Classical capacity
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.
Background
Mixed states and quantum channels
A mixed quantum state is a unit trace,positive operator known as a density operator, and is often denoted
by,,, etc. The simplest model for a quantum channel
is a classical-quantum channel
which sends the classical letter
at the transmitting end to a quantum state at the receiving
end, with noise possibly introduced in between. The receiver's task is to perform a measurement to determine the
input of the sender. If the states are perfectly
distinguishable from one another and the channel is noiseless, then perfect decoding is trivially possible. If the states all
commute with each other then the channel is effectively classical.
The situation becomes nontrivial only when the states
have overlapping support and do not necessarily commute.
Quantum measurements
The most general way to describe a quantum measurement is with apositive operator-valued measure, whose elements are typically denoted 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.
Classical communication over quantum channels
The above is sufficient to consider a classical classical communication scheme over a cq channel. The sender uses a cq channel to map a classical letter x to a quantum state, which is then sent through some noisy quantum channel, and then measured using some POVM by the receiver, who obtains another classical letter.Precise definition
The classical capacity can be defined as the maximum rate achievable by a coding scheme for classical information transmission, which can be defined as follows.Definition. A -coding scheme for classical information transmission using a quantum channel is given by pair of an encoding map and a decoding POVM such that with respect to the Hilbert-Schmidt inner product for all.
Definition. A rate is achievable for the channel if either or and for any there exists a -coding scheme such that and both hold.
Holevo-Schumacher-Westmoreland theorem
The Holevo information of a quantum channel can be defined aswhere is a classical-quantum state of the form
for some probability distribution and density operators which can be input to the given channel.
Schumacher and Westmoreland in 1997, and Holevo independently in 1998, proved that the classical capacity of a quantum channel can be equivalently defined as
Gentle measurement lemma
The gentle measurement lemma states that a measurement succeeding with high probability does not disturb the state too much on average.Lemma. Given an
ensemble with expected
density operator, suppose
that an operator with succeeds with
probability on the state :
Then the subnormalized state is close
in expected trace distance to the original state :
The gentle measurement lemma has the following analog which holds for any operators
,, such that :
The quantum information-theoretic interpretation of this inequality is
that the probability of obtaining outcome from a quantum measurement
acting on the state is bounded by the sum of the probability of obtaining
on summed and the distinguishability of
the two states and.
Non-commutative union bound
Lemma. For a subnormalized state such that and, and for projectors,..., we have
Intuitively, Sen's bound is a sort of "non-commutative union
bound" because it is analogous to the union bound
from classical probability theory:
where are events. The analogous quantum bound would be
if we think of as a projector onto the intersection of
subspaces. However, this only holds if the projectors,
..., commute. If the projectors are non-commuting, then one must use a non-commutative or quantum union bound.
Proof
We now prove the HSW theorem with Sen's non-commutative union bound. Wefirst describe how the code is chosen, then give the construction of Bob's POVM,
and finally analyze the error of the protocol.
Encoding map
We first describe how Alice and Bob agree on arandom choice of code. They have the channel and a
distribution. They choose classical sequences
according to the IID distribution.
After selecting them, they label them with indices as. This leads to the following
quantum codewords:
The quantum codebook is then. The average state of the codebook is then
where.
Decoding POVM construction
Sen's bound from the above lemmasuggests a method for Bob to decode a state that Alice transmits. Bob should
first ask "Is the received state in the average typical
subspace?" He can do this operationally by performing a
typical subspace measurement corresponding to. Next, he asks in sequential order,
"Is the received codeword in the
conditionally typical subspace?" This is in some sense
equivalent to the question, "Is the received codeword the
transmitted codeword?" He can ask these
questions operationally by performing the measurements corresponding to the
conditionally typical projectors.
Why should this sequential decoding scheme work well? The reason is that the
transmitted codeword lies in the typical subspace on average:
where the inequality follows from. Also, the
projectors
are "good detectors" for the states because the following condition holds from conditional quantum
typicality:
Error analysis
The probability of detecting thecodeword correctly under our sequential decoding scheme is equal to
where we make the abbreviation. Thus, the probability of
an incorrect detection for the 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 gives an upper bound of
We then apply Sen's bound to this expression with and the sequential
projectors as,,...,. This gives the upper bound
Due to concavity of the square root, we can bound this expression from above
by
where the second bound follows by summing over all of the codewords not equal
to the codeword.
We now focus exclusively on showing that the term inside the square root can
be made small. Consider the first term:
where the first inequality follows from and the
second inequality follows from the gentle operator lemma and the
properties of unconditional and conditional typicality. Consider now the
second term and the following chain of inequalities:
The first equality follows because the codewords and
are independent since they are different. The second
equality follows from. The first inequality follows from
. Continuing, we have
The first inequality follows from and exchanging
the trace with the expectation. The second inequality follows from
. The next two are straightforward.
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.