Q4L13b.txt

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The entropy of quantum systems has
some peculiar and unexpected behavior
compared with the classical.
We define this entropy the Von Neumann entropy as minus trace
of RHO log RHO for density matrix RHO.
For example, the entropy of a pure state zero is zero,
and the entropy of this fully mixed qubit is one bit.
Evidently, the Von Neumann entropy of RHO
is simply the Shannon entropy of the eigenvalues of RHO.
So what could be surprising about this entropy?
Well, let's look at its properties.
From the classical case, we expect
that the entropy is greater than or equal to zero,
the entropy is bounded above by the size of the Hilbert space
just like the Shannon entropy is bounded
by the size of the alphabet.
And now for some quantum behavior.
If a joint system ab is pure, then the entropy of either half
is going to be equal to each other.
The proof of this is straightforward.
Take a bipartite state PSI expressed
in terms of the basis for a and b
and apply the Schmidt decomposition.
This gives a single coefficient djj and Schmidt basis terms
psi j and phi j for systems a and b.
Note that when either system is then traced over,
the same eigenvalues are obtained and thus
the entropies of the two halves are identical.
A fourth property of the Von Neumann entropy,
which will be useful to us shortly,
is the entropy of a probabilistic combination
of density matrices.
This particular form may result when half of a bipartite state
is measured, giving result k and leaving the other half
in density matrix RHO sub k.
Let us simplify by expressing RHO sub k in its diagonal basis
with elements lambda kj.
The Von Neumann entropy then reduces
to a form which looks a great deal like a Shannon
entropy of a probability distribution
given by p sub k lambda kj.
Let us break up the log of this product
into the sum of the logs.
And thus we have a term with log p sub k
and a second term with log lambda sub kj.
Lambda kj drops out of the first term in this expression,
leaving us with h of p, and the second term
becomes a convex combination of s of RHO sub k.
Please keep this property in mind for the proof
of Holevo's theorem.
Let us turn now to the definition
of mutual information for the quantum case.
We build on the conditional entropy
defined as s of p given q, being s of p q minus s of q.
Be careful, though, because conditional entropy
in the quantum case can be negative.
That is not true in the classical case.
The mutual information between p and q
is given by s of p plus s of q minus s of pq
just as is the case with the classical mutual information
i of xy.
It is very helpful to know some fundamental properties
of this quantum mutual information first.
Just like in the classical case, conditioning reduces entropy.
So s of a given b and c is less than s of a given b.
Discarding a system never increases
the mutual information.
Again, just like in the classical case.
So s of a and b and c is greater than or equal to s of a and b
with no c.
Next, quantum operations can never
increase the mutual information if acting
on just one of the subsystems.
Specifically, for systems p and q,
if the quantum operation acts, say,
only on the q subsystem, here shown as a channel e,
then the mutual information of the original systems p and q
is greater than or equal to the mutual information
between the transformed systems p prime and q prime.
This is true for any trace preserving quantum operation e,
and it makes sense because e cannot increase any correlation
between p and q acting only on the q subsystem.
These properties all have classical analogs,
but let me stress that it is important to be
careful of the differences with the classical.
The Shannon entropy of a single random variable x
is always less than or equal to the Shannon entropy
of a joint between x and y.
This is not true in the quantum case, though.
For example, consider a bipartite quantum state
pq, which is an entangled qubit state, 0 0 plus 1 1.
The Von Neumann entropy of p is equal to the Von Neumann
entropy of q, which is one qubit,
but the Von Neumann entropy of the total system
is 0 because it is a pure state.
The Von Neumann entropy and quantum mutual information
thus have direct classical analogs but also differences
to keep in mind.