And then the rules of the, of the game are going to be completely analogous. So, the
outcome is going to be you with probability. Let's call this angle theta
prime. So, so, the outcome may be u with probability. Cosine squared theta prime
and [inaudible] with probability. Sine squared theta prime. And of course the
result of the measurement is that, if the outcome is U, then the new state is in
fact this state U and in the other case the outcome is the new state is really U.
So, here's another way of saying, of saying this whole thing. So, let's say
that u was. And beta zero, zero + beta one, one. So [inaudible] might be - beta
ten + beta 01. So these are orthogonal states. And now. The probability. You have
this outcome cosine squared theta prime. This is exactly. The inner product between
the state psi and u square. So if we were to write then a product this way, it's
inner product of u. And sine. Squared. And this is in the product that we [inaudible]
and sine squared. Okay, so let's see some examples and let's see how to work with
all this. So. Let's first talk about a very special basis called the Hadamard
Basis. This is this is a very important basis. It consist of, of these two states,
the plus state which is this super position of zero and one with, with equal
magnitudes one over square root of two, zero + one over square root of one. And
the minus state which is. One / square root of zero - one / square root of one.
So, what distinguished is that these two states is the, is the sign, the phase here
in front of the, in front of the one. So now you could ask. Can we really, can we
distinguish this plus state from the minus state through a measurement. Also, let's
do a measurement in the 0,1 basis, in the standard basis so if you do a measurement
in the standard basis, we end up saying, and if he, if he stop in the state, state
plus. Then you see zero with probability one half and one with probability one
half. In this case, the new state is zero. In this case the new state is one. What
about if we start with the mindless state. Well once again, we see zero with
probability. Square of this which is one-half and one with probability square
of this of the phase goes away with probability one-half. And again in each of
these cases the new status, zero, one. So, so the results are completely identical
when we do a measurement. And the reason is clear, it's because, because the
probability is determined by the square of the amplitude. And so whether it's
positive or negative, it makes no difference to the outcome. So, standing
there we can actually distinguish plus. The plus state from the minus state. Well,
so. Here's an idea, why not measure in a different basis, so measure. In a
different basis and some [inaudible] basis but which basis would you pick? Well, it's
clear, right? If you want to distinguish plus or minus, why not measure it in the
plus, minus basis. That's a perfectly good basis. And so, if you are mentioning the
+,,,, - basis and you start with the plus state, what do you see? Well, the outcome
is plus with probability one minus the probability zero. And on this case a new
status plus, well this never happens but if it were to happen the new state would
be minus. If you start with the minus state, you see plus with probability.
Well, zero. And minus the probability of one. So, you end up distinguishing these
two completely, it's probability one. You know for sure if you are measuring the
plus minus basis and you start with the plus state, you always get plus and if
it's other than minus state, you always get [inaudible] of minus. Okay. But now,
suppose we start from a general state psi and you want to measure it in the plus,
minus basis. How do you figure out what's the probability you, you see plus and
what's probability you see minus? So this is this is an exercise in linear algebra
and we are going to go through it just to make sure that, that you remember your
linear algebra. And also, when we go through it. You know, we are actually
going to be doing this in this [inaudible] notation. S o, so, not only the review of,
of Linear Algebra for you but it will also be a way to familiarize yourself with this
[inaudible] notation. Okay, so, so, here's a picture. There's, this is the standard
basis, the 0,1 basis, has a +,,,, - basis to make my four angle. Okay? Here's,
here's plus in terms of zero and one and so on. And now what we want to do is, we
are given some state psi. Let's say it happens to be. You know, it, it happens to
be a state psi which makes 60 degree angle. So this is what psi looks like.
It's one-half zero + square root three over two, one. And now we want to measure
the state in the plus, minus spaces. So, how do we figure out what's the
probability of seeing the outcome plus? Okay. So, one way to compute is, is just
by computing in a product, right? Remember, the probability that [inaudible]
plus is the square in the product over that psi makes with plus. So we write it
out, size this state pluses that state and we take the inner product. Take the square
off of the magnitude so when you taking off products you get one-half times one
over square root two + square root 3/2 times one over square root two. And so you
add that, take the square and you get. Similarly, you can carry out and disperses
also for probability of minus, it's the inner product that sine make at the minus
state squared. But you can also, you know, you can work it out exactly like this so
you can, you can also say that this is. The probability of plus which is just.
Four - two square root three = eight. Okay. So, that's one way of computing
these, these probabilities. Here's, here's another we, we can do with this. So, what
we can do is, we can do a change of bases. So to figure out what the state si. What's
the probability of plus. Minus when we measure psi to +,,,, - basis, what we'll
do is first, we'll rewrite psi in the +,,,, - basis. So, we want to express it
as beta one+ but beat 1- and then, if you are doing the measurement in the +,,,, -
basis, we can just [inaudible] it off, the probability of + is beta bl magnitude
squared and probability of - is beta one [inaudible] t squared. Okay, so how do we
do this, this change of basis. Well, so, what we want to do is, we want to rewrite
the zero state in terms of plus and minus and the one state in terms of plus and
minus. So you can do that either by inspection, by seeing either zero must be.
Equals to position of + and - or you can just, just solve, right. If you add these
two together you will get the +,,,, +,,, -. Well, these two vectors will cancel. =
two over square root two zero. Or zero is. Early you're just transposing. You get
one / square root of two+ + one / square root of 2- and similarly, if you subtract
these two. You get square root two x one and so one as square of two x plus, minus,
minus. Okay, so now, we'll just go back and substitute this in, in here. So, psi
which is, which is this super position, we just substitute for zero as being on
number square of two+ + one / two squared- and similarly for one. And then we
simplify so we gather the + together and the - together. And so this is beta
[inaudible] that's beta one and now the probability that we see a plus is the
outcome of the measurement is plus is just beta [inaudible] squared magnitude squared
which again gives us the same values before. Okay. So now let's extend our
understanding of the measurement principle to doing a measurement in an arbitrary
basis. So, so let's, let's go back and review what we had. So. Again we are
thinking about a two level system which is you know, the state of an electron of
bonded energy so it's either in this ground state or excited state. Represented
by zero and one so, okay so, so in general, the state of the system psi. Is
some super position, alpha zero, zero + alpha one, one. And we already saw that we
can view this as, as a vector and a two-dimensional vector space. It's a
complex vector space but I'll, for convenience and drawing it out, I'll
pretend it's all real. So this is zero and one and now I'll state psi. Is, is just
some vector like t his. Okay. And we also saw what it meant to do a measurement. So
when we do a measurement, this state collapses into either the zero state or
the one state, either see the ground state or the excited state with probability
cosine square theta. Theta is the angle that psi makes with, with the zero state.
Okay and the new state is either the ground state or the excited state. But now
in general, what we can do is we can measure not just to see that the state is
a ground state or excited state. We can pick any orthogonal pieces for this, for
this space so in other words what we can do is we can pick. Some basis consisting
of, of the state u. [inaudible] And, and we can do a measurement in this arbitrary
basis [inaudible].