Q1L2d.txt

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The operator sum representation is
one of the most important tools in quantum error correction.
This theorem connects what we have
done with the system environment model and the quantum
operations formalism, which we just saw.
Recall that the system environment interaction led
to a epsilon map defined using operation elements as follows,
with one constraint that is the square sum of the operation
elements is identity.
The theorem states that this map is a valid quantum operation.
I encourage you to verify it for yourself.
Thus what we may do is to employ this representation of quantum
operations for all quantum operations, that
is, all black box operations or maps taking rho to rho prime.
And there will exist a set of quantum operation elements
to describe any such black box transformation.
We do not prove that here, but it is fairly straightforward.
Furthermore, there also exists by dint
of the system environment model-- which
we considered-- a system environment model
for any quantum operation.
Recall that what this means is that any quantum operation may
be represented by the input state interacting unitarily
with an environment, which may start in its pure state,
initial state, and then the environment
is measured producing the resulting output density matrix
rho prime.
Due to the fact that this environment may
be measured in any basis, the operation elements
are not necessarily unique.
This is an important degree of freedom for our exploration
of quantum error correction.
Let us consider some examples of the operator sum
representation.
A very simple example is the simple unitary transform.
Here the operation elements are clearly just U.
A second example is measurement.
Consider a projective measurement,
where the measurement operators are the projectors, P sub k.
The operator sum representation of this measurement operation
simply has operation elements given by the projector.
Notice, by the way, the projectors
conjugate to produce themselves.
A third example is given by this quantum circuit.
And this will be the first illustrative example
of a system environment interaction
model, which is very physically meaningful in our discussion.
So the black box contains two qubits
interacting via a controlled rotation operator.
The second qubit is our environment.
The first qubit is our system, which produces
E of rho as the output.
What is the quantum operation described by this?
Recall that the rotation operation acting
on a single qubit is a 2 by 2 matrix, which is
a rotation around the y-axis.
The elements are thus real valued,
and are given by cosines and sines of
theta divided by 2.
For simplicity of notation, let lambda equal sine squared
theta divided by 2.
If we denote this box by U, then we may write the transform.
Again, this is a two qubit transform as a four
by four matrix.
This four by four matrix has basis states
given by the environment and system labels,
where I am putting the environment label first
and the system label second.
The matrix elements may be understood
by realizing when the system is 0,
then nothing happens to the environment.
That is what happens in this quadrant where s is equal to 0,
the identity matrix appears.
In the quadrant where the system is 1,
we find that the target, the environment qubit
is rotated by the Ry operation.
Here I write it in terms of lambda simply for conciseness.
The rest of the terms are off diagonal,
and thus are simply going to be 0's.
So let us fill those in.
That gives us the four by four unitary matrix representing
this black box operation.
Given this unitary operation, our next natural step
is to write down the operation elements
E0 and E1 corresponding to the two possible measurement
outcomes of the environment qubit.
Recall from our study of the system environment model
that the operation element E sub k
is defined as the matrix element of U between the environment
state e sub k and the initial environment state.
E0 is therefore the matrix element of U between 0 and 0,
that is, the environment qubit does not change.
Remember this is not a number, but rather an operator which
acts on the system qubit.
And we can read out what happens by looking at all the matrix
elements of U which have 0 and 0 for the environment qubit,
and that gives us this upper left hand corner block of U.
Copying these, we find that the operation on the system qubit
is this matrix, 1 0 0 square root of 1 minus lambda.
Similarly, we may find E1.
And this is the matrix element of U
when e changes between 0 and 1.
That is this bottom left corner of U.
And reading off from this submatrix,
we find that E1 is the operation 0 0 0 square root of lambda
acting on the system qubit.
It turns out that this operation is
an important physical process known as phase damping.
We may see why that is true by applying this quantum
operation to an input density matrix, which is arbitrary,
a b b-star, c.
E of rho, the quantum operation acting on this input state
is given by a sum over two terms, one with E0
and the other with E1 as the operation element.
Let's do this step-by-step by multiplying the first two
matrices in each of the elements of the sum first.
For E0 times rho, we find this matrix.
And then we're left over with E0 dagger on the right.
And for the second term in the sum
we obtain this matrix, with E1 dagger on the right.
Completing this little bit of matrix algebra,
we find that the output density matrix is simply
the input density matrix with the off-diagonal terms
multiplied by square root of 1 minus lambda.
Moreover, if square root of 1 minus lambda
is an exponentially decaying function of time--
which it often is in real practice--
then we find that the off-diagonal elements
decay with a rate which we call T2, the phase damping time.
So this is a very interesting example.
Let us build on this example with a similar quantum circuit
and see how the operation elements change.
So recall that the phase damping black box quantum circuit
had a controlled rotation about the y-axis,
happening from the system to the environment.
Let us add to the circuit a controlled NOT gate, which
happens between the environment and the system
after the controlled rotation.
So both the controlled rotation by y, and the controlled NOT
operations are in this black box.
Let us now write the operation elements.
For E0, we find the same thing as we obtained for phase
damping because the environment does not change, only
when the system qubit is in the 0 state.
And when the environment is in the 0 state,
the second controlled NOT does nothing.
So E0 is the same as it was in the phase damping case.
For E1, the operation element is the same
as it was for phase damping except that a X operation,
a NOT gate has to happen to the system
after the phase damping E1 operation element.
That is because the controlled NOT
acts due to the environment being in the 1 state.
Therefore E1 can be read off simply
as being 0, square root of lambda, 0 0.
This quantum operation is known as amplitude damping.
It reflects the physical process of energy loss from the system.
We can see that by doing the same thing we did with phase
damping-- namely apply this quantum
operation to an arbitrary single qubit input density matrix--
the result is that the output of this quantum operation
is this transformed density matrix state.
Recall that a plus c is equal to 1.
And therefore, we can write the diagonal elements just
with terms with c.
The off-diagonal terms are just as we found for phase damping.
The diagonal terms now also change.
And the diagonal terms have a very meaningful change. In terms
of an exponential for lambda,
now using T1 instead of T2, what we find
is that the diagonal elements decay
such that the left top element goes to 1,
and the bottom right element goes to 0.
That is the qubit in the limit of infinite time decays away.
Also the off-diagonal terms decay.
Now that is phase damping.
The diagonal term decay is the loss of energy from the system.
That is the amplitude decay term of this process.
Note also, that the off-diagonal terms
decay at twice the rate of the diagonal terms.
That is, the effective T2 is 2 times T1.
This process is the physical process
of loss of energy from a qubit.
We have thus modeled phase damping and amplitude damping,
two fundamental physical processes causing errors
to qubits using the operator sum representation.
We will next see how we may reinterpret
these representations in terms of errors, so that we may correct them with quantum error correction.