:57:15,430
So it's tempting
to think that this

1343
00:57:15,430 --> 00:57:17,420
is a philosophical
question, but it turns out

1344
00:57:17,420 --> 00:57:20,003
not to be in a way that will be
made sharp in about 10 minutes

1345
00:57:20,003 --> 00:57:21,040
with Bell's Inequality.

1346
00:57:21,040 --> 00:57:25,050
But a complete answer
to that question

1347
00:57:25,050 --> 00:57:27,246
remains open, and,
probably, always will.

1348
00:57:27,246 --> 00:57:29,496
But let me rephrase the
question slightly, and tell me

1349
00:57:29,496 --> 00:57:32,860
if this an accurate statement.

1350
00:57:32,860 --> 00:57:36,340
Look, at the end of the
day, what we're doing

1351
00:57:36,340 --> 00:57:38,330
is we're going to
develop a model where

1352
00:57:38,330 --> 00:57:41,969
quantum mechanical calculation
does something in particular.

1353
00:57:41,969 --> 00:57:44,260
And that may or may not be
a good model the real world.

1354
00:57:44,260 --> 00:57:46,267
And in particular,
whatever the actual thing,

1355
00:57:46,267 --> 00:57:48,100
the actual system, that
we're studying does,

1356
00:57:48,100 --> 00:57:51,391
may or may not be
well described by

1357
00:57:51,391 --> 00:57:53,345
that quantum mechanical model.

1358
00:57:53,345 --> 00:57:56,230
So can we check
whether or not it is?

1359
00:57:56,230 --> 00:57:59,860
Is that more or
less the question?

1360
00:57:59,860 --> 00:58:04,740
Yeah, and so the problem
is all we can ever do

1361
00:58:04,740 --> 00:58:07,430
is say that our model
is a good or bad model.

1362
00:58:07,430 --> 00:58:09,564
On the other hand, we
can do the following.

1363
00:58:09,564 --> 00:58:10,980
And this is the
really neat thing.

1364
00:58:10,980 --> 00:58:12,850
You might say, look,
underlying quantum mechanics

1365
00:58:12,850 --> 00:58:14,420
is going to be something
more fundamental

1366
00:58:14,420 --> 00:58:16,440
that's going to lead to
slightly different results

1367
00:58:16,440 --> 00:58:17,590
in exactly the
sort of situations

1368
00:58:17,590 --> 00:58:19,215
where we're going to
care about quantum

1369
00:58:19,215 --> 00:58:22,210
computation of large
numbers and bits.

1370
00:58:22,210 --> 00:58:26,750
And if you tell me just a little
tiny bit about what properties

1371
00:58:26,750 --> 00:58:29,660
that underlying
description will have,

1372
00:58:29,660 --> 00:58:31,370
that becomes an
empirical question.

1373
00:58:31,370 --> 00:58:32,470
So, for example,
if you say, look,

1374
00:58:32,470 --> 00:58:34,469
I suspect that underlying
the quantum mechanical

1375
00:58:34,469 --> 00:58:37,270
probabilities is some classical
probability distribution

1376
00:58:37,270 --> 00:58:39,735
over a hidden variable that
you have not actually measured.

1377
00:58:39,735 --> 00:58:41,090
And what we're
going to find out is

1378
00:58:41,090 --> 00:58:42,798
that we can rule that
out experimentally.

1379
00:58:42,798 --> 00:58:44,360
Just that extra
little assumption

1380
00:58:44,360 --> 00:58:46,227
that there's an underlying
hidden variable--

1381
00:58:46,227 --> 00:58:48,310
a secret probability
distribution on some variable

1382
00:58:48,310 --> 00:58:49,950
we just haven't
observed yet-- that

1383
00:58:49,950 --> 00:58:53,540
is enough information about the
system to rule out that model,

1384
00:58:53,540 --> 00:58:54,280
amazingly.

1385
00:58:54,280 --> 00:58:58,200
So I think we'll never have
a full answer your question.

1386
00:58:58,200 --> 00:59:03,170
But all we can do is work and
see how well our models fits.

1387
00:59:03,170 --> 00:59:04,670
And so far, nothing's
ever disagreed

1388
00:59:04,670 --> 00:59:07,100
with the quantum
mechanical description.

1389
00:59:07,100 --> 00:59:10,612
Let me hold off on
questions just now.

1390
00:59:10,612 --> 00:59:12,350
But it's a good and
interesting question

1391
00:59:12,350 --> 00:59:15,160
that's a hard one
deal with, by which I

1392
00:59:15,160 --> 00:59:16,370
mean it's an open question.

1393
00:59:16,370 --> 00:59:19,042
So Matt provides
for us an operator

1394
00:59:19,042 --> 00:59:20,500
that allows us to
calculate f of x.

1395
00:59:20,500 --> 00:59:22,440
Now you might have
said, well look,

1396
00:59:22,440 --> 00:59:25,390
why not just take x, and why not
have Matt build a machine that

1397
00:59:25,390 --> 00:59:26,970
takes x and gives you f of x.

1398
00:59:31,260 --> 00:59:32,545
Could you have done that?

1399
00:59:32,545 --> 00:59:34,000
AUDIENCE: [INAUDIBLE]

1400
00:59:34,000 --> 00:59:36,872
PROFESSOR: Well, it's
not exactly no cloning.

1401
00:59:36,872 --> 00:59:39,330
But let me leave this to you
as a fun thing to think about.

1402
00:59:39,330 --> 00:59:41,970
Why do we need this
carrier bit, as well?

1403
00:59:45,630 --> 00:59:48,010
OK.

1404
00:59:48,010 --> 00:59:49,110
So there's our set up.

1405
00:59:49,110 --> 00:59:51,310
Matt provides this
function for us, U.

1406
00:59:51,310 --> 00:59:53,620
And here's the algorithm.

1407
00:59:53,620 --> 00:59:55,510
So the algorithm.

1408
00:59:55,510 --> 00:59:58,850
And it's a series of steps,
one by one we do them.

1409
00:59:58,850 --> 01:00:01,180
We perform these operations
are on our qubit.

1410
01:00:03,744 --> 01:00:04,910
So here's what Deutsch says.

1411
01:00:04,910 --> 01:00:11,970
Deutsch says start input, a
state psi, is equal to 0, 1.

1412
01:00:11,970 --> 01:00:13,870
First qubit is 0,
in the state 0.

1413
01:00:13,870 --> 01:00:15,850
The second qubit is in
the state 1, for sure.

1414
01:00:15,850 --> 01:00:17,440
We implement that
with our boxes,

1415
01:00:17,440 --> 01:00:19,106
or however we want
to implement it.

1416
01:00:19,106 --> 01:00:21,230
So we find ourselves in a
definite state, you know,

1417
01:00:21,230 --> 01:00:22,320
hard-soft.

1418
01:00:22,320 --> 01:00:24,470
So we take a hard box,
and we take a soft box,

1419
01:00:24,470 --> 01:00:26,386
and we pull out the hard
one and the soft one.

1420
01:00:30,150 --> 01:00:34,360
One, Hadamard each.

1421
01:00:34,360 --> 01:00:35,747
Hadamard on both bits.

1422
01:00:35,747 --> 01:00:38,480
Both qubits.

1423
01:00:38,480 --> 01:00:38,980
OK.

1424
01:00:38,980 --> 01:00:40,188
So what does this take us to?

1425
01:00:40,188 --> 01:00:42,950
It takes us to psi
is equal to-- well,

1426
01:00:42,950 --> 01:00:49,275
the 0 goes to 1 over
root 2, times 0 plus 1.

1427
01:00:49,275 --> 01:00:50,150
Did I erase Hadamard?

1428
01:00:50,150 --> 01:00:51,620
No, good.

1429
01:00:51,620 --> 01:00:53,710
There's Hadamard.

1430
01:00:53,710 --> 01:00:55,300
So it does this-- does this.

1431
01:00:55,300 --> 01:00:57,860
So it takes the first
one to 0 plus 1,

1432
01:00:57,860 --> 01:01:03,320
and it takes the second one
to 1 over root 2, 0 minus 1.

1433
01:01:07,660 --> 01:01:08,534
Cool?

1434
01:01:08,534 --> 01:01:10,450
So at this point, this
isn't very interesting.

1435
01:01:10,450 --> 01:01:12,616
What we've done is we take
it from one superposition

1436
01:01:12,616 --> 01:01:14,210
to a different superposition.

1437
01:01:14,210 --> 01:01:16,168
And doesn't seem to have
anything to do with f.

1438
01:01:16,168 --> 01:01:17,830
In fact, we haven't measured f.

1439
01:01:17,830 --> 01:01:18,860
Two.

1440
01:01:18,860 --> 01:01:20,810
Apply f.

1441
01:01:20,810 --> 01:01:24,260
So we apply our
operation U sub f.

1442
01:01:24,260 --> 01:01:29,810
And well, this is a sort
of an entertaining one.

1443
01:01:29,810 --> 01:01:31,580
If we take this 1
of root 2-- so I'm

1444
01:01:31,580 --> 01:01:33,840
going to rewrite this in
a slightly simpler form--

1445
01:01:33,840 --> 01:01:43,340
this is 1/2, 0, times--
0 times 0 minus 1.

1446
01:01:43,340 --> 01:01:44,450
That's 1.

1447
01:01:44,450 --> 01:01:49,100
Plus 1 times 0 minus 1.

1448
01:01:53,310 --> 01:01:55,720
And the reason I'm doing that
is we're going to apply Uf.

1449
01:01:55,720 --> 01:02:00,100
And Uf, our function
f, uses that first bit

1450
01:02:00,100 --> 01:02:03,800
as a control bit for the second.

1451
01:02:03,800 --> 01:02:06,140
So here's the control
bit for the second.

1452
01:02:06,140 --> 01:02:10,820
So we apply Uf, and
this gives us 1/2.

1453
01:02:13,310 --> 01:02:15,310
I'm going to actually do
this on the next board,

1454
01:02:15,310 --> 01:02:16,910
because it's going
to be gigantic.

1455
01:02:28,030 --> 01:02:31,944
So this gives us 1/2.

1456
01:02:31,944 --> 01:02:35,500
0-- so that first one, this is
going to take this and give it

1457
01:02:35,500 --> 01:02:38,825
0 plus f of 0,
and 1 plus f of 0.

1458
01:02:38,825 --> 01:02:45,896
So times f of 0, plus 0.

1459
01:02:45,896 --> 01:02:52,010
Minus f of 0 plus 1.

1460
01:02:52,010 --> 01:03:02,905
Plus for the 1, this going to
be times f of 1, now, plus 0.

1461
01:03:02,905 --> 01:03:10,130
Minus f of 1, plus 1.

1462
01:03:10,130 --> 01:03:12,540
OK?

1463
01:03:12,540 --> 01:03:16,040
Now, here's a crucial step.

1464
01:03:16,040 --> 01:03:18,430
This is equal to, and
note the following,

1465
01:03:18,430 --> 01:03:21,870
look at this particular guy.

1466
01:03:21,870 --> 01:03:24,950
So for that particular
guy, suppose f of 0 is 0.

1467
01:03:24,950 --> 01:03:28,940
If f of 0, is 0, then this
is 0 plus 0, which is 0.

1468
01:03:28,940 --> 01:03:32,860
So f of 0 is equal to 0.

1469
01:03:32,860 --> 01:03:37,160
And this gives me 0, and this
gives me 0 plus 1, which is 1.

1470
01:03:37,160 --> 01:03:38,610
0 minus 1.

1471
01:03:38,610 --> 01:03:43,980
On the other hand, if
f of 0 is equal to 1,

1472
01:03:43,980 --> 01:03:47,920
then we get 1 plus
0, which is 1.

1473
01:03:47,920 --> 01:03:52,930
And here we get 1 plus
1, which is 0 minus 0,

1474
01:03:52,930 --> 01:03:56,230
which is equal to
minus 0 minus 1.

1475
01:03:59,591 --> 01:04:00,090
Yeah?

1476
01:04:00,090 --> 01:04:01,620
OK.

1477
01:04:01,620 --> 01:04:10,750
So I can write this as minus 1
to the f of 0 times 0 minus 1.

1478
01:04:14,200 --> 01:04:15,200
Everyone cool with that?

1479
01:04:15,200 --> 01:04:18,300
This is just a little
exercise in binary arithmetic.

1480
01:04:18,300 --> 01:04:21,950
So we can write this first term.

1481
01:04:21,950 --> 01:04:36,149
This is 1/2 minus 1 to the
f of 0, 0, times 0 minus 1.

1482
01:04:36,149 --> 01:04:38,440
So that's for the first one,
and exactly the same logic

1483
01:04:38,440 --> 01:04:39,773
is going to apply to the second.

1484
01:04:39,773 --> 01:04:42,580
But now f of 1,
instead of f of 0.

1485
01:04:42,580 --> 01:04:54,370
Plus minus 1 to the f of 1,
times 1, times 0, minus 1.

1486
01:04:58,620 --> 01:05:00,420
Now, I want to point
something out to you.

1487
01:05:00,420 --> 01:05:03,840
If f of 0 is equal to f of
1, than what's true of f of 0

1488
01:05:03,840 --> 01:05:04,840
plus f of 1?

1489
01:05:09,364 --> 01:05:10,780
Well, if they're
the same exactly,

1490
01:05:10,780 --> 01:05:13,050
then either it's 0 plus
0, in which case we get 0,

1491
01:05:13,050 --> 01:05:16,630
or it's 1 plus 1, in
which case we get 0.

1492
01:05:16,630 --> 01:05:23,241
So this is 0, if it's the
same, and 1, if it's not.

1493
01:05:23,241 --> 01:05:23,740
OK.

1494
01:05:23,740 --> 01:05:26,440
So we could either know them
both, or we can measure f of 0

1495
01:05:26,440 --> 01:05:28,790
plus f of 1.

1496
01:05:28,790 --> 01:05:30,830
So notice what happens here.

1497
01:05:30,830 --> 01:05:34,180
This is equal to
1/2, and now I'm

1498
01:05:34,180 --> 01:05:37,120
just going to pull out
a factor of f of 0,

1499
01:05:37,120 --> 01:05:42,990
minus 1 to the f of 0, times--
well, both of these terms

1500
01:05:42,990 --> 01:05:45,320
have a 0 minus 1
on the second bit,

1501
01:05:45,320 --> 01:05:47,910
so the second qubit is
in the state 0 minus 1.

1502
01:05:47,910 --> 01:05:49,940
Right, everyone cool with that?

1503
01:05:49,940 --> 01:05:56,680
So this is equal to, for
the first qubit, 0 plus 1,

1504
01:05:56,680 --> 01:06:01,860
times minus 1 to the f of 0,
that I pulled out to square it,

1505
01:06:01,860 --> 01:06:09,800
plus f of 1, times 0 minus 1.

1506
01:06:13,110 --> 01:06:15,370
And here's the quantity
we wanted to measure.

1507
01:06:15,370 --> 01:06:17,970
If this is 0, then they're even.

1508
01:06:17,970 --> 01:06:19,980
Then they're the same.

1509
01:06:19,980 --> 01:06:23,570
If it's 1, then
they're not the same.

1510
01:06:23,570 --> 01:06:29,120
So at this point we
just forget three,

1511
01:06:29,120 --> 01:06:32,370
forget about the second qubit.

1512
01:06:37,180 --> 01:06:39,240
Oh, lord.

1513
01:06:39,240 --> 01:06:41,894
Forget about the
second qubit, and so

1514
01:06:41,894 --> 01:06:43,310
forget about the
second qubit just

1515
01:06:43,310 --> 01:06:46,780
does this, just
focus on this guy.

1516
01:06:46,780 --> 01:06:51,520
And now, if f of 0 plus f of 1
is 0, so that they're the same,

1517
01:06:51,520 --> 01:06:55,100
this is 0, minus 1 to the
0, 0, so we get 0 plus 1.

1518
01:06:55,100 --> 01:07:00,340
So same, then our
state is 0 plus 1.

1519
01:07:03,600 --> 01:07:06,220
And if they're
different, then we

1520
01:07:06,220 --> 01:07:14,411
get the state 0 minus 1
Everyone agree with that?

1521
01:07:14,411 --> 01:07:16,660
But if they're the same we
get 0 plus 1, and different

1522
01:07:16,660 --> 01:07:17,330
we get 0 minus 1.

1523
01:07:17,330 --> 01:07:18,996
Still doesn't work
for us, because if we

1524
01:07:18,996 --> 01:07:21,630
measure, what's the
probability we get 0 here?

1525
01:07:21,630 --> 01:07:22,250
1/2.

1526
01:07:22,250 --> 01:07:23,540
And the probability
that we get 1 is 1/2.

1527
01:07:23,540 --> 01:07:24,706
Similar, if we measure here.

1528
01:07:24,706 --> 01:07:25,292
It was 0, 1/2.

1529
01:07:25,292 --> 01:07:26,170
1, 1/2

1530
01:07:26,170 --> 01:07:29,257
On the other hand these
states are familiar to us

1531
01:07:29,257 --> 01:07:31,090
because they're what
you get by Hadamarding.

1532
01:07:33,787 --> 01:07:36,120
So why don't we take these,
from these states to these--

1533
01:07:36,120 --> 01:07:37,953
by doing the inverse
of the Hadamard, which,

1534
01:07:37,953 --> 01:07:40,150
as it turns out,
is Hadamard itself.

1535
01:07:40,150 --> 01:07:41,180
So four.

1536
01:07:44,930 --> 01:07:45,920
Hadamard the first bit.

1537
01:07:50,950 --> 01:07:57,310
And the output is
the state, psi out,

1538
01:07:57,310 --> 01:08:03,770
is equal to 1/2, 1 plus,
minus 1 to the f of 0

1539
01:08:03,770 --> 01:08:14,690
plus f 1, 0 plus
1/2, 1 minus, minus 1

1540
01:08:14,690 --> 01:08:21,479
to the f of 0 plus f 1, 1.

1541
01:08:21,479 --> 01:08:25,970
And now, if f of 0 and f of
1 are the same, this is a 0.

1542
01:08:25,970 --> 01:08:27,899
We get 1 plus 1.

1543
01:08:27,899 --> 01:08:30,600
We get just 0, and this is 0,
because this is 1, this is 1.

1544
01:08:30,600 --> 01:08:31,600
They subtract we get 0.

1545
01:08:31,600 --> 01:08:34,529
They're same you get this
state 0, properly normalized.

1546
01:08:34,529 --> 01:08:36,847
If they're not the same,
you get this state 1,

1547
01:08:36,847 --> 01:08:37,680
properly normalized.

1548
01:08:37,680 --> 01:08:40,500
Now if we measure 0, we
know they're the same,

1549
01:08:40,500 --> 01:08:42,920
and if we measure 1, we
know they're different.

1550
01:08:42,920 --> 01:08:46,810
And so with absolute
certainty, now five.

1551
01:08:46,810 --> 01:08:49,270
Measure the first qubit.

1552
01:08:53,689 --> 01:09:00,729
And we get 0 implies the
same, and 1 implies different.

1553
01:09:00,729 --> 01:09:03,740
And we did all of this
with a single evaluation

1554
01:09:03,740 --> 01:09:06,521
of our function f, right here.

1555
01:09:06,521 --> 01:09:08,479
This is where we apply
our function evaluation.

1556
01:09:08,479 --> 01:09:10,210
We apply the function
evaluation once,

1557
01:09:10,210 --> 01:09:12,210
and we deterministically
get the result,

1558
01:09:12,210 --> 01:09:14,310
whether they're the
same or different.

1559
01:09:14,310 --> 01:09:18,080
So with one call to
Matt, to my Oracle,

1560
01:09:18,080 --> 01:09:21,970
with one call to Matt, which
cost me one million dollars,

1561
01:09:21,970 --> 01:09:25,120
we get the answer to whether
it's the same or different.

1562
01:09:25,120 --> 01:09:27,890
And that's a factor of
2 faster than the best

1563
01:09:27,890 --> 01:09:28,773
classical algorithm.

1564
01:09:31,890 --> 01:09:33,270
But that's not so satisfying.

1565
01:09:33,270 --> 01:09:35,819
This was supposed to be
exponentially better.

1566
01:09:35,819 --> 01:09:41,180
And so that's where Jozsa comes
in, and together with Deutsche,

1567
01:09:41,180 --> 01:09:43,180
Deutsche and Jozsa the
show the following.

1568
01:09:43,180 --> 01:09:47,370
That there's an exactly
analogous problem for n qubits.

1569
01:09:47,370 --> 01:09:48,010
Wow.

1570
01:09:48,010 --> 01:09:51,778
There's exactly analogous
from for n qubits,

1571
01:09:51,778 --> 01:09:53,319
the Deutsche-Jozsa problem.

1572
01:09:56,680 --> 01:10:00,000
And now, how many different
strings of integers

1573
01:10:00,000 --> 01:10:01,660
could you put in?

1574
01:10:01,660 --> 01:10:04,490
There are now 2 to
the n possible states.

1575
01:10:04,490 --> 01:10:07,240
And if you want to know whether
f is the same for all of them,

1576
01:10:07,240 --> 01:10:09,320
the worst case scenario
is you evaluate

1577
01:10:09,320 --> 01:10:12,250
f on the first possible
combination, 0, 0, 0, 0,

1578
01:10:12,250 --> 01:10:13,870
and you get some number.

1579
01:10:13,870 --> 01:10:17,590
You measure f on 0, 0, 0, 0,
1, and get the same number,

1580
01:10:17,590 --> 01:10:18,970
and just keep
doing that forever.

1581
01:10:18,970 --> 01:10:21,011
And you still don't know
if they're all the same,

1582
01:10:21,011 --> 01:10:24,370
until you get to
the very last one.

1583
01:10:24,370 --> 01:10:29,020
So, order 2 to the n is
the worst case scenario.

1584
01:10:29,020 --> 01:10:32,070
But technical scales are
of a order 2 to the n.

1585
01:10:32,070 --> 01:10:36,340
So classically, it
takes an enormous number

1586
01:10:36,340 --> 01:10:38,150
of observations.

1587
01:10:38,150 --> 01:10:40,610
But in the quantum
Deutsche-Jozsa algorithm--

1588
01:10:40,610 --> 01:10:47,780
and now in the n qubit
Deutsch-Jozsa problem-- one

1589
01:10:47,780 --> 01:10:54,360
quantum operation, and you
get a deterministic result.

1590
01:10:54,360 --> 01:10:58,020
And all of this, you evaluate
it once, and you know.

1591
01:10:58,020 --> 01:10:59,120
You've solved the problem.

1592
01:10:59,120 --> 01:11:00,900
So instead of 2 to
the n operations,

1593
01:11:00,900 --> 01:11:03,780
it takes a single one.

1594
01:11:03,780 --> 01:11:06,040
And now, for a large number
n of bits, for example,

1595
01:11:06,040 --> 01:11:09,530
for large integers-- dealing
with very large numbers--

1596
01:11:09,530 --> 01:11:12,410
this is dramatically,
exponentially more efficient

1597
01:11:12,410 --> 01:11:15,570
than the classical algorithm.

1598
01:11:15,570 --> 01:11:18,577
So at this point, people start
really seriously thinking

1599
01:11:18,577 --> 01:11:20,160
about quantum
computation, whether you

1600
01:11:20,160 --> 01:11:21,280
could get it to work.

1601
01:11:21,280 --> 01:11:24,190
And how much we
juice you can get out

1602
01:11:24,190 --> 01:11:26,340
of actually building
such a quantum computer.

1603
01:11:26,340 --> 01:11:28,090
And this has developed
into a whole theory

1604
01:11:28,090 --> 01:11:31,750
in the whole field of the
theory of computation.

1605
01:11:31,750 --> 01:11:34,900
The thing I want to emphasize
is that the crucial move

1606
01:11:34,900 --> 01:11:36,660
is observing that,
in quantum mechanics,

1607
01:11:36,660 --> 01:11:39,032
you can entangle
degrees of freedom.

1608
01:11:39,032 --> 01:11:41,490
The crucial move is observing
that you can entangle degrees

1609
01:11:41,490 --> 01:11:42,700
of freedom, quantum
mechanically.

1610
01:11:42,700 --> 01:11:44,830
And that's what gave us
all of the nice effects.

1611
01:11:44,830 --> 01:11:46,288
We have these
interference effects.

1612
01:11:46,288 --> 01:11:47,790
And these interference
effects lead

1613
01:11:47,790 --> 01:11:50,900
to the deterministic
outcome being correlated

1614
01:11:50,900 --> 01:11:53,020
with the result of
the computation.

1615
01:11:53,020 --> 01:11:55,260
The interference is crucial.

1616
01:11:55,260 --> 01:11:58,730
And this brings us
to the last point,

1617
01:11:58,730 --> 01:12:03,080
which is exactly what's so
troubling about entanglement.

1618
01:12:03,080 --> 01:12:05,600
And so here is
where EPR come in.

1619
01:12:05,600 --> 01:12:08,384
And Einstein, Podolsky, and
Rosen say the following.

1620
01:12:08,384 --> 01:12:10,550
They say, look, there are
two things that are deeply

1621
01:12:10,550 --> 01:12:14,060
upsetting about this
entanglement story.

1622
01:12:14,060 --> 01:12:16,229
Let me just give you a
precise experiment, they say.

1623
01:12:16,229 --> 01:12:18,270
They say, let me give you
precise experiment that

1624
01:12:18,270 --> 01:12:19,940
embodies all the
weirdness of this.

1625
01:12:19,940 --> 01:12:21,830
Suppose I take two
of these qubits.

1626
01:12:21,830 --> 01:12:25,400
And I put the qubits
in an entangled state,

1627
01:12:25,400 --> 01:12:28,150
up, up, plus down, down.

1628
01:12:28,150 --> 01:12:28,670
OK.

1629
01:12:28,670 --> 01:12:30,420
Let's normalize this
with a 1 upon root 2.

1630
01:12:30,420 --> 01:12:32,290
So there's our state.

1631
01:12:32,290 --> 01:12:35,330
And then we take the first
qubit, so there's our two bits,

1632
01:12:35,330 --> 01:12:38,900
we take the first qubit, we
send it somewhere faraway.

1633
01:12:38,900 --> 01:12:41,451
And someone named
Alice is sitting here

1634
01:12:41,451 --> 01:12:42,700
and is holding on to that bit.

1635
01:12:42,700 --> 01:12:44,890
And we take the
second bit far away.

1636
01:12:44,890 --> 01:12:47,690
And someone named Bob,
conventionally, is sitting here

1637
01:12:47,690 --> 01:12:49,180
and holds a second bit.

1638
01:12:49,180 --> 01:12:52,260
Now given this initial
of configuration,

1639
01:12:52,260 --> 01:12:54,060
what is the
probability that Alice

1640
01:12:54,060 --> 01:12:56,470
will measure the spin to be up?

1641
01:12:56,470 --> 01:12:57,570
Her spin to be up.

1642
01:12:57,570 --> 01:12:58,260
1/2, right?

1643
01:12:58,260 --> 01:12:59,060
And 1/2 down.

1644
01:12:59,060 --> 01:13:00,860
Similarly, Bob 1/2 up and down.

1645
01:13:00,860 --> 01:13:05,260
Once Alice has measured
the state to be up,

1646
01:13:05,260 --> 01:13:08,130
immediately she knows
something about Bob's spin.

1647
01:13:08,130 --> 01:13:11,212
Bob's state will be up,
because I chose this one.

1648
01:13:11,212 --> 01:13:13,545
I could have chosen the other,
which is the more popular

1649
01:13:13,545 --> 01:13:15,180
[INAUDIBLE].

1650
01:13:15,180 --> 01:13:17,557
So Bob's will also be up.

1651
01:13:17,557 --> 01:13:19,890
Now if you look at this list--
you do this over and over

1652
01:13:19,890 --> 01:13:22,130
and over again-- their list
just some random list of ups

1653
01:13:22,130 --> 01:13:24,040
and downs, ups and downs,
but they're exactly

1654
01:13:24,040 --> 01:13:25,420
correlated amongst each other.

1655
01:13:25,420 --> 01:13:29,074
So at this point, EPR
were really upset.

1656
01:13:29,074 --> 01:13:31,240
Because they say, look,
there are two possibilities.

1657
01:13:31,240 --> 01:13:33,740
Either there was an answer to
the question all the way along

1658
01:13:33,740 --> 01:13:36,450
of whether Alice's was
up and Bob's was up,

1659
01:13:36,450 --> 01:13:38,910
or Alice's was down
and Bob's was down.

1660
01:13:38,910 --> 01:13:43,700
Or there's some deep
non-locality in the universe,

1661
01:13:43,700 --> 01:13:47,700
such that a distant measurement,
causally disconnected,

1662
01:13:47,700 --> 01:13:51,620
can have an influence
on Bob's state,

1663
01:13:51,620 --> 01:13:53,320
such that they're correlated.

1664
01:13:53,320 --> 01:13:55,662
This may seem random, but
it's certainly not random,

1665
01:13:55,662 --> 01:13:57,120
because it's
correlated with Alice,

1666
01:13:57,120 --> 01:13:58,990
even though Alice is
wildly disconnected,

1667
01:13:58,990 --> 01:14:01,600
a distant observer.

1668
01:14:01,600 --> 01:14:04,940
Nothing could have traveled
across that distance

1669
01:14:04,940 --> 01:14:08,464
in the time it took
to do the measurement.

1670
01:14:08,464 --> 01:14:09,880
So they're sort
of three responses

1671
01:14:09,880 --> 01:14:10,838
you could take to this.

1672
01:14:10,838 --> 01:14:13,327
The first response is, look,
there isn't a problem here.

1673
01:14:13,327 --> 01:14:15,660
It's just saying that quantum
mechanics is insufficient.

1674
01:14:15,660 --> 01:14:17,661
There's secretly
a hidden variable,

1675
01:14:17,661 --> 01:14:19,160
a variable you
haven't observed yet,

1676
01:14:19,160 --> 01:14:21,620
a property of an electron
that determines whether it's

1677
01:14:21,620 --> 01:14:23,020
going to be up or down early on.

1678
01:14:23,020 --> 01:14:24,820
And the fact that it
looks probabilistic just

1679
01:14:24,820 --> 01:14:26,570
means that there's
some classical dynamics

1680
01:14:26,570 --> 01:14:29,510
for this hidden variable that
effectively is probabilistic,

1681
01:14:29,510 --> 01:14:30,980
like a particle
moving in a fluid.

1682
01:14:30,980 --> 01:14:32,880
Like dust pollen
grains in a fluid,

1683
01:14:32,880 --> 01:14:34,460
it just moves around randomly.

1684
01:14:34,460 --> 01:14:38,110
Bu it just looks random, and
it's not actually random.

1685
01:14:38,110 --> 01:14:40,710
That's because there's an
underlying classical mechanism

1686
01:14:40,710 --> 01:14:43,160
controlling the
probability distribution.

1687
01:14:43,160 --> 01:14:45,580
The second version is a
quantum mechanical version.

1688
01:14:45,580 --> 01:14:48,060
The second interpretation
is to say that, look,

1689
01:14:48,060 --> 01:14:49,744
this may look upsetting.

1690
01:14:49,744 --> 01:14:51,410
And I grant you that
it looks upsetting,

1691
01:14:51,410 --> 01:14:52,730
but I'm a quantum mechanic.

1692
01:14:52,730 --> 01:14:54,490
And quantum mechanics
works like a champ.

1693
01:14:54,490 --> 01:14:56,060
And I'm not about to throw
it out, and say that there's

1694
01:14:56,060 --> 01:14:57,710
some secret, hidden variables.

1695
01:14:57,710 --> 01:14:58,740
It just works.

1696
01:14:58,740 --> 01:15:02,220
So just give up on your naive
notions of locality, let it go,

1697
01:15:02,220 --> 01:15:05,030
and just do the quantum
mechanical calculation.

1698
01:15:05,030 --> 01:15:07,442
Practicing physicists look
at this, and just yawn.

1699
01:15:07,442 --> 01:15:09,650
If you're a practicing
physicist, you just forget it.

1700
01:15:09,650 --> 01:15:12,330
Like, obviously, it works, so
there's no more conversation

1701
01:15:12,330 --> 01:15:13,236
to be had.

1702
01:15:15,790 --> 01:15:17,470
So meanwhile, there's
a second version

1703
01:15:17,470 --> 01:15:19,960
of this, which is
slightly more disturbing.

1704
01:15:19,960 --> 01:15:21,530
Suppose Alice
measures up-- and this

1705
01:15:21,530 --> 01:15:23,153
is all on the z-direction--
but Alice measures up

1706
01:15:23,153 --> 01:15:24,040
in the z-direction.

1707
01:15:24,040 --> 01:15:26,210
She thus knows that
Bob's particle is up

1708
01:15:26,210 --> 01:15:27,200
in the z-direction.

1709
01:15:27,200 --> 01:15:30,740
But simultaneously,
Bob could measure spin

1710
01:15:30,740 --> 01:15:33,230
in the x-direction,
and determine

1711
01:15:33,230 --> 01:15:36,470
that his spin is up in
the x-direction as well.

1712
01:15:36,470 --> 01:15:40,245
At that point, EPR say, look,
we measured, empirically,

1713
01:15:40,245 --> 01:15:42,450
that the particle is both
up in the z-direction

1714
01:15:42,450 --> 01:15:44,010
and up in the x-direction.

1715
01:15:44,010 --> 01:15:46,910
It's just that we did that
measurement using entanglement.

1716
01:15:46,910 --> 01:15:51,850
But at the beginning
of the day, we

1717
01:15:51,850 --> 01:15:55,590
had that Sx and
Sz don't commute.

1718
01:15:55,590 --> 01:15:59,000
So you can't have a state with
definite Sx and definite Sz.

1719
01:15:59,000 --> 01:15:59,870
You cannot possibly.

1720
01:15:59,870 --> 01:16:01,746
It doesn't mean
anything to say so.

1721
01:16:01,746 --> 01:16:04,370
Einstein wants to say that this
is because quantum mechanics is

1722
01:16:04,370 --> 01:16:05,650
great, but incomplete.

1723
01:16:05,650 --> 01:16:07,610
The rest of us want to
say that, no it's not,

1724
01:16:07,610 --> 01:16:13,525
but that sounds like a
philosophical question.

1725
01:16:13,525 --> 01:16:14,900
And that's the
way it was treated

1726
01:16:14,900 --> 01:16:18,517
for a very long time, until
you come along to Bell.

1727
01:16:18,517 --> 01:16:20,100
And Bill made a
beautiful observation.

1728
01:16:20,100 --> 01:16:22,340
Bell said, look,
telling me that there's

1729
01:16:22,340 --> 01:16:27,980
an underlying probabilistic
classical description tells

1730
01:16:27,980 --> 01:16:30,630
me enough to make this
an empirical question.

1731
01:16:30,630 --> 01:16:33,284
Because it's saying
that the statistics,

1732
01:16:33,284 --> 01:16:34,950
the random statistics
for Bob and Alice,

1733
01:16:34,950 --> 01:16:37,900
are correlated by a
classical dynamics

1734
01:16:37,900 --> 01:16:39,639
rather than independent.

1735
01:16:39,639 --> 01:16:40,680
So here's Bell's version.

1736
01:16:40,680 --> 01:16:41,650
So remember at the
very beginning,

1737
01:16:41,650 --> 01:16:43,066
we talked about
Bell's experiment.

1738
01:16:43,066 --> 01:16:47,000
We said, consider three binary
properties, a, b, and c.

1739
01:16:47,000 --> 01:16:50,610
The number of some set
that are a and not b,

1740
01:16:50,610 --> 01:16:54,157
plus the number that
are b but not c,

1741
01:16:54,157 --> 01:16:56,240
is always greater than or
equal to the number that

1742
01:16:56,240 --> 01:16:57,092
are a but not c.

1743
01:16:57,092 --> 01:16:58,550
And the way we
proved this was just

1744
01:16:58,550 --> 01:17:01,008
by noting that, if these are
classical deterministic binary

1745
01:17:01,008 --> 01:17:04,370
properties, then a not
b means a not b and c,

1746
01:17:04,370 --> 01:17:06,220
or a not b and not c.

1747
01:17:06,220 --> 01:17:08,050
And ditto for each
of these other guys.

1748
01:17:08,050 --> 01:17:09,530
And we ended up
with an expression

1749
01:17:09,530 --> 01:17:13,150
which was, number that are
a-- using that logically to,

1750
01:17:13,150 --> 01:17:17,530
number that are a not b
and c plus the number that

1751
01:17:17,530 --> 01:17:22,080
are not a, b, and not c, is
greater than or equal to 0.

1752
01:17:22,080 --> 01:17:23,200
And that's clearly true.

1753
01:17:23,200 --> 01:17:26,451
You can't have a number
of elements be negative.

1754
01:17:26,451 --> 01:17:28,450
So this is trivially true,
quantum mechanically.

1755
01:17:28,450 --> 01:17:30,241
But now here's the
experiment I want to do.

1756
01:17:30,241 --> 01:17:33,010
I want to do actually
the EPR experiment.

1757
01:17:33,010 --> 01:17:34,810
And here's the
experiment I want to run.

1758
01:17:34,810 --> 01:17:38,170
Alice is going to
measure up or down at 0,

1759
01:17:38,170 --> 01:17:41,720
and Bob is going to measure
up or down at theta.

1760
01:17:41,720 --> 01:17:44,440
Alice is then going to
measure up and down at theta,

1761
01:17:44,440 --> 01:17:48,030
and Bob is going to measure
up and down at 2 theta.

1762
01:17:48,030 --> 01:17:49,490
And the third
experiment is going

1763
01:17:49,490 --> 01:17:51,930
to be Alice going to
measure up and down at 0,

1764
01:17:51,930 --> 01:17:54,450
and Bob is going to measure
up and down at 2 theta.

1765
01:17:54,450 --> 01:18:02,800
So a is up or down at--
up at 0, b is up at theta,

1766
01:18:02,800 --> 01:18:07,360
b is up at theta-- and, sorry,
this could be down at theta,

1767
01:18:07,360 --> 01:18:10,350
not b is down a theta. b
is going to be up at theta,

1768
01:18:10,350 --> 01:18:13,740
and c is going to
be down at 2 theta.

1769
01:18:13,740 --> 01:18:16,910
And a, again, up at 0 and
not c is down at 2 theta.

1770
01:18:19,810 --> 01:18:22,980
So we can rephrase this
as the probability that

1771
01:18:22,980 --> 01:18:27,110
given that we are up at 0,
what is the probability that we

1772
01:18:27,110 --> 01:18:29,750
are subsequently down at theta?

1773
01:18:29,750 --> 01:18:34,394
Plus the probability
that we are up at theta,

1774
01:18:34,394 --> 01:18:36,310
what is the probability
that we're up at theta

1775
01:18:36,310 --> 01:18:40,370
and subsequently
down at 2 theta?

1776
01:18:40,370 --> 01:18:44,950
And then the probability
that we are up at 0 and down

1777
01:18:44,950 --> 01:18:47,450
to theta, using an
EPR measurement, where

1778
01:18:47,450 --> 01:18:50,510
one is performed by Alice and
the other is performed by Bob.

1779
01:18:50,510 --> 01:18:52,280
Exactly as EPR wanted.

1780
01:18:52,280 --> 01:18:54,300
And we computed this
last time-- in fact

1781
01:18:54,300 --> 01:18:58,575
I just erased, because
I'm excited about this,

1782
01:18:58,575 --> 01:19:00,950
I guess-- I just erased the
wave function that we needed,

1783
01:19:00,950 --> 01:19:02,150
the state we needed.

1784
01:19:02,150 --> 01:19:06,250
And the state that we needed
was that if we are down

1785
01:19:06,250 --> 01:19:11,560
at the angle theta, then this is
equal to cosine of theta upon 2

1786
01:19:11,560 --> 01:19:19,750
down at 0, plus i sin
theta upon 2, up at 0.

1787
01:19:19,750 --> 01:19:21,790
And this is enough to
answer our question.

1788
01:19:21,790 --> 01:19:23,740
This is the quantum
mechanical prediction.

1789
01:19:23,740 --> 01:19:25,781
What's the probability
that given that we're down

1790
01:19:25,781 --> 01:19:28,890
at the angle theta,
we're up at the angle 0?

1791
01:19:28,890 --> 01:19:31,000
Well, if we're down
at theta, the probably

1792
01:19:31,000 --> 01:19:33,190
that we're up at
0-- the coefficient

1793
01:19:33,190 --> 01:19:35,427
is i sine theta upon
2, and the probability

1794
01:19:35,427 --> 01:19:37,510
is the norm squared of
that expansion coefficient.

1795
01:19:37,510 --> 01:19:41,449
So the probability is sine
squared of theta upon 2.

1796
01:19:41,449 --> 01:19:43,490
Similarly, the probability
that we're up at theta

1797
01:19:43,490 --> 01:19:46,580
and down the 2 theta, well,
by rotation by theta, this

1798
01:19:46,580 --> 01:19:48,320
gives me exactly the same thing.

1799
01:19:48,320 --> 01:19:50,850
So it's, again, going
to be sine squared,

1800
01:19:50,850 --> 01:19:54,904
sine squared theta upon 2.

1801
01:19:54,904 --> 01:19:56,320
The probability
that we're up at 0

1802
01:19:56,320 --> 01:19:59,660
and down at 2 theta, well, just
taking a factor of 2 for theta

1803
01:19:59,660 --> 01:20:01,010
everywhere.

1804
01:20:01,010 --> 01:20:04,910
And that gives me
sine squared of theta.

1805
01:20:08,117 --> 01:20:10,700
Now, I ask you, is left the left
hand side always greater than

1806
01:20:10,700 --> 01:20:12,033
or equal to the right hand side?

1807
01:20:14,620 --> 01:20:15,850
And this is easy to check.

1808
01:20:15,850 --> 01:20:17,435
Let's do this for
a very small theta.

1809
01:20:17,435 --> 01:20:20,210
For a very small theta,
sine squared theta.

1810
01:20:24,310 --> 01:20:28,460
So for small theta,
much less than 1,

1811
01:20:28,460 --> 01:20:33,020
sine theta squared is theta
upon 2 the angle squared,

1812
01:20:33,020 --> 01:20:36,150
which is equal to
theta squared upon 4.

1813
01:20:36,150 --> 01:20:37,670
And the next one
is the same thing,

1814
01:20:37,670 --> 01:20:41,950
plus theta upon 2 squared,
which is equal to theta squared

1815
01:20:41,950 --> 01:20:45,900
upon 4, theta squared upon
4, so theta squared upon two.

1816
01:20:45,900 --> 01:20:48,520
And the right hand side
is sine squared theta,

1817
01:20:48,520 --> 01:20:50,890
which is theta squared.

1818
01:20:50,890 --> 01:20:54,410
And is theta squared upon 2
greater than or equal to theta?

1819
01:20:54,410 --> 01:20:55,654
Certainly not.

1820
01:20:55,654 --> 01:20:57,320
So quantum mechanics
predicts that if we

1821
01:20:57,320 --> 01:21:00,830
do the EPR experiment, using
these observables repeatedly,

1822
01:21:00,830 --> 01:21:02,980
and built up statistics,
what we'll find

1823
01:21:02,980 --> 01:21:06,059
is an explicit violation
of the Bell Inequality.

1824
01:21:06,059 --> 01:21:08,350
And what that would represent
if it were actually true,

1825
01:21:08,350 --> 01:21:09,725
if we actually
observed it, would

1826
01:21:09,725 --> 01:21:12,240
be a conclusive empirical
proof that there

1827
01:21:12,240 --> 01:21:16,700
are no classical definite
configurations underlying

1828
01:21:16,700 --> 01:21:19,750
the probability of
quantum mechanical events.

1829
01:21:19,750 --> 01:21:22,070
It would say that it's
impossible to build

1830
01:21:22,070 --> 01:21:24,910
a classical theory with hidden
variables that are randomly

1831
01:21:24,910 --> 01:21:26,980
distributed such
that you reproduce

1832
01:21:26,980 --> 01:21:28,890
the predictions of
quantum mechanics.

1833
01:21:28,890 --> 01:21:32,070
We see already that it doesn't
agree with quantum mechanics.

1834
01:21:32,070 --> 01:21:34,260
The question is does it
agree with the real world?

1835
01:21:34,260 --> 01:21:37,630
So someone has to build
this experiment and check.

1836
01:21:37,630 --> 01:21:39,380
And this was done
by Alain Aspect.

1837
01:21:45,010 --> 01:21:48,230
And it violates
Bell's Inequality.

1838
01:21:48,230 --> 01:21:49,652
There is no
classical description

1839
01:21:49,652 --> 01:21:50,860
underlying quantum mechanics.

1840
01:21:50,860 --> 01:21:54,020
The universe around you is
inescapably probabilistic.

1841
01:21:54,020 --> 01:21:55,562
It evolves in a
deterministic fashion

1842
01:21:55,562 --> 01:21:56,811
through Schrodinger evolution.

1843
01:21:56,811 --> 01:21:58,780
But when we measure
things, we measure results

1844
01:21:58,780 --> 01:22:00,250
with probabilities.

1845
01:22:00,250 --> 01:22:03,020
And those probabilities cannot
be explained through some

1846
01:22:03,020 --> 01:22:04,479
underlying classical dynamics.

1847
01:22:04,479 --> 01:22:06,770
If there's something else
underlying quantum mechanics,

1848
01:22:06,770 --> 01:22:09,660
whatever else we know about
it, is it is not classical.

1849
01:22:09,660 --> 01:22:12,290
And this property of
probabilistic evolution,

1850
01:22:12,290 --> 01:22:16,160
or probabilistic measurement, is
an inescapable and empirically

1851
01:22:16,160 --> 01:22:19,752
verified property of
the reality around us.

1852
01:22:19,752 --> 01:22:20,960
And that's quantum mechanics.

1853
01:22:20,960 --> 01:22:22,930
Thanks guys.