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