Okay. So, one of the advantages of
studying the basic principles of Quantum
Mechanics in terms of qubits is that it
makes things much simpler and you can
actually get to fairly interesting
concepts in quantum physics very quickly.
So, what I'll do is go through a very
simple explanation in the setting of
qubits of the Uncertainty Principle. Okay,
so, what's the Uncertainly Principle? I'm
sure some of you may have heard about it
already. His you know, this was discovered
by Werner Heisenberg, it's called
Heisenberg's Uncertainty Principle, and
the most common usage of it is with
respect to position and velocity. So what
Heisenberg discovered is that, in quantum
mechanics, you cannot really pinpoint both
the position and the velocity of a
particle with perfect position. So if you,
if you know its position perfectly, then,
you, you end up being very, very uncertain
about its velocity. And if you know it's
velocity, then you end up being very
uncertain about it's position. And what
you can do is, you can, you can try to
narrow down both but then there's a limit
to how well you, you, you know each of
them and there's this, and that's this
uncertainty principle. It says, the
uncertainty in the position times the
uncertainty in the velocity is at least
some, some quantity. Okay. So now, what
we're going to do is we're going to study
an analogous principle, uncertainty
principle for qubits, okay? And what this
is going to do is, it's going to
illustrate the basic principles, you know,
why is there an uncertainty principle at
all for position and velocity? Well,
we'll, we'll look at that later in the
semester but for now in the context of
qubits, we are going to look at these two
quantities. So, remember you know, a qubit
is, is the, the state of this electron in
the hydrogen atom around to excite it. And
we might want to measure the bit you know,
whether, whether the, whether the qubit is
in the state zero or one. Of course, the
bit has a physical interpretation, its the
energy of this, it corresponds to the ene
rgy whether its in the ground state or in
excited, whether the energy is low or
high. We could also measure a different
quantity, the sign whether the qubit is in
state plus or minus, okay? These are
strange states of the, of, of this
electron but you could imagine that this
could have some physical meaning. So let's
pretend for the moment that these are two
different physical quantities that we're
interested in, the Bit and the Sign. Okay.
So, we could ask the question, can we know
both the bit and the sign of the qubit
with perfect position? So, what do I mean
by this? So let's say that, we have our
qubit, it's in the, it's in the state psi
which is Alpha0 (zero) + Alpha1 (one). So
now, what its bit value? Well, we know
that if you're going to do a measurement,
it would be in the state zero with
probability Alpha-nought squared and the,
and the state one with probability Alpha
one squared. And so if Alpha-nought and
Alpha one are both non-zero then we'd be
somewhat uncertain about the actual bit
value. So the only way to be perfectly
sure about the bit value is if either
Alpha0 = one or if Alpha1 = one. So there
are only two possible states of this, of
this qubit where we'd be perfectly certain
about the bit value. If it's either in the
state zero or in the state one, okay? So
this is for the bit value. Of the qubit.
But now, I suppose we were interested in
the sign. Well then, we could write psi as
B0(+) + B1(-). And now, if you we are to
again measure, we'd see it in the plus
state that's probability Beta-nought
squared and minus with probability Beta
one squared. Again, if we want to be
certain about the, about the sign value,
the only way this could be is if
Beta-nought is one or if Beta one is one
which is the same thing as saying, if it's
either in the state plus or in the state
minus. But now, what if the qubit person,
the state plus? How certain would be,
we'll be about the bit value? Well the
answer is, if a person state plus, then we
are maximally uncertain about the bit
value because if we were to measure it on
the bit basis, we'll get 50, 50, zero and
one. And so, what this does is that the
bit basis and the sign basis are
incompatible with each other. If you are
certain about one then you're maximally
uncertain about the other and vice versa.
And this is the origin of the uncertainty
principle. It turns out that the bit and
the sign basis are for each transforms of
each other. This is something we'll, we'll
talk about later. So, you know, it's just
to give you a heads up. And in a similar
way, the position on the velocity for a
transforms of each other. And so, the
uncertainty principle just comes from the
fact that these two basis incompatible of
each other, you know, they are maximally
far from each other. Okay, so here's a way
you can actually quantify this. So, we can
define the spread of a quantum state. So,
if a state psi is written like this and
the standard basis like this and the, and
the sign basis. And now let's define the
spread in the standard basis to be the
absolute value of Alpha-nought plus the
absolute value of Alpha one and similarly
the spread in the sign basis is just the
absolute value of Beta-nought + B1. So now
let's, let's look at the spread for the
basis state zero. Well in this case,
Alpha-nought is one and Alpha one is zero
and so this spread is, is one. We don't
know how it spread. If you look at the, at
the plus state, its spread is, well, the
plus state is one over √2 one over √2
so it's spread is one over √2 + one over
√2 which is √2. So its, it has a large
spread. On the other hand, if you were to
look at the spread in this, in the sign
basis, then the, the spread of the zero
state in the sign basis, well in the sign
basis the zero state looks like one over
√2 one over √2. And so, its spread is,
is large it's √2. Whereas in the sign
basis, the spread of the plus state, well
that's just Beta ≠ one Beta one = zero,
so its spread is just one. So, what this
shows is that, what you can do is you can
try to make the, the spread and the
standard basis small say by using you
know, the zero state b ut then the spread
in the sign basis becomes large. It
becomes as large as √2. On the other
hand, you can try to make the spreads
small in the, in the sign basis but then
it becomes larger than the standard basis.
And in fact, this spread is always between
one and √2. And so what you can show, is
that no matter what state you pick, no
matter what Alpha-nought , Alpha one are,
Beta-nought, Beta one, the spread in the
standard basis times the spread in the, in
the sign basis is always at least √2 for
any state sign. Okay, so this shows that
there's, there's, there's going to be
uncertainty. No matter what you do, you
can just trade off the uncertainty in the,
in the bit and the sign. So, that's a
very, very simple illustration of the
uncertainty principle in this context of