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.