Okay. So now let's, let's, let's start
with studying what are the allowable
states of a 2 qubits system? So as
before, what we are going to do is
consider two atomic qubits which means
that, let's say, we have two hydrogen
atoms and we are, we are thinking of the,
the states of the electron. In fact, two
particular states, the ground state and
the excited state which we used to encode
for zero and one. And so now since we have
two such atoms, the two electrons can be
used to represent, classically, two bits
of information. So the possible states are
|00>, |01>, |10>, and |11>. What about
quantumly? Well the superposition axiom
tells us that, the state is in general, a
superposition of all these four
possibilities, that the system doesn't
make up its mind which of these four
states its in. And so, you have a complex
amplitude for each of these four possible
classical states. Say for example, α00 is 
the complex amplitude of |00>. And as
before, we have this normalization
condition. The sum of the squares of the
magnitudes of these amplitudes adds up to
1. The normalization condition is
important because it tells us what happens
during a measurement. So if you actually
look and see what the state of the system
is, you will see each of these four
possibilities with probability the square
of the magnitude of that corresponding
amplitude. Moreover, if you happen to
observe |00>, then the new state of the
system will exactly be |00>. So, let's do an
example. So suppose for example that the,
that the state of the system was
1/√2|00> + <i>i<i>/2|01> - 1/2|11>. So</i></i>
you can check that this is a valid state
of the system because it satisfies this
normalization condition. So now what
happens when you measure this system?
Well, what happens is, that you see |00>
that both of the electrons are in the
ground state with probability. So that the
probability that you see |00> is 1/2,
the square of this. And then the new
state is |00>. The probability that you see
|01> is a 1/4, and then the new state
is |01>, etc.
Okay. But now you could ask a different 
question. What happens if instead if you measure
only the first qubit? What's the result of
that measurement? Okay. So, let's try to
understand that. So again we, we start
with some quantum state of the two qubits,
and let's say we have our same example
that, that |Ψ> happens to be 1/√2|00> +
<i>i<i>/2|01> - 1/2|11>. So what happens if
you, if you measure just the, the first</i></i>
qubit? What's the probability you see
zero? So, so the answer is, the
probability that you see zero for the
first qubit is exactly the same as the
probability that you would see a zero if
you measured both qubits. Okay. So if you
measured both qubits, what's the
probability you'd see a zero on the first
qubit? Well, you would only see a zero if
you had this outcome, or this outcome.
Well this happens with probability
|α00|², and this happens with
probability |α01|². And so that's the,
that's the chance you see a zero. So in
this example, it would be 1/2 +
1/4 = 3/4. Now what's the
new state if you happen to see a zero?
Well, the answer is, is, is really quite
elegant. What you do is, if you happen to
see a zero then what, what you end up
doing is discarding all the parts of the
superposition which are incompatible with
the answer. So you discard these parts of
the superposition. And so what you are
left with is this superposition but it's
not normalized, so you just normalize it.
So the new state will be α00|00> +
α01|01> / √(|α00|² + |α01|²). So now
what, what you should do is check that the
amplitudes, well the amplitude for zero
is, is what? Well the amplitude for zero
is, is this fraction. What you should do
is check that the square of the magnitude
of this, plus the square of the magnitude
of the amplitude for 01, they add up to
one, okay? This is something that's easy
to check. So going back to our example,
what's the new state if you happen to see
a zero for the first qubit, when you
measure only the first qubit? Well, we
cross off this and then we'd renormalize.
So we'd get 1/√2|00> + i/2|01>, and then
you'd have to divide this by the square
root of this squared magnitude plus this
squared magnitude. Right? What's the
magnitude of this square plus this
squared? It's 3/4, okay? So
that's the answer. Which you can further
simplify, but we are more concerned here
with, the actual logic of the situation.