Vol.4 No.6&7
December 31,
2004
Quantum Information and
Underlying Mathematics
Editorial note
( pp409410)
Nonstabilizer
quantum codes from Abelian subgroups of the error group
(pp411436)
V. Arvind, P.P. Kurur and K.R.
Parthasarathy
This paper is motivated by the computergenerated
nonadditive code described in Rains et al\cite{rains97nonadditive}. We
describe a theory of nonstabilizer codes of which the nonadditive code
of Rains et al is an example. Furthermore, we give a general strategy of
constructing good nonstabilizer codes from good stabilizer codes and
give some explicit constructions and asymptotically good nonstabilizer
codes. Like in the case of stabilizer codes, we can design fairly
efficient encoding and decoding procedures.
Instrumental processes, entropies, information in quantum continual
measurements (pp437449)
A. Barchielli and G. Lupieri
In this paper we will give a short presentation of
the quantum L\'evyKhinchin formula and of the formulation of quantum
continual measurements based on stochastic differential equations,
matters which we had the pleasure to work on in collaboration with
Prof.\ Holevo. Then we will begin the study of various entropies and
relative entropies, which seem to be promising quantities for measuring
the information content of the continual measurement under consideration
and for analysing its asymptotic behaviour.
Optical
demonstrations of statistical decision theory for quantum systems
(pp450459)
S.M.
Barnett
The work of Holevo and other pioneers of quantum
information theory has given us limits on the performance of
communication systems. Only recently, however, have we been able to
perform laboratory demonstrations approaching the ideal quantum limit.
This article presents some of the known limits and bounds together with
the results of our experiments based on optical polarisation.
A
resourcebased view of quantum information (pp460466)
C.H. Bennett
We survey progress in understanding quantum
information in terms of equivalences, reducibilities, and asymptotically
achievable rates for transformations among nonlocal resources such as
classical communication, entanglement, and particular quantum states or
channels. In some areas, eg source coding, there are straightforward
parallels to classical information theory; in others eg
entanglementassisted communication, new effects and tradeoffs appear
that are beginning to be fairly well understood, or the remaining
uncertainty has become focussed on a few simple open questions, such as
conjectured additivity of the Holevo capacity. In still other areas,
e.g. the role of the back communication and the classification of
tripartite entanglement, much remains unknown, and it appears unlikely
that an adequate description exists in terms of a finite number of
resources.
On the
quantumness of a Hilbert space (pp467478)
C.A. Fuchs
We derive an exact expression for the quantumness
of a Hilbert space (defined in C.A. Fuchs and M. Sasaki, Quant. Info.
Comp. {\bf 3}, 377 (2003)), and show that in composite Hilbert spaces
the signal states must contain at least some entangled states in order
to achieve such a sensitivity. Furthermore, we establish that the
accessible fidelity for symmetric informationally complete signal
ensembles is equal to the quantumness. Though spelling the most trouble
for an eavesdropper because of this, it turns out that the accessible
fidelity is nevertheless easy for her to achieve in this case: Any
measurement consisting of rankone POVM elements is an optimal
measurement, and the simple procedure of reproducing the projector
associated with the measurement outcome is an optimal output strategy.
Statistical estimation of a quantum operation
(pp479488)
A. Fujiwara
This paper reviews some recent developments in
quantum channel estimation theory, putting emphasis on an active
interplay between noncommutative statistics and information geometry.
Classical capacity of freespace optical communication
(pp489499)
V. Giovannetti, S. Guha, S.
Lloyd, L. Maccone, J.H. Shapiro, B.J. Yen and H.P. Yuen
The classicalinformation capacity
of lossy bosonic channels
is studied, with emphasis on the
farfield free space channel.
Comments on multiplicativity of maximal
pnorms when p=2
(pp500512)
C. King and M.B. Ruskai
We consider the maximal
pnorm associated with a completely positive map
and the question of its multiplicativity under
tensor products. We give a condition under which
this multiplicativity holds when p = 2,
and we describe some maps which satisfy our
condition. This class includes maps for
which multiplicativity is known to fail for large p.
Our work raises some questions of independent interest
in matrix theory; these are
discussed in two appendices.
On
consistency of quantum theory and macroscopic objectivity
(pp513522)
L. Lanz, B. Vacchini and O.
Melsheimer
We argue that the consistency problem between
quantum theory and macroscopic objectivity must be placed inside a
quantum description of macroscopic nonequilibrium systems. Resorting to
thermodynamic concepts inside quantum field theory seems to be
necessary.
Toward
implementation of coding for quantum sources and channels
(pp526536)
M. Sasaki
We review our experiment on quantum source and
channel codings, the most fundamental operations in quantum
infocommunications. For both codings, entangling letter states is
essential. Our model is based on the polarizationlocation coding, and a
quasisingle photon linear optics implementation to entangle the
polarization and location degrees of freedom. Using singlephoton events
in a subset of possible cases, we simulate quantum codingdecoding
operations for nonorthogonal states under the quasipure state
condition. In the quantum channel coding, we double the spatial
bandwidth (number of optical paths), and demonstrate the information
more than double can be transmitted. In the quantum source coding, we
halve the spatial bandwidth to compress the data and decompress the
original data with the high fidelity approaching the theoretical limit.
The
classical capacity achievable by a quantum channel assisted by limited
entanglement
(pp537545)
P.W. Shor
We give the tradeoff curve showing the capacity
of a quantum channel as a function of the amount of entanglement used by
the sender and receiver for transmitting information. The endpoints of
this curve are given by the HolevoSchumacherWestmoreland capacity
formula and the entanglementassisted capacity, which is the maximum
over all input density matrices of the quantum mutual information. The
proof we give is based on the HolevoSchumacherWestmoreland formula,
and also gives a new and simpler proof for the entanglementassisted
capacity formula.
The
uncertainty relation for joint measurement of position and momentum
(pp546562)
R.F. Werner
We prove an uncertainty relation, which imposes a
bound on any joint measurement of position and momentum. It is of the
form (\Delta P)(\Delta Q)\geq C\hbar, where the `uncertainties'
quantify the difference between the marginals of the joint measurement
and the corresponding ideal observable. Applied to an approximate
position measurement followed by a momentum measurement, the
uncertainties become the precision \Delta Q of the position
measurement, and the perturbation \Delta P of the conjugate
variable introduced by such a measurement. We also determine the best
constant C, which is attained for a unique phase space covariant
measurement.
Quantum
and classical message protect identification via quantum channels
(pp564578)
A. Winter
We discuss concepts of message identification in the
sense of Ahlswede and Dueck via general quantum channels, extending
investigations for classical channels, initial work for
classicalquantum (cq) channels and ``quantum fingerprinting''. We show
that the identification capacity of a discrete memoryless quantum
channel for classical information can be larger than that for
transmission; this is in contrast to all previously considered models,
where it turns out to equal the common randomness capacity (equals
transmission capacity in our case): in particular, for a noiseless qubit,
we show the identification capacity to be 2, while transmission and
common randomness capacity are 1. Then we turn to a natural concept of
identification of quantum messages (i.e. a notion of
``fingerprint'' for quantum states). This is much closer to quantum
information transmission than its classical counterpart (for one thing,
the code length grows only exponentially, compared to double
exponentially for classical identification). Indeed, we show how the
problem exhibits a nice connection to visible quantum coding.
Astonishingly, for the noiseless qubit channel this capacity turns out
to be 2: in other words, one can compress two qubits into one and this
is optimal. In general however, we conjecture quantum identification
capacity to be different from classical identification capacity.
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