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PROFESSOR: Superposition is very
unusual and very interesting.

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Now we've said
about superposition

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that in classical physics, when
we talk about superposition

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we have electric fields, and
you add the electric fields,

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and the total electric field
is the sum of electric fields,

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and it's an electric field.

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And there's nothing
strange about it.

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The nature of superposition
in quantum mechanics

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is very strange.

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So nature of superposition--

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I will illustrate it in a
couple of different ways.

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One way is with a device that
we will get accustomed to.

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It it's called the
Mach-Zehnder interferometer,

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which is a device with
a beam splitter in here.

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You send in a beam of light--

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input- beam splitter
and then the light--

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indeed half of it gets
reflected, half of it

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gets transmitted.

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Then you put the mirror here--

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mirror 1, you put
the mirror 2 here,

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and this gets recombined
into another beam splitter.

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And then if there would
be just a light going in,

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here there would be
two things going out.

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There's another one
coming from the bottom.

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There will be two.

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There will be interference.

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So you put a detector
D0 here and a detector

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E1 here to detect the light.

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So that's the sketch of the
Mach-Zehnder interferometer--

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beam splitters and mirrors.

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Take a beam, spit the light, go
down, up, and then recombine it

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and go into detectors.

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This was invented
by these two people,

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independently, in the 1890s--

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'91 to '92 apparently.

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And people did this with light--

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beams of light before they
realized they're photons.

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And what happens with a beam
of light-- it's interesting--

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comes a beam of light.

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The beam splitter sends half
of the light one way half

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of the light the other way.

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You already know with
quantum mechanics

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that's going to be probabilistic
some photons will go up maybe

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some photons will go down
or something more strange

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can happen.

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If you have a
superposition, some photons

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may go both up and down.

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So that's what can happen
in quantum mechanics.

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If you send the beam,
classical physics,

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it divides half and
half and then combines.

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And there's an
interference effect here.

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And we will design this
interferometer in such a way

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that sometimes we can produce
an interference that everything

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goes to D0 or
everything goes to D1

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or we can produce suitable
interferences that we

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can get fractions of the
power going into D1 and D2--

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D0 and D1.

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So we can do it
in different ways,

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but we should think of
this as a single photon.

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Single photos going
one at a time.

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You see, whatever light you
put in here, experimentally,

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the same frequency
goes out here.

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So what is interference?

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You might think, intuitively,
that interference

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is one photon interfering with
another one, but it can't be.

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If two photos would interfere
in a canceling, destructive

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interference, you will
have a bunch of energy.

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It goes into nothing.

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It's impossible.

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If they would interfere
constructively,

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you would add the
electric fields

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and the amplitude would be
four times as big because it's

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proportional to the square.

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But two photos are not
going to go to four photons.

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It cannot conserve energy.

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So first of all, when you
get light interference,

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each photon is
interfering with itself.

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It sounds crazy, but it's
the only possibility.

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They cannot interfere
with each other.

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You can send the
photons one at a time

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and, therefore, each
photon will have

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to be in both beams
at the same time.

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And then, each photon
as it goes along,

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there will be an
interference effect,

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and the photon may end
up here or end up there

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in a probabilistic way.

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So you have an example
of superposition.

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Superposition.

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A single photon
state a single photon

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is equal to superposition of
a photon in the upper beam

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and a photon in the lower beam.

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It's like two different states--

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a little different
from here, you

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had photons in two
different polarizations

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states superposed.

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Here you have photons
in two different beams--

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a single photon is in both
beams at the same time.

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And unless you have that, you
cannot get a superposition

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and an interference that is
consistent with experiment.

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So what does that mean
for superpositions?

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Well, it means something
that we can discuss,

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and I can say things
that, at this moment,

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may not make too
much sense, but it

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would be a good idea that you
think about them a little bit.

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We associated
states with vectors.

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States and vectors
are the same thing.

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And it so happens that
when you have vectors,

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you can write them as
the sum of other vectors.

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So the sum of these two
vectors may be this vector.

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But you can also write it as
the sum of these two vectors--

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these two vectors
add to the state.

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And you can write any vector
as a sum of different vectors,

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and that's, actually,
quite relevant.

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You will be doing that
during the semester--

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writing a state a superposition
of different things.

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And in that way
you will understand

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the physics of those states.

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So for example, we can
think of two states--

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A and B. And you see, as I
said, states wave functions,

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vectors--

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we're all calling
them the same thing.

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If you have a superposition
of the states A and B,

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what can happen?

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All right, we'll do
it the following way.

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Let's assume if you
measure some property on A,

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you always get value A. So you
measure something-- position,

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momentum, angular momentum,
spin, energy, something--

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on A, it states
that you always get

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A. Suppose you measured the
same property on B. You always

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get B as the value.

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And now suppose you have a
quantum mechanical state,

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and the state is
alpha A plus beta B--

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it's a superposition.

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This is your state.

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You superimpose A and B. And
now you measure that property.

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That same property you
could measure here,

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you measure it in your state.

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The question is,
what will you get?

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You've now superimpose
those states.

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On the first state, you always
get A; on the second state,

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you always get B. What do you
get on the superimposed states,

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where alpha and
beta are numbers--

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complex numbers in general?

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Well the most, perhaps,
immediate guess

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is that you would get
something in-between

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maybe alpha A plus beta B
or an average or something.

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But no, that's not what
happens in quantum mechanics.

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In quantum mechanics, you always
get A or you always get B.

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So you can do the
experiment many times,

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and you will get A many times,
and you may get B many times.

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But you never get
something intermediate.

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So this is very different
than in classical physics.

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If a wave has some
amplitudes and you

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add another wave of
different amplitudes,

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you measure the energy you
get something intermediate.

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Here not!

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You make the superposition and
as you measure you will either

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get the little a
or the little b but

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with different probabilities.

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So roughly speaking, the
probability to get little a

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is proportional to the number in
front of here is alpha squared,

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and the probability
to measure little b

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is proportional to beta squared.

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So in a quantum superposition,
a single measurement

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doesn't yield an average result
or an intermediate result.

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It leads one or the other.

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And this should
connect with this.

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Think of the photon we
were talking about before.

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If you think of the photon
that was at an angle

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alpha in this way, you could say
that the polarizer is measuring

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the polarization of the object.

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And therefore, what
is the possible result

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it may measure the
polarizations say oh, if it's

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in the x direction
you get it right,

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and what is the
probability that you get

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it to be in the x direction is
proportional to cosine squared

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alpha-- the coefficient
here squared.

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So the probability that
you find the photon

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after measuring
in the x direction

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is closer in squared
alpha, and the probability

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that you'll find that here
is sine squared alpha.

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And after you measure,
you get this state which

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is to say the following thing.

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The probability to get the
value A is alpha squared,

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but if you get A,
the state becomes A

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because this whole state
of the system becomes that.

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Because successive measurements
will keep giving you the value

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A. If you get B,
the state becomes

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B. So this is what is called
the postulate of measurement

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and the nature of superposition.

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This is perhaps the
most sophisticated idea

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we've discussed today, in which
in a quantum superposition

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the results are
not intermediate.

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So when you want to figure
out what state you have,

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you have to prepare many
copies of your state

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in this quantum system and
do the experiment many times.

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Because sometimes you'll
get A, sometimes you'll

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get B. After you've
measured many times,

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you can assess the probabilities
and reconstruct the state.