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.