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PROFESSOR: Mach-Zehnder--

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interferometers.

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And we have a beam splitter.

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And the beam coming
in, it splits into 2.

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A mirror--

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another mirror.

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The beams are recombined
into another beam splitter.

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And then, 2 beams come out.

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One to a detector d0--

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and a detector d1.

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We could put here any kind
of devices in between.

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We could put a little
piece of glass,

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which is a phase shifter.

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We'll discuss it later.

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But our story is a
story of a photon

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coming in and somehow leaving
through the interferometer.

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And we want to describe this
photon in quantum mechanics.

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And we know that the
way to describe it

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is through a wave function.

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But this photon can live
in either of 2 beams.

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If a photon was
in 1 beam, I could

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have a number that tells me the
probability to be in that beam.

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But now, it can be
in either of 2 beams.

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Therefore, I will
use two numbers.

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And it seems reasonable to
put them in a column vector.

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Two complex numbers that give
me the probability amplitudes--

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for this photon to be somewhere.

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So you could say, oh, look here.

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What is the probability that
we'll find this photon over

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here?

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Well, it may depend on the time.

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I mean, when the photon
is gone, it's gone.

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But when it's crossing here,
what is the probability?

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And I have 2 numbers.

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What is the probability
here, here, here, here?

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And in fact, you could
even have 1 photon

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that is coming in through 2
different channels, as well.

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So I have 2 numbers.

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And I want, now, to do
things in a normalized way.

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So this will be the probability
amplitude to be here.

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This is the probability
amplitude to be down.

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And therefore, the probability
to be in the upper one--

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you do norm squared.

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The probability to
be in the bottom one,

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you do norm squared.

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And you get 1.

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Must get 1.

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So if you write 2 numbers,
they better satisfy that thing.

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Otherwise, you are not
describing probabilities.

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On the other hand, I may have
a state that is like this.

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Alpha-- oh, I'll
mention other states.

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State 1-0 is a photon
in the upper beam.

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No probability to be
in the lower beam.

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And state 0-1 is a
photon in the lower beam.

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So these are states.

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And indeed--

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think of superposition.

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And we have that the
state, alpha beta--

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you know how to
manipulate vectors--

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can be written as alpha 1-0.

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Because the number goes
in and becomes alpha 0.

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Plus beta 0-1.

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So the state, alpha
beta, is a superposition

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with coefficient alpha of
the state in the upper beam

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plus the superposition
with coefficient beta

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of the state in the lower beam.

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We also had this
little device, which

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is called the beam
shifter of face delta.

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If the probability
amplitude completing

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is alpha to the left
of it, it's alpha

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e to the i delta to the right
of it, with delta a real number.

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So this is a pure phase.

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And notice that alpha is
equal to e to the i delta.

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The norm of a complex
number doesn't change when

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you multiply it by a phase.

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The norm of a complex
number times a phase

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is the norm of
the complex number

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times the norm of the phase.

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And the norm of any phase is 1.

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So actually, this doesn't
absorb the photon,

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doesn't generate more photons.

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It preserves the probability
of having a photon there,

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but it changes the phase.

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How does the beam
splitter work, however?

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This is the first thing
we have to model here.

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So here is the beam splitter.

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And you could have
a beam coming--

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A 1-0 beam hitting it.

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So nothing coming from below.

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And something coming from above.

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And then, it would reflect
some and transmit some.

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And here is a 1--

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is the 1 of the 1-0.

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And here's an s and a t.

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Which is to mean that this beam
splitter takes the 1-0 photon

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and makes it into an st photon.

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Because it produces a
beam with s up and t down.

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On the other hand, that
same beam splitter-- now,

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we don't know what those
numbers s and t are.

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That's part of designing
a beam splitter.

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You can ask the
engineer what are s

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and t for the beam splitter.

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But we are going to figure
out what are the constraints.

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Because no engineer would be
able to make a beam splitter

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with arbitrary s and t.

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In particular, you
already see that if 1-0--

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if a photon comes in,
probability conservation,

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there must still be a photon.

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You need that s squared
plus t squared is equal to 1

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because that's a photon state.

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Now, you may also have a
photon coming from below

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and giving you uv.

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So this would be a 0-1
photon, giving you uv.

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And therefore, we
would say that 0-1--

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gives you uv.

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And you would have u plus v
norm squared is equal to 1.

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So we need,
apparently, 4 numbers

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to characterize
the beam splitter.

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And let's see how
we can do that.

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Well, why do we need,
really, 4 numbers?

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Because of linearity.

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So let's explore that
a little more clearly.

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And suppose that I ask you,
what happens to an alpha beta

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state--

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alpha beta state if it
enters a beam splitter?

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What comes out?

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Well, the alpha beta
state, as you know,

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is alpha 1-0 plus beta 0-1.

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And now, we can use our rules.

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Well, this state, the beam
splitter is a linear device.

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So it will give you alpha times
what it makes out of the 1-0.

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But out of the 1-0 gives you st.

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And the beta times 0-1
will give you beta uv.

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So this is alpha s plus
beta u times alpha t

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plus beta v. And I can write
this, actually, as alpha beta

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times the matrix, s u t v.

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And you get a very nice thing,
that the effect of the beam

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splitter on any photon
state, alpha beta,

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is to multiply it by
this matrix, s u t v.

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So this is the beam splitter.

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The beam splitter acts
on any photon state.

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And out comes the matrix
times the photon state.

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This is matrix action,
something that is going

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to be pretty important for us.

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How do we get those numbers?

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After all, the beam
splitter is now

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determined by these
4 numbers and we

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don't have enough information.

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So the manufacturer can tell
you that maybe you've got--

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you bought a balanced
beam splitter.

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Which means that if you have
a beam, half of the intensity

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goes through, half of the
intensity gets reflected.

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That's a balanced beam splitter.

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That simplifies things
because the intensity

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here, the probability,
[INAUDIBLE]

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must be the same as that.

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So each norm squared
must be equal to 1/2,

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if you have a balanced--

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beam splitter.

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And you have s squared equal
t squared equal u squared

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equal v squared equal 1/2.

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But that's still far from enough
to determine s, t, u, and v. So

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rather than determining,
them at this moment,

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might as well do a guess.

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So can it be that the
beam splitter matrix--

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Could it be that the
beam splitter matrix

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is 1 over square root of
2, 1 over square root of 2,

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1 over the square root of 2,
and 1 over square root of 2.

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That certainly satisfies
all of the properties

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we've written before.

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Now, why could it be wrong?

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Because it could be pluses
or minuses or it could be i's

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or anything there.

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But maybe this is right.

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Well, if it is right, the
condition that it be right

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is that, if you take a
photon state, 1 photon--

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after the beam splitter,
you still have 1 photon.

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So conservation of probability.

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So if you act on a
normalized photon

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state that satisfies this
alpha squared plus beta squared

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equal 1, it should still give
you a normalized photon state.

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And it should do
it for any state.

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And presumably, if you get
any numbers that satisfy that,

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some engineer will be able
to build that beam splitter

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for you because it
doesn't contradict

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any physical principle.

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So let's try acting on
this with on this state--

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1 over square root of 2,
1 over square root of 2.

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Let's see.

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This is normalized--
1/2 plus 1/2 is 1.

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So I multiply.

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I get 1/2 plus 1/2 is 1, and 1.

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Sorry, this is not normalized.

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1 squared plus 1
squared is 2, not 1.

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So this can't be
a beam splitter.

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No way.

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We try minus 1 over
square root of 2.

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Actually, if you try this for
a few examples, it will work.

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So how about if we
tried in general.

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So if I try it in general,
acting on alpha beta,

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I would get 1 over square root
of 2 alpha plus beta and alpha

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minus beta.

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Then, I would check
the normalization.

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So I must do norm
of this 1 squared.

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So it's 1/2 alpha
plus beta squared

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plus 1/2 alpha minus
beta norm squared.

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Well, what is this?

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Let me go a little
slow for a second.

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[INAUDIBLE] plus beta star.

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Plus alpha minus beta.

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Alpha star minus beta star.

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Well, the cross terms vanish.

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And alpha alpha star, alpha
alpha star, beta beta star,

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beta beta star add.

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So you do get alpha
squared plus beta squared.

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And that's 1 by
assumption because you

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started with a photon.

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So this works.

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This is a good beam
splitter matrix.

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It does the job.

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So actually--

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Consider this beam splitters.

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00:14:35,890 --> 00:14:39,840
Actually, it's not the
unique solution by all means.

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But we can have 2 beam splitter
that differ a little bit.

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So I'll call beam splitter
1 and beam splitter 2.

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Beam splitter or 1
will have this matrix.

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And beam splitter 2 will have
the matrix were found here,

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which is a 1 1 1 minus 1.

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So both of them work, actually.

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And both of them are
good beam splitters.

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I call this--

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beam splitter 1.

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And this, beam splitter 2.

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And we'll keep that.

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And so we're ready, now, to
think about our experiments

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with the beam splitter.