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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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magnitude of the corresponding aptitude.