- [Elisa] So, as you can see,
we will have three different parts
that I'm gonna go through with you today.
So the first one is from bits to qubits
where we'll give you the very basics
of what a qubit is,
how we mathematically describe it.
So we're gonna tell you
about the Dirac notation.
Then how measurements work
and how we can illustrate quantum states
on the Bloch sphere.
Then we will have a second
part on quantum circuits,
where I show you what the basic
single and two-qubit gates are,
and how we can describe
states on multiple qubits.
And then in the last part,
we will talk about entanglement.
So I will show you what Bell states are,
we will talk about our very
first quantum protocol,
which is the teleportation protocol.
And then we will discuss
how to illustrate much,
illustrate states on logical
qubits on the Q-sphere
between each of these parts or somewhere
round in between there we will have
two to 10 minute breaks hopefully.
And also I will,
because I cannot
look at the questions
all the time obviously.
So every like around 15 minutes
I will have a look at the questions,
I will check through the
most uploaded questions
and try to answer one or two of them.
Yeah, I hope,
I hope you're all okay
you can now understand
that I will not be able
to answer all questions.
If sometimes I might not answer
the most important one just
because I think it's
something that takes longer
to explain or so,
but then you will be able
to ask that later on in the
channels,
search the links that I provided.
So let's start with the first part
from bits to qubits.
So maybe something else that
I should mention is that
my lecture notes will be
available for all of you
like an hour or so after the livestream.
So if you're like me,
and you cannot multitask that well on
trying to listen and focus
and at the same time writing
down everything I'm saying,
you will, don't worry you will
get the lecture notes anyway.
So you can fully concentrate
on what I'm trying to tell you now.
So, what we know from classical computing,
if classical computers,
then you know that they're
programmed to work only with states
that are either in zero or one.
Everything is based on
bit strings that are
bit strings of just zeros and one,
as the name suggests,
so for quantum for classical computing.
We have states that
are either zero or one.
However, in quantum mechanics,
it's different.
I guess most of you have heard
that word, superposition.
So states can be
what we call a superposition.
Which means that they can be
simultaneously in zero and one.
This superpositions
allow us that we can
then make calculations
not only on one state,
but actually on multiple
states at the same time.
For example, you can think about
if you have, on classical computers,
if you have a bit string
that contains three bits,
then with three bits,
you're able to give one
number between zero and seven.
So one out of eight numbers,
you can make a computation with that one.
If you're now on a quantum computer,
you can be in, every qubit can in
an equal superposition of
zero and one for example.
And that way, we can be in a superposition
of all eight states at the same time.
What this means is that
if we can calculate on all
these states at the same time,
we can get a big speed up.
And actually we have,
we know some quantum algorithms
that are exponentially faster
than any classical algorithm
that we know so far.
Which comes from the fact that we can
do the calculations on
superposition states.
However, now this sounds very trivial.
Oh, nice we just get
everything done much faster
because we just always
go into superposition
and then make awkward
calculations at once.
But it's not that easy.
So the problem is
that once we measure
the superposition state,
it will collapse.
So what that means
it collapses to one of its states
to one of the (mumbles) states
of the operator that we are measuring.
So it means that we can
only get one answer.
So let's say we do a calculation,
we do it on the superposition
on all different states at the same time,
but then we do a measurement
and during this measurement,
we will only get one answer.
So we will only get one answer
to one of the states
for the calculations that we did,
but we will not get all states
that are in superposition.
So this is why it is not that easy,
to design quantum algorithms
but we one, actually has to come up
with smart methods to
in the end be able to get some speed ups.
So for example, one thing that we can use
are so called interference effects.
So I guess most of you have had
in their Physics high school
lectures, they might have heard
about these interference effects.
For example, when you have
this double slit experiment with light,
so you have two slits that
light could go through.
But then in the end,
you will see an interference patterns,
interference pattern.
You will have,
you have some photons that interfere
constructively and some that
interfere destructively.
So you have either,
if you have different amplitudes
and the amplitudes can be
positive and negative, for example,
and then they can cancel each other out.
So you won't see anything like there,
or they can, well add up
and then you get an even higher peak.
So these interference effects
are things that we will
use for quantum states
as well as you will see
in like the next few hours.
Because each quantum state
we can have different amplitudes
for each of the states
and then these states
can as well interfere.
And this is something so
we can do calculations
on all states at the same time,
we're only able to get one
state output in the end.
But for example,
if we have a quantum algorithm
where we're looking for
and we're researching a big
database for one element
that we wanna find,
then we can use these interference effects
where we say, okay,
we somehow design a quantum
algorithm very smartly,
in a way that all the
elements that we don't want,
they somehow, destructively interfere,
while the element that we want to get,
we make sure that we get some
constructive interference there.
So then all the other,
the wrong answers basically
cancel each other out.
And the right answer is
the one that remains.
And then if we get the measurement,
we're able to get the right answer
faster than we would get
it on a classical computer.
So this is the main idea.
It's still not that easy.
That's of course why it is one of the
big research areas right now,
and a very hot topic
and well a lot of people
are trying to find
more and more good quantum algorithms.
So just as this was just for motivation,
lets start with a Dirac notation,
which is our mathematical
way to describe quantum states.
So, we use it
to describe quantum states.
So let us choose two
numbers, no two strings A and B,
that are both two dimensional
arrays with complex entries.
And then we can look at
what we call a ket.
You will see that a lot
in the lecture today.
So we put around our vector A,
we put around this like,
well, the things that you just see.
And this means we just look at the vector.
So if we have these two complex numbers,
A one and A two,
we will just look at a column vector.
A ket, so a ket is simply column vector.
It's really not more than that.
Then we can also look at the bra.
The bra, we do,
the sign is the opposite way.
And what it means is we
take the ket,
and we take the complex conjugate
and transpose the vector.
Which means,
so if we have our vector B one, B two,
that would be just a ket.
Now we have the (mumbles)
is what we call this T kind of thing.
And it means we,
instead of a column vector,
we will get row vector
and the elements will
be complex conjugated,
which I indicate by a star.
So we have B one star and B two star.
So maybe just to illustrate it,
if I have some number B,
which equals C plus D times I,
where I is the complex, the complex I now.
The imaginary unit,
then the star.
So the complex conjugated of that will be
C minus D times I.
C and D would be now real numbers.
So yeah, this is just
so the bra is very similar to the ket,
just complex, conjugated and transposed.
Now we can take both of them,
the bra and the ket and combine them.
So I put the bra here.
And then the ket.
And this, what I'm doing with that
is I'm taking the inner product.
Not sure how much linear algebra
knowledge all of you have.
But so what the inner product means
is that I take the first
element of the first vector,
in this case, well,
A one or B one,
and then, and the first
element of the second vector.
So I'm taking A one and B one star,
multiply them and I add the other one.
A two and B two star.
And this is by the way,
the same as if I would take
the inner product of A and B,
and then complex conjugate it.
So taking the inner product,
but it gives me,
because I'm just multiplying
and adding numbers
in the end I will get a complex number.
So here I have a column vector,
then a row vector,
and then a complex number.
And now as a last thing,
what we can also do
is we can combine the bra
and ket the opposite way.
So we start with the ket,
and then put at the bra.
So we have A
and then B.
And maybe now as a side note,
people kept asking me
why I'm putting an X in between there,
this is not an X.
But I on purpose did now here
because it's just,
one part belongs to the ket,
the other one belongs to the bra.
And people if they're lazy
or wanna save time, they
just do this instead.
But yeah, it's not an X.
And so what this means
if I have a column vector
and multiply it with a row vector,
I'm getting in this case
a two times two dimensional matrix.
So the first element is A one, B one star.
Then I have A one, B two star,
A two B one star,
and A two B two star.
Yeah, and that's it already.
So these are the most important
things you should remember
about the Dirac notation.
We will use it a lot during the day today.
And then also,
I think, yeah, I'll use it tomorrow,
and maybe also in the future lectures.
So but yeah, it's really just vectors.
Now, when we talk about qubits,
we usually look at two different states
we define some states are always,
defined in the same convention.
So one state is our state zero,
which is just a vector one zero.
And the other one is the state one,
which is the state zero, one.
These two states
are orthogonal.
Which we can very easily verify
if we take the inner product.
Now we just learned what
the inner product is,
inner product means I'm taking
the first element off.
On this case,
the state zero, which is one
times the first element of
the state one which is zero.
Plus,
well, maybe I should write
that down so it's clear.
So we have this,
times that.
So I take the one times the zero,
plus zero times one,
which gives me zero.
And if the inner product
of two states is zero,
then that means that these
two states are orthogonal.
Now, another important
feature of quantum states
or our convention actually
but quantum states,
is that we always normalize them.
Which mathematically can be described as
the inner product of twice
the same quantum state,
no matter for all quantum
states is always one.
So if I'm, for example,
looking at a superposition state,
let's say I'm looking at the state psi
which is an equal
superposition of zero and one,
then in order to get the
normalization constraint satisfied
that I get the inner product is one,
I have to add this factor
of one over square root of two.
So if I write it as a vector,
the state would be one
over square root two,
then one over square root two here.
So maybe now have a look
at the first questions.
Okay, the most uploaded
question is on labs.
- [Brian] Yeah, so I--
- [Elisa] I think
there's no questions yet, right?
- [Brian] Elisa are you able to hear me?
- [Elisa] I can hear you, yes.
- [Brian] Okay, cool.
So I was looking through them as well.
Looks like many of them
are related to the labs.
We're gonna ask--
- [Elisa] Okay.
- [Brian] Lab questions.
I hope I answered them earlier today.
But if not,
when we email around one o'clock today
to all the lab students,
please look for the information there.
So please don't lab ask questions here,
because we're gonna send
that invite out later today.
- [Elisa] I actually just noticed now
that in the chat, there
are some questions.
So someone is for example asking
whether we can only discriminate
orthogonal quantum states?
That's a very good question.
So, (chuckles) we will
look at measurements later.
And what we will consider is
for now only measurements,
projective measurements,
which are measurements
on orthogonal states,
but in general, they don't
have to be orthogonal.
There's also in general,
if we look at arbitrary POVMs,
which is something that we're not gonna
talk in the lecture here
because that's too, well
goes too deep and (chuckles)
too much content to be covered.
Then you can also discriminate
non orthogonal quantum states.
But,
so another question is--
(Brian mumbles)
- [Brian] Sorry Elisa,
I was gonna say,
I was gonna continue to
go through the questions
that have been officially submitted.
And I'll
monitor these down
to make sure that they're
all relevant to your lecture.
So next time you do a question, break,
all the ones in the
official question channel
should be appropriate.
- [Elisa] Okay, great.
I just saw another question
that I'm gonna answer now,
which is something that
anyway I wanted to mention.
So someone asked, why are
quantum states always normalized?
Which is a very good question.
So we do normalize quantum states,
it's actually just a convention.
So in theory, one could also use
non normalized quantum states.
But so if we,
we will see in like 20 minutes (mumbles)
or actually very soon,
you will see how we can
determine for example, the
probabilities of measurements
of getting different outcomes.
And for that one it's very,
for that it's very useful
if we have normalized quantum states,
it's much easier to
determine the probability
of getting a specific quantum state.
And then all the probabilities
add up to one of course.
Also, if we describe the
evolution of quantum states,
and we describe it by unitaries,
well, this is also something,
yeah, that helps if we have
we always look at normalized states,
we always, but then everything
just stays simple and nice.
And yeah, it's a convention.
It's not necessary.
But if you wouldn't do it,
you would need to make
sure people know that
you're not looking at,
I mean, one would notice
immediately but the most it's very,
everyone does it.
Everyone does normalize
the quantum states.
But yeah, it's not.
Physically it does not
really have a meaning
if we scale a vector by something else.
But you also later on you will see the,
how we plot states on the Bloch sphere
and that's also important
that they are all normalized.
But yeah, good question.
So now I will continue with measurements.
So as I just mentioned before,
what we choose now for this lecture
is we're only gonna
look at orthogonal bases
to describe and measure quantum states
which are so called
projective measurements.
So the generalizations of those are,
also exist, but we will
not cover them now.
Most in what we do in practice is usually
a projective measurement
anyway, so that's the most
important thing to know.
So, if you have a measurement
onto the basis zero, one.
So you just introduce this basis,
the basis where we choose
the two states zero and one
which are orthogonal states.
When during such a measurement
the state will collapse,
which is something that I
mentioned in the beginning
when I told you that it's not
that easy to design quantum algorithms
because our state will always collapse
into either of the states.
So if I do a measurement
onto the basis zero, one,
and it will come to collapse
either into state zero,
or into state one.
The reason why it collapses
into one of these two states
is that those are the eigenstates
of the sigma Z operator.
So sigma Z is a Pauli operator.
It's the Pauli-Z operator.
Sometimes it's also just
called, just called Z.
And I will go on later into,
I would give you more
details about Pauli matrices
and when we look at dates
we will discuss all these Pauli matrices
in more detail, so don't worry for now,
but just so you know
these are the eigenstates
of the Pauli-Z operator.
And so we call this,
that's why we call this Z measurement.
However, in general,
there are infinitely many different basis.
But there's some more common ones,
some that we use a lot.
So the other common ones
I'm gonna give you
one is the plus minus basis.
So we have to state plus,
which is defined as
the equal superposition
of the state zero and one.
And then we have to state minus,
which is also a superposition
of zero and one.
However, we have a minus sign
between the two states zero and one.
So I told you earlier that
there's different amplitudes.
And that for interference effects,
we can have different amplitudes
in different what we call relative phases
between the different states.
So we can have zero plus one,
but we can also have zero
minus one for example.
And so these two states
are indeed also orthogonal.
So if you take the inner product
now of these two states,
you would see that it gives zero again,
which means that we have
two orthogonal states.
And then another common basis
is the plus I and minus I basis.
Unfortunately, that one has very,
has a lot of different names.
So some people also call
it the plus Y and minus Y
basis or the left and right basis
before now call it plus I and minus I.
In this case,
we again have a superposition
of zero and one.
However, the phase is
actually a complex one now,
so we have the I, the imaginary unit here.
We have zero plus I times one
and then minus I,
the orthogonal state to that
is zero minus I.
And again you can check
that these two states are orthogonal.
Now, these two basis,
they also correspond to eigenstates.
And they correspond to the eigenstates
of the sigma X.
So, the Pauli X operator
and the Pauli Y operator respectively.
So,
we have sigma X and
sigma Y with eigenstates
plus minus and plus I minus I
and we have sigma Z with
eigenstate zero one.
So as you can imagine now I guess,
if I'm doing a measurement
onto the plus minus basis,
I can then call this as an X measurement,
or on the plus I minus I basis,
I would call it a Y measurement.
Now in order to determine
the probabilities of different outcomes,
we have the so called Born rule.
So if I'm giving a state
psi in the beginning,
and let's say I measure it
in some basis that I call X,
so I do a projective measurement
and I project onto the
state X and X orthogonal,
so this is just the state
that is orthogonal to X.
And now the probability that my state psi
actually collapses to the state X.
Is then given by
the probability to measure X
is the projection of the state psi
onto the state X,
which is the inner product
of these two states,
then I take the absolute
value and I square it.
And what one could easily show then,
thanks to the normalization, this is,
if we would not have
normalized states now,
this would be a bit more complicated.
We would need to divide below
divide by the normalized,
by the normalization of that
state basically to normalize
divide by the, inner
product of the two states
but we can forget about that
because we said we will only look at
normalize states for now.
So we can just do this
and then also the probability
of all the different states add up to one.
Now, I will give you some examples.
So that hopefully becomes clear.
So let's say we have to state,
some state psi, which is,
now I need to normalize with the factor
one over square root of three,
zero plus square root of two times one
and I measure it
in the basis zero, one.
Now I want you to determine
the probability to measure state zero.
As I just told you,
it's the inner product.
So I have the projection onto zero
to have the state psi.
And then I shall take the
absolute value and square it.
And now it's, we can just
write it as two terms.
It's all linear.
So we write first the first term,
I can get the factor,
one over squared of three (indistinct)
so that it's
easier to look at it.
And then the factor of
square root of 2/3
and also get it out and I
have the term zero, one here.
(mumbles) squared.
Now, what we know is that
since we have normalized states,
this inner product of
the state zero and zero,
it's just one.
(mumbles) normalized.
On the other hand,
the inner product of
the states zero and one
is zero because these two
states are orthogonal.
So I'm only getting the first term
and then squared,
which gives me 1/3.
In the same way,
I can determine the
probability to get state zero.
Which will be 2/3.
Which we can also see, easily see,
because we know that both probabilities
need to add up to one.
Now, I'll give you a second example.
Let's consider the state zero minus one.
Then we measure it,
this time in a different basis.
We will measure it in
the basis plus minus.
And we are interested in the probability
of getting the output plus.
So the probability to get the plus state
is just given by the
projections of the state psi
to the state plus, squared.
So we
write it out in all the different terms.
We have here the plus
state is zero plus one.
And then we have the state psi,
which is one over square root of two.
Zero minus one,
take the absolute value squared.
And now we can get the two factors
of one over square root of two,
we can multiply them
and get them out of the bracket already.
Out of the absolute
(mumbles) and square them
so we have one quarter.
And then everything else stays
inside this absolute values.
So we have two terms then
we have zero, zero minus zero, one.
Plus one, zero,
minus one, one.
And, squared.
Now we can again look at this four terms
just as before, we know,
in a product of two times
the same state is one,
thanks to normalization.
Where these terms,
just add zero because they are orthogonal.
However, now what we see is that
due to the different phases,
before we take the square,
we will have one minus
one in that expression.
So we will get zero as a probability.
So, the probability to
measure sate plus is zero,
which means the probability
to measure state minus is one.
And this is actually very expected.
It's not surprising,
because when I looked at
the inner product of psi and plus psi,
is exactly the,
maybe you have noticed that
psi actually the state minus
that I defined earlier when
I told you about the basis
I told you one basis, was
the plus minus basis here.
So, we have hit the plus
state and the minus state
and we did a measurement in on the,
we projected onto the plus minus basis
and the state that we were measuring
was exactly the minus state.
So if I'm measuring the plus minus,
the minus state and the plus minus basis,
since plus and minus are orthogonal.
It's clear that this will get zero
and that we will for sure
measure that minus state.
Okay, so now I think I will,
again, check on the questions.
If I can enable the video camera.
Well I'm not sure whether this helps,
I mean, I can,
well Brian what do you think?
Do you think it makes sense if I...
I'm a bit worried about
because so many people--
- [Brian] Yeah.
- [Elisa] Are already
complaining about that.
That the stream breaks down in between.
So I'm wondering whether--
- [Brian] Though, we
saw the same question.
I commented on it and said,
for that person to reach
out to us via Discord.
We're looking into alternatives
for what we may be able to do.
It's a great question.
For the exact reason you just said.
Elisa, I'm not sure if
turning the video on right
now is gonna be possible
because the stream was already
struggling just what we had before.
But please reach out to us over Discord
and we'll continue talking with you about
what some options are.
- [Elisa] Okay, yeah, good.
Also, I mean, honestly,
because I'm writing down on the iPad,
I'm not even sure whether
you would see a lot
because you would just see my forehead
and I'm looking at the iPad.
So you might not
probably wouldn't even see my lips moving.
And I think it would,
I'm not sure where that would even help.
(Elisa and Brian laughs)
Someone is asking where
do we get the T shirt
that Elisa is wearing?
So I'm wearing Qiskit T shirt
that I actually got at the at a lecture,
a Qiskit lecture two years ago, I think.
Almost two years ago.
I don't know.
Maybe someone else can
answer that question
I unfortunately cannot.
- [Brian] Yeah.
- [Elisa] Okay someone said--
- [Brian] I can jump in on that one.
When we are able to
resume in person events.
That's a great place to look for some.
But in the meantime,
I think helping out or participating in
Qiskit events that happen
on the community level
where you're doing something to help
organize other members
of your own community.
That is a great way to earn
yourself some Qiskit swag.
- [Elisa] Yes, and then
maybe last question that I
quickly will answer now.
Someone's asking,
which is the best place to work on Qiskit.
So if you want to work
with Qiskit, there's,
I've actually also
maybe I should have noticed that and
I should have mentioned
that in the beginning.
I'm gonna,
today's lecture will be very theoretical,
as you might have noticed already.
So I will give you a lot
of formulas and stuff.
And I will only in the very end
show you a bit of Qiskit code
on how to implement some basic states
just to illustrate some
quantum states that we
will talk about today.
And tomorrow,
on the other hand, we will
also have very theoretical
and a very theoretical analysis
of different algorithms,
but I will always after each algorithm,
I will immediately show
how it works in Qiskit
so we will have both.
And now to answer the question
how to work with Qiskit,
Well yeah, one has two options.
Basically, either you install
it and run it locally.
The installation process depending on,
yeah, if you don't have
the stable version,
it should not be too hard.
And I think there's YouTube
videos on how to do it.
But of course, it takes some time,
if you just quickly wanna run something
or also make sure that
you always have the newest
version of Qiskit.
Then I would recommend to use
the IBM Quantum Experience,
which you can just online access
by quantumminuscomputing.ibm.com,
you can create an account there
and just create notebooks
and work with Qiskit.
So I think in the beginning,
that might be a good start.
At some point, you might
prefer to run things locally,
especially if your
internet connection is bad
or something like that.
Yeah.
But yeah, you can test it today.
What you, today in the labs,
you can test whatever works for you best.
I think actually, for the labs today,
it's recommended to install
Node and install a Qiskit
and run it everything locally
because we have some qubit
notebooks that need be prepared
so, people should download Qiskit there.
Okay so, let me continue
with the Bloch sphere.
So, if we consider any quantum states,
any normalized quantum state
and for now we will restrict ourselves
on pure quantum states.
Pure quantum states are states
that are in a known state
where the entropy is zero
so we know which state that is in.
Which can be a superposition state.
We know that it's in
the state zero plus one
but we well,
if it's a very well defined state
and we know the answer if we
do this specific measurement.
However, there's also mixed states,
which I will not go into today,
because we just don't
have time to treat them
in more detail.
But that is states that have high entropy,
where you don't know whether it's in zero,
the state zero or in the state one.
Even if you no matter
what measurement you do,
you can never be sure
which outcome you will get.
And but we will not treat those states.
We'll just look at pure states today,
which are states that you
can always write as a ket,
'cause if they're not pure anymore,
you cannot write them this way.
So we just consider these states
and then the most general pure states,
the most general way to write it
at any pure state
is as cosine of theta plus two
times the state zero
plus E to the I phi sine is,
theta over two times one.
And so, in order to make sure that
we can have all possible relative phases,
phi can be anywhere here
between zero and two Pi.
So this describes what we
call the relative phase.
So it's maybe important to mention
there is the relative phase
and the global phase.
The relative phase is
what you have between states zero and one,
in this case if I have zero plus one,
or zero minus one,
or zero plus I times one.
This E to the I phi
is what determines the relative phase
and this is very important.
This is how we get all the
interference effects how we can
basically how we do quantum computing
is because we make use
of these relative phases.
And there's also a global phase,
which would be maybe you've noticed
there's now nothing in front of that part
if I would multiply the whole
state psi with minus one
and would still be normalized
and would still be a valid state.
However, physically,
it does not make any difference
from when I don't do that.
All expectations where
I use will be the same
whenever I do a measurement I get the same
so these global phases,
what do we call them?
We don't care about.
That's why in general, I do not need any,
E to the I phi in front of the first term,
but just on the second term
just to see the relative phase.
So we have the relative phase,
and then we have our angle theta.
Theta is between, can
be between zero and Pi.
And it determines the probability
to measure the state zero
or the state one in this case,
if we do a zero, one measurement,
so the probability to
measure zero, in this case,
would be if we take the
projection of the state zero,
we would see that it's cosine
of theta over two squared,
and the probability to measure one
equals sine as squared of theta over two,
and we know that cosine squared
plus sine squared gives one
so they add up as expected.
Now all normalized states,
all normalized pure states
can be illustrated on what
we call the Bloch sphere.
So, they can all be illustrated
on the surface of a sphere
with radius one
and so, I just
before I said we also had these mix tapes,
these mix tapes would actually then be
illustrated as some states
inside the Bloch sphere.
But if we have just pure
states, for now we will,
all these states will be on
the surface of the sphere
and now we can determine of course,
we somehow interested
and want to know where,
what the coordinates are
of any given state.
And for that we have the
so called Bloch vector.
So the Bloch vectors,
sine given by sine is theta, cosine theta,
sine theta, sine phi and
cosine of theta. (sighs)
So let's look at examples.
Some of the most important states
or states that we will look at a lot,
now we can plot them on the Bloch sphere.
So I try to
shape your circle.
So this is our Bloch sphere,
we have our three axis.
So we have here the X,
the x-axis, the y-axis and the z-axis.
And they all have,
so this is a sphere with radius one.
So they all and here
now we have our angles.
Theta will actually be this angle.
So we'll see.
And this angle is gonna be phi.
So let's look at the different states.
Some examples,
we start with a state zero.
For the state zero,
if we look at the
general way that I gave you before,
here, the one in pink, this one,
on how to describe general states,
if we want to get the state zero,
we see that theta would need to be zero.
If theta is zero, which
immediately get phi equal to zero,
because cosine of zero is one
and sine of zero is zero.
So we will not get any part
on the one on state one.
Actually, in this case,
it also does not matter at all what
our angle phi is,
because the second part,
this part here would anyway be zero.
So now if we look at the
Bloch vector, in this case,
we get sine theta, which is already zero.
So we can zero here and one here.
So the coordinates of the state zero
are just zero, zero, one.
So in X direction, zero in Y direction
and one in Z direction.
So here we will have our state zero,
basically on the North
Pole of the Bloch sphere.
In a very similar way,
we can determine where the state one lies.
So we know that we want
now the general expression
to give us state one,
so we want sine of theta
over two to be one.
So we need to choose theta equal to Pi.
Pi can again be arbitrary.
And our Bloch vector,
in this case will be zero, zero minus one.
So it will be down here, state one.
Now let's look at state plus.
You've seen that state before
it's the equal to the opposite
position of zero and one.
And we have zero phase between them.
So first of all,
if we want to get the equal superposition
we need to choose in order to have
zero and one with equal probability,
we need to choose theta to be Pi over two.
Because then we have one over squared,
the one over squared of two vector
and then phi since we have zero plus one,
E to the I phi just needs to be zero.
So it needs to be one.
So phi needs to be zero,
so that we do not get any phase.
And then our Bloch vector is given by
one, zero, zero. (mumbles)
One zero, zero means the
state lies on the x-axis.
So we have it here.
And in the same way,
we can also determine the minus state,
theta again needs to be Pi over two,
because we can get one
over square root of two
as vector in front both
of, both zero and one.
But now we have zero minus one.
So E to the I phi needs to be minus one,
which means that phi needs to be Pi.
So I hope you're all familiar
with complex numbers.
If not, I can very quickly
comment on that later.
So in this case, we get minus zero, zero.
State, we'll be here.
And now I don't wanna bore anyone.
So I'm just quickly gonna give you the
other two states that
I think are important
to plot on the Bloch sphere,
I guess you can
already guess where they are.
So again, because we
have equal superposition
of we have equal probability
to measure zero and one.
So theta needs to be Pi over two again,
but now the angles are
actually Pi over two
and three times Pi over two,
which you can determine
when you look at the
general description of the state again,
just as before.
And then the states
will be, zero, one, zero
and zero minus one zero.
So we have them
surprise on the way X is now.
So we have the state plus I on the right,
and minus I on the left.
Now, one needs to be careful though,
because if we look at
states on the Bloch sphere,
the angles are twice as large
as they would be in actual Hilbert space.
So for example the states zero and one,
we showed before that zero
and one are orthogonal.
Orthogonal in Hilbert space,
means that they have a
right angle 90 degrees.
However, here on the Bloch sphere,
we have one on the North Pole
and one on the South Pole.
So they actually have
an angle of 180 degrees.
So, when I gave you the
description for a general state,
just at the beginning of
the Bloch sphere chapter,
I told you that we can write psi as
cosine of theta over two times zero,
and so on.
And now this theta over two
is exactly why we have,
why I wrote it with theta over two,
I could also just have
written it as cosine of theta.
But so theta is the angle that
we have on the Bloch sphere.
Which you can see in the,
in the plot when I plotted it here.
You see the theta is just
the angle that we have,
basically towards the
North Pole to the z-axis.
Well, of course,
by the description if we
mathematically look at it
and write it as a vector,
we can tell that theta over two
is the actual angle that we
have in the Hilbert space.
(mumbles) So don't confuse that.
It's yeah,
just make sure you always remember that
if states are orthogonal
on the Bloch sphere
that means that they are not
orthogonal in Hilbert space
but only if they are on opposite
sides on the Bloch sphere
that means that they are orthogonal.
So now when,
something that one could
very nicely illustrate
on the Bloch sphere, I think,
is the measurements.
Because we look at Z measurement now.
A Z measurement
corresponds to a projection
onto the z-axis.
So if we have a state,
let's say I put a state here,
okay, I have a state that
shows here to this point,
then,
then the probability to measure this,
if I do a Z measurement,
that means I'm projecting onto the z-axis.
So with a high probability,
I would actually get the output one.
And with a smaller probability,
I would get the output zero.
Sorry, the opposite way.
With a high probability,
it's gonna collapse here to the,
towards the zero state and
with a small probability,
it's gonna collapse to
the one state down here.
Similar also to if I
measure in the plus I minus
I state, then with a...
You can tell from the state here,
with a high probability,
it's gonna collapse to this side
and the small probability
it's gonna go to the plus I state.
So while the C measurement
corresponds to projection onto the z-axis.
So the states zero and one,
which are the two eigenstates
of the Pauli set operator.
We also have the same then for X and Y.
So analogously for X and Y.
If we look at an X measurement,
I told you before,
that's on the plus minus basis,
because the plus and the minus state,
we defined them as the eigenstates
of the Pauli X operator,
and they lie on the x-axis,
as you can tell,
so if I measure on the X measurement,
I'm gonna, the state is gonna collapse to,
onto the x-axis.
And if we do it,
if we do a Y measurement,
we can see the Y,
two eigenstates of the Y operator
all on the y-axis as well.
So yeah, I think one can
nicely illustrate then
on the Bloch sphere,
how these projective
measurement works in this case.
But as I've mentioned in
the beginning as well,
we have infinitely many phases
because we can basically
project onto any given
state plus its orthogonal state.
We can do a projective measurement onto
this X, oops should be in
straight line of course,
onto this axis, for example,
and it should go through the middle.
Let's try again.
We could do a projective measurement
after the state for example,
that would be a possibility
as well, of course.
But yeah, usually we do
C measurements or what
the experiment usually does and then
theoretically, it's
also very common to use
plus minus measurements for example.
Good so now I'm,
think I'm done with the first part.
