1
00:00:01,000 --> 00:00:05,007
Okay. So, one of the advantages of
studying the basic principles of Quantum

2
00:00:05,007 --> 00:00:12,002
Mechanics in terms of qubits is that it
makes things much simpler and you can

3
00:00:12,002 --> 00:00:17,009
actually get to fairly interesting
concepts in quantum physics very quickly.

4
00:00:17,009 --> 00:00:23,001
So, what I'll do is go through a very
simple explanation in the setting of

5
00:00:23,001 --> 00:00:28,009
qubits of the Uncertainty Principle. Okay,
so, what's the Uncertainly Principle? I'm

6
00:00:28,009 --> 00:00:34,003
sure some of you may have heard about it
already. His you know, this was discovered

7
00:00:34,003 --> 00:00:40,044
by Werner Heisenberg, it's called
Heisenberg's Uncertainty Principle, and

8
00:00:40,044 --> 00:00:48,007
the most common usage of it is with
respect to position and velocity. So what

9
00:00:48,007 --> 00:00:55,007
Heisenberg discovered is that, in quantum
mechanics, you cannot really pinpoint both

10
00:00:55,007 --> 00:01:02,002
the position and the velocity of a
particle with perfect position. So if you,

11
00:01:02,002 --> 00:01:07,008
if you know its position perfectly, then,
you, you end up being very, very uncertain

12
00:01:07,008 --> 00:01:12,001
about its velocity. And if you know it's
velocity, then you end up being very

13
00:01:12,001 --> 00:01:17,001
uncertain about it's position. And what
you can do is, you can, you can try to

14
00:01:17,001 --> 00:01:22,002
narrow down both but then there's a limit
to how well you, you, you know each of

15
00:01:22,002 --> 00:01:26,009
them and there's this, and that's this
uncertainty principle. It says, the

16
00:01:26,009 --> 00:01:31,001
uncertainty in the position times the
uncertainty in the velocity is at least

17
00:01:31,001 --> 00:01:37,006
some, some quantity. Okay. So now, what
we're going to do is we're going to study

18
00:01:37,006 --> 00:01:44,008
an analogous principle, uncertainty
principle for qubits, okay? And what this

19
00:01:44,008 --> 00:01:49,055
is going to do is, it's going to
illustrate the basic principles, you know,

20
00:01:49,055 --> 00:01:54,018
why is there an uncertainty principle at
all for position and velocity? Well,

21
00:01:54,018 --> 00:01:59,064
we'll, we'll look at that later in the
semester but for now in the context of

22
00:01:59,064 --> 00:02:06,015
qubits, we are going to look at these two
quantities. So, remember you know, a qubit

23
00:02:06,015 --> 00:02:11,057
is, is the, the state of this electron in
the hydrogen atom around to excite it. And

24
00:02:11,057 --> 00:02:16,074
we might want to measure the bit you know,
whether, whether the, whether the qubit is

25
00:02:16,074 --> 00:02:23,046
in the state zero or one. Of course, the
bit has a physical interpretation, its the

26
00:02:23,046 --> 00:02:28,027
energy of this, it corresponds to the ene
rgy whether its in the ground state or in

27
00:02:28,027 --> 00:02:33,097
excited, whether the energy is low or
high. We could also measure a different

28
00:02:33,097 --> 00:02:41,027
quantity, the sign whether the qubit is in
state plus or minus, okay? These are

29
00:02:41,027 --> 00:02:46,027
strange states of the, of, of this
electron but you could imagine that this

30
00:02:46,027 --> 00:02:50,095
could have some physical meaning. So let's
pretend for the moment that these are two

31
00:02:50,095 --> 00:02:58,027
different physical quantities that we're
interested in, the Bit and the Sign. Okay.

32
00:02:58,027 --> 00:03:06,012
So, we could ask the question, can we know
both the bit and the sign of the qubit

33
00:03:06,012 --> 00:03:12,036
with perfect position? So, what do I mean
by this? So let's say that, we have our

34
00:03:12,036 --> 00:03:20,012
qubit, it's in the, it's in the state psi
which is Alpha0 (zero) + Alpha1 (one). So

35
00:03:20,012 --> 00:03:28,058
now, what its bit value? Well, we know
that if you're going to do a measurement,

36
00:03:28,058 --> 00:03:33,045
it would be in the state zero with
probability Alpha-nought squared and the,

37
00:03:33,045 --> 00:03:38,036
and the state one with probability Alpha
one squared. And so if Alpha-nought and

38
00:03:38,036 --> 00:03:43,070
Alpha one are both non-zero then we'd be
somewhat uncertain about the actual bit

39
00:03:43,070 --> 00:03:48,098
value. So the only way to be perfectly
sure about the bit value is if either

40
00:03:48,098 --> 00:03:58,006
Alpha0 = one or if Alpha1 = one. So there
are only two possible states of this, of

41
00:03:58,006 --> 00:04:06,077
this qubit where we'd be perfectly certain
about the bit value. If it's either in the

42
00:04:06,077 --> 00:04:17,080
state zero or in the state one, okay? So
this is for the bit value. Of the qubit.

43
00:04:17,080 --> 00:04:30,043
But now, I suppose we were interested in
the sign. Well then, we could write psi as

44
00:04:30,043 --> 00:04:39,098
B0(+) + B1(-). And now, if you we are to
again measure, we'd see it in the plus

45
00:04:39,098 --> 00:04:44,060
state that's probability Beta-nought
squared and minus with probability Beta

46
00:04:44,060 --> 00:04:49,041
one squared. Again, if we want to be
certain about the, about the sign value,

47
00:04:49,041 --> 00:04:55,030
the only way this could be is if
Beta-nought is one or if Beta one is one

48
00:04:55,030 --> 00:05:02,074
which is the same thing as saying, if it's
either in the state plus or in the state

49
00:05:02,074 --> 00:05:10,064
minus. But now, what if the qubit person,
the state plus? How certain would be,

50
00:05:10,064 --> 00:05:16,063
we'll be about the bit value? Well the
answer is, if a person state plus, then we

51
00:05:16,063 --> 00:05:23,041
are maximally uncertain about the bit
value because if we were to measure it on

52
00:05:23,041 --> 00:05:29,080
the bit basis, we'll get 50, 50, zero and
one. And so, what this does is that the

53
00:05:29,080 --> 00:05:34,025
bit basis and the sign basis are
incompatible with each other. If you are

54
00:05:34,025 --> 00:05:38,088
certain about one then you're maximally
uncertain about the other and vice versa.

55
00:05:38,088 --> 00:05:44,089
And this is the origin of the uncertainty
principle. It turns out that the bit and

56
00:05:44,089 --> 00:05:50,033
the sign basis are for each transforms of
each other. This is something we'll, we'll

57
00:05:50,033 --> 00:05:55,010
talk about later. So, you know, it's just
to give you a heads up. And in a similar

58
00:05:55,010 --> 00:06:00,073
way, the position on the velocity for a
transforms of each other. And so, the

59
00:06:00,073 --> 00:06:06,043
uncertainty principle just comes from the
fact that these two basis incompatible of

60
00:06:06,043 --> 00:06:12,007
each other, you know, they are maximally
far from each other. Okay, so here's a way

61
00:06:12,007 --> 00:06:18,053
you can actually quantify this. So, we can
define the spread of a quantum state. So,

62
00:06:18,053 --> 00:06:25,038
if a state psi is written like this and
the standard basis like this and the, and

63
00:06:25,038 --> 00:06:30,085
the sign basis. And now let's define the
spread in the standard basis to be the

64
00:06:30,085 --> 00:06:35,019
absolute value of Alpha-nought plus the
absolute value of Alpha one and similarly

65
00:06:35,019 --> 00:06:42,058
the spread in the sign basis is just the
absolute value of Beta-nought + B1. So now

66
00:06:42,058 --> 00:06:48,016
let's, let's look at the spread for the
basis state zero. Well in this case,

67
00:06:48,016 --> 00:06:53,089
Alpha-nought is one and Alpha one is zero
and so this spread is, is one. We don't

68
00:06:53,089 --> 00:07:02,052
know how it spread. If you look at the, at
the plus state, its spread is, well, the

69
00:07:02,052 --> 00:07:10,062
plus state is one over √2 one over √2
so it's spread is one over √2 + one over

70
00:07:10,062 --> 00:07:18,093
√2 which is √2. So its, it has a large
spread. On the other hand, if you were to

71
00:07:18,093 --> 00:07:23,096
look at the spread in this, in the sign
basis, then the, the spread of the zero

72
00:07:23,096 --> 00:07:30,052
state in the sign basis, well in the sign
basis the zero state looks like one over

73
00:07:30,052 --> 00:07:38,047
√2 one over √2. And so, its spread is,
is large it's √2. Whereas in the sign

74
00:07:38,047 --> 00:07:44,039
basis, the spread of the plus state, well
that's just Beta ≠ one Beta one = zero,

75
00:07:44,039 --> 00:07:52,016
so its spread is just one. So, what this
shows is that, what you can do is you can

76
00:07:52,016 --> 00:07:59,015
try to make the, the spread and the
standard basis small say by using you

77
00:07:59,015 --> 00:08:05,007
know, the zero state b ut then the spread
in the sign basis becomes large. It

78
00:08:05,007 --> 00:08:10,068
becomes as large as √2. On the other
hand, you can try to make the spreads

79
00:08:10,068 --> 00:08:17,078
small in the, in the sign basis but then
it becomes larger than the standard basis.

80
00:08:17,078 --> 00:08:23,056
And in fact, this spread is always between
one and √2. And so what you can show, is

81
00:08:23,056 --> 00:08:29,019
that no matter what state you pick, no
matter what Alpha-nought , Alpha one are,

82
00:08:29,019 --> 00:08:35,021
Beta-nought, Beta one, the spread in the
standard basis times the spread in the, in

83
00:08:35,021 --> 00:08:41,030
the sign basis is always at least √2 for
any state sign. Okay, so this shows that

84
00:08:41,030 --> 00:08:45,080
there's, there's, there's going to be
uncertainty. No matter what you do, you

85
00:08:45,080 --> 00:08:51,064
can just trade off the uncertainty in the,
in the bit and the sign. So, that's a

86
00:08:51,064 --> 00:08:59,032
very, very simple illustration of the
uncertainty principle in this context of