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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 :
where is a classical-quantum state of the following form:
is a probability distribution, and each is a density operator that can be input to the channel .
Achievability using sequential decoding
We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate for communicating classical data over a quantum channel). 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.
Review of quantum mechanics
In order to prove the HSW coding theorem, we really just need a few basic things from quantum mechanics. First, a quantum state is a unit trace, positive operator known as a density operator. Usually, we denote it by , , , etc. The simplest model for a quantum channel is known as a classical-quantum channel:
The meaning of the above notation is that inputting the classical letter at the transmitting end leads to a quantum state at the receiving end. It is the task of the receiver to perform a measurement to determine the input of the sender. If it is true that the states are perfectly distinguishable from one another (i.e., if they have orthogonal supports such that for ), then the channel is a noiseless channel. We are interested in situations for which this is not the case. If it is true that the states all commute with one another, then this is effectively identical to the situation for a classical channel, so we are also not interested in these situations. So, the situation in which we are interested is that in which the states have overlapping support and are non-commutative.
The most general way to describe a quantum measurement is with a positive operator-valued measure (POVM). We usually denote the elements of a POVM as . 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 using a measurement device corresponding to the POVM , then the probability for obtaining outcome 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.
Quantum typicality
The reader can find a good review of this topic in the article about the typical subspace.
Gentle operator lemma
The following lemma is important for our proofs. It demonstrates that a measurement that succeeds with high probability on average does not disturb the state too much on average:
Lemma: Given an ensemble with expected density operator , suppose that an operator such that succeeds with high probability on the state :
Then the subnormalized state is close in expected trace distance to the original state
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