Good morning. In today's lecture, we'll
talk about how to describe the quantum
state of a system of two qubits. And in
the process we'll, we'll talk about a very
fundamental concept, that of entanglement,
quantum entanglement. Quantum entanglement
is, is, is probably the most important
concept in, in quantum information,
quantum computation. It's the key resource
that makes exponential speedups possible
in quantum computation. In fact, one way
one can think, describe it is quantum
computation, quantum information, an
exploration of quantum entanglement.
Something that was discovered in the early
days of quantum mechanics and then, not
really pursued too, too much, much depth.
So, entanglement was discovered by.
Einstein Podolsky Rosen, and Einstein later
described it once derisively as spooky
action at a distance. So, entanglement is
a very counter intuitive concept and you
know, in today's lecture and the next
lecture we'll try to come to grips with
this concept. Now, actually there's a part
from Erwin Schrödinger which I found very
interesting because it, you know, this is
from the 1930's, where he already talked
about, he says, I would not call
entanglement one but rather the
characteristic trait of quantum mechanics,
the one that enforces its entire departure
from classical lines of thought. So this
is a remarkably modern viewpoint on
entanglement. Okay, but before we do all
this, I'd like to do a review of what
we've seen so far. So, what we've studied
so far is, what's a qubit. And a model of the
qubit is, well, we've been describing it
in terms of atomic qubits, in terms of the
energy levels of an electron. And we are
thinking of two distinct energy levels,
the ground state and the first excited
state which, if this was a classical
system, we could use to store a bit of
information and quoting zero as a ground
state and one by the excited state. But of
course, in quantum mechanics, the electron
doesn't make up it's mind, whether it's in
the ground or excited state, in general.
And it can be in a superposition of the
ground and excited state where it has some
complex amplitude of being in the ground
state and some other complex amplitude of
being in the excited state. The
interesting thing being of these, these,
these amplitudes are complex numbers and
they're, and they're normalized okay so,
so in general, we understate, α0|0> +
α1|1> where α0 and α1 are complex
numbers. And the sum of the squares of
their magnitude add up to one, so it's
normalized. Okay, the normalization
condition is, is interesting because,
because when we actually go to look and
see which state the electron is, is in, it
quickly makes up it's mind and it ends up
in the ground state with amplitude alpha
zero, magnitude squared. And in the
excited state with amplitude alpha one,
magnitude squared. Moreover, as soon as we
may look, the state of the system is, is
disturbed and the new state is exactly
consistent with what, what we measured.
So, if we were to look at this example
that we had earlier, the probability that
we'd see zero would be one over square
root squared which would be one-half and
then the new state would be exactly the
zero state. And the probability that we
see one would also be half because that
would be one over two squared plus one
over two squared. And the new state would
now be the one state. Okay, so all this
we, we saw last time. We also saw that
there's a geometric interpretation of, of
what it means for, to have the state of
the qubits. So for example, let me just
use real numbers. So let's say that, that
the state of the qubit was
1/√2|0> - √3/2|1>, sorry, that doesn't add up, so,
I should actually, let's say it's
1/2|0> - √3/2|1>. So you can see
it's normalized. Now, the geometric
interpretation says, that the state of
this qubit is a unit vector in a two
dimensional vector space where we have the
axis, the coordinating axis are labeled
with zero and one, the ground and excited
state. And, the state of the system sits
on a, on a unit circle and we can plot it
out here and it, it might look something
like this. So this would be |Ψ>, okay. And
now there's, there's also an
interpretation of, of this in terms, when
we do a measurement. What we're saying is
that, that the state collapses to the zero
state with, probability cos²θ
and it collapses to the one state
with probability, well, cosine square of
the angle it makes with the one step
which, which actually ends up being
sin²Θ. So this is the probability
of |0>, and sin²θ is the
probability of |1>. Okay, now, okay, let,
let me just give you a couple of other
examples of qubits. So it turns out that
photons, particles of light, have a qubit
associated with them, which is called the
polarization which is roughly the
orientation of the electric field
oscillations associated with, with this,
you know, and we can think of these,
these, oscillations as either being
horizontal or vertical, and that
corresponds to qubit. The spin of particle
like an electron is a quantized version of
its, its angular momentum and it, it also
forms a qubit where the spin is either
pointing up or down. So now, how would you
measure these, these qubits? Well, in the
case of photon polarization, this is
particularly easy. It's you know, it's in
terms you do this with the polarizing
filter. Whether filter, depending upon
it's orientation, it might allow photons
that are polarized vertically but not
horizontally. So think of the, you know,
of the photon sort of, moving
horizontally, the polarizing filter is, is
orthogonal to it and it's aligned let's
say, vertically. So then, it only allows
photons through that are polarized
vertically and blocks photons that are
aligned horizontally. But of course, in
general, the, the state of the photon, so,
so let's say this is, this is horizontally
polarized, that's vertically polarized. So
in general, the state of the photon will
be some superposition of vertical and
horizontal. This is going to be α
times horizontal plus, α0 times
horizon, horizontal plus α1 times
vertical. And so, if you, i f you were to
pass such a photon through these filters
then the net effect is it, it gets blocked
with probability |α0|², and it's
transmitted with probability |α1|². If
it's transmitted, then it's new state is
vertically polarized. Okay, so this is
what you see when you, when you, hold up a
polarizing filter. You see photons coming
through, and they are polarized in various
different directions. But of course, you
have the ones that are horizontally
polarized are blocked, ones the vertically
are, are transmitted. The ones that are
polarized diagonally, they are transmitted
with some probability but then what you
see is vertically polarized photons, only.
Okay. Well, there's another question you
could ask which is, what happens if you
orient this filter so that it's not
aligned either vertically or horizontally
but rather diagonally at a 45 degree
angle? Well, it makes sense that what it
would do is allow through those photons
which are at a 45 degree orientation, but
block everything which is at a -45 degree
orientation. So in terms of the qubit
picture, here's what it corresponds to.
What it corresponds to is that, we are now
measuring in this diagonal basis, this is
|+> and this is |->. So now if we
started with a photon in this, in this
particular state, the probability that it
will go through is given by cos²θ, 
where θ is this angle. And the
probability that it's blocked is
sin²θ. Moreover, if it, if it goes
through, then it must be orientated in a
diagonal, at a 45 degree angle. Okay? So
this is how measurement at an arbitrary in
an arbitrary orthogonal basis works. So
this is what it means to measure at an
arbitrary angle. Now, what happens if you,
if you take two of these polarizers and
you align one horizontally and one
vertically? Well, you know, if you look
through it, well, you have lots of light
coming through. Some of it, some of it
passes through the horizontally polarized
one, some of it passes through the
vertically polarized one. But where the
two intersect , you see a dark, dark
patch. And the reason is, simple. The
light that is coming through, it might be
lets say, you know, lets say, we have only
one photon coming through. Well, you know,
it might be polarized diagonally
α0|→> + α1|↑>.
And lets say, this first polarized that
goes through is horizontally, it's
horizontally aligned. So then, with
probability |α0|², it goes
through and now the light is horizontally
polarized. But now it's blocked with
probability one by the second filter. And
so, the net effect is no matter what it,
what it's original orientation, it's
definitely blocked by the two filters in
conjunction. But now you can do, do a very
interesting experiment. What you could do
is, between these two filters, you could
insert a third one which is, which is
aligned at a 45 degree angle. Or π/4
So this is our third filter, which
is inserted between the two. And now what
happens? Well, the interesting thing that
happens is, that this middle patch is no
longer dark. And the reason is, some of
the light that comes in, it goes through
the first filter. But now it's
horizontally polarized. None of it would
get through this, this second filter, but
on the other hand, we have dispersed this
third filter in the middle and that's at
45 degree angle. And so, these
horizontally polarized photons get through
this middle filter with probability half.
Okay, so, so, so through the middle
filter, they get through with probability,
half. And then, the output, the, the
photons that get through are polarized at
a 45 degree angle. Now when they try to
get through the, the second filter, which
was vertically polarized, they are going
to get through with probability half,
because they, they are oriented at 45
degrees to the, to the orientation of this
filter. And of course the new, the
photons, that get through are now
vertically polarized. Okay, so, so that's
it as far as, as far as measurement goes.
Now if you remember, we, we also have this
notion of the uncertainty principle for
qubits where we, we could measure the
qubit either in the bit basis |0>,|1>, or in
the sign basis. So remember this is, this
is our bit basis, |0>, |1>. This is |+> and
|->. |+> is where, where |+> is, and
equals to superposition of |0> and |1>,
and |-> is an opposite superposition of
both |0> and |1>. What the uncertainty
principle tells us is that, we cannot know
both the bit value and the sign value
simultaneously. In other words, suppose we
were to measure in the bit basis, well
then, we know that, that the qubit is
either in the state |0> or in the state
|1>. But now it's maximally uncertain with
respect to the sign basis, and vice versa.
If we were to measure in the sign basis,
it's maximally uncertain with respect to
the bit basis.