17-11-2012, 02:20 PM
Quantum Information Theory
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Abstract
We survey the field of quantum information theory.
In particular, we discuss the fundamentals of the field, source
coding, quantum error-correcting codes, capacities of quantum
channels, measures of entanglement, and quantum cryptography.
INTRODUCTION
RECENTLY, the historic connection between information
and physics has been revitalized, as the methods of
information and computation theory have been extended to
treat the transmission and processing of intact quantum states,
and the interaction of such “quantum information” with traditional
“classical” information. Although many of the quantum
results are similar to their classical analogs, there are notable
differences. This new research has the potential to shed light
both on quantum physics and on classical information theory.
In retrospect, this development seems somewhat belated,
since quantum mechanics has long been thought to underlie
all classical processes. But until recently, information itself
had largely been thought of in classical terms, with quantum
mechanics playing a supporting role of helping design the
equipment used to process it, setting limits on the rate at
which it could be sent through certain quantum channels.
Now we know that a fully quantum theory of information and
information processing offers, among other benefits, a brand
of cryptography whose security rests on fundamental physics,
and a reasonable hope of constructing quantum computers that
could dramatically speed up the solution of certain mathematical
problems. These feats depend on distinctively quantum
properties such as uncertainty, interference, and entanglement.
QUANTITATIVE THEORY OF ENTANGLEMENT
Since entanglement appears to be responsible for the remarkable
behavior of information in quantum mechanics, a
means of quantifying it would seem useful. We will discuss
several such measures in this section. There is one “best”
measure for entanglement of pure states, which is defined using
entropy. However, there does not appear to be a unique best
measure of entanglement for mixed states; which measure is
best depends on what the measure is being used for. We will
discuss several measures, all of which agree for pure quantum
states