M1L7p.txt

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The first example, which we pretty much covered
in the last class, was an entangled state interferometer.
So instead of having the Mach-Zehnder interferometer
as we had before, we have sort of a Bell state creation
device.
Then we provide the phase shift.
And then we have a Bell analysis device.
And I want to use here the formalism and the symbols
we had introduced earlier.
So just as a reminder, what we need is I need two gates,
one is the Hadamard gate, which in matrix representation
has this matrix form.
And that means if you have a qubit which is either up
or down and you apply the matrix to it,
you put it in a superposition state of up and down.
It's a single cubit rotation.
You can say for the block on the block sphere,
it's a 90-degree rotation.
The second gate we need is the controlled NOT.
And the controlled NOT, we discussed it previously,
can be implemented by having an interferometer
and using a nonlinear Kerr medium coupling another photon
to the interferometer in such a way
that if you have a photon in mode C,
it creates a phase shift to the interferometer.
If there is no photon in mode C, it
does not create an additional phase shift.
And as we have discussed, this can
implement the controlled NOT.

So these are the ingredients.
And at the end of the last class,
I just showed you what those quantum gates can do for us.
If you have two qubits at the input, we apply the--
and we assume they are both in logical zero--
the Hadamard gate makes the coherent superposition.
And then we have a controlled NOT
where this is the target gate.
Well, if this is 0, if the control bit is 0,
the target would stay 0.
So we get 0 0.
If the control bit is 1, the target bit is flipped to one,
so if we 1 1.
So the result is that we have now
created a state 0 0 plus 1 1.
And if we apply a phase shift to all of the photons coming out
on the right-hand side, we get a phase shift which is 2 phi.
So where do you get the idea?
If you take advantage of a state 1 1,
it has twice the phase sensitivity as a single photon,
and we may, therefore, get the full benefit
of the factor of 2, and not just square root 2.
And this is what this discussion is about.

Questions?

OK.
So that's where we ended.
We can now use another controlled NOT
and bring in the third qubit.
So what we recreate here, the state which is either all bits
are 0, all bits are 1.
And then the phase shift gives us three times the phase shift
phi.
And therefore, by bringing in more and more qubits,
I've shown you n equals 1, n equals 2, n equals 3,
so now you can go to n, we obtain states which have
n times the phase sensitivity.
Let me just dimension in passing that for n equals 3,
the superposition of 0 0 0 and 1 1 1
goes by the name, not gigahertz, GHZ,
Greenberger-Horne-Zeilinger state.
And those states play an important role
in test of Bell's inequality.
Or more generally, states which are microscopically distinct
are also regarded cat states, or Schrodinger cat states.
OK.
If we apply a phase shift and go now
through the entangler in reverse,
just the reverse sequence of C NOT gates and Hadamard gates,
then we have all the other n minus-- have the n minus 1
qubits, all the qubits except for the first, reset to 0.
But the first qubit is now in a superposition state
where we have the phase to the power n.
And now we can make a measurement,
and P is now the probability to find a single photon
in the first qubit.
So in other words, our measurement
is exactly the same as we had before,
where we had the normal Mach-Zehnder interferometer.
We put in one photon at a time and we
determined the probability, what is
the photon at one of the outputs of the interferometer.
But the only difference is that we
have now a factor of n in the exponent for the phase shift.
And I want to show you what is caused by this factor of n.
Let me first remind you how we analyzed
the sensitivity of an interferometer
for single photon input before.
So for a moment, set n equals 1 now.
Then you get what we discussed two weeks ago.
The probability just scales as the interferometer,
sinusoidal, cosinusoidal fringes with cosine phi.
If you do measurements with the probability P,
that's a binomial distribution.
The standard deviation, or the square root
of the variance in the binomial distribution is P times 1
minus P. And by inserting P here, we get sine phi over 2.

Derivative of P, which is our signal, with a phase
is given by that.
And then just using error propagation,
the uncertainty in the phase is the uncertainty
in P divided by dp dt.

So then we repeat the whole experiment n times.
And if you have a binomial distribution,
and we do n tosses of our coin with probability P,
we get an m under the square root.
And then we get 1 over square root m.
And this was what I showed you two weeks ago.
The standard shot noise limit.
But now we have this additional factor of n
here, which means that our probability has
a cosine which goes with n phi.
When we take the derivative, we get a factor of n here.
And then dp d phi has the derivative.
But then we have to use the chain rule, which
has another factor of n.
And therefore, we obtain now that the phase sensitivity
goes as 1/n.
And therefore, we reach the Heisenberg limit.
So if you have now your n photon entangler,
the entanglement operation with these n qubits
is special-- being spread or replaced by the entangler,
we have now a sensitivity which goes 1/n.
And then, of course, if you want you
can repeat the experiment n times.
And whenever you repeat an experiment,