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