U1L5a.txt

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Last time, we talked about tensor products.
And we've also talked about measurement.
What we haven't talked about is how measurement
combines with tensor products--
how to combine measurement and tensor products.

And that's essentially what I'll be talking about today
and also going into measurement in more detail.

So suppose we have this state 1 over root 3, 0, 0
plus 0, 1 plus 1, 1.

So this is an entangled state.

So there's two qubits here.
And we know what happens if we measure it in the basis 0, 0,
0, 1, 1, 0 1, 1.
We get 0, 0 with probability 1 over root 3 squared.
That's 1/3.
We get 0, 1, probably same, 1 over root 3 squared.

And there's something I think--
I don't remember whether I told you or not.
But after the measurement, the system
acts as if it had been in the measured state all along.

Now, some kinds of quantum mechanical measurements,
you know, depending on the experiment,
will actually destroy the state.
And some of them won't.
But if you make a measurement, you cannot tell--
and you get 0, 0--
you cannot tell whether you were in this state to begin with,
and you happened to measure 0, 0 with probability 1/3,
or whether you've just been in the 0, 0 state to begin with,
in which case the probability of getting that measurement would
be 1.
And you would still be in the same--
well, so the traditional description
of the measurement as your measurement
projects you onto the stage.
OK, so we know how to measure in this basis.
Suppose we just measure the first qubit.

What happens?
Well, I want to say we can rewrite this as equal to 0,
root 2/3 or rather root 1/3 0, 0, 0 plus 1 plus 1,
say root 1/3, 1.

So if we measure 0, the remaining qubits
should be in 0, 0 or rather in 0 plus 1.
And if we measure 1, the remaining qubit will be in 1.

So 0 probability, well, that's going to be probably 2/3.
And qubit b will be in 0 plus 1, 1 over root 2.

And with probably 1/3, we will measure the first qubit
in the state 1.
And the second qubit will be in the state 1.

So probably, I need to describe this further.

Actually, there are a bunch of different ways
of describing this.
And we'll go through several of them today.
But one way of doing it is measure--

OK, you have a combined system.
So there are two pieces of your system A, B. One way of doing--
and want to measure in let's say U e1, e2, through e sub d.

You want to measure A in the basis e0 through ed minus 1.

So you can write--

so state cal it psi A, B, you can write psi
is equal to sum j equals 0 through d minus 1 e sub j
A, v sub j B. And here, these are not going to be normalized.
Because over here, if you look at this,
these two things are not normalized.

So what you get--
e sub k with probability v sub k squared.

And system B is in state v sub k over v sub k.

So we get the result k with probably v sub k squared.

So yeah, and system B is in state--
is in the state v sub k when you divide by its length
to make it a unit vector.

So I should say this is the measurement result.
Is this clear?

Well, over here, we could have measured the second qubit--

qubit.
And we would get 0b with probability 1/3.
And then you would get--
and state system A would be in state 0.

And you would get 1 sub b with probability 2/3.
And A would then be in 0 plus 1.

If you take this thing and measure the second qubit,
you will get exactly the same probabilities here.
So it doesn't matter whether you made your system
B first or system A first.
What will happen is that you will get the same results
as if you'd measured both of them at the same time.
And this should be clear because here you
know you get 0 sub A probability of 1/2 and 0
sub B probability 1/2.
And when you multiply 2/3 by 1/2, you get 1/3.