Q4L12a.txt

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An insightful perspective for quantum communication
is provided by starting with the scenario
of classical communication.
Here, the goal is to transmit a message, say x, described
by its probability distribution function.
What is received is a noisy message
from the output of the channel.
Call that y.
The input to the channel is x.
Two principal ideas describe this scenario mathematically.
First, h of x, known as the entropy of x,
is the number of bits needed to faithfully represent
the message x on average.
Second, the capacity c is the max
over all probability distributions of something
called the mutual information, I of xy,
which is the maximum error-free communication
rate achievable over the noisy channel.
The two key concepts here are the entropy h
of x and the mutual information I of xy.
There are parallels to this in quantum communication.
In this scenario, the source messages
are described by a density matrix.
This is a distribution which may be
viewed as a distribution over pure states.
For example, these may be non-orthogonal states,
in contrast to the classical case.
The received message is also described by a density matrix.
And in contrast to the classical case,
this is possibly decoded quantum mechanically,
say by a quantum computer.
Analogous to the classical case, the Von Neumann entropy
describes the number of qubits needed to faithfully represent
the density matrix rho, and this is asymptotically.
This entropy is similar to, but different from,
the Shannon entropy, as we will see.
Also, there are several distinct scenarios
for noisy communication.
After all, classical data may be the source, as well as
the received data.
Alternatively, the source may be quantum data.
Or in fact, the receiver may receive and decode
quantum mechanically.
Or both the sender and receiver may be quantum.
It is known that these channel capacities
are ordered CCC is less than CQC and CCQ, and is less than CQQ.
There are even richer scenarios beyond these four.
Entanglement may also be used to assist, for example,
classical communication.
When the sender and receiver have
pre-shared entangled qubits, this
may change the channel capacity.
Another set of capacity measures result when quantum information
is allowed to be sent, but only one way or two ways,
and possibly in addition to having a classical side
channel.
Beyond this, quantum communication
may also involve multiple parties. Quantum communication is thus a fascinating field of study.