Q1L2c.txt

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Let us now turn to a formal definition
of what a quantum operation is.
Recall that our goal is to describe
in general what a legal map is for rho going to E of rho,
not just including unitary operations.
We find that a map E is a valid quantum operation if
and only if three criteria are met.
First, the output density matrix must have a trace of one.
Second, E must be a linear and convex operation
that is acting on a statistical mixture of density matrix
inputs.
It must also produce a statistical mixture
and do so linearly.
These two constraints are very natural from the standpoint
of quantum mechanics being a linear matrix theory and also
that the output must be a density matrix.
So therefore, the map must be trace preserving.
A third criterion has two parts and that
is that the output density matrix must also be positive.
It is natural to require that the output of the map
be positive.
But there is more.
It turns out that one must also allow for the output
to be positive, even when the map is only operating
on part of the system.
Suppose we have a reference and a quantum system R
and Q. Then if E only acts on Q, then the output still
must be positive, even though nothing
was done to the reference part of the system.
This concept is known as complete positivity.
And it is much more general than just positivity.
The complete positivity requirement
can be understood and appreciated
by looking at the quantum circuit representation
of this criterion.
The initial density matrix has E acting on one part, namely
the quantum part.
Whereas the reference part has nothing acting on it.
Despite E only acting on this subsystem of the density
matrix, it must still produce a valid output density matrix.
And thus this output state must be positive.
We can further understand this criterion
by looking at a specific example.
Consider this particular map which acts on a single qubit,
say abcd, and it transforms that qubit
by transposing the off-diagonal elements.
It is straightforward to see that a positive density
matrix is transformed to a positive density matrix
by this transpose map.
However, if we look at the operation of this map on a 2
qubit system where the first qubit has nothing happening to it,
we may then, by looking at the 4 by 4 matrix which corresponds
to this transform I tensored with E,
see that the operation of the trace acting
on the second qubit flips these off diagonal elements
in each local quadrant of the 4 by 4 matrix.
This is known as a partial transpose operation.
Now, let us apply this partial transpose operation
to a specific density matrix, which is this 1001, 0000, 0000,
1001 state.
That is, the density matrix for the two qubit
state 0 0 plus 1 1 normalized.
The partial transpose produces this output density matrix
state.
The density matrix just has 1 and 0s,
and in fact, looks tantalizingly legal and has a trace of 1.
However, in fact, it is not positive.
It is an illegal state, not a density matrix.
And that is because the inner block matrix
has an eigenvalue of minus 1.
The transpose is thus a non-physical operation.
But it is still mathematical useful
as a negative partial transpose test for entanglement.