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Phys. Rev. A 76 (2007) 062320; arXiv:0707.3752
Types of Quantum Information
Robert B. Grifths
*
Department of Physics, CarnegieMellon University,
Pittsburgh, PA 15213, USA
Quantum, in contrast to classical, information theory, allows for diFerent incompatible types (or
species) of information which cannot be combined with each other. Distinguishing these incompat
ible types is useful in understanding the role of the two classical bits in teleportation (or one bit in
onebit teleportation), for discussing decoherence in informationtheoretic terms, and for giving a
proper de±nition, in quantum terms, of “classical information.” Various examples (some updating
earlier work) are given of theorems which relate diFerent incompatible kinds of information, and
thus have no counterparts in classical information theory.
PACS numbers:
I. INTRODUCTION
Despite an enormous number oF publications in the
±eld oF quantum inFormation (see [1, 2] For useFul intro
ductions), neither the Fundamental principles underlying
the subject, nor its connection with classical inForma
tion theory as developed by Shannon and his successors
[3], is altogether clear. On the one hand there has been
some dispute [4, 5] about whether Shannon’s ideas can
be applied at all in the quantum domain. On the other
hand there have been suggestions that the connection
with Shannon’s ideas occurs only For macroscopic sys
tems or asymptotically large
N
(number oF transmissions,
or whatever) limits, as in what is sometimes called “quan
tum Shannon” theory [6, 7]. The author’s position is
that, to the contrary, there are perFectly consistent ways
oF applying the basic ideas oF classical inFormation the
ory to small numbers (even one) oF microscopic quantum
systems provided attention is paid to the Hilbert space
structure oF quantum theory, and probabilities are intro
duced in a consistent Fashion. And, Further, that this
approach has advantages in that simple systems are sim
pler to think about than complicated systems, so it is
useFul to develop some intuition as to how they behave.
One goal is to understand both classical and quantum in
Formation theory in Fully quantum terms, since the world
is (most physicists believe) Fundamentally quantum me
chanical.
The basic strategy oF this paper is based on the idea
that quantum inFormation comes in a variety oF incom
patible
types
or
species
.
Each type or species reFers
to a certain class oF (typically microscopic) mutually
compatible properties oF a quantum system. As long as
the discussion oF inFormation is limited to a single type,
all the usual Formalism and intuition provided by clas
sical inFormation theory apply directly to the quantum
*
Electronic address:
rgrif@cmu.edu
domain. On the other hand, incompatible types oF quan
tum inFormation cannot be combined, as this makes no
sense in the context oF standard Hilbert space quantum
mechanics. Since in classical inFormation theory there is
only a single type or species oF inFormation, or, equiva
lently, all di²erent types are compatible with each other,
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 Spring '08
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 Physics

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