Q4L13c.txt

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Let us now discuss and prove Holevo's theorem,
one of the earliest results in quantum information theory.
The scenario here starts with a classical message
X, which is then mapped by some encoding
to a quantum state rho.
And this rho or rho sub X may be an alphabet
with non-orthogonal states.
The states are sent over a noiseless channel-- no noise--
and received, then measured to produce the channel output
Y, a classical variable.
We will model this measurement in the operator sum
representation with operator elements E sub Y.
The reason why this is a noisy communication scenario
is because the source signals may be
non-orthogonal with each other.
Holevo's theorem puts a bound on the maximum rate at which X
can communicate with Y. That is that the mutual information is
bounded above by the entropy of the density matrix
minus the probabilistically weighted combination of the von
Neumann entropy of the signal alphabet states.
Here rho is the sum of P sub X times rho sub X.
Let us explicitly prove Holevo's theorem
because it is very instructive to see
how such quantum information theoretic proofs are
constructed.
We introduce fictitious systems P and M,
helpers to Alice and Bob, alongside the quantum system Q
sent over the channel.
Specifically, the initial state can then
be written as this density matrix over the Hilbert
space of PQM where the first system denotes
Alice's classical message, the second system
denotes the message sent-- Q-- and the third system, M,
holds Bob's measurement result.
The final state of this system may also
be written in the same form as rho P prime Q prime M prime.
Alice's classical message state P remains the same,
Q changes to become the post-measurement state
after Bob's operation with E sub Y,
and M becomes the measurement result Y.
So here we have P prime, Q prime, and M prime.
We observe that the entropy between P and Q
is related to the entropy between P prime
and M prime as follows.
First, this entropy between P and Q
is equal to the entropy between P and Q and M.
And that is at least as large as the entropy between P prime
and Q prime M prime because the measurement cannot increase
mutual information between P prime and Q prime M prime.
We may then discard Q prime because discarding cannot
increase the mutual information between P prime and M prime.
OK.
Now we have an expression linking the mutual information
between P and Q and the mutual information
between P prime and M prime.
The left-hand side of this expression, S of PQ,
can be simplified by noting that the density matrix of P and Q
is given by this labeled stochastic combination, which
we have seen previously.
The quantum mutual information between P and Q
is thus found to be H of P plus S of rho-- remember,
Q is the state rho-- minus the entropy of P sub X
and the convex combination of the entropies of rho sub X
weighted by P sub X.
S of rho is the entropy of the state sent over the channel Q,
and this term here is the result from property number four
from the previous lecture.
The two H of P terms cancel, thus leaving us with S of rho
minus this stochastic combination
on the left-hand side.
For the right-hand side, we note that, given X, the probability
of Y is simply the measurement result probability trace
of rho sub X EY dagger EY.
Also, the reduced density matrix rho of just P prime M prime
is given by tracing over Q. This gives a diagonal density matrix
where we rewrite the measurement probabilities using
the conditional probability expression above.
This output density matrix allows
us to now express the right-hand side of our entropy equation
by direct substitution into the definition
of the quantum mutual information,
namely S of P prime plus S of M prime minus S of P prime M
prime.
We find H of X plus H of Y-- that's
P prime M prime-- minus the joint entropy of P prime M
prime.
And from our density matrix, we see
that is simply the classical Shannon entropy of X and Y.
Thus we find the right-hand side is
no more than the mutual information between X and Y.
Put all together, we now have a proof of Holevo's theorem,
namely that the mutual information is bounded
above by this expression.
For a single qubit, Holevo's theorem
stipulates that the mutual information between X and Y
is at most the entropy of the density matrix, and that is 1.
You will explore more interesting cases in the problem set.