Okay. Good morning. Today we are going to
start with the central object of our study

which is Qubits. So, in this lecture,
we'll, we'll start with an introduction to

qubits and to two of the very basic axioms
of quantum mechanics, the Super Position

Principle and the Measurement Principle.
And then as an exercise, what, what we'll

do is talk about a very simple uncertainty
principle for qubits. Okay so, let's,

let's, let's get started. So quantum
mechanics where the name comes from is

this, this concept of quantization which
is that certain quantities such as energy,

get quantized when you, when you're
working with you know, for instance with

electrons and an atom. What it means for
this quantities to be quantized is unlike

classical physics where the energy of a
particle is, is just an arbitrary,

arbitrary real number. Here, when the,
when the electron is in an atom, its

energy can only take on certain discrete,
prescribed values. So, here's, here's a

simple sort of picture of it. So, imagine
that this is a picture of a, of the

electron, of a hydrogen atom. So the, the
solid dot here is a nucleus and, and then

these dotted lines here represent possible
orbits of the electron. And of course,

this is a very simple schematic but its
meant to represent of this factor. The

electron cannot just be on any state, it
must be either on the ground state or in

this first excited state or the second
excited state and so on. And each of these

has a discrete energy level. Okay, so this
phenomenon of quantization occurs when the

particle is confined and you know, we'll,
we'll say a little bit about this later in

the course but for now, we want to think
of a system which can be in one of

finitely many different states. So,
imagine that you have a hydrogen atom and

the, the energy of the electron is, is
bounded so it can be in let's say, in one

of these, these three states, ground first
exited or second exited. Okay. So in

general, we have a key level system you
know , so, we work in the example you

know, I think you know, k is three and,
and the possible states of the system we

are labeling as zero, one through k - one,
okay? So, so in other words, for example,

if k were two then we could use this,
this, the state of the electron

classically to represent a bit of
information. Okay. So, what are the

possible states of the quantum system? And
this is where this very strange principle

comes into play, it's called the
Superposition Principle. And what it says

is that the electron in general doesn't
make up it's mind which of these key

distinct, distinguishable states sits in.
In fact, what it is in is some sort of

superposition of all k simultaneously.
Now, the superposition is very hard to

picture and, you know, mathematically,
it's very easy to describe. So, what the

superposition is, is its described by, by,
by specifying a complex number for each of

these possibilities. So, it's in the
ground state with probability amplitude,

Alpha zero where Alpha zero is some
complex number. And it's in the first

excited state that probability amplitude
Alpha one and so on. Now we'll also

require that this probability amptitudes
are normalized so the sum of the squares

of the magnitudes of these amplitudes is
one. So, let's, let's do a simple example.

So for example, if k = three. And let's
say all these amplitudes were equal, so

for example, we could have a superposition
one over √3 zero + one over √3 one +

one over √32. And you can check that
this is normalized because the some of the

squares this one. Now of course, you can,
you can also make, make one of these you

know, these can be arbitrary complex
numbers so you can have a negative sign

here or you could make, make this i/√3
where, where i is the √-1. That's sits

in a imaginary number here. Or you could
make it for example be the case that the

superposition looks like this, it's
one-half (zero) - one-half (one) +

one-half + i/2 (two). And of course, you
know just, just make sure you understand,

you, you remind yourself how complex
numbers works, the magnitude of this c

omplex number is a half, the magnitude of
this one is a half so when you look at the

square of the magnitude, you get, in this
case, you get a quarter, you get a

quarter. And the square root of the
magnitude of this complex number is just

the square of the real part plus the
square of the imaginary part so you get a

quarter plus a quarter is a half and all
that sums up to one. Okay, so that's,

that's the superposition principle. Now,
you know, mathematically its very simple,

all it say is you describe the state as
this linear superposition. But, what does

it really mean? And, you know, what does
it mean that there are these complex

numbers associated with each of these
possibilities and what does it mean that

the state of the system is, is this linear
superposition. One way to think about it

is, you know, this is what we saw last
time that the double slit experiment where

the electron went through each of the
slits with some complex amplitude,

alright? So these are the same complex
amplitudes so that electron was in a

superposition going through slit one and
slit two. This electron is in a

superposition of being on the ground
state, first excited, second excited

state. Another way to make sense of this
is by understanding what happens when you

actually measure this electron. So, when
you go to measure this electron, in fact,

you don't see it in a superposition. What
happens is the electron quickly makes up

his mind which of the key classical states
itself. And, and the way it does that, the

rule according to which it does, does that
is that, what you end up seeing is that

it's in this j state with probability
Alpha j magnitude squared. I guess you

take the square of the magnitude of this
amplitude and that's what the, what's the

probability. So, it's a good thing that
these, these squares of the magnitude

summed up to one because these are, these
are really mutually exclusive probability,

possibilities and the probabilities add up
to one. The second thing to know about

this which is very strange about the set
up is as soon as you make the measurement,

if the outcome is j, that's the new state
of the system. So the superposition

disappears and what you are left with is
that the new state of the system is j. So

for example, if ask, if the state of our
system was one-half (zero). What did we

have last time? We had one-half (zero) -
one-half (one) + one-half + i/2 (two) and

we make a measurement. Then the
probability that the outcome is zero is a

quarter and if this is the outcome, then
the new state is zero. The probability of

one is also a quarter and new state is
zero, is one. And its probability, and the

outcome is two with probability one-half.
And if so, the new status two. Okay? So,

and you can, you can ask what you know,
this is, this is sort of a funny thing,

you know that, that quantum system is in
some, some state which is a linear

superposition which we have a hard time
picturing. And then, it's only when we

look at it, if we measure it, that you
know, the state collapses into one of

these distinct possibilities, one of the
key possibilities. With this simple rule,

you know, so the rule is simple but how to
interpret it is, is, is very, it's very

problematic. And the reason it's
problematic is that nothing in you know,

nothing in our daily lives, nothing in our
study of classical physics or in the world

around us in interactions with the world,
physical world around us prepares us for,

for something this strange. So the great
physicist Richard Fineman once said that,

nobody understands quantum mechanics and
what he meant is that it's hard to

understand this intuitively. Now we have
the beauties of, of, of learning quantum

mechanics this way in terms of qubits is
that, it's a very simple system. There's

very little going on this k level systems.
And so, it exposes this paradoxical

elements of quantum mechanics right
upfront. Okay? So, for those of you who

have seen it for the first time. This is
the mysterious part of quantum mechanics

with nothing hidden from you. Okay,
finally, you know, I said that we were

introducing qubits. Well qubits just
happened to be the special case of k level

systems where k = two. So, imagine that
the electron can either be in the ground

state or the first excited state which
will represent by zero and one. You know,

you, you can use this or a bit of
information translate to zero, first

excited state is one. And then the general
state of the system as before is some

linear superposition like this zero and
one-half some complex amplitudes.

Normally, suitably, and if you make a
measurement, the system quickly makes up

its mind whether its on the ground of the
excited state with probability square of

magnitude of the corresponding aptitude.