In quantum information theory, the classical capacity of a quantum channel is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel.
Holevo, Schumacher, and Westmoreland proved the following least upper bound on the classical capacity of any quantum channel
We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate
We first review the minimal amount of quantum mechanics needed for the theorem.
We then cover quantum typicality, and finally we prove the theorem using a recent sequential decoding technique.
In order to prove the HSW coding theorem, we really just need a few basic things from quantum mechanics.
It is the task of the receiver to perform a measurement to determine the input of the sender.
These operators should satisfy positivity and completeness in order to form a valid POVM: The probabilistic interpretation of quantum mechanics states that if someone measures a quantum state
is equal to and the post-measurement state is if the person measuring obtains outcome
These rules are sufficient for us to consider classical communication schemes over cq channels.
The reader can find a good review of this topic in the article about the typical subspace.
is close in expected trace distance to the original state
: The quantum information-theoretic interpretation of the above inequality is that the probability of obtaining outcome
is upper bounded by the probability of obtaining outcome
If the projectors are non-commuting, then Sen's bound is the next best thing and suffices for our purposes here.
We now prove the HSW theorem with Sen's non-commutative union bound.
We divide up the proof into a few parts: codebook generation, POVM construction, and error analysis.
We first describe how Alice and Bob agree on a random choice of code.
Sens' bound from the above lemma suggests a method for Bob to decode a state that Alice transmits.
Bob should first ask "Is the received state in the average typical subspace?"
This is in some sense equivalent to the question, "Is the received codeword the
He can ask these questions operationally by performing the measurements corresponding to the conditionally typical projectors
The reason is that the transmitted codeword lies in the typical subspace on average: where the inequality follows from (\ref{eq:1st-typ-prop}).
codeword correctly under our sequential decoding scheme is equal to where we make the abbreviation
codeword is given by and the average error probability of this scheme is equal to Instead of analyzing the average error probability, we analyze the expectation of the average error probability, where the expectation is with respect to the random choice of code: Our first step is to apply Sen's bound to the above quantity.
But before doing so, we should rewrite the above expression just slightly, by observing that Substituting into (3) (and forgetting about the small
We now focus exclusively on showing that the term inside the square root can be made small.
Consider now the second term and the following chain of inequalities: The first equality follows because the codewords
Putting everything together, we get our final bound on the expectation of the average error probability: Thus, as long as we choose
, there exists a code with vanishing error probability.