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Okay. So now let's, let's, let's start
with studying what are the allowable

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states of a 2 qubits system? So as
before, what we are going to do is

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consider two atomic qubits which means
that, let's say, we have two hydrogen

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atoms and we are, we are thinking of the,
the states of the electron. In fact, two

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particular states, the ground state and
the excited state which we used to encode

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for zero and one. And so now since we have
two such atoms, the two electrons can be

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used to represent, classically, two bits
of information. So the possible states are

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|00>, |01>, |10>, and |11>. What about
quantumly? Well the superposition axiom

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tells us that, the state is in general, a
superposition of all these four

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possibilities, that the system doesn't
make up its mind which of these four

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states its in. And so, you have a complex
amplitude for each of these four possible

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classical states. Say for example, α00 is 
the complex amplitude of |00>. And as

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before, we have this normalization
condition. The sum of the squares of the

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magnitudes of these amplitudes adds up to
1. The normalization condition is

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important because it tells us what happens
during a measurement. So if you actually

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look and see what the state of the system
is, you will see each of these four

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possibilities with probability the square
of the magnitude of that corresponding

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amplitude. Moreover, if you happen to
observe |00>, then the new state of the

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system will exactly be |00>. So, let's do an
example. So suppose for example that the,

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that the state of the system was
1/√2|00> + <i>i<i>/2|01> - 1/2|11>. So</i></i>

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you can check that this is a valid state
of the system because it satisfies this

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normalization condition. So now what
happens when you measure this system?

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Well, what happens is, that you see |00>
that both of the electrons are in the

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ground state with probability. So that the
probability that you see |00> is 1/2,

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the square of this. And then the new
state is |00>. The probability that you see

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|01> is a 1/4, and then the new state
is |01>, etc.

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Okay. But now you could ask a different 
question. What happens if instead if you measure

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only the first qubit? What's the result of
that measurement? Okay. So, let's try to

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understand that. So again we, we start
with some quantum state of the two qubits,

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and let's say we have our same example
that, that |Ψ> happens to be 1/√2|00> +

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<i>i<i>/2|01> - 1/2|11>. So what happens if
you, if you measure just the, the first</i></i>

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qubit? What's the probability you see
zero? So, so the answer is, the

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probability that you see zero for the
first qubit is exactly the same as the

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probability that you would see a zero if
you measured both qubits. Okay. So if you

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measured both qubits, what's the
probability you'd see a zero on the first

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qubit? Well, you would only see a zero if
you had this outcome, or this outcome.

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Well this happens with probability
|α00|², and this happens with

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probability |α01|². And so that's the,
that's the chance you see a zero. So in

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this example, it would be 1/2 +
1/4 = 3/4. Now what's the

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new state if you happen to see a zero?
Well, the answer is, is, is really quite

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elegant. What you do is, if you happen to
see a zero then what, what you end up

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doing is discarding all the parts of the
superposition which are incompatible with

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the answer. So you discard these parts of
the superposition. And so what you are

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left with is this superposition but it's
not normalized, so you just normalize it.

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So the new state will be α00|00> +
α01|01> / √(|α00|² + |α01|²). So now

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what, what you should do is check that the
amplitudes, well the amplitude for zero

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is, is what? Well the amplitude for zero
is, is this fraction. What you should do

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is check that the square of the magnitude
of this, plus the square of the magnitude

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of the amplitude for 01, they add up to
one, okay? This is something that's easy

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to check. So going back to our example,
what's the new state if you happen to see

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a zero for the first qubit, when you
measure only the first qubit? Well, we

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cross off this and then we'd renormalize.
So we'd get 1/√2|00> + i/2|01>, and then

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you'd have to divide this by the square
root of this squared magnitude plus this

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squared magnitude. Right? What's the
magnitude of this square plus this

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squared? It's 3/4, okay? So
that's the answer. Which you can further

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simplify, but we are more concerned here
with, the actual logic of the situation.