Q1L2e.txt

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Recall that a density matrix has an infinite number
of possible unravelings into stochastic mixtures
of pure states.
It turns out that, fascinatingly,
quantum operations also have an infinite number of unravelings
into different operation elements representing
the same quantum operation.
Why?
What is the physical intuition behind this infinity
of possible choices of quantum operation representations,
and how might we exploit this, for example,
for quantum error correction?
Recall the quantum circuit we employed for the system
environment model.
We have a system and an environment
going into a black-box unitary operation.
The output is our E of rho, and we usually
measure the environment immediately.
But, in fact, the environment may be measured
after a change of basis.
That may be implemented by a unitary operation shown here
as U prime.
Moreover, it should be evident that the resulting map should
not change when the basis of the measurement of the environment
changes.
Thus, we have the following lemma-- if the operation
element set E sub k describes a valid operator sum
representation E, then a different operation element
set F, related to E by a unitary transform,
will also describe the same quantum operation.
We may show that this is true simply
by writing out the operator sum representation of the quantum
operation using elements F. That is, F rho F dagger.
And now, let us substitute in the relationship with E. So,
we have a unitary transform of E times rho
times the daggered version of this.
So, we have E dagger k prime, U complex conjugated.
Here, we have U dagger U, and that simply
is the identity represented here as a Dirac delta
function over k and k prime.
That causes the sum to keep only the diagonal terms of the sum,
and thus, the result is the same operation, E, acting on rho.
This unitary degree of freedom in
the operator-sum representation is explored further
in the problem set.
It turns out to be crucial in the insight which
allows phase damping to be turned from a contiguous
process into one in which one has,
with some probability, nothing happening to the system.
That is, alpha I. Or, with probability 1 minus alpha,
a complete phase flip of the qubit,
namely, a Pauli Z operation.
This is a natural interpretation made for errors and the formalism of error correction.